The discovery of the quantum spin Hall effect and topological insulators more than a decade ago has revolutionized modern condensed matter physics. Today the field of topological states of matter is one of the most active and fruitful research areas for both experimentalists and theorists. The physics of topological insulators is typically well described by band theory and systems of noninteracting fermions. In contrast several of the most fascinating effects in condensed matter physics merely exist due to electronelectron interactions examples include unconventional superconductivity the Kondo effect and the MottHubbard transition. P The aim of this review article is to give an overview of the manifold directions which emerge when topological bandstructures and correlation physics interfere and compete. These include the study of the stability of topological bandstructures and correlated topological insulators. Interactioninduced topological phases such as the topological Kondo insulator provide another exciting topic. More exotic states of matter such as topological Mott insulator and fractional Chern insulators only exist due to the interplay of topology and strong interactions and do not have any bandstructure analogue. Eventually the relation between topological bandstructures and frustrated quantum magnetism in certain transition metal oxides is emphasized.
['Condensed Matter - Strongly Correlated Electrons', 'Condensed Matter - Materials Science', 'Condensed Matter - Quantum Gases']
Interacting topological insulators a review
{'Interacting topological insulators: a review': 'Stephan Rachel  \nSchool of Physics, University of Melbourne, Parkville, VIC 3010, Australia  \n(Dated: October 18, 2018)  \nThe discovery of the quantum spin Hall effect and topological insulators more than a decade ago has revolutionized modern condensed matter physics. Today, the field of topological states of matter is one of the most active and fruitful research areas for both experimentalists and theorists. The physics of topological insulators is typically well described by band theory and systems of non-interacting fermions. In contrast, several of the most fascinating effects in condensed matter physics merely exist due to electron-electron interactions, examples include unconventional superconductivity, the Kondo effect, and the Mott-Hubbard transition.  \nThe aim of this review article is to give an overview of the manifold directions which emerge when topological bandstructures and correlation physics interfere and compete. These include the study of the stability of topological bandstructures and correlated topological insulators. Interaction-induced topological phases such as the topological Kondo insulator provide another exciting topic. More exotic states of matter such as topological Mott insulator and fractional Chern insulators only exist due to the interplay of topology and strong interactions and do not have any bandstructure analogue. Eventually the relation between topological bandstructures and frustrated quantum magnetism in certain transition metal oxides is emphasized.', 'CONTENTS': 'References  \n- 30 tially anticipated the rich field of topological Weyl semimet-', 'I. INTRODUCTION': "Condensed matter physics is the broad field of physics research which is devoted to understand solids and liquids. In the last century, condensed matter physics was a success story for both theorists and experimentalists due to the LandauGinzburg paradigm (Ginzburg and Landau, 1950) and the concept of spontaneous symmetry breaking. Most aspects of quantum magnetism, superconductivity, and superfluidity can be well understood in this context, just to mention a few. All these examples have in common that at high temperatures the physical system is in a disordered phase. Upon decreasing the temperature, the system acquires order due to spontaneous symmetry breaking below a critical temperature T c . For instance, the spins of localized electrons in a magnet will randomly point in arbitrary directions at high temperatures, but are aligned along a certain direction at low temperatures leading to a finite magnetization acting as the local order parameter. Despite the fact that other important paradigms such as Anderson localization (Anderson, 1958) were available, physics associated with spontaneous symmetry breaking dominated the condensed matter research.  \n- 22 23 24 25 27 29 A mentionable exception is provided by the BerezinskyKosterlitz-Thouless transition (Berezinskii, 1972; Kosterlitz and Thouless, 1973) in two spatial dimensions: below a critical temperature the system does not order as expected but forms a quasi-ordered state with vortex-antivortex pairs instead. This exception is particular worth mentioning since vortices themselves are regarded as topologically stable objects. The different phases of 3 He(Vollhardt and Wölfle,\n- 29 1990) provide another exception, and that line of research par-  \nals (Bevan et al. , 1997; Volovik, 2003; Wan et al. , 2011; Xu et al. , 2015a) (which is, however, not the focus of this review). Athird exception is the Su-Schrieffer-Heger (SSH) model intended to describe solitons in polyacetylene (Su et al. , 1979). It was acknowledged early that these solitons are 'topological objects' and the SSH model is an ideal test ground for fractionalization , i.e. , that a particle carries parts of an elementary quantum number (Jackiw and Rebbi, 1976; Laughlin, 1999). What was not recognized is that the SSH model realizes a one-dimensional (1D) topological insulator characterized by non-local invariants. The simple reason is that the aforementioned concepts were not available at the time.  \nCertainly the most striking event in the context of 'nonLandau-Ginzburg physics' was the discovery of the integer quantum Hall effect (QHE) featuring the exact quantization of the Hall conductivity (1980) (Klitzing et al. , 1980). In the quantum Hall experiments, a two-dimensional electron gas as realized in gallium arsenide heterostructures is placed in a strong perpendicular magnetic field. While Drude theory predicts a linear dependence of the transverse resistivity on the magnetic field strength, at sufficiently low temperatures and strong magnetic fields plateaus were observed instead (Klitzing et al. , 1980). This result is even more striking when it is expressed in the inverse quantity, the transverse (Hall) conductivity, and in units of e 2 /h : the plateaus correspond to integer numbers with an unprecedented accuracy. Thouless and coworkers showed that this integer number is in fact a topological invariant (Thouless et al. , 1982), the first Chern number, which also correspond to the number of chiral edge states being responsible for the transport in this otherwise insulating system. At the same time, various experimental groups performed QHE experiments which eventually led to the discovery of the fractional quantum Hall effect (FQHE) (Tsui et al. , 1982) featuring a subtile interplay of non-trivial topology of the Landau levels, disorder, and electron-electron interactions. The elementary excitations, quasi-holes, carry charge e/ 3 (for the first observed plateau at ν = 1 / 3 (Tsui et al. , 1982)) as demonstrated in the famous shot noise experiments more than 20 years ago (de Picciotto et al. , 1997; Saminadayar et al. , 1997). The FQHE with its quasi-hole excitations provides one of the most impressive examples of electron fractionalization (Arovas et al. , 1984; Laughlin, 1983) in nature. Today, we also consider the FQHE as the only widely accepted realization of a system with intrinsic topological order (see Sec. II.D).  \nIt was more than 20 years later, when C. Kane and E. Mele(Kane and Mele, 2005a,b) and independently B. A. Bernevig and S.-C. Zhang (Bernevig and Zhang, 2006) predicted a new state of matter (2005) dubbed quantum spin Hall (QSH) effect which has revolutionized the field of topological phases and made it to one of the main research directions of modern condensed matter physics: topological condensed matter physics. QSH insulators are spinful extensions of integer QHE systems and protected by time-reversal (TR) symmetry. First proposed to be realized in two-dimensional (2D) graphene monolayers (Kane and Mele, 2005a) and strained  \nzinc blende semiconductors (Bernevig and Zhang, 2006), it was shortly after generalized to three spatial dimensions (Fu et al. , 2007; Moore and Balents, 2007; Roy, 2009), today known as 'Topological Insulators'. Due to the vanishingly small amount of spin-orbit coupling present in graphene (Min et al. , 2006; Yao et al. , 2007), the QSH effect was eventually observed in HgTe/CdTe heterostructures for the first time (Bernevig et al. , 2006; König et al. , 2007). Unlike the QHEs, topological insulators do not require any magnetic field. Instead the intrinsic spin-orbit coupling leads to the non-trivial band topology. Moreover, instead of having chiral edge modes, topological insulators exhibit spin-filtered counterpropagating pairs of chiral edge modes, so-called helical edge modes. These are only two out of various reasons why topological insulators are believed to have a bright future regarding applications similar to spintronics (sometimes referred to as 'topotronics').  \nBoth the integer QHE as well as topological insulators can be described as free fermion models (in contrast to the FQHE). This raises the question to what extent are such topological band structures stable with respect to electron-electron interactions? And if they are not stable, what happens otherwise? As we well know Coulomb interactions are always present in solids, sometimes screened and weak, sometimes strong and causing new phenomena such as the metal-to-insulator transition (Mott, 1968) or the Kondo effect (Kondo, 1964).  \nIn addition to condensed matter systems, tremendous progress has been made in the past years in realizing topological matter using quantum simulators (Cooper et al. , 2018; Goldman et al. , 2014; Lu et al. , 2016; Ozawa et al. , 2018). Most notably, ultra-cold quantum gases also allow to tune interactions (Cooper et al. , 2018; Goldman et al. , 2014), thus providing an alternative and controlled approach to investigate interacting topological insulators.", 'A. Outline': "This review aims to shed some light on the various aspects of the interplay of topological band structures and (strong) electron-electron interactions, simply referred to as Interacting Topological Insulators . In the literature, there are already several excellent review articles including the general reviews about topological insulators (Hasan and Kane, 2010; Qi and Zhang, 2011) as well as the insightful textbook Topological Insulators and Topological Superconductors (Bernevig, 2013). Several slightly more specialised review articles about TIs are available (Hasan and Moore, 2011; Maciejko et al. , 2011; Wehling et al. , 2014; Zhang and Zhang, 2013) as well as reviews about topological materials (Ren et al. , 2016; Yan and Zhang, 2012), about correlation effects in topological bandstructures (Hohenadler and Assaad, 2013), about the edge physics of 2D topological insulators (Dolcetto et al. , 2016), and about fractional Chern insulators (Bergholtz and Liu, 2013; Neupert et al. , 2015a) as well as reviews about spin-orbit coupling in quantum gases covering the develop-  \nment of topological phases in cold atoms (Cooper et al. , 2018; Galitski and Spielman, 2013; Goldman et al. , 2014). Also G. E. Volovik's textbook addresses many aspects of topology which are directly relevant for the physics reviewed in this article (Volovik, 2003) although it was written before the discovery of topological insulators. There are several points which might come to the reader's mind when thinking about topology and interactions . Some of them are discussed in the following.  \nIn Section II the most important topological insulator (TI) models, Chern insulators and Z 2 TIs, are reviewed. A short discussion of the effective theory and the spin texture of helical edge states of Z 2 TIs is provided and the influence of a broken axial spin symmetry emphasized. The section is completed by briefly reviewing and comparing the concepts symmetry protected topological order and intrinsic topological order . In Section III.B, various paradigmatic models for interacting TIs at half filling are discussed: examples of 1D correlated TIs (Sec. III.B.1), the Haldane-Hubbard model as a prototype of an interacting Chern insulator (Sec. III.B.2), the Kane-Mele-Hubbard model as a prototype of a correlated Z 2 TI (Sec. III.B.3), its bond-dependent, multidirectional generalization dubbed sodium-iridate-Hubbard model (Sec. III.B.4), the Bernevig-Hughes-Zhang-Hubbard model (Sec. III.B.5), the TR invariant Hofstadter-Hubbard model (Sec. III.B.6) and other correlated topological insulators in higher dimensions (Sec. III.B.7). For all of these models the stability towards electron correlations and, eventually, the breakdown of the TI phases into conventionally ordered or sometimes even into other exotic phases are considered. A completely different perspective is discussed in Section III.C which reviews interaction-induced topological insulator phases. The idea is to find scenarios where a topologically trivial band-structure experiences a phase transition into a topological insulator phase driven by electronelectron interactions. The considered mechanisms contain mere bandstructure renormalization effects, fluctuation induced phases (Sec. III.C.1), and spontaneous symmetry breaking (Sec. III.C.2). Eventually, a brief discussion about the most promising class of candidate materials, topological Kondo insulators, is presented (Sec. III.C.3). In Section III.D topological Mott insulators as well as fractional Chern insulators and frational topological insulators are discussed as prototype systems which merely exists due the interplay of topology and strong interactions and do not have any band structure analogue. In Section III.E the physics of strongly interacting surface states of a three-dimensional strong TI is reviewed. Eventually in Section IV the connection between TI physics and the honeycomb iridates A 2 IrO 3 (A=Na, Li) and other so-called Kitaev materials, which have been proposed to host various topological states of matter such as QSH insulators, strong TIs, and quantum spin liquids, is emphasized. Section V provides a summary and an outlook.", 'II. NON-INTERACTING TOPOLOGICAL INSULATORS': "Given the enormous interest in topological band insulators there are already several notable reviews (Bernevig, 2013; Hasan and Kane, 2010; Hasan and Moore, 2011; Maciejko et al. , 2011; Ren et al. , 2016; Yan and Zhang, 2012; Zhang and Zhang, 2013). This section about non-interacting TIs is not meant to replace them but rather give the necessary introduction to render this review self-contained. First, we will briefly introduce the integer quantum Hall effect. Then we will review the most important band insulators with a nontrivial topology, Chern insulator and Z 2 topological insulators. Both have in common that they can be formulated as Bloch Hamiltonians, i.e. , tight-binding models of free particles on a lattice. This formulation has the benefit that the models can be easily extended to Hubbard-type models perfectly suited for the investigation of electron-electron interactions discussed in the following sections. At the end of this section, we further elaborate on the role of 'topology'. We point out that there are different concepts of topology floating around in the literature. Moreover, we briefly review the most important characterization schemes for topological phases and give a very brief overview of candidate materials.  \nA topological insulator (TI) is a band insulator in the bulk featuring metallic edge or surface states which are topologically protected against elastic single-particle backscattering (Qi and Zhang, 2011). The term 'topological insulator' is often used or meant synonymously with 'timereversal invariant Z 2 TI'. We will use, however, the more relaxed definition that any band insulator featuring a 'non-trivial topology' is a TI, and that also includes Chern and quantum Hall insulators.", 'A. Integer Quantum Hall Effect': "The integer QHE (Klitzing et al. , 1980) can be considered as the mother state of all topological states of matter. The QHE is realized in a two-dimensional electron gas which is subject to a perpendicular, homogeneous magnetic field. The resulting system is a bulk-insulator, being reflected in a vanishing longitudinal conductivity σ xx = σ yy = 0 where σ is the conductivity tensor. At the same time the transverse or Hall conductivity σ xy is finite and quantized,  \nσ xy = C e 2 h (1)  \nwith C ∈ Z , the first Chern number, which will be discussed below. Since the system is bulk-insulating the finite Hall conductivity must be associated with edge transport. In fact, the integer number C corresponds to the number of chiral edge modes where each mode carries a unit of conductance e 2 /h . This intuitively explains why σ xy shows a step-like behavior as a function of magnetic field: as long as a bulk gap is present C is integer-valued and constant; only due to a finetuned magnetic field value the bulk might become metallic  \nallowing C to switch to another value which must be again integer-valued when the bulk-gap reopens.  \nThouless, Kohmoto, Nightingale, and de Nijs (TKNN)(Thouless et al. , 1982) used the Kubo-formula to compute the Hall conductivity and indeed found the result (1) with  \nC = 1 2 π ∫ BZ d k ( ∂A y ( k ) ∂k x -∂A x ( k ) ∂k y ) (2)  \nand  \nA µ = -i ∑ occ . bands α 〈 α k | ∂ ∂k µ | α k 〉 (3)  \nwhere 'BZ' stands for the first Brillouin zone and 'occ. bands' refers here to filled Landau levels or energy bands, respectively. The vector A is the Berry connection or Berry vector potential of the Bloch state | α k 〉 and the integrand in (2) is the corresponding Berry curvature F xy ( k ) . The expression (2) is formally equivalent to the Berry phase (Berry, 1984) [see also (Niu et al. , 1985; Zak, 1989)] with the difference that the latter can have a range of integration which is a smaller subset of the BZ. C is the first Chern number, sometimes also called TKNN invariant. Efficient methods to numerically compute the Chern number have been proposed (Fukui et al. , 2005). Chern numbers appear in algebraic topology and differential geometry in the context of Chern classes which are characteristic classes associated with complex vector bundles. In (2) the range of integration covers the full BZ, i.e. , C is a highly non-local object and some people find it helpful to consider the Chern number as a nonlocal order parameter. This provides a complete explanation of the integer QHE. Each Landau level carries a Chern number C = 1 . The magnetic field strength controls how many Landau levels are filled. As long as the magnetic field strength is such that the Fermi level is in the gap between two Landau levels, the Chern number corresponds to this number of occupied Landau levels and stays constant (explaining the quantized and perfectly flat Hall plateaus). And only at a certain fine-tuned magnetic field strength when the Fermi level hits a Landau level, the system becomes metallic (reflected in finite σ xx ) and looses its quantized Hall conductivity.  \nThe second important physical effect is the fractional quantum Hall effect (FQHE) (Laughlin, 1983; Tsui et al. , 1982) which, unlike the integer effect, results in a strongly correlated state of matter: additional plateaus can be observed which correspond to fractions of the Hall conductivity in units of e 2 /h , i.e. , rational numbers such as 1/3, 1/5, 2/7 etc., while the system is incompressible. Since the Chern number C needs to be integer-valued, Eq. (1) in conjunction with (2) cannot provide an explanation for the observed phenomenon (Girvin and Prange, 1990). In order to observe FQHE one has to tune the magnetic field strength such that the Fermi level lies inside a Landau level. Instead of showing metallic behavior strong electron-electron interactions drive the system into a gapped, incompressible phase which is not adiabatically connected  \nwith the physics of the integer QHE. The reason why strong interactions can be present at all in a 2D electron gas stems from the flatness of the Landau levels. Since the Landau levels are essentially dispersionless, the kinetic energy is heavily quenched and even the smallest electron-electron interactions, which are always present, turn out to be effectively large. That is, not the interactions but the ratio 'interactions / kinetic energy' is large. We note that the FQHE is a state of matter which exists due to a subtle interplay between the topologically non-trivial Landau levels and strong interactions. It is worth mentioning that also disorder is a necessary ingredient to provide full understanding of this effect (Girvin and Prange, 1990). It furthermore turns out that the elementary excitations in FQHE systems are exotic quasi particles which carry only fractions of the electron charge -e . To give a famous example: when the magnetic field is tuned such that the transverse conductivity takes the value σ xy = 1 3 e 2 h then the elementary excitations are quasi-holes with charge -e 3 (Laughlin, 1983). One should keep in mind that every FQH state will be the groundstate of a corresponding many-body Hamiltonian involving some type of Coulomb interactions (Haldane, 1983a) and a single-particle description is insufficient. Later we will discuss the lattice version of FQH systems, the socalled frational Chern insulators in Sec. III.D.2. In contrast to the FQHE, the integer QHE is described by a free fermion theory.  \nThe formation of Landau levels in a 2D electron gas assumes free fermionic particles in a magnetic field with kinetic energy T = ∑ i p 2 i 2 m . Now we will address the interesting question how non-interacting lattice fermions are affected by a strong (orbital) magnetic field. This problem was first successfully analyzed by Hofstadter in 1976 (Hofstadter, 1976), even a few years before the discovery of the integer QHE. Here we briefly illustrate the Hofstadter problem on the square lattice which will be the starting point for the study of interaction effects in Hofstadter bands, the Hofstadter-Hubbard model (Cocks et al. , 2012; Goldman et al. , 2010), and fractionalization in Hofstadter bands (Möller and Cooper, 2015). Spinless fermions on the square lattice are described by a hopping Hamiltonian H = -∑ 〈 ij 〉 ( tc † i c j + h . c . ) . In order to include a finite magnetic field we choose Landau gauge and modify the hopping term using Peierls substitution,  \nt c † i c j -→ t c † i exp ( i 2 π φ 0 ∮ S A dglyph[lscript] ) c j , (4)  \nwith the Dirac flux quantum φ 0 = hc/e . For a vector potential of strength α = p/q (we do not consider irrational values here) being the number of flux quanta per lattice cell yields  \nA ( r ) = p q φ 0 A LC y ˆ x . (5)  \nThe area of an elementary lattice cell is A LC = a 2 for a square lattice as considered in the following ( a being the lattice spacing). For a square lattice we find in the chosen gauge the hopping pattern, which is shown in Fig. 1 (a) for α = 1 / 3 , a typ-  \nJ. Phys. B: At. Mol. Opt. Phys.  \n46  \n(2013) 134004  \nPPOrth  \net al  \n(a)  \n(b)  \n(c)  \nFigure 2 Spectrum of (a) a massless ( m = 0 ) and (b) a massive Dirac Hamiltonian ( m = 0 ).  \n<!-- image -->  \n<!-- image -->", 'B. Chern Insulators': "Figure 1. Energy spectrum for a cylinder geometry as a function of magnetic flux is α = 1 / 10, and γ and λ x staggering λ x > 0. Then in section 3, we investigate the role of the Hubbard interaction in detail using RDMFT. We present the interacting phase diagram in section 3.1, and explicitly determine the robustness of the topological phase Figure 1 The Hofstadter problem. (a) Square lattice with modified hoppings for vertical bonds according to (4) for a magnetic field of strength α = 1 / 3 in units of φ 0 . The red box indicates the unit cell. (b) Spectrum for α = 1 / 10 computed on a cylinder geometry. Bulk bands are shown in blue, chiral edge modes localized at both cylinder edges are shown in red (Orth et al. , 2013).  \nwith respect to the Hubbard-  \nU  \ninteractions may drive a transition from a trivial to a topological  \nphase. Finally in section 3.3, we discuss the magnetic order  \nexhibited by the system at large interactions and half-filling. The magnetic phase diagram we observe is rich, containing various quantum phase transitions due to the TR-invariant ical spectrum solved on a cylinder for α = 1 / 10 is shown in panel (b).  \nU . We opposite directions and thus ∑ i filled C | ↑〉 i = -∑ i filled C | ↓〉 i . The bulk spectrum in figure 1(a) is invariant under translation ky → k ' y = ky + 2 π / q . This follows from the fact that for γ = λ x = 0 the Hamiltonian H in equation (1) commutes with the magnetic translation operator Tx , which is defined by Txcj = ω -y σ z cj +ˆ x Tx with ω = exp ( i2 πα ) [75, 76]. The phase factor ω -y σ z accounts for the gauge field To understand Haldane's seminal paper it is most instructive to consider the physics of a Dirac Hamiltonian in 2+1 dimensions. For two states | + 〉 and |-〉 (which might be spin, isospin, orbital degree of freedom, etc.) a Dirac theory is governed by the Hamiltonian  \nky .Bulkspectrumisshowninblue,edgestatesinred.Artificial take values as indicated. Doubly degenerate helical edge states (red) are traversing the bulk gaps. . In section 3.2, we show that γ . Both i.e., for cylindrical geometry, we present the energy spectrum for α = 1 / 10 in figure 1(a). Within the gaps of the bulk spectrum (blue), we find a varying number of helical edge modes (red), specifically Nh = { 1 , 2 , 3 , 4 , 4 , 3 , 2 , 1 } from bottom to top. This can be understood from an analogy to the corresponding QHE problem where the TR invariant magnetic field is replaced by a real magnetic field, which breaks TR symmetry. There, each bulk band carries a Chern number of C = 1, except the two touching bands around zero energy which in total carry a Chern number of C = -8. Indeed, the number of chiral edge modes per edge is given by the sum over the Chern numbers of the filled bands Nc = ∑ i filled Ci . The sign of Nc denotes the direction of the chiral edge mode. In the presence of TR symmetry, the sum over the Chern numbers for each gap of course vanishes, since up and down spins move in k y A Chern insulator is defined as a band insulator which possesses a quantized Hall conductivity σ xy but no net-magnetic field. Alternatively, one can think of a translation invariant band insulator with quantized Hall conductivity σ xy . Behind the concept of a Chern insulator stands F. D. M. Haldane's insight that not the orbital magnetic field but the broken time-reversal symmetry is essential for the realization of QHE(Haldane, 1988). Such a QHE without orbital magnetic field or Landau levels is usually referred to as quantum anomalous Hall effect (QAHE). Haldane proposed a simple tight-binding model on the honeycomb lattice with real nearest-neighbor hoppings and complex-valued secondneighbor hoppings (Haldane, 1988). This is accomplished by a staggered flux pattern for the different sublattices A and B such that the total flux enclosed by a hexagon, the smallest lattice cell, is zero. But the second neighbor hoppings acquire magnetic phases which leads to breaking of time-reversal symmetry and a QAHE phase with Chern numbers C = +1 or -1 , respectively. Most remarkably, recent progress in ultracold quantum gases and optical lattices has led to the experimental realization of Haldane's model (Jotzu et al. , 2014).  \nthat breaks translational symmetry along x . The magnetic translation operator Ty remains Tycj = cj +ˆ y Ty .Notethatwhile [ H , Tx ] = [ H , Ty ] = 0, if γ = λ x = 0, the translation operators do not commute between themselves [ Tx , Ty ] = 0, H = ∑ k ( c † k . + c † k , -) h ( k x , k y ) ( c k , + c k , -) (6)  \n/negationslash  \n/negationslash  \n[64]. Fermions acquire a magnetic phase that depends on x when tunnelling along the y -direction. Opposite spins acquire opposite phases due to the factor of σ z . The system remains periodic on the lattice if we choose the magnetic flux per plaquette to be α = p / q with coprime integers p , q . The unit cell contains q sites, and consequently the magnetic Brillouin zone (MBZ) extends between ( kx , ky ) ∈ [ -π q , π q ) × [ -π , π ) . The energy spectrum consists of q bands. The bands are separated by finite energy gaps, except for the two middle bands in case of even q , which touch each other at distinct wave-vectors in the MBZ, where they form Dirac cones. Specifically, this occurs at k = ( -π q , ± ( 2 m + 1 ) π q ) for 2 4 n and at k = ( 0 , ± 2 m π q ) for q = 4 + 4 n with n , m Considering periodic boundary conditions (PBC) along the y -direction and open boundary conditions (OBC) along but only [ T q x , Ty ] = 0. From Txckx , ky = ckx , ky + 2 πα Tx follows the additional degeneracy of the bulk spectrum (see figure 1(a)). Let us now investigate the effect of non-zero γ , λ x = 0 on the spectrum. If either of the two is non-zero, the Hamiltonian does not commute with Tx anymore. While H λ x = ∑ j ( -1 ) x λ xc † j cj still commutes with even powers of Tx ,i.e.,[ H λ x , T 2 n x ] = 0for n ∈ N , one finds that the tx -hopping term only commutes with T q / 2 x for 0 < γ < 0 . 5. This follows directly from ∑ j [ Tx , c † j +ˆ x e -i2 πγσ x cj ] = ∑ j c † j +ˆ x i σ x sin ( 2 πγ ) For α = p/q there are q bands; each of the isolated bands carries a Chern number ∆ C = 1 . According to the TKNN formula, in order to obtain the Hall conductance or total Chern number, associated with a given energy gap, one has to sum up the Chern numbers of the occupied bands. For instance, let us consider the case q = 10 where the lowest gap exhibits a Chern number C = 1 , the second lowest gap C = 2 etc. As mentioned previously, in each gap there must be C chiral edge modes. This is illustrated in Fig.1 (b) where a cylinder geometry has been used; thus there are even C chiral edge modes per edge , which are shown in red for the case q = 10 . where the summation runs over k = ( k x , k y ) corresponding to momentum quantum numbers accounting for the twodimensionality of the system. The 2 × 2 matrix h expressed in terms of Pauli matrices σ i is then given by h ( k x , k y ) = glyph[vector] k · glyph[vector]σ ≡ k x σ x + k y σ y (7) leading to a linearly dispersing spectrum glyph[epsilon1] ± = ±| k | with band-touching at k = 0 at zero energy, see Fig. 2 (a).  \nglyph[negationslash]  \nNote that for any α = 1 / 2 time-reversal symmetry is broken due to the orbital magnetic field. The Hofstadter problem is a realization of the quantum Hall effect on a lattice. When plotting the energy spectrum (without any momentum quantum numbers) vs. the magnetic field strength α ∈ [0 , 1] it shows for the square lattice a remarkable fractal structure having the shape of a moth or a butterfly (Hofstadter, 1976). On other lattices similar fractal structures appear.  \nmagnetic field term α and the spin-mixing term terms may be tuned experimentally. Apart from N«eel and collinear magnetic order we find various spiral phases. To understand the magnetic phase diagram we analytically derive the effective quantum Heisenberg Hamiltonian in this regime. We analyse this Hamiltonian in the classical limit using MonteCarlo simulations and recover the same phases that we found in our RDMFT analysis of the itinerant model at large end with a conclusion in section 4. 2. Non-interacting phase diagram: spectrum and non-Abelian Hofstadter butterfly In this section we consider the non-interacting system and set U = 0. For γ = 0, the two spin species decouple and we obtain two time-reversed copies of the Hofstadter problem Note that for α = p/q Peierls phase factors exp( mi 2 π/q ) appear in the m th row or line, respectively. When m = q , the phase becomes (a multiple of) 2 π which corresponds to the field strength of one flux quantum φ 0 . The corresponding phase factor yields +1 1 . It is further worth mentioning that translational symmetry is broken in the Hofstadter problem. This might appear to be counterintuitive since the applied magnetic field is homogeneous. One can show, however, that the applied vector potential explicitly breaks translation symmetry (Bernevig, 2013; Kohmoto, 1985). The system remains invariant under translation of q sites along the direction of increased unit cell, see for instance the y direction in Fig. 1 (a).  \nq  \n=  \n+  \n0  \n,  \n1  \n,...  \n∈  \n{  \nq  \n.  \n[cos  \n(  \n4  \nπα  \ny  \n)  \n-  \nz  \n<!-- image -->  \nglyph[negationslash]  \n/negationslash  \nBy adding a mass term to (7), h m = mσ z , the Dirac node opens a gap of size 2 m , see Fig. 2 (b). Such a gapped or mas-  \n1  \n-  \ni  \nσ  \nsin  \n(  \n4  \nπα  \ny  \n)  \n]  \ncjTx  \n.  \n(2)  \nsive Dirac theory is always associated with a Chern number C = sgn( m ) / 2 displaying half-quantum Hall effect. The result can be readily derived (Bernevig, 2013; Volovik, 1988). The Berry connection reads  \nA x = -k y 2 √ k 2 + m 2 [ √ k 2 + m 2 -m ] (8)  \nand  \nA y = k x 2 √ k 2 + m 2 [ √ k 2 + m 2 -m ] (9)  \nleading to the Berry curvature  \nF xy = m 2( m 2 + k 2 ) 3 / 2 (10)  \nwhich agrees with the general expression F xy = glyph[epsilon1] abc d a ∂ x d b ∂ y d c / (2 d 3 ) if the Hamiltonian is written as h ( k x , k y ) = glyph[vector] d · glyph[vector]σ with glyph[vector] d T = ( k x , k y , m ) . Integrating F xy over the infinite plane eventually leads to (Bernevig, 2013; Volovik, 1988)  \nC = 1 2 π ∫ d 2 k F xy = m 2 ∫ ∞ 0 kdk ( m 2 + k 2 ) 3 / 2 = m 2 1 √ m 2 . (11)  \nThis half-QHE stems from the continuum description of the Dirac fermion. On a lattice, the bands which have finite band width need to bend down at some point; these contributions of the band will then add another half of a Chern number leading to either a trivial ( i.e. , total C = 0 ) or a topological phase ( i.e. , total C = ± 1 ). Apparently the continuum is not sufficient to determine whether or not a phase is topologically non-trivial; what can be determined, however, is that the Chern number will change by ∆ C = ± 1 when m changes it sign from ± m to ∓ m through a gap-closing transition at m = 0 . The presence of this half-QHE is sometimes also referred to as a meron which is in high-energy physics defined as half an instanton.  \nRegularizing the massive Dirac Hamiltonian and defining it on a lattice brings down the high-energy modes which contain 'the other half of a Chern number'. Depending on the sign ( i.e. , depending on the topology) the two halves of the Chern number cancel or add up to an integer Chern number. Apparently the realization of a topological phase of a lattice Dirac Hamiltonian depends on the model details; nonetheless the two most important lattice realizations of the massive Dirac Hamiltonian are certainly the Haldane model (Haldane, 1988) and the two-orbital square lattice Chern insulator (Qi et al. , 2008b) which corresponds to a single spin-channel of the Bernevig-Hughes-Zhang model (Bernevig et al. , 2006). In the literature these models are referred to as lattice Chern insulators or simply Chern insulators.  \nThe two-orbital square lattice model (Qi et al. , 2008b) is a straight-forward regularization of h = k x σ x + k y σ y + mσ z and reads  \nH 2 -orb . = ∑ k ( c † s, k c † p, k ) h ( k ) 2 -orb . ( c s, k c p, k ) (12)  \nFigure 3 Phase diagrams of the Chern insulator models on the (a) square (Qi et al. , 2008b) and (b) honeycomb lattices (Haldane, 1988). The blue dots (lines) correspond to gap-closing points (lines) where the Chern number changes by | ∆ C | = 1 . At the red dots, at two points in the Brillouin zone the gap closes simultaneously associated with a change of Chern number | ∆ C | = 2 .  \n<!-- image -->  \nwith the orbital indices s and p and the Bloch matrix  \nh ( k ) 2 -orb . = t sin k x σ x + t sin k y σ y + ( m +cos k x +cos k y ) σ z . (13)  \nHere in this simple form there is only one free parameter m (assuming t = 1 ). Possible gap-closing positions in the Brillouin zone are restricted to momenta k x = 0 , π and k y = 0 , π . Assuming that the system is topologically trivial for m → ±∞ ('atomic limit') one readily derives the phase diagram shown in Fig. 3 (a). At m = 0 , two diabolic points ( i.e. , Dirac points) appear at k 0 = (0 , π ) and ( π, 0) which allows the system to change the Chern number by | ∆ C | = 2 . The other gap-closing points at m = ± 2 involve only a single Dirac point changing the Chern number only by | ∆ C | = 1 .  \nTo regularize a massive Dirac Hamiltonian on a honeycomb lattice is rather simple because already a real nearest-neighbor hopping model realizes a spectrum with two band-touching points effectively described by the Dirac Hamiltonian. These band-touching points are denoted as K and K ' located at the corners of the hexagonal Brillouin zone. The honeycomb lattice features a two-atomic basis where we label the different basis sites as A and B , respectively. Adding a potential imbalance between A and B sites correspond to a mass term λ v σ z (Semenoff, 1984). It turns out, however, that the gapped phase caused by λ v σ z is topologically trivial. Duncan Haldane pointed out in his famous paper from 1988 that the mass term leading to the Chern insulator phase must contain not only a sign-change between the sublattices A and B but also between the two Dirac points (or 'valleys') K and K ' , λσ z τ z , where τ corresponds to valley degree of freedom. These conditions can be realized on a honeycomb lattice due to imaginary second-neighbor hopping breaking time-reversal but preserving translation symmetry.  \nThe Haldane model (Haldane, 1988) is governed by the Hamiltonian  \nH H = t ∑ 〈 ij 〉 c † i c j + λ ∑ 〈〈 ij 〉〉 c † i e iφν i j c j + λ v ∑ i ξ i c † i c i . (14)  \nHere t denotes the amplitude for real nearest-neighbor hopping, λ for second neighbor hopping and φ the phase which  \nFigure 4 Spectra computed on a cylinder geometry ('nano-ribbon'). (a) Two-orbital square lattice model (Qi et al. , 2008b) for parameter m = 0 . 75 t . (b) Haldane model (Haldane, 1988) for parameters λ = 0 . 03 t , φ = π/ 2 , and λ v = 0 . Both spectra correspond to the C = +1 phase and feature a single chiral edge mode per edge. Both spectra were computed for a cylinder containing 48 unit cells.  \n<!-- image -->  \nmixes real and imaginary second neighbor hopping, λ v the Semenoff mass, and ξ i = ± 1 on sublattice A or B , respectively. The three parameters λ , λ v , and φ lead to a richer phase diagram than that of the square lattice CI. The phase φ essentially mixes real and imaginary second-neighbor hopping; while the former breaks particle-hole symmetry but leaves the Dirac points unchanged otherwise, the latter breaks TR symmetry and opens the gap into the topological phase. In contrast, λ v opens the gap into a trivial phase. The interplay of φ and λ v leads to the phase diagram shown in Fig. 3 (b). The Bloch matrix corresponding to (14) contains close to the Dirac points K and K ' the mass term d z σ z with  \nd z = M ± 3 √ 3 λ sin φ . (15)  \nglyph[negationslash]  \nPlus and minus signs correspond to valley K and K ' , respectively. In analogy to the limit m → ∞ in case of the square lattice CI, here we consider the atomic limit λ v → ∞ where all particles are localized on one of the sublattices, clearly being topologically trivial ( C = 0 ). This phase remains insulating for 3 √ 3 λ sin φ < M < ∞ and due to adiabaticity the Chern number cannot change. Further decreasing M closes the gap for M = 3 √ 3 λ sin φ at K ' in the Brillouin zone. The sign change of the mass at the Dirac point K ' is again associated with a change of Chern number | ∆ C | = 1 . Upon further decreasing M one passes another gap-closing for M = -3 √ 3 λ sin φ at K and the Chern number changes back to C = 0 . Along the blue lines in the phase diagram Fig. 3 (b) only a single gapless Dirac cone is present. This is by no means a violation of the fermion doubling theorem (Nielsen and Ninomiya, 1981) since TR symmetry is broken for φ = 0 , ± π . Only at φ = 0 or φ = ± π (marked by the red dots in the phase diagram in Fig. 3), TR is intact and for these parameters the upper and lower bands touch simultaneously at both K and K ' points in the Brillouin zone. By analogy, we see that only for m = 0 the square lattice Chern insulator model (12) can be TR invariant. For a deeper discussion about the relationship of TR, fermion doubling, and lattice Dirac fermions we refer the reader to (Herbut, 2011).  \nChern insulator models were also discussed on other lattices - mainly motivated by the search for fractional Chern  \ninsulators (see Sec. III.D.2) - including the kagome lattice (Bergholtz and Liu, 2013; Tang et al. , 2011), the triangular lattice (Venderbos et al. , 2012), the checkerboard lattice (Neupert et al. , 2011a; Sun et al. , 2011), the dice lattice (Wang and Ran, 2011), and the ruby lattice (Hu et al. , 2011).  \nIn the previous section, the first Chern number as a topological invariant was introduced for systems with broken timereversal symmetry (note that the Chern number must vanish whenever time-reversal invariance is enforced). The 'quantization' of the Chern number and its stability is guaranteed by the bulk gap: since the band above the gap carries a topological defect ( i.e. , a monopole in its Berry curvature) and the band below the gap another one (an antimonopole) there is no way to remove these defects (this is a rephrasing of the earlier statement that the sign of the mass in the Dirac Hamiltonian cannot change without gap-closing). Now let us consider a heterostructure consisting of an insulator with finite Chern number and another insulator with zero Chern number ( e.g. vacuum). At the interface the topological defects of the Chern bands ( i.e. , the monopoles in the Berry curvature) must recombine. This is accomplished by locally closing the gap at the interface ( i.e. , the edge of the Chern insulator). This 'local gap-closing' corresponds to the edge states traversing the bulk gap in the energy spectrum. Whenever a system carries a finite Chern number C , there must be C chiral edge modes at its boundaries to trivial insulators or vacuum. That is the bulk boundary correspondence . In Fig. 4 we show the energy spectra of both discussed CI models for parameters corresponding to the C = +1 phase computed on a cylinder geometry. Since a cylinder exhibits two edges the spectra feature two chiral edge modes which cross at k = π where k is the momentum quantum number due to translation invariance along the circumference of the cylinder. Analyzing the corresponding eigenvectors clearly reveals that the two edge modes are localized on opposite edges of the cylinder (not shown here). A chiral edge mode is considered to be stable against not too strong disorder. Loosely speaking the inherent chirality of the system forbids the edge mode to stop or even 'turn around'.  \nIn principle, the quantum Hall and Chern insulators are systems which do not require any symmetry, only TR must be broken. Apart from the fact the TR is a discrete symmetry, note that TR does not need to be spontaneously broken. Moreover, a Chern insulator does not possess a local order parameter. The Chern number which is sometimes considered a 'topological order parameter' is a highly non-local object as it involves summation of the full Brillouin zone, see Eq. (2). Eventually one might ask whether the U(1) symmetry associated with particle number or charge conservation is required which we refer to in the following as U(1) charge . While the quantized Hall conductivity σ xy looses its quantized value when U(1) charge is broken (Volovik, 1988) the Chern number does not change and the chiral edge mode does not acquire a gap. Note that the part of the Hall conductance related to the edge states is in topological superconductors generally expected to be quantized (Volovik, 1992). The persistence of the  \nchiral edge mode (and thus the Chern number) in the presence of a superconducting pairing term has explicitly been shown(Rachel, 2016).", 'C. Z 2 Topological Insulators': "In 2004 and 2005, Kane and Mele(Kane and Mele, 2005a,b) and independently Bernevig and Zhang (Bernevig and Zhang, 2006) proposed the Quantum Spin Hall (QSH) effect in two spatial dimensions. The QSH effect is very similar to the integer QHE with the major differences that no external field is required and time-reversal symmetry is not broken. The usual way most people introduce the QSH effect is by considering Bloch electrons with spin-1/2 degree of freedom: while the ↑ -spins are described by an integer QHE with positive chirality ( C = +1 ), the ↓ -spins by an integer QHE with negative chirality ( C = -1 ) which is the time-reversal conjugate copy of the ↑ -spin system. This construction is manifest in the Bloch-matrix (16) of a typical QSH system: the 2 × 2 block describing the ↑ -spin system is odd under TR, and the 2 × 2 block of the ↓ -spin system is its TR conjugate. The Bloch matrix thus reads  \nH ( k ) QSH = ( H ( k ) QHE 0 0 H ∗ ( -k ) QHE ) (16)  \nwhere the basis is defined via the spinor Ψ † = { c † ↑ ,O 1 , c † ↑ ,O 2 , c † ↓ ,O 1 , c † ↓ ,O 2 } and O 1 / 2 refers to different orbitals or sublattices, respectively. This can be visualized as two quantum Hall layers with opposite chirality, see Fig. 5. By  \nFigure 5 The QSH system as a merger of two QH layers for the opposite spin components with opposite chirality.  \n<!-- image -->  \nconstruction, the resulting system is TR invariant and while the bulk still is insulating the ↑ -spins circulate, say, clockwise and the ↓ -spins counter-clockwise around the sample edge: The QSH effect features helical , spin-filtered metallic edge modes(Murakami, 2011).  \nBefore discussing more of the phenomenology of the QSH effect and its microscopic realizations let us take a different perspective. Following Kane and Mele, in two spatial dimensions and in the presence of time-reversal symmetry, there are precisely two different types of band insulators (Kane and Mele, 2005b): topologically trivial and topologically nontrivial ones, the latter displaying the QSH effect. This is a very strong and a rather unexpected result; these two phases  \nare distinguished or classified by a Z 2 invariant originally introduced by Kane and Mele. Note that this insight in conjunction with the Z 2 invariant directly leads to the finding that in three spatial dimensions there are 16 different band insulators when TR symmetry is conserved (Fu et al. , 2007; Moore and Balents, 2007; Roy, 2009), see below. At that time, Moore and Balents also coined the name 'topological insulator'. Today it is also common in two spatial dimensions, with the names 'QSH insulator' and ' Z 2 topological insulator' often used as synonyms.  \nKane and Mele introduced the QSH effect as a possible ground state for graphene as they realized that TR invariant 'intrinsic' spin-orbit coupling (SOC) of the type glyph[vector] L · glyph[vector] S is allowed by symmetry. This SOC leads to a gap in the energy spectrum of graphene. It is instructive to consider only its z component, L z · S z , as it reveals that the tight-binding version of L z · S z corresponds to a spinful version of the Haldane mass term (Kane and Mele, 2005a; Konschuh et al. , 2010). The Kane-Mele SOC reads  \nH SOC = iλ ∑ 〈〈 ij 〉〉 ∑ αβ = ↑ , ↓ c † i,α ν ij s z αβ c jβ . (17)  \nThe minimal version of the Kane-Mele (KM) model is complemented by the Semenoff mass term H v (Semenoff, 1984) and by a Rashba SOC H R which might be caused by an external field or substrate (Kane and Mele, 2005a),  \nH v = λ v ∑ iσ ξ i c † iσ c iσ , H R = λ R ∑ 〈 ij 〉 c † i ( s × d ) z c j . (18)  \nHere ξ i = +1 for i ∈ sublattice A and ξ i = -1 for i ∈ sublattice B. The vector d points from site i to j . More importantly, while H SOC and H v preserve the U(1) spin symmetry associated with spin conservation, H R breaks it down to Z 2 ; it further breaks particle-hole symmetry and the mirror z →-z symmetry. Kane and Mele showed that the QSH effect remains stable for finite Rashba SOC; that is, the QSH effect is not only a singular point where the U(1) spin symmetry protects two 'non-communicating' quantum Hall systems which are fully characterized by two Chern numbers for the different spin channels, C ↑ and C ↓ = -C ↑ . Instead, the QSH phase remains intact when the U(1) spin symmetry is broken due to H R (see the phase diagram as a function of λ R and λ v as inset between Fig. 6 (a) and (b) (Kane and Mele, 2005b)). In addition, Kane and Mele introduced a new Z 2 invariant (previously mentioned) which does not rely on spin-symmetry and distinguishes the QSH phase from the trivial insulating phase. The helical edge states of the QSH phase remain stable as long as the bulk gap remains finite. Note that the KM phase diagram [inset between Figs. 6 (a) and (b)] is valid only for small λ < 0 . 1 t . Increasing λ further stabilizes an additional phase between TI and the (semi-) metallic phase (at λ v = 0 ) where due to bending of the bulk bands only an indirect gap with helical edge states still persists (Laubach et al. , 2014), see Fig. 6 (c) and (d).  \nAs beautiful the Kane-Mele proposal for QSH effect in  \nconnected  \nthe energy  \nin a zigzag  \nthe insulat-  \nenergy gap  \nedge states  \nbetween  \nthe edge  \ndifference  \neach energy  \nx  \n/.0136  \nk  \nx  \na=  \n2  \n/.0134  \nx  \nsin  \ny  \n4sin  \nx  \ncos  \ny  \nx  \nsin  \ny  \nx  \ncos  \ny  \n146802-2  \nnot have the same spin, spin accumulates at the edge.  \nThe rate of spin accumulation defines a spin Hall conduc-  \ntance  \nd  \nη  \nS  \nz  \nι  \n=dt  \n/.0136  \nG  \ns  \nxy  \nE  \n, where  \nFIG. 10. (Color online) Heat map of the grand potential as a  \n<!-- image -->  \nPHYSICAL REVIEW B 90 ,165136(2014) a cylindric to a toroidal geometry, the bulk spectra should be unchanged with the only difference being that the edges have disappeared, which is exactly what we find. 2006). Since the quantized conductance was independent of sample width it was attributed to edge transport, and today it is widely accepted as the first observation of the QSH effect. This short sequence of theoretical prediction and experimental confirmation is certainly one of the main reasons why the field of topological insulators became so popular. In the meantime, QSH effect was also proposed in InAs/GaSb quantum wells (Knez et al. , 2011) and is today considered as a candidate material (F. Nichele et al. , 2016; Knez et al. , 2014). Topological insulating band structures can be defined on any lattice, early examples include the Lieb and perovskite lattices (Weeks and Franz, 2010b) and the ruby lattice (Hu et al. , 2011).  \nFIG. 8. (Color online) (a) λ R -λ phase diagram for the noninFigure 6 (a) Ribbon spectrum of the Kane-Mele model for finite λ , λ R , and λ v within the QSH phase. (b) Same model but with parameters corresponding to the topologically trivial phase. Inset: λ R -λ v phase diagram with trivial insulator (I) and QSH phases; thick black lines indicate (semi-) metallic regions. Reprinted with permission from (Kane and Mele, 2005b). Copyright (2005) by the American Physical Society. (c) At λ v = 0 , an additional topological semiconductor (TS) phase appears between QSH and metallic phases for sufficiently large λ . (d) Ribbon spectra within the TS phase: topological edge states exist within the indirect bulk gap. Panels (c)+(d) are reprinted from (Laubach et al. , 2014).  \nteracting  \nKane-Mele model displaying the TI, metal (M), and  \ntopological-semiconductor (TS) phase. (b) Zoom into the edge  \nspectrum for λ = 0 . 2, λ R = 0 . 6, U = 0 shown in Fig. 7. (c) U -λ R phase diagram for λ = 0 . 2 in the nonmagnetic regime: the weak TS phase persists in the presence of interactions. graphene might be, subsequent ab initio calculations predicted that the intrinsic SOC is tiny (too small to be experimentally observable) and graphene remains a semi-metal (Konschuh et al. , 2010; Min et al. , 2006; Yao et al. , 2007). Ever since then material scientists are searching for other honeycomb lattice materials with possibly heavier elements to guarantee a stronger SOC necessary to realize the Kane-Mele scenario. A brief discussion of candidate materials is added below.  \nShortly after the initial proposal of Kane and Mele, also Bernevig and Zhang proposed the QSH effect in zinc blende semiconductors under strain (Bernevig and Zhang, 2006). In a subsequent paper, Bernevig, Hughes, and Zhang (BHZ) predicted that the QSH effect should be realized in HgTe/CdTe quantum wells (Bernevig et al. , 2006). The bands in vicinity of the Fermi level become inverted leading to a change of the sign of the mass gap (as it happens for SOC). Using k · p theory BHZ showed that the Bloch matrix around the points in the Brillouin zone where the band inversion occurs is effectively described by a spinful and TR invariant version of the two-orbital CI model (12) in agreement with the QSH structure (16). They further proposed that the most likely source of U(1) spin breaking is due to a bulk inversion asymmetry. Rashba SOC has been considered as another possible source (Rothe et al. , 2010).  \nIn 2007, a quantized and sample-width independent conductance of 2 e 2 /h in HgTe/CdTe quantum wells was measured in the group of L. Molenkamp (König et al. , 2007), in agreement with the predictions of BHZ (Bernevig et al. ,  \nFIG. 9. (Color online) Spectral function  \nA  \n(  \nk,  \nω  \n) on cylindrical  \ngeometry [as defined in Eq. (6)] for  \nλ  \n0  \n.  \n2,  \nλ  \nR  \n0  \n.  \n6, and various  \nB. Strong interactions and magnetic order For finite λ > 0 and λ R = 0, the magnetic region of the phase diagram is an XY antiferromagnet as discussed above. Treating the Rashba term as a small perturbation leaves the magnetic phase unchanged. Thus we expect an XY -AFM in the weakλ R region. First, we use the six-site cluster and compute the grand potential /Omega1 as a function of h x and h y . As expected we find the XY -AFM. /Omega1 as a function of h x and h y shows a perfect circle at finite Weiss fields h x/y (Fig. 10). For the six-site cluster, the saddle point associated with the XY -AFM phase is found at decreasing Weiss fields h x/y when we increase the Rashba coupling. For λ R = 0 . 3(atfixed λ = 0 . 1), we do not find any magnetic solution anymore (see lower panels in Fig. 10). This implies that there is either a true nonmagnetic insulator phase or there is a magnetically ordered phase which cannot be detected within VCA. For instance, this is the case for incommensurate spiral order, where the Weiss field is incompatible with the cluster partitioning. A spiral phase is likely to occur since the spin Hamiltonian [i.e., the Hamiltonian obtained in the strong-coupling limit U →∞ of Eq. (1)] contains terms of Dzyaloshinskii-Moriya type [66]. Recently, spiral order was also found in a Kane-Mele-type model [16], with multidirectional SO coupling in the presence of strong interactions [66,81,82]. In principle, we cannot rule out the existence of the nonmagnetic insulator phase for large U and large Rashba spin-orbit coupling. The existence of such a phase would be The QSH effect is today considered as a paradigm for a symmetry protected topological (SPT) phase. Although it was clear that the QSH effect is protected by TR symmetry ( i.e. , in the absence of TR symmetry backscattering at the helical edges is possible) it was not anticipated at the time that QSH insulators belong to a much bigger class of physical systems. In the meantime it is common knowledge that QSH insulators are protected by TR and U(1) charge symmetry. If either of these symmetries is broken the QSH phase can be adiabatically connected to a trivial band insulator (assuming no other symmetry such as U(1) spin or point group symmetries are present). Usually the absence of topological protection is signalled by a gap opening at the helical edge states. This can be explicitly tested by applying an antiferromagnetic Zeeman field or a superconducting term to the BHZ or KM models. In all cases the edge modes start to gap out immediately; an exception is the KM model for λ R = 0 when the staggered Zeeman field points in the z direction. Due to the particular choice of the Zeeman field U(1) spin is still preserved and takes over the topological protection of the QSH phase (Rachel, 2016). This state has been dubbed 'spin Chern insulator' (Ezawa et al. , 2013). These ideas can be applied to situations where the symmetry breaking perturbations are acting more locally on the QSH system. For instance, only half of the sample or only the sample edges or even a few edge sites can be exposed to such a perturbation and making the edge states disappear accordingly (Rachel and Ezawa, 2014).  \nexciting, in particular, since it could be related to a recently proposed fractionalized quantum spin-Hall phase (dubbed QSH /star )[83]. Inhomogeneous and locally varying perturbations have been used to predict applications and devices, see e.g. (Qi et al. , 2008a; Rachel and Ezawa, 2014; Timm, 2012); the underlying fundamental physics base, however, on the concept of SPT phases which will be discussed in Sec. II.D  \nThe edge states of a QSH insulator can be seen as two counterpropagating spin-filtered chiral edge modes and provide a new symmetry class of one-dimensional liquids: the 'helical Luttinger liquid' (Wu et al. , 2006). In a conventional spinful Luttinger liquid right- and left-moving electrons carrying ↑ -spin or ↓ -spin, respectively, are the constituents of the system. In case of the helical edge theory of a QSH insulator where U(1) spin symmetry is preserved the right-mover is a pure spin-↑ state while the left-mover is a pure spin-↓ state. In the literature this phenomena has been dubbed spin filtering or spin-momentum locking . Since spin and momentum are  \nlength scale  \n1  \n=  \n/C21  \nlator, e.g.,  \n/C30  \nk  \nk  \nx; y  \nspin-rotation matrices  \nB  \nð  \nÞ  \n/  \ninto the bulk of the topological insu-  \ne  \n/C21  \n/C0  \nk  \nð  \nk  \nÞ  \ny  \ne  \nikx  \n.  \nAs a consequence, the  \ninvolve convolutions over  \ny  \nof  \nthe two-dimensional eigenstates along with the 2D inter-  \nV  \nx; y  \n.  \nnot independent degrees of freedom anymore, the helical liquid is reminiscent of a spinless Luttinger liquid despite its spin index. In a helical liquid, the topological protection due to TR symmetry is reflected in the absence of all terms related to elastic single-particle back-scattering (Wu et al. , 2006). This can intuitively be understood in a simplified picture where we denote the right-moving (left-moving) edge state as | R, ↑〉 ( | L, ↓〉 ). Any back-scattering term should contain a matrix element of the form 〈 L, ↓ ∣ ∣ R, ↑ 〉 which is identically zero if both states are fully polarized in opposite spin-projection. Assuming that both | L, ↓〉 and | R, ↑〉 are eigenstates to the same energy, 〈 L, ↓ ∣ ∣ R, ↑ 〉 = 0 even when the axial spin symmetry U(1) spin is broken; this follows from Kramers theorem. action potential U ð x; y Þ and impurity potential ð Þ Importantly, however, the smallk behavior of B k is always compatible with the expansion (5), since the latter follows from TR invariance. In order to justify our effective 1D model (1)-(4), we determined the momentum-dependent rotation of the spin quantization axis explicitly for the BHZ model [10] in the presence of Rashba SOI. This model provides a description of 2D topological insulators realized in HgTe = CdTe quantum wells. We used exact diagonalization to solve the Hamiltonian on a cylinder of width W , with periodic boundary conditions in the x direction and edges at y ¼ 0 and y ¼ W . In the following, we shall outline the procedure and the results; details will be published elsewhere. The 2D topological insulators realized in HgTe = CdTe quantum wells can be modeled using a Hamiltonian  \nIt is worthwhile to further elaborate on the role of the axial spin symmetry, U(1) spin . In experimental situations it will usually be broken since external electrical fields and the effect of a substrate are unavoidable (Kane and Mele, 2005b). In the absence of the U(1) spin symmetry the spin quantization axes will be tilted away from the ˆ z direction. Moreover, as demonstrated explicitly in (Rod et al. , 2015; Schmidt et al. , 2012), the spin quantization axes rotates as a function of momentum (see Fig. 7): H ¼ P k H ð k Þ , where H ð k Þ has a 4 /C2 4 in a basis containing two spin-degenerate  \n½  \nB  \ny  \nk  \n1  \nB  \nk  \n2  \n/C138  \n¼  \nZ  \ndy  \n/C30  \ny  \n/C11  \n1  \n;U;k  \n1  \nð  \nx  \n0  \n;y  \nÞ  \n/C30  \n/C11  \n2  \n;U;k  \n2  \nð  \nx  \n0  \n;y  \nÞ  \n:  \nmatrix structure orbitals, FIG. 3 (color online). Off-diagonal component ½ B y k 1 B k 2 /C138 /C0þ of the spin-rotation matrix determined from the numerical solution of the BHZ model with Rashba SOI using Eq. (13). 156402-4 Figure 7 Rotation of spin quantization axis as function of momenta, K ( k 1 , k 2 ) (reprinted from (Schmidt et al. , 2012)). The plot corresponds to the BHZ model with finite bulk inversion asymmetry which breaks the U(1) spin symmetry.  \n<!-- image -->  \nK ( k 1 , k 2 ) = ∫ dxψ † -,k 2 ( x, y 0 ) ψ + ,k 1 ( x, y 0 ) . (19)  \nglyph[negationslash]  \nHere ψ ± ,k ( x, y ) denotes the wave function of the right (left) moving edge state with momentum k , evaluated at real space position ( x, y ) ( x is the coordinate perpendicular to the edge). At long wavelengths one finds K ( k 1 , k 2 ) ≈ k -2 0 ( k 2 1 -k 2 2 ) with the material-specific parameter k 0 . The corresponding edge theory of a QSH insulator where the U(1) spin symmetry is broken has been named generic helical liquid (Kainaris et al. , 2014; Schmidt et al. , 2012). The physics of TR invariant backscattering potentials with broken U(1) spin symmetry or a 'Rashba impurity' have been extensively discussed (Crépin et al. , 2012; Geissler et al. , 2014; Kharitonov et al. , 2017; Lezmy et al. , 2012; Ström et al. , 2010; Xie et al. , 2016). For an elastic scattering process, i.e. , initial and final state have the same energy, left- and right-mover still have orthogonal spin components due to Kramer's theorem. In contrast, in case of an inelastic process right- and left-mover are not orthogonal anymore and the matrix element which enters the back scattering Hamiltonian will in general be finite, 〈 L ∣ ∣ R 〉 = 0 . In the presence of electron-electron interactions and an impurity, it has been shown that the combinations of these ingredients give rise to a temperature dependent correction of the conductance, δG ∝ T 4 . It turned out that this is at low temperatures the strongest correction to the otherwise quantized spin Hall conductance (Kainaris et al. , 2014; Schmidt et al. , 2012). To derive the dependence of the spin quantization axes on momentum we originally considered a cylindrical geometry where momentum is associated with translation symmetry around the circumference. That is, the Brillouin zone is effectively one-dimensional. In order to approach geometries as used in real experiments, e.g. Hall bars, one can also consider circular disks where the rotation of spin quantization axes can  \nbe obtained as a function of angular momentum (Rod et al. , 2015). Since the energy spectrum is in the vicinity of the crossing point of the edge modes approximately linear the (angular) momentum is locked to the energy values. This allows to define the rotation of spin quantization axes depending on the energy of the initial and final states in the absence of any translational or rotational symmetries (Rod et al. , 2015). As discussed in Sec. III.B.4, the broken U(1) spin symmetry does not only alter the spin texture of the helical edge states but it also influences the phase diagrams of the topological Hubbard models drastically.  \nHelical Luttinger liquids (Wu et al. , 2006) represent an important subject as it opens the research topic 'topological insulators' to the field of one-dimensional electron systems and the powerful methods developed therein. Also, helical liquids allow to study interesting scenarios such as helical quantum edge gears (Chou et al. , 2015) or, in combination with disorder and interactions, the emergence of gapless glassy edge states spontaneously breaking time-reversal symmetry (Chou et al. , 2017), just to mention a few. A recent review article discusses the edge physics of 2D TIs and helical Luttinger liquids (Dolcetto et al. , 2016).  \nNow let us consider the three-dimensional (3D) case: while the integer quantum Hall effect does not exist in odd space dimensions, shortly after the prediction of the 2D QSH insulating state, three groups indepedently introduced the 3D generalization (Fu et al. , 2007; Moore and Balents, 2007; Roy, 2009). The central idea is to apply the Z 2 invariant previously introduced for 2D systems to 3D Bloch matrices H ( k x , k y , k z ) . By fixing one of the momentum quantum numbers k x , k y , or k z to take the constant value 0 or π , the remaining Bloch matrix is effectively two-dimensional. Let us name the new invariants x 0 := Z 2 [ H (0 , k y , k z )] , x π := Z 2 [ H ( π, k y , k z )] , y 0 := Z 2 [ H ( k x , 0 , k z )] etc. Each of these six new invariants can take the values even and odd, but from the 2 6 = 64 possible combinations only 16 are inequivalent. Eventually we define the 'strong' invariant ν = x 0 · x π =  \nð  \nÞ  \n/C11  \n1  \n/C11  \nV  \n2  \ny 0 · y π = z 0 · z π 2 and simply write ( ν ; x 0 , y 0 , z 0 ) in order to fully characterize the phase. Thus there are 16 time-reversal invariant and insulating phases in 3D (Fu et al. , 2007; Moore and Balents, 2007; Roy, 2009). Eight of them are so-called strong topological insulators (STIs) with ν = 1 and seven are weak topological insulators (WTIs) with ν = 0 ; the phase with (0; 000) is the trivial insulating state. An efficient way to compute these invariants numerically has been derived (Fukui and Hatsugai, 2007). STIs possess at the surface or interface to a topologically distinct phase an odd number of surface states which realize a 2+1-dimensional Dirac theory (Qi et al. , 2008b). In contrast, WTIs feature surfaces with an even number (including zero) of such surface states. In a topological quantum field theory there is only the distinction between topological insulators with θ = π and trivial insulators with θ = 0 (Qi et al. , 2008b). While the latter corresponds to STIs, the former to topologically trivial states of matter including WTIs. That is the main reason why WTIs did not attract much attention in the early years of topological insulators. First detailed studies of WTIs in the presence of disorder or interactions revealed, however, that WTIs are not much less robust than their strong cousins and similarly interesting (Kobayashi et al. , 2013; Liu et al. , 2012a; Mong et al. , 2012). Recently, also the topological terms of interacting topological insulators were derived (Guo et al. , 2018; Wang et al. , 2015b) consistent with Witten's approach (Witten, 2016).  \nAfter the prediction and discovery of 2D and 3D TIs about ten years ago, several extensions and generalizations of these states have attracted much attention: for instance, topological crystalline insulators (Fu, 2011), Weyl semimetals (Bevan et al. , 1997; Volovik, 2003; Wan et al. , 2011; Xu et al. , 2015a) (sometimes referred to as gapless topological insulators), and 'new fermions' (Alexandradinata et al. , 2014). In addition, a full classification of topological bandstructures based on quantum chemistry considerations has been derived (Bradlyn et al. , 2017).  \nIn the following, we will give a very short overview of candidate materials for 2D and 3D TI phases. A more complete and thorough discussion of materials can be found in the relevant review articles (Ando and Fu, 2015; Hasan and Kane, 2010; Hasan and Moore, 2011; Qi and Zhang, 2011; Ren et al. , 2016; Yan and Zhang, 2012).  \nIn two spatial dimensions, besides the previously discussed quantum well systems HgTe/CdTe (König et al. , 2007) and InAs/GaSb (Knez et al. , 2014) the main focus has been on other honeycomb-lattice materials involving sufficiently heavy atoms to produce a significant spin-orbit gap. Most interestingly is the family of silicene, germanene, and stanene (Geissler et al. , 2015; Liu et al. , 2011; Rachel and Ezawa, 2014; Xu et al. , 2018, 2013), where the latter is the most promising system. Also molecular  \ngraphene (Ghaemi et al. , 2012) has been suggested and coldatom settings (Goldman et al. , 2010) which might feature the QSH effect. Recently, the adatom system bismuthene on a SiC substrate (Reis et al. , 2017) has been reported to feature a signficantly large bulk gap suggesting even the presence of a room temperature QSH phase.  \nIn three spatial dimensions, after the discovery of the first STI material Bi 1 -x Sb x (Hsieh et al. , 2008) a countless number of candidate materials has been proposed and many of them were claimed to be successfully measured and identified as TIs. Here we only review the first few experiments. The discovery of the first STI phase was reported in the compound Bi 0 . 9 Sb 0 . 1 (Hsieh et al. , 2008; Teo et al. , 2008). Angle-resolved photoemission spectroscopy (ARPES) measurements on the (111) surface of Bi 0 . 9 Sb 0 . 1 demonstrated the presence of unusual surface states which cross the Fermi energy five times between ¯ Γ and ¯ M . Another feature of the topological surface state is their inherent π Berry phase since spin and momentum are locked. Within spin resolved ARPES also this feature could be observed (Hsieh et al. , 2009b).  \nThe second generation of materials includes the prominent systems Bi 2 Se 3 , Bi 2 Te 3 , and Sb 2 Te 3 . In contrast to the experimental challenges to handle BiSb compounds, in particular Bi 2 Se 3 shows topological insulating behavior even at room temperature and in the absence of magnetic fields (Hasan and Kane, 2010). Moreover, Bi 2 Se 3 features a single Dirac cone on its surface (Hsieh et al. , 2009a). At the same time, pioneering ab initio calculations identified Bi 2 Se 3 , Bi 2 Te 3 , and Sb 2 Te 3 as single-Dirac-cone materials (Zhang et al. , 2009a).  \nFurther important TI materials are HgTe and α -Sn, as proposed already in 2007 (Fu and Kane, 2007). Strained HgTe is a strong TI as shown through magneto transport and ARPES measurements (Brüne et al. , 2011). Similarly, strained α -Sn realized a strong 3D TI phase as revealed through ARPES measurement combined with ab initio calculations; α -Sn represents the first elemental TI which is promising for engineering future devices (Barfuss et al. , 2013; Ohtsubo et al. , 2013).  \nToday, topological insulators represent a major research direction for the fields of spectroscopy, material growth, and transport measurements (Ando and Fu, 2015; Hasan and Kane, 2010; Hasan and Moore, 2011; Qi and Zhang, 2011; Ren et al. , 2016; Yan and Zhang, 2012).", 'D. Symmetry Protection vs. Topological Order': "The discovery of the quantum spin Hall effect and topological insulators accelerated the field of topological phases and turned it into one of the most active fields of contemporary condensed matter research. As a consequence, a large amount of scientific publications related to condensed matter physics contain nowadays the word 'topological'. Unfortunately, there are (at least) two important concepts which are often mixed up in the literature: symmetry protected topological (SPT) phases and topologically ordered phases. In the following, we will briefly define them, list the most impor-  \ntant examples, and emphasize the difference between them. Note that both concepts are very active fields of research and this section only aims to give a brief overview but neither a detailed review nor a complete list of references. For further reading we refer to recent review articles (Senthil, 2015; Wen, 2017).  \nAn SPT phase (Chen et al. , 2012, 2010; Pollmann et al. , 2010; Volovik, 2003; Wang et al. , 2015b; Wen, 2012, 2014) has short-range entanglement, is protected by one or several symmetries, and carries gapless modes at its edges or boundaries, respectively. The central idea is that such a phase is stable against (small) perturbations as long as the protecting symmetry is preserved. When stronger perturbations are applied, a change of the ground state is only possible via closing of the bulk gap. We previously discussed the QSH effect as a paradigm for an SPT phase (the discussion also applies to 3D TIs). Note that free fermion phases in arbitrary dimensions were classified even before SPT phases became popular: Schnyder, Ryu, Furusaki, and Ludwig (Schnyder et al. , 2008) and independently Kitaev (Kitaev, 2009) proposed a table ('ten-fold way') corresponding to the random matrix classification of Altland and Zirnbauer (Altland and Zirnbauer, 1997; Zirnbauer, 1996). This classification uses anti-unitary symmetries, time-reversal and particle-hole, to distinguish the different phases. The TR invariant Z 2 topological insulators belong to class AII in this table, the integer QHE to class A (the class which does not require any symmetry agreeing with our previous statement). Topological insulators which rely on other symmetries such as inversion - and thus fall out of this classification scheme - were proposed (Hughes et al. , 2011). Also topological crystalline insulators - TIs which are protected by a space group symmetry instead of TR were proposed (Fu, 2011) and subsequently discovered experimentally (Liu et al. , 2014a), representing today another exciting research direction (Kargarian and Fiete, 2013; Liu et al. , 2013a, 2014a; Slager et al. , 2013; Xu et al. , 2012).  \nAnother instructive example of an SPT phase is the spin 1 antiferromagnetic chain (aka Haldane chain) (Haldane, 1983b) including the exactly soluble AKLT model (Affleck et al. , 1987). Here the protecting symmetries are TR, bondinversion, and dihedral symmetries (Pollmann et al. , 2010). One needs to break all these symmetries in order to loose its character, i.e. , only then it is possible to adiabatically transform it to the atomic limit, a trivial product state, without closing of the bulk gap. The spin 1 chain features gapless zero modes at its edges, the so-called dangling spins. Other SPT phases of interests are bosonic TIs (Liu et al. , 2014b).  \nThe concept of SPT phases has been extensively applied to classify quantum phases in the past years. In one spatial dimension, SPT phases were claimed to be completely classified by the elements in the second group cohomology class (Chen et al. , 2011; Pollmann et al. , 2010). Also for non-interacting fermionic systems in d spatial dimensions a complete characterization has been derived based on both group cohomology theory and K -theory (Kitaev, 2009; Schnyder et al. , 2008; Wen, 2012). Classification based on SPT phases for interact-  \ning bosonic states of matter in d > 1 have recently been explored (Chen et al. , 2012).  \nNote that the previously discussed example of the spinChern insulator demonstrates that also other symmetries can protect quantum phases. TR symmetry and particle hole ( i.e. , charge conjugation) are certainly the most robust symmetries - because they are anti-unitary symmetries. But inversion, mirror or other point group symmetries can in principle lead to 'topological' protection, the previously mentioned 'crystalline topological insulators' constitute a prominent example (Fu, 2011). In such cases, the presence of edge states is, however, not guaranteed (Hughes et al. , 2011). It is worth emphasizing that SPT phases can be realized in non-interacting theories, i.e. , systems of free fermions, but are not limited to them.  \nThis is in stark contrast to systems exhibiting (intrinsic) topological order. First introduced by X.-G. Wen (Wen, 1990) in the context of the fractional QHE, today several families of strongly correlated electron systems are known to be topologically ordered . In principle, no symmetry is required to be preserved for this type of order - but no symmetry is required to be spontaneously broken either 3 . Topologically ordered states possess long-range entanglement (which immediately rules out free fermion theories), exhibit a ground state degeneracy when defined on a torus but no degeneracy when defined on a sphere, and the elementary excitations of such systems are anyons, exotic quantum particles obeying fractional statistics. Examples are the ν = 1 /m Laughlin fractional quantum Hall states ( m being an odd integer) featuring an m -fold degenerate ground state on the torus and the toric code (Kitaev, 2003) as a prototype of an Z 2 spin liquid featuring a four-fold degenerate ground state on the torus geometry. Different topological orders can be characterized either by their topological quantum field theories or by their modular matrices (Zhu et al. , 2014). Recently, a hierarchy construction which allows deriving all topological orders in two spatial dimensions was developed (Lan and Wen, 2017). In this review, we are only touching topologically ordered phases when discussing the CI glyph[star] and QSH glyph[star] phases in Sec. III.B.2 and in Sec. III.B.4, respectively, topological Mott insulators (in Sec. III.D), fractional Chern and topological insulators (Sec. III.D.2), Kitaev materials (Sec. IV) and when discussing the effect of electronelectron interactions on the surface of a strong TI in Sec. III.E.  \nSeveral comments are in order.  \n- (i) Topologically ordered phases always involve (strong) electron-electron interactions.\n- (ii) In some cases in the literature, topological insulators are referred to as systems with 'topological order' - which has nothing to do with the previously discussed concept of intrinsic topological order.  \n(iii) The role of interactions in the context of SPT phases is somewhat ambivalent - on the one hand they can stabilize SPT phases which do not have a free fermionic analog, a nice example is provided by parafermionic chains (Alicea and Fendley, 2016; Motruk et al. , 2013); on the other hand they can destroy SPT phases, best illustrated by the reduction of the BDI symmetry class to the Z 8 classification in one spatial dimension (Fidkowski and Kitaev, 2011; Turner et al. , 2011). To test this reduction experimentally was suggested both in the superlattice material CeCoIn 5 /YbCoIn 5 (Yoshida et al. , 2017a) and in a cold atom system (Yoshida et al. , 2017b).  \n(iv) It is possible to construct states of matter which exhibit topological order and possess a protecting symmetry simultaneously. Such states of matter are named symmetry enriched topological (SET) phases (Lu and Vishwanath, 2016). A straight-forward example are TR-invariant fractional topo-  \nlogical insulators (Levin and Stern, 2009), see Sec. III.D.2. (v) The integer QHE is neither topologically ordered (because it is a free fermion theory) nor protected by any symmetry. On of all topological phases. The QHE simply realizes a chiral phase. The example of the QHE nicely demonstrates the dilemma of characteriztopological phases in a simple fashion. This issue is,  \nthe other side, it is clearly the mother state ing all however, beyond the scope of this review.", 'III. INTERACTING TOPOLOGICAL INSULATORS': 'The previously discussed Chern and topological insulators are fascinating states of matter in all the varieties they appear. Most notably, they are all described by theories of free fermions. In condensed matter physics, many of the striking phenomena are due to (strong) electron-electron interactions. Examples include unconventional superconductivity (Bednorz and Müller, 1986; Steglich et al. , 1979; Stewart, 2017), the Kondo-effect (Kondo, 1964), or the Mott-Hubbard transition (Mott, 1968), just to mention a few. Furthermore, in all solids Coulomb interactions are unavoidable, sometimes they might be screened and weak, but sometimes they are strong and can even dominate the physical properties of a material. Therefore it is both a natural and important question to ask what the effect of interactions in these topological insulators might be. One can group the most relevant aspects into the following list of questions:  \n- · Is a topological bandstructure stable with respect to electron-electron interactions and what happens when the TI phase breaks down?\n- · The conceptually inverse question is whether electronelectron interactions can turn a topologically trivial ground state into a non-trivial one? With other words, are interaction-induced topological phases possible?\n- · Can more exotic states of matter emerge when the nontrivial band topology competes with strong electronelectron interactions? How does the band filling affect this competition?  \n- · How are the topological edge or surface states influenced by strong interactions?\n- · Eventually one can ask to what extent the topological properties of non-interacting bandstructures become relevant for the corresponding strong coupling phases of certain frustrated quantum magnets?  \nIn the following, we will focus on all these aspects - some of them are discussed in more detail, others are briefly mentioned.', 'A. Early Considerations of Interacting Topological Insulators': "Already in the first Kane-Mele paper (Kane and Mele, 2005a) the effect of long-ranged Coulomb interactions in graphene is considered: in order to estimate the value of the spin-orbit potential Kane and Mele took into account the renormalization of the spin-orbit gap ∆ SO due to the interaction of electrons with the exchange potential induced by ∆ SO . They claimed that Coulomb interactions may increase the energy gap. The renormalization causes a divergent correction to ∆ SO which together with other contributions can be summed using the renormalization group. Once the renormalization group equations are derived and solved one can indeed see that the spin-orbit potential is enhanced due to Coulomb interactions. Note that these considerations provide an argument for the stability of the TI phase with respect to electronelectron interactions but they are limited to the perturbative regime where Coulomb interactions are weak.  \nMost other works focussing on correlation effects included the interactions exclusively for the edge states on the level of helical Luttinger liquids (Wu et al. , 2006). Another notable early work addresses the effects of interactions in combination with disorder for the topological edge states of TIs (Xu and Moore, 2006).  \nAnother direction where interactions were considered enclose the attempts to generalize or extend the KaneMele invariant (Kane and Mele, 2005b) to interacting systems (Gurarie, 2011; Wang et al. , 2010, 2012d; Wang and Zhang, 2014); several of these papers are inspired by earlier work of Volovik (Volovik, 2003). An interesting discussion can be found in Ref. (Budich and Trauzettel, 2013). The main idea is to express the invariant in terms of the single-particle Green's function and its derivatives. It can be shown that these formulations are equivalent to the KaneMele(Kane and Mele, 2005b) or Fu-Kane (Fu and Kane, 2007) invariants in the absence of interactions. They allow, however, easily to implement electron correlations encoded in the self-energy (Gurarie, 2011; Wang et al. , 2010; Wang and Zhang, 2014) and thus presumably extend the classification of non-interacting topological bandstructures to their correlated counterparts. Another interesting concept is the topological Hamiltonian (Wang and Yan, 2013) which is defined as the Bloch matrix of the noninteracting system plus the self-energy  \nevaluated at zero frequency, h topo = h ( k ) + Σ( ω = 0 , k ) . In particular for numerical methods which rely on strategies to efficiently compute the full Green's function ( e.g. dynamical mean-field theory, quantum cluster appraoches etc.) it has the major advantage to be easily accessible in contrast to other invariants.", 'B. Correlated Topological Insulators': 'Testing the stability of topological bandstructures and deriving interacting phase diagrams of model Hamiltonians has stimulated a large amount of research in the past years. Here we consider several paradigmatic two-dimensional TI models supplemented by Coulomb interactions. In order to simplify the discussion we always consider local Hubbard interactions for spinful models,  \nH I = U ∑ i n i ↑ n i ↓ , (20)  \nand nearest-neighbor interactions for spinless models,  \nH (1) I = V 1 ∑ 〈 ij 〉 n i n j , (21)  \nif not mentioned otherwise. Although the Hubbard term (20) represents the simplest interaction term for spinful electrons, in two and three spatial dimensions there is no exact solution for Hubbard-type models. The focus of this section will be mainly on the stability of the TI phase, on the emergence of conventional orders due to spontaneous symmetry breaking, but also on novel, exotic phases. There is an earlier review (Hohenadler and Assaad, 2013) about correlated TIs covering some of the material discussed in the following.', '1. One-dimensional Correlated Topological Insulators': "As discussed earlier, topological (Chern) insulators are lattice realizations of the Dirac Hamiltonian. Originally, theoretical and experimental interest was primarily focussed on 2D and 3D systems. In the meantime, also 1D systems have been considered with and without inclusion of electron-electron interactions and experimental setups proposed and even realized. Details can be found in the brief review article (Guo, 2016).  \nThe simplest model realizing a 1D TI is the Su-SchriefferHeger (SSH) model which was introduced in 1979 to describe polyacetylene (Heeger et al. , 1988; Su et al. , 1979). It is a Peierls-type model with alternating weak and strong hoppings governed by the Hamiltonian  \nH SSH = ∑ i ( t -( -1) i δt ) [ c † i +1 c i +H . c . ] . (22)  \nFor δt = 0 the model is gapless and describes free fermions with cos k dispersion. For finite δt , the energy spectrum is  \ngapped being topologically non-trivial (trivial) for δt < 0 ( δt > 0 ). Often also the spinful version of (22) is discussed. Topological invariants are available, either based on Berry phase, single-particle Green's functions, or entanglement entropy (Guo and Shen, 2011; Manmana et al. , 2012; Wang et al. , 2015a). Other 1D TIs are based on superlattices or Creutz-type models (Gholizadeh et al. , 2018; Hetenyi and Yahyavi, 2018). As mentioned before, the topological properties can be best understood from the perspective of gapped Dirac Hamiltonians (Shen et al. , 2011).  \nSeveral of these models have been discussed in the presence of electron-electron interactions: spinless SSH Hubbard model (Sirker et al. , 2014), spinful SSH Hubbard model (Barbiero et al. , 2018; Manmana et al. , 2012; Wang et al. , 2015a) as well as other 1D topological Hubbard models (Guo and Shen, 2011; Jünemann et al. , 2017). All of them turn out to be robust against weak or moderate interaction strength. Also a bosonic SSH model with repulsive interactions has been analyzed including topologically non-trivial Mott phases (Grusdt et al. , 2013).  \nExperimental progress in realizing 1D TIs has been made using ultra-cold atoms or photonic crystals (Atala et al. , 2013; Kraus et al. , 2012; Verbin et al. , 2013). At least ultra-cold quantum gases allow to tune finite interactions between the atoms and thus realize interacting TIs in 1D.", '2. Correlated Chern Insulators and Haldane-Hubbard model': 'The first paper addressing the effect of electron-electron interactions in the spinless Haldane model was an exact diagonalization study using the Lanczos method (Varney et al. , 2010). Triggered by the nearest-neighbor Coulomb term (21) the transition from the weakly-correlated Chern insulator phase with circulating edge modes into the Mott phase with charge density wave order was investigated. The analogous study for hardcore bosons did not show any topological features (Varney et al. , 2010). Later, the bosonic Haldane-Hubbard model was reconsidered; superfluid and Mott insulating phases supporting local plaquette currents were found (Vasi\'c et al. , 2015).  \nMotivated by the experimental realization of the Haldane Chern insulator in an optical lattice (Jotzu et al. , 2014), \'spinful\' versions of the Haldane model were investigated where both spin channels feature a Chern number leading to a Chern insulator phase with C = ± 2 (\'doubled Haldane model\'). The effect of interactions was studied using slaverotor method (Hickey et al. , 2015) as well as slave-spin technique (Maciejko and Rüegg, 2013; Prychynenko and Huber, 2015). Besides the variety of magnetically ordered phases, also correlated CI and Chern metal phases were found; most remarkably, it was suggested that an exotic CI glyph[star] phase could possibly be realized given that the topological as well as the interaction term are sufficiently large (Maciejko and Rüegg, 2013). This phase corresponds to the \' Z 2 fractionalized Chern insulator" found in an exactly solvable model (Zhong et al. ,  \nPRL  \n109,  \n229902 (2012)  \nPHYSICAL REVIEW LETTERS  \nPHYSICAL REVIEW LETTERS  \nJOHANNES REUTHER, RONNY THOMALE, AND STEPHAN RACHEL  \nPHYSICAL REVIEW B  \n86  \n,155127(2012)  \nPRL  \n109,  \n229902 (2012)  \n2013). Properties of the exotic phase include topological order, chiral edge states, a quantized Hall conductance. It can be seen as the chiral, time-reversal broken version of the QSH glyph[star] phase or as an \'orthogonal Chern insulator\', further discussed in Sec. III.B.4. In contrast to the CI glyph[star] phase found within slavespin method, slave-rotor theory led to the prediction of chiral spin liquid phases for sufficiently strong interactions (Hickey et al. , 2015). XY - AFM TI SM U/t λ/t Neel (a) 0.6 0.8 0.4 0.2 0 0 2 4 U/t (a) λ =0 . 2 ,U =2 . 0 ω 2 λ/t FIG. 2 (color online). measured along the edge. The parameters are WEI WU, STEPHAN RACHEL, WU-MING LIU, AND KARYN LE HUR × α /star -( k 1) α -( k 2) α /star + ( k 3) α + ( k 4) 〈 l † k 1 ↑ l k 2 ↑ l † k 3 ↓ l k 4 ↓ 〉 = 1 N 2 ∑ | α -( k 1) | 2 | α + ( k 3) | 2 , (4) λ =0 . 2 ,U =2 . 0 0 2 0 2 WEI WU, STEPHAN RACHEL, WU-MING LIU, AND KARYN LE HUR × α /star -( k 1) α -( k 2) α /star + ( k 3) α + ( k 4) 〈 l † k 1 ↑ l k 2 ↑ l † k 3 ↓ l k 4 ↓ 〉 = 1 N 2 ∑ | α -( k 1) | 2 | α + ( k 3) | 2 , (4)  \n(HLL).  \nPHYSICAL REVIEW B  \n85  \n=0  \n.  \n2  \n,U  \n=3  \n.  \n0  \n˜  \nλ/t  \ndiagram of the sodium-iridate Hubbard model as obtained in Ref. 33. The transition from TI to a valence bond solid (VBS) phase that links  \n0  \n1.0  \nIn the meantime, also numerical studies for the spinful Haldane-Hubbard model are available. Within Dynamical Cluster Approach a direct transition from the the Chern insulator phase into an antiferromagnet was found which is likely to be of first order (Imriška et al. , 2016). Using dynamical meanfield theory, the model has been studied with additional staggered sublattice potential H v as defined in Eq. (18). Besides the trivial band insulating and trival (Mott) insulating phases (magnetism was not considered), an intermediate phase with C = 1 was found (Vanhala et al. , 2016). Another example of a correlated Chern insulator was discussed in Ref. (Doretto and Goerbig, 2015) where a staggered π -flux lattice leads to explicit time-reversal breaking (the same bandstructure was the starting point in Ref. (Neupert et al. , 2011a)). Using a bosonization formalism, the quarter filled system can be mapped to an interacting boson model; interestingly, the spin-wave excitation spectrum above the correlated Chern insulator remains gapless while for the analogous TR-invariant version it features a gap. insulator (TI) to XY -plane antiferromagnet (AFM) was derived within slave-rotor theory which underestimates Uc . (b) Mean-field phase to AFM was derived within slave-spin theory which overestimates Uc . The phase diagram of (b) is qualitatively similar to (a) apart from the additional \'QSH /star phase.\' At λ = ˜ λ = 0 and not too large U the semimetal (SM) phase of graphene is present. See main text for details. instead obtains a valence bond solid (VBS) phase. In the limit ˜ λ → 0itisobviousthatoneshouldfindNeelorderinsteadand that the VBS order is an artifact of the specific slave-particle approach. Regarding the values of Uc (e.g., for λ = ˜ λ = 0), one should keep in mind that the microscopic Uc ∼ 4 . 3 as found within QMC 45 is underestimated by slave-rotor theory ( Uc = 1 . 68) while it is overestimated by the slave-spin approach ( Uc ∼ 8) (Fig. 2). IV. STRONG-COUPLING LIMIT We consider the limit of infinitely strong electron-electron interactions. As a result, charge fluctuations are frozen out and we obtain a pure spin Hamiltonian at half filling. Most importantly, the complex next-nearest-neighbor spin-orbit hoppings result in anisotropic and more complicated second-neighbor spin exchange terms which we analyze in the following. A. Kane-Mele spin model Taking the limit U →∞ of the Kane-Mele-Hubbard model (4) results in the effective spin model 13 H KM = J 1 ∑ 〈 ij 〉 S i S j + J λ ∑ 〈〈 ij 〉〉 [ -S x i S x j -S y i S y j + S z i S z j ] , (6) where J 1 = 4 t 2 /U and J λ = 4 λ 2 /U . The second-neighbor to compare other numerical approaches against the PFFRG method, which we will use in the following. B. Sodium iridate spin model The strong-coupling limit of the SIH model is given by the spin Hamiltonian H SI = J 1 ∑ 〈 ij 〉 S i S j -J ˜ λ ∑ 〈〈 ij 〉〉 S i S j + 2 J ˜ λ γ Note that the γ links are the second-neighbor links [the green, red, and blue lines in Fig. 1(b)]. It is structurally similar to the Heisenberg-Kitaev (HK) Hamiltonian which has been found to adequately describe the A2IrO3 iridates from a spin-orbit Mott picture (A = Na or Li). 38 Whereas the SI model assumes the nearest-neighbor hybridization to be trivial and to be essentially given by real Ir-Ir hybridization, the kinetic theory underlying the HK model more carefully resolves the emergent terms from a multiorbital Ir-O cluster superexchange model. 48,49 Depending on the Ir-O-Ir angle, these terms are either more or less relevant than the next-nearest-neighbor exchange terms which are considered in the SI model. 37 For the links in vertical direction (links with σ z ), one obtains the same term as for H KM,whileforthelinks associated with σ x one finds + S x i S x j -S y i S y j -S z i S z j and so on. For H KM we have seen that the magnetization turns into the XY plane. Here, however, the term + S x i S x j -S y S y -S z S z will force the magnetization into the YZ plane while the term -S x i S x j + S y i S y j -S z i S z j favors the XZ plane and so on. Since λ =0 . 2 ,U =5 . 2 λ =0 . 02 ,U =3 . 6 2 0 4 6 ω 0 0 -2 2 -2 0 0 -2 0 0 1.0 -1.0 1.0 -1.0 k y a k y a 0 0 1.0 1.0 -1.0 -1.0 -2 2 FIG. 2 (color online). Dynamical spectral functions in the (a) charge sector, (b) single-particle sector, and (c),(d) spin sector, measured along the edge. The parameters are U=t ¼ 2 , L ¼ 24 , L 0 ¼ 64 , /C21 =t ¼ 0 : 25 , and /C12 t ¼ 40 . Dashed lines are fits to the edge state dispersion in (b). (b) (a) (d) (c) FIG. 2 (color online). Dynamical spectral functions in the (a) charge sector, (b) single-particle sector, and measured along the edge. The parameters are U=t ¼ 2 , L ¼ 24 , L 0 ¼ 64 , /C21 =t ¼ 0 : 25 , and /C12 t ¼ state dispersion in (b). (b) (a) (d) (c) U=t ¼ 2 , L ¼ 24 , L 0 ¼ 64 , /C21 =t ¼ 0 : 25 , and /C12 t ¼ 40 . Dashed lines are fits to the edge state dispersion in (b). (b) (a) (d) (c) FIG. 3 (color online). As in Fig. 2 but for U=t ¼ 5 . λ =0 . 2 ,U =5 . 2 λ =0 . 2 ,U =2 . 0 λ =0 . 2 ,U =3 . 0 λ =0 . 02 ,U =3 . 6 2 0 4 6 ω ω 0 0 -2 2 2 -2 0 0 2 -2 0 0 1.0 -1.0 1.0 -1.0 k y a k y a 0 0 1.0 1.0 -1.0 -1.0 -2 2 k 1 k 3 where we used in the second last line 〈 l † k 1 ↑ l k 2 ↑ l † k 3 ↓ l k 4 ↓ 〉 = δ k 1 k 2 δ k 3 k 4 as well as 〈 u † k σ u k σ 〉 = 0, 〈 u † k σ l k σ 〉 = 0, and 〈 l † k σ l k σ 〉 = 1. N refers to the number of unit cells. Using the relations ∑ k | α ± ( k ) | 2 = ∑ k | β ± ( k ) | 2 = N/ 2 , (5) we confirm that D occ | U = 0 = 〈 TBI | ni ↑ ni ↓ | TBI 〉 = 1 4 , (6) independent of λ in agreement with CDMFT; see Fig. 5. Above we considered without loss of generality ni σ = a † i σ a i σ . Identical calculations performed with the bi σ operators which belong to the other sublattice yield, of course, the same result. In the opposite limit, U →∞ , we clearly find D occ = 0 as a fingerprint of Mott physics since we impose half filling. III. HELICAL EDGE STATES To describe the edge states associated with QSH insulators, first we apply the concept of the helical Luttinger liquid λ =0 . 2 ,U =5 . 2 λ =0 . 02 ,U =3 . 6 2 0 4 0 -2 2 -2 0 -2 0 0 1.0 -1.0 1.0 -1.0 k y a k y a 0 0 1.0 1.0 -1.0 -1.0 -2 2 FIG. 6. (Color online) Spectral function A ( ky, ω ) for a cylinder geometry with armchair edges ( L = 96, T = 0 . 05). Top: helical edge states of the QSH phase with λ = 0 . 2 and U = 2 . 0 or 3 . 0, respectively. Bottom, left: for U = 5 . 2 >Uc (SDWphase) the helical edge states disappear while the single-particle gap remains finite. Bottom, right: A ( ky, ω ) inside the SL phase; no edge states cross the gap. The color bar corresponds to the intensity of A ( ky, ω ). SDW transition. We can also check that the spin sector at the edges is described by an Ising order characterized by 〈 S x i 〉 /negationslash = 0 and 〈 S y i 〉 = 0. This shows that the SDW transition affects the charge sector of the HLL in a nontrivial way, resulting in a k 1 k 3 where we used in the second last line 〈 l † k 1 ↑ l k 2 ↑ l † k 3 ↓ l k 4 ↓ 〉 = δ k 1 k 2 δ k 3 k 4 as well as 〈 u † k σ u k σ 〉 = 0, 〈 u † k σ l k σ 〉 = 0, and 〈 l † k σ l k σ 〉 = 1. N refers to the number of unit cells. Using the relations ∑ k | α ± ( k ) | 2 = ∑ k | β ± ( k ) | 2 = N/ 2 , (5) we confirm that D occ | U = 0 = 〈 TBI | ni ↑ ni ↓ | TBI 〉 = 1 4 , (6) independent of λ in agreement with CDMFT; see Fig. 5. Above we considered without loss of generality ni σ = a † i σ a i σ . Identical calculations performed with the bi σ operators which belong to the other sublattice yield, of course, the same result. In the opposite limit, U →∞ , we clearly find D occ = 0 as a fingerprint of Mott physics since we impose half filling. III. HELICAL EDGE STATES To describe the edge states associated with QSH insulators, first we apply the concept of the helical Luttinger liquid λ =0 . 2 ,U =5 . 2 λ =0 0 -2 2 -2 0 -2 0 1.0 -1.0 -1.0 k y a 0 1.0 -1.0 -1.0 -2 2 FIG. 6. (Color online) Spectral function A ( ky, ω ) f geometry with armchair edges ( L = 96, T = 0 . 05). edge states of the QSH phase with λ = 0 . 2 and U respectively. Bottom, left: for U = 5 . 2 >Uc (SDWpha edge states disappear while the single-particle gap r Bottom, right: A ( ky, ω ) inside the SL phase; no edge gap. The color bar corresponds to the intensity of A ( ky SDW transition. We can also check that the spin edges is described by an Ising order characterized and 〈 S y i 〉 = 0. This shows that the SDW charge sector of the HLL in a nontrivial way, (c) (d) Figure 8 (a) Phase diagram of the Kane-Mele-Hubbard model as a function of U/t and λ/t (figure taken from Ref. (Reuther et al. , 2012)). (b) Single-particle spectral function A ( q, ω ) of the same model for U/t = 2 and λ/t = 0 . 25 obtained within Quantum Monte Carlo. Reprinted with permission from (Hohenadler et al. , 2011, 2012a). Copyright (2012) by the American Physical Society. (c)+(d) Single particle spectral function A ( k y , ω ) computed within cluster dynamical mean-field theory for λ/t = 0 . 2 and (c) U/t = 2 and (d) U/t = 3 (figures are taken from Ref. (Wu et al. , 2012a)).  \n15  \nrespectively; we obtain for the noninteracting part  \n<!-- image -->  \nweek ending  \n30 NOVEMBER 2012  \n30  \n.  \n02  \n,U  \n=3  \n.  \n6  \n40  \n. Dashed lines  \nlinks  \nk  \n-  \n∑  \n37,48  \ni  \nj  \ni  \nj  \n3. Kane-Mele-Hubbard Model The most important model to capture both non-trivial band topology and correlation physics in 2D is the Kane-MeleHubbard (KMH) model(Rachel and Le Hur, 2010) governed by the Hamiltonian H KMH = -t ∑ 〈 ij 〉 ∑ σ c † iσ c jσ + iλ ∑ 〈〈 ij 〉〉 ∑ αβ = ↑ , ↓ c † i,α ν ij s z αβ c jβ + U ∑ i n i ↑ n i ↓ (23) consisting of a real hopping term with amplitude t , the beforementioned SOC (17) with amplitude λ , and the Hubbard term (20) with amplitude U . In the meantime, the KMH model has been extensively studied both analytically (Griset and Xu, 2012; Hamad et al. , 2016; Lee, 2011; Mardani et al. , 2011; Rachel and Le Hur, 2010; Soriano and Fernández-Rossier, 2010) and numerically (Hohenadler et al. , 2011, 2012b; Hung et al. , 2013; Laubach et al. , 2014; Meng et al. , 2014; Wu et al. , 2012a; Yamaji and Imada, 2011; Yu et al. , 2011; Zeng et al. , 2017b; Zheng et al. , 2011) and its phase diagram is wellestablished [see Fig. 8 (a)]. It contains in the weak-coupling regime up to moderate interactions U > ∼ t the topological band insulator phase; at some critical U c a phase transition into a magnetically ordered phase occurs. We will see below exchange term (indicated by 〈〈 · 〉〉 )actingmerelyonindividual, i.e., triangular, sublattices partially frustrates the system. The XY -spin terms prefer ferromagnetic order on the individual sublattices, which is consistent with antiferromagnetic order on the original honeycomb lattice; in contrast, the Ising term S z i S z j favors antiferromagnetic order on the sublattice competing with both the XY terms and the J 1 term. The magnetization, which might point in any direction for J λ = 0 due to spin rotational invariance, turns into the XY plane in order to avoid the frustrating part of the J λ term. 13 These findings were confirmed within QMC, 15,16 the variational cluster approximation (VCA), 43 and cluster dynamical meanfield theory (CDMFT) 44 calculations at intermediate U/t ≈ 5 ... 9 and small λ . For small J λ , one can thus employ H KM all the terms (links) are equally distributed over the lattice, a priori no plane or direction is preferred. As the system sets out to be N«eel ordered for J ˜ λ = 0, it is conceivable that the competing ordering tendencies might at first compensate each other and allow for a persistent N«eel order at small J ˜ λ . V. METHOD The PFFRG approach 34-37,50 starts by reformulating the spin Hamiltonian in terms of a pseudofermion representation of the spin-1 / 2 operators S µ = 1 / 2 ∑ αβ f † α σ µ αβ f β ( α , β = ↑ , ↓ , µ = x,y,z ) with fermionic operators f ↑ and f and Pauli matrices σ µ .SucharepresentationenablesustoapplyWick\'s theorem, leading to standard Feynman many-body techniques. 155127-4 FIG. 3 (color online). As in Fig. 2 but for U=t ¼ 5 . 229902-2 FIG. 3 (color online). As in Fig. 2 but for U=t ¼ 5 . 229902-2 229902-2 Hence, we linearize the spectrum around the Fermi points and switch from lattice operators to field operators ψ R ↑ ( x ) and ψ L ↓ ( x ), which are right- and left-moving fields, H 0 = vF ∫ dx ( ψ † R ↑ i ∂ x ψ R ↑ -ψ † L ↓ i ∂ x ψ L ↓ ). Note that a standard single-particle (disorder) backscattering term ψ † R ↑ ψ L ↓ + H . c . , which opens up a mass gap in the spinless Luttinger liquid, is not allowed since the model is odd under time-reversal symmetry, T 2 = -1. Only two time-reversal-invariant interactions are allowed: the forward scattering as well as the umklapp scattering ∼ ψ † R ↑ ψ † R ↑ ψ L ↓ ψ L ↓ which is not intrinsically present in the Hubbard model. Instead we shall include the forward interaction HI = U ∫ dx ( ψ † R ↑ ψ R ↑ ψ † L ↓ ψ L ↓ ). As long as there is no magnetic order in the bulk, ( H 0 + HI ) can be solved exactly by resorting to bosonization which results in power-law decaying spin and charge correlations in the HLL. This result is also obtained through a spin-wave analysis at the edges for weak interactions. 39 It is worth mentioning that, in contrast, spin-spin correlation functions in the bulk decay very rapidly. 8 At the SDW transition, HI turns into HI ≈ -Um ∫ dx ψ † L ↓ ψ R ↑ + H . c . with m = 〈 ψ † R ↑ ψ L ↓〉 ; applying the bosonization procedure and introducing the Luttinger parameter K as usual, the Hamiltonian becomes H = ∫ dx v 2 [ 1 K ( ∂ x φ ) 2 + K ( ∂ x θ ) 2 ] -Um sin √ 4 πφ ( π a ) 2 . (7) Here, v is the renormalized plasmon velocity and a the lattice spacing, the field φ contains both spin and charge degrees of freedom, and ∂ x θ and φ are conjugate variables. 15 The sine-Gordon term is a relevant perturbation for repulsive interactions (since K /lessmuch 2) and hence, through the pinning of the φ field, the edge modes acquire a charge gap at the metal-insulator transition of Kosterlitz-Thouless type. Note that the disordering-ordering transition in the spin sector also influences the charge properties of the edges. In addition, we performed CDMFT simulations on honeycomb cylinders with length x = (3 a/ 4) L with both armchair and zigzag edges ( L is the number of sites in the x direction). By computing the spectral function A ( ky, ω ) we extract the edge state spectrum in the presence of interactions. In contrast to previous work 21 we treat the full microscopic Hamiltonian. We observe the following: (i) The plasmon velocity v associated with the edge modes slightly decreases for increasing U as expected from the HLL. (ii) The intensity of the spectral function decreases with increasing U . (iii) The edge modes gap out when 〈 S x i 〉 becomes finite. In Fig. 6 we show exemplarily the edge modes for fixed λ = 0 . 2, armchair boundary conditions, and U = 2 . 0, 3 . 0, and 5 . 2. We further performed computations for the spectral function on zigzag ribbons. Now, in contrast to the armchair case, the periodic boundary conditions are imposed in the x direction, yielding kx as a good quantum number. For zigzag edges one extracts the more familiar spectrum 2 of the KM model at small λ (see Fig. 7, top): the edge states connect the upper band of a gapped Dirac cone K with the lower band of the other gapped Dirac cone K \' , and vice versa. In addition, we have also shown the situation for stronger SOC λ = 0 . 2 (see Fig. 7,bottom)wherethebulkspectrumisrelativelyflat. Both spectral functions are computed for U = 2 . 0. The main findings (i)-(iii) obtained for armchair ribbons also apply for zigzag ribbons. We also computed the spectral function on a cylinder for parameters U and λ which belong to the spin liquid phase (see Fig. 6). Although the single-particle gap is very small we did not find edge states inside the gap. While it is not known 205102-4 (HLL). 15 Hence, we linearize the spectrum around the Fermi points and switch from lattice operators to field operators ψ R ↑ ( x ) and ψ L ↓ ( x ), which are right- and left-moving fields, respectively; we obtain for the noninteracting part H 0 = vF ∫ dx ( ψ † R ↑ i ∂ x ψ R ↑ -ψ † L ↓ i ∂ x ψ L ↓ ). Note that a standard single-particle (disorder) backscattering term ψ † R ↑ ψ L ↓ + H . c . , which opens up a mass gap in the spinless Luttinger liquid, is not allowed since the model is odd under time-reversal symmetry, T 2 = -1. Only two time-reversal-invariant interactions are allowed: the forward scattering as well as the umklapp scattering ∼ ψ † R ↑ ψ † R ↑ ψ L ↓ ψ L ↓ which is not intrinsically present in the Hubbard model. Instead we shall include the forward interaction HI = U ∫ dx ( ψ † R ↑ ψ R ↑ ψ † L ↓ ψ L ↓ ). As long as there is no magnetic order in the bulk, ( H 0 + HI ) can be solved exactly by resorting to bosonization which results in power-law decaying spin and charge correlations in the HLL. This result is also obtained through a spin-wave analysis at the edges for weak interactions. 39 It is worth mentioning that, in contrast, spin-spin correlation functions in the bulk decay very rapidly. 8 At the SDW transition, HI turns into HI ≈ -Um ∫ dx ψ † L ↓ ψ R ↑ + H . c . with m = 〈 ψ † R ↑ ψ L ↓〉 ; applying the bosonization procedure and introducing the Luttinger parameter K as usual, the Hamiltonian becomes H = ∫ dx v 2 [ 1 K ( ∂ x φ ) 2 + K ( ∂ x θ ) 2 ] -Um sin √ 4 πφ ( π a ) 2 . (7) Here, v is the renormalized plasmon velocity and a the lattice spacing, the field φ contains both spin and charge degrees of freedom, and ∂ x θ and φ are conjugate variables. 15 The sine-Gordon term is a relevant perturbation for repulsive interactions (since K /lessmuch 2) and hence, through the pinning of the φ field, the edge modes acquire a charge gap at the metal-insulator transition of Kosterlitz-Thouless that the disordering-ordering transition in the influences the charge properties of the edges. In addition, we performed CDMFT s comb cylinders with length x = (3 a/ 4) L with and zigzag edges ( L is the number of sites in the By computing the spectral function A ( ky, ω the edge state spectrum in the presence of contrast to previous work 21 we treat the full Hamiltonian. We observe the following: (i) velocity v associated with the edge modes for increasing U as expected from the HLL. (ii) of the spectral function decreases with increasing edge modes gap out when 〈 S x i 〉 becomes finite. I show exemplarily the edge modes for fixed λ = 0 boundary conditions, and U = 2 . 0, 3 . 0, and 5 . 2. We further performed computations for the sp tion on zigzag ribbons. Now, in contrast to case, the periodic boundary conditions are direction, yielding kx as a good quantum number edges one extracts the more familiar spectrum 2 model at small λ (see Fig. 7, top): the edge states upper band of a gapped Dirac cone K with the lo the other gapped Dirac cone K \' , and vice versa. we have also shown the situation for stronger (see Fig. 7,bottom)wherethebulkspectrumisr Both spectral functions are computed for U = 2 . findings (i)-(iii) obtained for armchair ribbons zigzag ribbons. We also computed the spectral function on a parameters U and λ which belong to the spin (see Fig. 6). Although the single-particle gap is v did not find edge states inside the gap. While it 205102-4 that this is a rather generic behavior for TIs. The KMH model in the absence of any Rashba SOC λ R = 0 features above U c an XY antiferromagnet with in-plane magnetization. The presence of an antiferromagnetic XY phase can be best seen by deriving the strong coupling limit of (23) leading to the Kane-Mele spin model (Rachel and Le Hur, 2010), H = 4 t 2 U ∑ 〈 ij 〉 S i S j + 4 λ 2 U ∑ 〈〈 ij 〉〉 ( -S x i S x j -S y i S y j + S z i S z j ) . (24) The antiferromagnetic nearest-neighbor Heisenberg model on the honeycomb lattice exhibits a Neel ordered ground state since the lattice is bipartite. While the second-neighbor spin exchange for x and y components is compatible with the antiferromagnetic nearest-neighbor exchange, the secondneighbor exchange S z i S z j is competing with the nearestneighbor contributions. Instead of resulting in a situation of frustrated magnetism, the system finds, however, an elegant way to circumvent it. By turning the magnetization vector into the XY plane (and avoiding any finite component in the ˆ z direction) the S z i S z j exchange is neutralized. This argument is strictly valid only for large U but meanfield treatment of different magnetizations shows that already at U c the in-plane magnetized AFM is energetically favorable compared to an easy axis order (Wu et al. , 2012a). These findings are in agreement with numerically exact quantum Monte Carlo simulations (Hohenadler et al. , 2011, 2012b; Zheng et al. , 2011), with pseudo-fermion functional RG analysis (Reuther et al. , 2012), cluster dynamical mean-field the-  \nS  \nγ  \ni  \nS  \n0  \n1.0  \ny  \nγ  \nj  \n.  \n(7)  \na  \n↓  \n)  \nory (Wu et al. , 2012a), variational cluster approach (Laubach et al. , 2014), and density-matrix renormalization group (Zeng et al. , 2017b). Also the slave-rotor analysis points towards an XY antiferromagnetically ordered phase (Rachel and Le Hur, 2010), see Sec. III.D.1. In several works, it has been argued and shown that the phase transition from the TI into the inplane antiferromagnet is of three-dimensional XY universality class (Griset and Xu, 2012; Hohenadler et al. , 2011; Wu et al. , 2012a; Zeng et al. , 2017b).  \nWhile the minimal version of the KMH model (23) has been studied by many authors, only few works (Hohenadler and Assaad, 2014; Laubach et al. , 2014) investigated the additional effect of the Rashba term. While the physics for not too large Rashba-SOC is similar to that of the pure KMH model, the combination of larger values of intrinsic and Rashba SOC even changes the non-interacting phase diagram (Laubach et al. , 2014) as mentioned in Sec. II.C [see Fig. 6 (c)]. In contrast to the λ v -λ R phase diagram shown in the inset between Figs. 6 (a) and (b), between QSH phase and trivial insulator a metallic phase with topological edge states is present. Band bending due to large Rashba leads to an indirect band gap; locally in momentum space for each wavevector k the gap is still preserved. It was shown that this \'topological semiconductor\' phase persists for finite U (Laubach et al. , 2014). Eventually the magnetism at large U is influenced because the Rashba-term generates Dzyaloshinsiki-Morya spin exchange which might favor spiral orders over collinear magnetic states.  \nThat a topological insulator is stable with respect to weak interactions is certainly not surprising since it possesses an energy gap like any other insulator. The interesting question one might ask is whether or not the metallic edge states persist in the presence of interactions. Apart from meanfield attempts, numerical methods including dynamical meanfield theory, its cluster extension, and variational cluster approach where the full Green\'s function G ( k , ω ) is (approximately) computed is particularly suited to answer this question. One can impose a cylinder geometry and compute the single-particle spectral function of the respective topological Hubbard model,  \nA ( k y , ω ) = -1 π Im {G ( k y , ω ) } (25)  \nwhere k y is the good momentum quantum number of the cylinder which is finite in the x direction. In Ref. (Hohenadler and Assaad, 2012; Hohenadler et al. , 2011; Wu et al. , 2012a; Yu et al. , 2011), A ( k y , ω ) is computed in the correlated TI regime of the KMH model, see Fig. 8 (c) and (d).  \nAmongst the manifold works on the KMH model, particular attention was given to the edges of the topological regime. While essentially all papers agree on the bulk phase diagram as shown in Fig. 8 (a) (apart from actual numbers for the U values where phase transitions occurs, which are somewhat method-dependent), the discussion of the edges turns out to be more diverse. Quantum Monte Carlo simulations (Zheng et al. , 2011) predicted two different non-  \nmagnetic phases: for zero to weak interactions the TI phase with helical edge states, but for slightly stronger interactions an analogous phase where the helical edges become unstable. The edge states can spontaneously break TR symmetry and acquire magnetic order due to two-particle backscattering (Zheng et al. , 2011). A similar observation has been made using variational Monte Carlo. Increasing Hubbard interactions lead to a strong suppression of the charge Drude weight in the helical edge channels. This mechanism drives a phase transition from the TI phase to an edge-Mott insulator phase (Yamaji and Imada, 2011; Yoshida and Kawakami, 2016). Extensions of the KMH model were also applied to describe the topological and magnetic phase transitions of silicene nanoribbons (Lü et al. , 2018).  \nA bosonic version of the KMH model was recently shown to host an emergent chiral spin state (Plekhanov et al. , 2018). Other correlated topological systems similar to the (fermionic) KMH model have also been studied (Lado and Fernández-Rossier, 2014; Parisen Toldin et al. , 2015). In particular, the interaction-driven phase transition from correlated topological insulator into an antiferromagnetic phase has been studied on the π -flux checkerboard lattice: in contrast to the KMH case discussed above, here the universality class of the transition seems to correspond to the one of the 2D Ising model (Zeng et al. , 2017b).', '4. Sodium-Iridate-Hubbard Model': "Kane and Mele proposed the quantum spin Hall effect originally for graphene (Kane and Mele, 2005a). Subsequent ab initio calculations clarified, however, that the spin-orbit gap is far too small to be observed (Min et al. , 2006; Yao et al. , 2007). Ever since then material scientists searched for other honeycomb-lattice materials with possibly heavier elements. One of the first proposals along these lines is the work by Shitade et al. (Shitade et al. , 2009) in which Na 2 IrO 3 , a 5 d transition metal oxide, is proposed as a layered correlated QSH insulator. Based on ab initio calculations an effective hopping model was derived which is reminiscent of the Kane-Mele model. Instead of isotropic SOC with amplitude iλσ z [for an illustration see Fig. 9 (a)] the three inequivalent secondneighbor hopping directions involve different Pauli matrices, see Fig. 9 (b). The band structure realizes a Z 2 TI (Shitade et al. , 2009) being in the same universality class as the KaneMele model; i.e. , when interpolating between both band structures the gap will not close and edge states (in case of a disc or cylinder geometry) will persist (Rachel, 2016). The authors of Ref. (Shitade et al. , 2009) argued that a correlated TI phase is present if Coulomb interactions are not too strong. The experimental status of Na 2 IrO 3 will be briefly discussed in Sec. IV.  \nWithin a slave-spin approach , Rüegg and Fiete investigated the aforementioned band structure supplemented with a Hubbard interaction, in the following referred to as sodium-iridate Hubbard (SIH) model (Rüegg and Fiete, 2012). Its Hamilto-  \nJOHANNES REUTHER, RONNY THOMALE, AND STEPHAN RACHEL  \nPHYSICAL REVIEW B  \n86  \n,155127(2012)  \n<!-- image -->  \nKMR  \nproduct form in pseudoparticles and slave spins, we find a noninteracting problem for the fermions, H f ¼ P i;j; /C11 ; /C12 t /C11/C12 ij g ij f y i /C11 f j /C12 , and the transverse-field Ising model PHYSICAL REVIEW LETTERS week ending 27 JANUARY 2012 transform into spiral magnetic order in the limit U →∞ .We note, however, that this necessarily implies that with increasing U the gap of the QSH /star phase closes at some point to form the Goldstone mode of the spiral order. In principle, the gap closure can occur at finite U (which would imply an additional phase boundary in Fig. 8)orat U →∞ .Inthelatterscenario, the QSH /star phase could extend up to U →∞ . Furthermore, one should keep in mind that the QSH /star phase as found in Ref. 33 might not be the only topological liquid candidate with similar properties, such as a doubled semion spin liquid. 55 The transport and thermal signatures ( i.e. , two-particle responses) as standard Fermi liquids but their single-particle spectrum is gapped. It seems that the slave-spin approach naturally generates phases such as QSH glyph[star] or CI glyph[star] (see Sec. III.B.2) which can be considered as 'orthogonal' phases.  \n[18,19,29,30] is not captured by our approach.) As dis-  \nTo test the stability of the uniform mean-field solution,  \nJ  \niλσ z i ˜ λσ x (a) (b) closest relative to an antiferromagnetic insulator (AFI) with spontaneously broken time-reversal symmetry [8]. Indeed, on the honeycomb lattice, VBS states are close in energy to the AFI [27,28]. The focus of this study is the intermediate phase at large spin-orbit coupling t 2 , where we find the strongly correlated QSH* insulator. (The spinliquid phase recently found in a small window around vanishing spin-orbit coupling and intermediate interactions for the slave spins, H s ¼/C0 1 =S 2 P ð i;j Þ J ij s x i s x j þ U= ð 4 S Þ P i s z i , supplemented with the self-consistency equations g ij ¼ h s x i s x j i =S 2 and J ij ¼/C0 P /C11/C12 h t /C11/C12 ij f y i /C11 f j /C12 þ H : c : i . The mean fields g ij and J ij are real and change sign under a gauge transformation: For f ð y Þ i /C27 ! /C15 i f ð y Þ i /C27 , s x i ! /C15 i s x i with /C15 i ¼/C6 1 , the mean fields transform according to g ij ! /C15 i g ij /C15 j and J ij ! /C15 i J ij /C15 j . Mean-field solutions which are related by a gauge transformation describe the same physical state after projection. By comparing the energies for different solutions of the self-consistency equations, we find the bulk phase diagram shown in Fig. 1(a). To compute g ij , we have used a 4-site cluster-mean-field approximation, which preserves the variational character of the total energy. The VBS is specified by g ij /C222 0 on one of the three nearest-neighbor bonds and zero otherwise. Both QSH and QSH* phases do not break any symmetry, and we can choose g ij ;J ij > 0 . They are distinguished by the fact that the ground state of the selfconsistent Ising model is either in the ferromagnetic phase with Z ¼ h s x i i 2 =S 2 > 0 for QSH or in the paramagnetic phase with Z ¼ 0 for QSH*. FIG. 1. (a) Phase diagram obtained in the Z 2 -mean-field approximation. QSH is a quantum spin Hall insulator and VBS a valence-bond solid. The focus of the current Letter is the strongly correlated QSH* at intermediate U and large spin-orbit coupling t 2 . (b) Topological ground-state degeneracy in the QSH*: logarithm of the energy difference between solutions with and without a global Z 2 flux on a L /C2 L torus for U ¼ 26 t and t 2 ¼ t . ˜ λ/t (c) TI SM 0.6 0.8 0.4 0.2 0 0 5 10 ˜ λ/t FIG. 8. (Color online) Schematic phase diagram for the SI Hubbard model as conjectured from our strong-coupling results of a N«eel to spiral transition at J ˜ λ = 0 . 53 (upper x scale), corresponding to ˜ λ /t ≈ 0 . 73 (lower x scale). While we cannot ultimately assess the nature of a possible intermediate phase aside from TI and AFM for strong interaction and strong spin-orbit coupling (yellow phase), its existence is likely. VII. DISCUSSION In view of our results for the SI model, we speculate about the implications for the phase diagram at intermediate U . We find the incommensurate phase for J ˜ λ /J 1 /greaterorequalslant 0 . 53 where J ˜ λ = 4 ˜ λ 2 /U and J 1 = 4 t 2 /U , implying a transition at ˜ λ t ≈ 0 . 73 for large U .InFig.8,wehavereplottedthephase diagram of R‹uegg and Fiete 33 in a slightly modified way. Our reasoning is the following: Since the spin model corresponds to U →∞ , we extrapolated the phase boundary between 'VBS (AFM)' and 'QSH /star 'ofthephasediagraminRef.33 to larger U . As the phase transition occurs for sufficiently large U approximately at ˜ λ t ≈ 0 . 73 (Fig. 8), we speculate that the observed phase transition from N«eel to spiral order in the spin model is a remnant of the phase transition into the QSH /star phase at intermediate U .Withinthisscenario,theQSH /star phase would alternative scenario-assuming that a QSH /star -type phase does not exist-would still require an additional phase comparable to the yellow phase of the schematic phase diagram in Fig. 8; in this case, the additional phase would most likely be a magnetically ordered phase (e.g., spiral order). Whether or not this phase is a topological liquid or just another magnetically ordered phase, we conjecture that in either case an additional phase of some kind should be present. In summary, we find that the physics of the SI spin model is much richer as compared to the KM spin model. This can be traced back to the different spin symmetries in both systems. The broken axial symmetry in the SI spin model prevents the spins from forming planar antiferromagnetic order and eventually leads to the emergence of a spiral phase. Note that this phase does not have any analog in similar models such as the Heisenberg-Kitaev model. 37 As such, we have identified multidirectional spin-orbit terms to be an interesting way to create new spin phases in the infiniteU limit and possibly even more exotic phases at intermediate U when charge fluctuations enter the picture. To give another direction of further investigation, it will be interesting to study the KM model in the presence of Rashba spin-orbit coupling H R = i λ R ∑ 〈 ij 〉 ∑ αβ c † i α [ˆ ez ( σ × d ij )] αβ c j β . The Rashba term breaks the remaining U(1) symmetry of the electron spin to Z 2, and also affects the z →-z mirror symmetry as well as particle-hole symmetry. Taking into account the Rashba spin-orbit coupling results in a more complicated spin Hamiltonian with some terms being of Dzyaloshinskii-Moriya type. The full Hamiltonian is given by x x y y z z Figure 9 (a) Illustration of the nearest-neighbor hopping (solid lines) and second-neighbor spin-orbit hopping (dashed lines) with amplitude iλ for the KM model. (b) Different second-neighbor bonds involve different Pauli matrices for the SIH model with amplitude i ˜ λ . (c) Phase diagram of the SIH model obtained within slave-spin method. Besides the QSH (TI) phase, an antiferromagnet (VBS / AFI) and the putative QSH ∗ phase are present; see main text for details. Reprinted with permission from (Rüegg and Fiete, 2012). Copyright (2012) by the American Physical Society. (d) Phase diagram of the strong-coupling spin limit of the same model (Reuther et al. , 2012), with an AFM Neel phase at weak ˜ λ and magnetic spiral phase at large ˜ λ . Remarkably, the phase boundary of the phase diagram in (c) is compatible with the one of the corresponding spin model in (d); magnetic exchange couplings are given by J x = 4 x 2 /U with x = ˜ λ, t . For details see main text. Panels (a), (b), and (d) are taken from (Reuther et al. , 2012). It turned out to be challenging to verify or further investigate the exotic QSH glyph[star] phase because one has to do nothing less than solving the 2D Hubbard model with broken spin symmetry at intermediate interaction strength. Due to the topological order, most mean-field type approaches cannot be used. At least there were two successful attempts to encircle the predicted QSH glyph[star] phase in the phase diagram: (i) by deriving the strong coupling limit, the ground state of the corresponding spin Hamiltonian has been solved (Reuther et al. , 2012); (ii) by deriving the limit of strong spin-orbit coupling the SIH model effectively decouples into two copies of triangular lattice Hubbard models with a peculiar band structure (Rachel et al. , 2015). The intrinsic SOC as illustrated in Fig. 9 (b) gives rise to the same spin exchange for the vertical bonds as in the KMH case ( ∝ σ z ), but the red bonds ( ∝ σ x ) lead to S x i S x j -S y i S y j -S z i S z j and the green bonds ( ∝ σ y ) to -S x i S x j + S y i S y j -S z i S z j . In a compact form one can write the sodium-iridate spin Hamiltonian as  \nλ  \nS  \ni  \nS  \nj  \nS  \ni  \nS  \nj  \nS  \ni  \nS  \nj  \ncussed below, the QSH* phase survives beyond the uniform mean-field treatment. Z 2 representation.The starting point of our formal discussion of the QSH* is a representation of the model Eq. (1) in an enlarged Hilbert space which consists of fermionic pseudoparticles denoted by f i /C11 and auxiliary S ¼ 1 = 2 slave spins s i [23]. This representation makes use of the particle-hole symmetry of the Hubbard interaction at half filling, and the Hamiltonian takes the form H ¼ 1 S 2 X i;j X /C11 ; /C12 t /C11/C12 ij s x i s x j f y i /C11 f j /C12 þ U 4 S X i ð S þ s z i Þ : (2) Equation (2) is a faithful representation of the original model Eq. (1) in the subspace where all the local operators u i : ¼ ð/C0 1 Þ s z i /C0 S þ ni (with n i ¼ P /C11 f y i /C11 f i /C11 ) act as identity u i ¼ 1 . Note that u i performs a local Z 2 gauge transformation: u i f ð y Þ i /C11 u i ¼/C0 f ð y Þ i /C11 and u i s x i u i ¼/C0 s x i . Hence, the physical subspace is the gauge-invariant subspace. Given a state j /C9 i in the enlarged Hilbert space, a physical state can be obtained by projection as long as s ¼ h /C9 j P 2 j /C9 i /C222 0 : j /C8 i ¼ s /C0 1 = 2 P j /C9 i , where P ¼ Q i ð 1 þ u i Þ = 2 . Note that the pseudoparticles f i /C11 carry both spin and charge of the original electron and no ad hoc assumption of their separation has been made. Nevertheless, we will see that spincharge separated quasiparticles emerge in the strongly correlated limit. Mean-field theory.To proceed, we study the model Eq. (2) in a mean-field approximation: Assuming a we have investigated inhomogeneous solutions on tori with L 1 unit cells in the a 1 direction and L 2 unit cells in the a 2 direction. The transverse-field Ising model has been solved in a semiclassical (large S ) approximation which is easily generalized to inhomogeneous configurations f J ij g (but it violates the variational character). The central observation is that in the QSH* there are self-consistent solutions where closed paths C encircling a Z 2 flux exist: Q ð ij Þ 2 C sgn ð g ij Þ¼/C0 1 . (On the other hand, in the QSH phase at small U , such solutions do not exist because g ij /C25 h s x i ih s x j i =S 2 . This requires that along any loop an even number of bonds are negative.) There are profound consequences of the above observation. In particular, we find that the ground state of the QSH* is fourfold degenerated in the thermodynamic limit as L 1 ; 2 !1 ; see Fig. 1(b). The four different ground states are characterized by the presence or absence of two global Z 2 fluxes. This degeneracy is robust against local perturbations of the system and is therefore a topological degeneracy. Numerically, the bond energies g ij J ij are uniform for any configuration of the global fluxes. Therefore, the difference in the ground-state energy observed for finite systems entirely results from the discrepancy of the k grid in the first Brillouin zone when periodic or antiperiodic boundary conditions are used to compute g ij and J ij . For gapped systems, it is expected that the difference depends exponentially on the circumference [31,32], which is consistent with our numerical results; see Fig. 1(b). Besides the fourfold topological degeneracy on the torus, 046401-2 〈〈 ij 〉〉 + ∑ δ 1 -links [ ( J 1 + J R) S x i S x j + ( J 1 -J R) ( S y i S y j + S z i S z j ) -√ J 1 J R ( S y i S z j -S z i S y j )] + ∑ δ 2 -links [ S x i/j →-1 2 S x i/j -√ 3 2 S y i/j and S y i/j → √ 3 2 S x i/j -1 2 S y i/j ] + ∑ δ 3 -links [ S x i/j →-1 2 S x i/j + √ 3 2 S y i/j and S y i/j →-√ 3 2 S x i/j -1 2 S y i/j ] , (12) where the third line in (12) is obtained from the second one by replacing S x i/j with -1 / 2 S x i/j -√ 3 / 2 S y i/j and so on. The different links denoted by δ i ( i = 1 , 2 , 3) are the three nearest-neighbor vectors of the honeycomb lattice. We expect the J λ -J R phase diagram to be interesting and to host some additional phases, which we defer to a future publication. 155127-8 nian is given by H SIH = -t ∑ 〈 ij 〉 ∑ σ c † iσ c jσ + i ˜ λ ∑ 〈〈 ij 〉〉 γ ∑ αβ = ↑ , ↓ c † i,α ν ij s γ αβ c jβ + U ∑ i n i ↑ n i ↓ (26) and γ = x, y, z as illustrated in Fig. 9 (b). Most notably, the corresponding slave-spin analysis revealed for sufficiently large SOC ˜ λ and Hubbard U an additional phase, dubbed QSH glyph[star] , which is absent in the KMH phase diagram. The phase diagram of the SIH model (Rüegg and Fiete, 2012) is shown in Fig. 9 (c); therein, QSH refers to the quantum spin Hall phase (the 2D TI) and VBS (AFI) to 'valence bond solid (antiferromagnetic insulator)'. Due to a limitation of the used slavespin approach, magnetic solutions cannot be found; instead, a valence bond solid is found as the state which comes energetically closest to the antiferromagnet. In other methods, this phase would clearly show up as the Neel ordered antiferromagnet. The time-reversal invariant QSH glyph[star] phase possesses topological order associated with a topological degeneracy on a torus geometry. It was further claimed to be described by the same field theory than the toric code (Kitaev, 2003); excitations were identified as Abelian anyons (Rüegg and Fiete, 2012). In Ref. (Zhong et al. , 2013), an exactly soluble model was proposed which features a fractionalized phase which is comparable to the QSH glyph[star] phase. It is worth emphasizing that one can interpret them also as an orthogonal QSH phase in analogy to the orthogonal metal phase (Nandkishore et al. , 2012; Zhong et al. , 2012). The orthogonal metal has the same H = 4 t 2 U ∑ 〈 ij 〉 S i S j -∑ 〈〈 ij 〉〉 ˜ λ 2 4 U S i S j +2 ∑ 〈〈 ij 〉〉 γ ˜ λ 2 4 U S γ i S γ j . (27) Note that γ is defined as for H SIH ; the bond-dependent Ising exchange S γ i S γ j is referred to as compass interactions (Nussinov and van den Brink, 2015) or Kitaev exchange (Kitaev, 2006). Both the antiferromagnetic nearest neighbor and the ferromagnetic second-neighbor exchange stabilizes Neel order on the honeycomb lattice. The nontrivial second-neighbor Kitaev exchange eventually turns the Neel state into an incommensurate spiral ordered state for sufficiently large ˜ λ (Reuther et al. , 2012). Interestingly, this magnetic phase diagram fits nicely to the mean-field phase boundaries of Ref. (Rüegg and Fiete, 2012), see the dashed line between panels (c) and (d); the combination of both phase diagrams is shown in Fig. 9 (c) and (d). Taking the limit of infinitely large SOC is more subtile as it is equivalent to setting the nearest-neighbor hoppings to zero, t → 0 . That is, the honeycomb lattice decouples into two independent triangular lattices. The 'new' nearest-neighbor hopping on such a decoupled triangular lattice is governed by the Hamiltonian H glyph[triangle] = iλ ∑ 〈 ij 〉 γ c † iα σ γ αβ c jβ . (28) In order to derive a more intuitive understanding it is helpful to apply Klein transformations to the creation and annihilation operators (Rachel et al. , 2015); Klein transformations are usually applied to spin operators (Chaloupka et al. , 2010; Kimchi and Vishwanath, 2014; Reuther et al. , 2012).  \nH  \n=  \n∑  \n[  \n-  \n-  \n+  \n]  \nFor a Klein map, the lattice is divided into four sublattices. All creation and annihilation operators defined on one of the sites remain untouched. The other three type of sites are transformed such that their spin component is rotated around the x , y , or z axes by π . Surprisingly, the formerly spinand bond-dependent imaginary hopping problem described by (28) appears to be real and spin-independent after the Klein map. The difference to an ordinary real tight-binding model on the triangular lattice are a few randomly distributed minussigns. Rewriting them in terms of π -flux Peierls phases reveals the underlying flux patterns consisting of alternating triangles threaded by π and 0 flux. Now an appropriate unit cell is easily chosen and the band structure computed: its lowenergy theory is a 2+1 dimensional Dirac theory, very similar to graphene.  \nA real spin-independent hopping problem with hoppings ± t results in the strong coupling limit to isotropic Heisenberg spin exchange because J = 4( ± t ) 2 /U is independent of the sign of t . The corresponding ground state, although frustrated, is known to be the 120-degree Neel state. This situation is quite interesting: a semi-metallic groundstate in the weak coupling, and a frustrated magnetically ordered groundstate in the strong coupling regime. In order to reveal the nature of the intermediate region of the phase diagram, a Variational Cluster Approach study suggested that a non-magnetic insulator phase is present (Rachel et al. , 2015). The nature of this quantum paramagnetic phase was claimed to be a spin liquid because the energy gain due to the formation of short-range resonating valence bonds (Anderson, 1973) was much higher compared to the ordinary triangular lattice Hubbard model. Since the latter is believed to host a spin liquid phase the same should be true for the triangular lattice π -flux Hubbard model (Rachel et al. , 2015).  \nIn order to close the 'loop' between the U → ∞ and the λ → ∞ result, one has to apply the same Klein map to the strong coupling groundstate and transform backwards. The 120 degree Neel state is transformed into a commensurate spiral ordered state with a 12-site unit cell; by weakly coupling two triangular lattices into a honeycomb lattice where each sublattice hosts such a spiral state, the spiral order becomes incommensurate as previously found within pseudo-fermion functional RG (Reuther et al. , 2012). Still the question about possible phases at intermediate U and intermediate spin-orbit coupling remains an open issue. Whether or not the QSH ∗ phase (Rüegg and Fiete, 2012) is realized remains an open question. Alternatively, it might be a magnetically ordered spiral phase (Liu et al. , 2013b; Reuther et al. , 2012)) or there could even multiple phases be hidden.", '5. Bernevig-Hughes-Zhang-Hubbard Model': "The BHZ model(Bernevig et al. , 2006) plays an important role as it effectively describes the physics of the HgTe quantum wells (König et al. , 2007), the first (and for a long time only) physical system displaying the QSH effect. On the other  \nside, HgTe/CdTe quantum wells are a heterostructure built of different materials which all realize a 3D zinc blende structure. Using k · p theory one can show that the BHZ model is indeed the true low-energy theory. In order to study interaction effects one need to regularize this model, usually it is done on a two-orbital square lattice. The BHZ model can be thought of as a spinful extension of the two-orbital square lattice Chern insulator (Sec. II.B). The Hamiltonian can simply be written using the spinor Ψ † = ( c † E, ↑ , c † H, ↑ , c † E, ↓ , c † H, ↓ ) as  \nH BHZ = ∑ k Ψ †       h ( k ) 2 -orb . -∆ ∆ ∆ -∆ h glyph[star] ( -k ) 2 -orb .       Ψ (29)  \nwhere h ( k ) 2 -orb . is the 2 × 2 Bloch matrix earlier introduced in Eq. 13 and ∆ quantifies the bulk inversion asymmetry (Bernevig et al. , 2006; Qi and Zhang, 2011). A real space formulation of (29) can be found, for instance, in Ref. (Rod et al. , 2015).  \nThis square lattice version of the BHZ model is then a good starting point to investigate the role of Hubbard interactions; it is, however, not directly relevant anymore for the HgTe/CdTe quantum wells. Nonetheless several authors studied the interplay of topology and interactions in this system (Budich et al. , 2013; Miyakoshi and Ohta, 2013; Tada et al. , 2012; Yoshida et al. , 2012, 2013). While some of the basic results are similar to those of the KMH model, for instance that the TI phase breaks down when a spontaneous magnetization sets in, the BHZ Hubbard model features a first-order phase transition (Amaricci et al. , 2015) not known for other topological Hubbard models. Moreover, the orbital structure nicely allows to study the effect of Hunds-coupling and intra-orbital interactions in addition to the inter-orbital Hubbard interactions (Amaricci et al. , 2015; Budich et al. , 2013):  \nH int = H U + H V + H J (30)  \nwith the inter-orbital Hubbard term, H U = U ∑ i ( n ( E ) i, ↑ n ( E ) i, ↓ + n ( H ) i, ↑ n ( H ) i, ↓ ) , intra-orbital repulsion H V = V ∑ i ( n ( E ) i, ↑ n ( H ) i, ↓ + n ( H ) i, ↑ n ( E ) i, ↓ ) , and the Hund's coupling H J = ( V -J ) ∑ i ( n ( H ) i, ↑ n ( E ) i, ↑ + n ( H ) i, ↓ n ( E ) i, ↓ ) . These additional interaction terms do not qualitatively change the general competition between topological weak-coupling regime and the strongly coupled Mott regime, but clearly they lead to a richer phenomenology. Ultimately, for a multi-orbital scenario such as in the BHZ model these terms will be present in real materials. The paramagnetic m -U phase diagram of the BHZ Hubbard model with H int is shown in Fig. 10 in Sec. III.C for the parameters V = U -2 J , J = 0 . 25 U and t = 0 . 3 .  \nThe role of interactions in the BHZ model becomes relevant in a completely different branch of physics: ultracold quantum gases and optical lattices instead of semiconductors. The tremendous progress to realize synthetic magnetic fields and SOC(Aidelsburger et al. , 2013; Dalibard et al. , 2011; Hauke et al. , 2012; Kennedy et al. , 2013) for ultracold atoms made in the past years allows to stabilize topological states of matter. Indeed, the Haldane Chern insulator model has recently been realized in an optical lattice (Jotzu et al. , 2014). Most ultracold atom systems have in common that a local particleparticle interaction can be tuned and controlled. With the help of a Feshbach resonance, HubbardU can even be tuned from the attractive to the repulsive regime. Thus the combination of synthetic gauge fields and arbitrary interactions makes ultracold quantum gases predestined as a playground for interacting topological insulators.  \nHere we will briefly discuss a specific cold atom proposal (Goldman et al. , 2010) where a spinful and time-reversal invariant Hofstadter system is discussed. The standard Hofstadter problem realized the QHE for spinless fermions on the square lattice as discussed in Sec. II. When the vector potential is created artificially, it is possible to choose the vector potential for both spin species individually. In particular, they can be chosen to possess opposite sign thus realizing a TR invariant magnetic field. The resulting state is a Z 2 TI exhibiting Hofstadter bands which can be interpreted as Landau levels in the limit of small magnetic field α = p/q . While this is expected for values of the Fermi energy within all 'Hofstadtergaps' with an odd Chern number per spin channel 4 , additional staggered sublattice potential and a Rashba-type spinorbit hopping make it possible to obtain a QSH phase even at half filling (Goldman et al. , 2010). The Hamiltonian can be defined on a square lattice as  \nH TRI -HH = λ ∑ j ( -1) j x c † j c j -∑ j ( tc † j +ˆ e x e 2 πiγσ x c j + tc † j +ˆ e y e 2 πiαj x σ z c j +H . c . ) (31)  \ndescribing spin-1/2 fermions subject to a synthetic gauge field. The σ z Pauli matrix in the second term corresponds to the spin-dependent field and guarantees TR invariance. The first term induces spin flips if the particles move along the x -direction. The λ term causes a staggering of the lattice in the x direction. Most importantly, all terms in (31) can be experimentally realized. This non-interacting model exhibits trivial and topological insulator phases as well as metallic regimes, depending on the filling.  \nTogether with local Coulomb interactions the rich phase diagram has been explored (Cocks et al. , 2012; Orth et al. , 2013). In particular, a variety of magnetically ordered phases has been found including different spiral states. In Ref. (Scheurer et al. , 2015) it has been shown that an anisotropic version of the BHZ model emerges as the lowenergy k · p theory of this time-reversal invariant Hofstadter model at half filling.  \nFurthermore, a 3D stacked version of this model can be tuned such that weak and strong TI phases emerge out of the critical point between normal and topological insulator phases (Scheurer et al. , 2015) when hoppings in z direction are turned on. Requirement is, however, that these hoppings in the third direction are attached to synthetic gauge fields. This setup is later discussed as the starting point for the investigation of the topological Mott insulator phase in a cubic lattice, see Sec. III.D.1.", '7. Higher-dimensional Correlated Topological Insulators': "Compared to the two-dimensional case, investigations of interaction effects in 3D bandstructures are rather rare. Notable exceptions are the interesting studies on the pyrochlore lattice (partially motivated by pyrochlore oxides and by the search for topological Mott insulators) (Go et al. , 2012; Wan et al. , 2011). As a consequence of electron-electron interactions, topological Weyl semimetals were predicted but also the more elusive axion insulator phase - essentially a 3D strong TI with broken TR symmetry - were found.  \nThe effect of Coulomb interactions and Hund's exchange coupling was studied in a cubic bandstructure with strong and weak TI phases (Amaricci et al. , 2016).  \nMost other works are focussed on interaction-induced topological phases such as the topological Kondo insulators (as discussed below). Other examples of interaction-induced topological phases are discussed in Refs. (Kurita et al. , 2011; Zhang et al. , 2009b). And a linear response theory of interacting topological insulators was derived in Ref. (Culcer, 2011).  \nRecent developments in the field of ultracold quantum gases in optical lattices has established the possibility of socalled 'synthetic' dimensions (Lohse et al. , 2018). This led to the realization of the four-dimensional integer quantum Hall effect (Kraus et al. , 2013; Lohse et al. , 2018). The idea is that an internal degree of freedom is used as if it was another spatial degree of freedom. This generally allows to investigate high-dimensional systems and led to the recent study of interacting topological insulators in d dimensional systems (Jian and Xu, 2018).", 'C. Interaction-Induced Topological Insulators': "In this section, we briefly consider systems where electronelectron interactions induce a topologically non-trivial phase. In the following we discuss three scenarios where this is real-  \nBUDICH, TRAUZETTEL, AND SANGIOVANNI  \nPHYSICAL REVIEW B  \n87  \n,235104(2013)  \nPOLE EXPANSION OF SELF-ENERGY AND INTERACTION  \nother.  \nCases  \n(a)  \n(b)  \n(  \nV,V  \n. We draw the  \n' U -only' interaction represents a very unreasonable choice for a multiorbital Coulomb vertex, because the interorbital matrix elements are entirely neglected. Cases (i) and (ii) are therefore much closer to a realistic Hamiltonian for a realizable material. We solve the model within dynamical mean-field theory (DMFT). DMFT-for a review see Ref. 43-corresponds to a mean-field approximation for the spatial degrees of freedom while preserving the local quantum dynamics of the interacting system. The self-energy /Sigma1 ( i ω ) depends therefore on the Matsubara frequencies only and the k dependence of the Green's function comes entirely from the one-particle Hamiltonian H BHZ. This is a good approximation whenever the local correlations are the dominant ones (e.g., for high dimensionality, or at high temperatures, or far away from quantum critical points). With these limitations in mind, we conclude that DMFT represents a very powerful tool for the present study which focuses on the way the self-energy-and hence the topology of the Green's function-is modified by dependent self-energies (fluctuation induced) can be responsible (Budich et al. , 2012; Wang et al. , 2012a,b). Examples of the former case (Cocks et al. , 2012; Orth et al. , 2013) but, in particular, also of the latter are nicely demonstrated within DMFT studies (Wang et al. , 2012a,b). Using the insight the additional poles in the self-energy can change the topology of the system's ground state motivated several works to study fluctuation-induced topological phases as an application of the pole-expansion method. A typical self-energy is given by Σ( ω ) = V 2 iω + P + V 2 iω -P with V , P being free parameters. It has been shown (Wang et al. , 2012b) that the topological index is trivial for P = 0 and non-trivial otherwise. In Fig. 11 the surface spectral function of a strong TI is shown for the case V = 1 and P = 2 of a system which was initially prepared in a trivial phase. ... PHYSICAL REVIEW B 85 ,235135(2012) Z 2 indices calculated by two methods (see text) are in accordance with each V † P Z 2 = 1 Z 2 = 0 V I 4 P I 4 | P | /Delta1 >V 2 | P | /Gamma1 <V 2 ) ⊗ I 4 diag( P, -P ) ⊗ I 4 P = 0 P = 0 because the ground state of ˜ H k is gapless while the interacting  \nversion of the continuous-time quantum Monte Carlo. The  \n/negationslash  \ndynamical effects of the electron-electron interaction. DMFT allows us to go one step beyond static mean-field treatments which, by construction, cannot address the ω dependence of the self-energy and, contrary to DMFT, cannot yield dynamicsinduced insulating states without long-range order. As a DMFT-impurity solver we use the hybridization-expansion A different example of a topologically trivial system which enters a topological Chern insulator phase due to interactions has recently been discussed in the context of a largeN exactly solvable model (Zhang and Zhai, 2018). system is gapful. This paradox is solved by noticing that for P = 0 the two poles of case (b) merge into one but we are still constructing the pseudo-Hamiltonian with NP = 8. When treating this particular point with NP = 4, case (b) reduced to case (a) with P = 0, which is topological trivial and consistent  \nwith its single-particle  \nWe start with the case (a):  \n/Sigma1  \n(  \nω  \n,  \n↦  \ninteracting pseudo-Hamiltonian  \nk  \n)  \nFIG. 3. (Color online) Surface spectral functions for self-energy  \n<!-- image -->  \ni  \nω  \n2  \nP  \nFIG. 1. (Color online) Phase diagram of the BHZ model with U , V = U -2 J , and J = 0 . 25 U, λ = 0 . 3. For m> 2 the noninteracting model is topologically trivial (green) and for m< 2 is in a topologically nontrivial QSH phase (red). At large values of U the system is in a Mott phase (blue). The red open circles at m = 0 indicate that the solution is metallic. The gray lines denote transition regions. The 'topological Hund insulator,' i.e., the nontrivial interacting phase in red, can be obtained for small-to-moderate values of U , starting from Figure 10 Phase diagram of the BHZ Hubbard model with additional Hunds coupling and intra-orbital repulsion. The balance between m causing a trivial band insulator and Hubbard U causing a Mott insulator leads to the survival of the topological band insulator up to rather large values of U . Reprinted with permission from (Budich et al. , 2013). Copyright (2013) by the American Physical Society. path defined by ω → G -1 imp ( ω ) = i ω -V i ω -P on the complex plane and find that the topological transition point corresponds to where the path touches the lower edge of the eigenenergy continuum, i.e., the band gap /Delta1 . This defines the phase boundary between the TI and metal phase V 2 P =± /Delta1 ; see Fig. 2(a). We then calculate the Z 2 index of the non-  \na trivial phase with  \nm  \n19  \n/greaterorsimilar  \n2.  \n˜  \nH  \nk  \nThe resulting phase diagram is shown in  \nwith the FDWN prediction. The lesson from this example is", 'details of the code used for the numerical results can be found in Ref. 44. The advantage of this flavor of impurity solver is that very low temperatures can be accessed (here we show 2. Topological Phases through Spontaneous Symmetry Breaking that when constructing the pseudo-Hamiltonian, degenerate poles should be treated carefully.': "results for  \nkBT  \n1  \n/  \n100 in units of the hopping parameter of  \nBHZ, but we even went down to 1 / 400) and the calculation is sign-problem free. We consider the half-filled case, i.e., two electrons on two orbitals. The E band lays 2 m above the H band, therefore, without interaction, the system is in the ( nH,nE ) = (2 , 0) configuration. U alone favors the (1 , 1) configuration. On the other hand, the interorbital repulsion tends to restore (2 , 0). The A completely different scenario of interaction induced topological phases was put forward by Raghu et al. (Raghu et al. , 2008) 5 . Additional longer-ranged Coulomb interactions shall spontaneously generate second-neighbor terms which either break TR symmetry (corresponding to a (doubled) IV. DISCUSSIONS Direct dealing with the frequency-momentum integration in Eq. (1) is the most general way of calculating the interacting topological index. It does not make any a priori assumptions on the form of the self-energies. However, as has been pointed  \nHund coupling  \nJ  \ntherefore plays the crucial role for favoring  \n45,46  \n1) in the strong coupling region.  \nThe effective reduction  \nof the energy difference between the orbitals is the important  \nprerequisite to get the topological phase in the intermediate  \n= -the diagonal matrix structure is in spin space, 37 h ( k ) = ( m -cos( kx ) -cos( ky ) ) σ z + λ sin( kx ) σ x + λ sin( ky ) σ y , and σ i are Pauli matrices in the E-H band pseudospin space. For the interacting part we take the most general two-orbital interaction parametrization H int = HU + HV + HJ . 38 HU = U ∑ i ( n ( E ) i, ↑ n ( E ) i, ↓ + n ( H ) i, ↑ n ( H ) i, ↓ ), represents an intraorbital Hubbard repulsion, HV = V ∑ i ( n ( E ) i, ↑ n ( H ) i, ↓ + n ( H ) i, ↑ n ( E ) i, ↓ ), and HJ = ( V -J ) ∑ i ( n ( H ) i, ↑ n ( E ) i, ↑ + n ( H ) i, ↓ n ( E ) i, ↓ ), an interorbital term for electrons with opposite and parallel spin, respectively. We also consider an S + S --like contribution, namely HS = -J ( c † H, ↓ c † E, ↑ c E, ↓ c H, ↑ + c † E, ↑ c † E, ↓ c H, ↑ c H, ↓ ), which makes the interaction fully SU(2) symmetric. In order to check to what extent our results depend on the details of the interaction, we consider three forms of H int: (i) HU + HV + HJ with V = U -2 J and J = 0 . 25 U ,(ii) HU + HV + HJ + HS with the same values of V and J , and (iii) HU alone. Hitherto, only the latter has been considered in the literature as interaction for the BHZ model (see Refs. 39-42). Yet, the region. Indeed, were the bands effectively getting further and further apart from each other, the system would just forever stay in the trivial band-insulating (2 , 0) configuration. This is, for instance, what would happen if we put J = 0. In Fig. 2, we show the evolution of the orbital occupations for all three interaction forms. We see how in all cases the system goes from the (2 , 0) to the (1 , 1) configuration upon increasing the interaction strength. Case (iii) gives an essentially linear decrease of the orbital polarization (confirming the oddness of such a choice of interaction) and, as a consequence, yields the topological transition in the weak-coupling regime where the frequency dependence of the self-energy is weak. As a consequence, the character of this transition is strongly static mean-field like. In contrast, cases (i) and (ii) are much richer and physically more interesting. Indeed, the topological transition is shifted to larger coupling, where the real part of the self-energy has a pronounced structure at low Matsubara frequencies. The dynamical fluctuations are therefore the driving force for the topological phase transition, as one can Nowlet us add local Coulomb interactions (20): as a consequence, doubly occupied sites cost an energy penalty U while singly occupied and empty sites do not. If U is sufficiently large, it will reverse the effect of λ v (for the sake of simplicity, we ignore the possible formation of magnetism due to U ). Reversing the effect of λ v drives the ground state back into the topologically non-trivial phase: this is the simplest example of an interaction-induced topological phase. Similarly, HubbardU can 'reverse' or renormalize the 'mass' term of the BHZ model; simultaneous increase of the mass term m leading to the orbital imbalance and Hubbard U forcing equal orbital occupancy can lead to a stable balance up to large energy scales. This has been explicitely shown for the BHZ Hubbard model (already discussed in Sec. III.B.5), see Fig. 10 (Budich et al. , 2013). We then move to case (b): /Sigma1 ( ω , k ) = V 2 i ω + P + V 2 i ω -P . 20 It has been shown that the interacting system is gapful and its Z 2 index equals 0 for P = 0and1otherwise. 12 Diagonalizing ˜ H k shows that the noninteracting system is gapful for P = 0, with Z 2 index equal to 1, which validates the aforementioned result. The gap of the pseudo-Hamiltonian closes at P = 0. At first look, this seems to be inconsistent with Eq. (3) 2 1 1 P 1 1 2 V Re G imp 1 Im G imp 1 Figure 11 The 3D bandstructure, originally prepared in a topologically trivial phase, is subject to an ω -dependent self-energy; the resulting system is a strong TI. Shown is the corresponding surface spectral function with the chemical potential at zero. Reprinted with permission from (Wang et al. , 2012b). Copyright (2012) by the American Physical Society.  \nthis makes two states per spin per unit cell since each unit cell consists of two sites. However, as an on-site Hubbard model, the Kane-Mele model can be seen as a spinful singleband model. The BHZ model in contrast takes into account one electronlike orbital (E) and one holelike orbital (H) for each spin. Hence, a lattice version based on this model should be naturally considered as a two-band Hubbard model. We consider the Hamiltonian, H = H BHZ + H int , (1) where H BHZ is the single-particle Hamiltonian associated with the Bloch-Hamiltonian H BHZ( k ) diag ( h ( k ) ,h ∗ ( k ) ) , where H (1 , ized. To begin with, we illustrate the simplest scenario possible. Consider the Kane-Mele model with staggered sublattice potential H v as defined in Eq. (18). The phase diagram coincides with the one of the Haldane model for φ = ± π/ 2 , Fig. 3, except that both CI phases are replaced by QSH phases. For sufficiently large λ v the system is in a trivial insulating phase and the majority of particles occupies B sites rather than A sites. At half filling, that implies that several A sites are doubly occupied while B sites are empty. Fig. 2(b). Phase boundaries are in accordance with the prediction based on FDWN. The surface Green's function could be directly calculated for the interacting system with given self-energy or from the surface Green's function associated to the pseudo-Hamiltonian ˜ H k , Eq. (4). We have numerically verified that the two approaches give identical results. The resulting surface spectral functions are shown in Fig. 3.Therearegaplesssurfacestates crossing the Fermi surface when the system is in the interacting topological phase; Fig. 3, upper panel.  \n/negationslash  \n235104-2  \n2  \n3", '1. Fluctuation-Induced Topological Phases FIG. 2. (Color online) Left panel: Paths of the mapping': "2  \n/Gamma1  \n<V  \n,  \nand results in a normal insulating phase.  \nω  \nQuite generally, Coulomb interactions can 'tune' topological insulator band structures from the trivial to the topological domain. This can already be understood in simplified meanfield pictures. Not only a 'true' self-energy Σ( ω, k ) , but in principle also k -dependent (static mean-fields) or purely ω -G -1 imp = i ω -V i ω -P with V = 1, P = 2(blueline),and P = 0 . 15 (red line). Gray line indicates the large circle we appended to close the contour. Crosses on real axis indicate the eigenenergy continuum of the noninteracting Hamiltonian, whose lower (upper) bound equals /Delta1 ( /Gamma1 ). Topological transition between TI and metal occurs when | P | /Delta1 = V 2 , where the blue line touches the lower bound. The gap 2  \nreopens for  \nP  \nRight panel: Phase diagram determined by calculating the  \nZ  \n|  \n|  \n2 index  \nof the pseudo-Hamiltonian (see the PE coefficients in Table I). Shaded  \nand white region have  \nZ  \n2  \n1 and 0, respectively, the yellow region  \ndenotes the metallic state. The phase boundaries  \nP  \n|  \n|  \n/Gamma1  \n=  \nV  \n2  \n=  \nP  \n/Delta1  \nare in accordance with the FDWN analysis.  \n|  \n|  \n=  \nV  \n2  \nand  \n↦  \n→  \n<!-- image -->  \ncase (a) with  \nV  \nP  \n=  \n0  \n.  \n15 is normal insulator. The chemical potential lies at zero.  \n=  \n1. Upper panel:  \nP  \n=  \n2 is in TI phase. Lower panel:  \n=  \nengenstates.  \n-  \n=  \na semimetal with two Fermi points  \nK  \n/.0006  \n/.0006  \n/.0006  \n/.0001  \n/.0136  \n2  \n/.0025  \n=  \n3  \nand the density of states vanishes linearly; the  \ndispersion in the vicinity of these so-called Dirac points  \nis governed by a 2D massless Dirac Hamiltonian in  \nk  \nspace. The CDW phase corresponds to an ordinary insula-  \ntor with a gap at the Fermi energy. As for  \n/.0031  \n,  \nits phase  \nrelative to the nearest-neighbor hopping amplitude plays  \nat a bicritical point. By integrating out the fermionic fields,  \nit is possible to construct a Landau-Ginzburg (LG) theory  \nnear the semimetallic region. Because of the linear disper-  \nsion of the Fermi points, the LG free-energy contains  \nanomalous terms of the form  \n/.0026  \n3  \nand  \nIm  \n/.0031  \n3  \n[16].  \nThus, even within mean-field theory, the CDW order pa-  \nHaldane term for spinless (spinful) fermions) or SU(2) spin symmetry (corresponding to the Kane-Mele term for spinful fermions only). In Ref. (Raghu et al. , 2008) both nearest and next-nearest neighbor repulsions were considered, but the main player clearly is the second-neighbor repulsion, an important role in determining its properties: while a nonzero Re /.0133 /.0031 /.0134 merely shifts the energy of the Dirac points, a nonzero imaginary part Im /.0133 /.0031 /.0134 opens a gap at the Fermi points. Thus, when the system remains at half-filling, it is more favorable to develop purely imaginary next-neighbor hopping amplitudes; such a configuration corresponds to a phase with spontaneously broken time-reversal symmetry. To see whether such a phase can be favored, we minimize the free-energy and arrive at the following self-  \nconsistent equations:  \nϕ  \nϕ  \nϕ  \n/.0133  \n/.0134  \nϕ  \nFIG. 3. Left: Phase diagram obtained with iDMRG calculations on an infinit  \n<!-- image -->  \nFIG. 2 (color online).  \nPhase diagram for spinless fermions  \nH (2) I = V 2 ∑ 〈〈 ij 〉〉 ( n i -1)( n j -1) , (32) /.0026 /.0136 1 6 L 2 X k V 1 /.0026 /.0135 2 V 2 /.0031 S k /.0135 /.0022 /.0030 S /.0030 /.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 ϕ t /.0133 k /.0134 ϕ 2 /.0135/.0133 V 1 /.0026 /.0135 2 V 2 /.0031 S k /.0135 /.0022 /.0030 S /.0030 /.0134 2 q ; (4)  \n<!-- image -->  \nis  \nwhere for spinful fermions n i = n i ↑ + n i ↓ . For the spinful case, one can define the operator χ µ ij = c † iα σ µ αβ c jβ , µ = 0 , 1 , 2 , 3 with σ µ = ( 1 , σ ) . The χ operators are related to the V 2 term by the identity ( n i -1)( n j -1) = 1 -1 2 ( χ µ ij ) † χ µ ij . (33) /.0031 /.0136 S /.0030 6 L 2 X k S k /.0135 /.0022 /.0030 /.0133 V 1 /.0026 /.0135 2 V 2 /.0031 S k /.0135 /.0022 /.0030 S /.0030 /.0134 /.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 ϕ t /.0133 k /.0134 ϕ 2 /.0135/.0133 V 1 /.0026 /.0135 2 V 2 /.0031 S k /.0135 /.0022 /.0030 S /.0030 /.0134 2 q : (5) Because of the vanishing density of states near the Fermi ( t /.0136 1 ). The semimetallic (SM) state that occurs at weak coupling is separated from the CDW and the topological QAH states via a continuous transition. The line separating the QAH and CDW marks a first-order transition, which terminates at a bicritical point. 156401-2 The phase boundaries, especially at second order transitions (see Sec. IV) a estimated from Figs. 4 to 8. We draw sharp lines here for better clarity. Ri patterns for the four phases with largest unit cell discussed in the main text. the particle number expectation value on the respective site given by Eq. (2). proportional to the amplitude t ij between nearest neighbors defined by Eq. (3) the red polygons. These correspond to (a) CMs phase with V 1 /t = 0 . 8 ,V 2 /t = 3 . (c) CDW II phase with V 1 /t = 5 . 6 ,V 2 /t = 3 . 2 , (d) CDW III phase with V 1 /t defect lines of rotated hexagons that have zero energetic cost in the classical li indeed it is severely reduced as the bond dimension χ increased. This suggests that the cylinder geometry implemented in iDMRG artificially differentiates bonds in its two perpendicular directions. The asymmetry induced by the finite bond dimension vanishes as χ is increased and thus it is not a physical effect. This is consistent Figure 12 (Left) Mean-field phase diagram for spinless fermions on the honeycomb lattice as originally proposed containing interactioninduced Chern insulator (QAH) and charge-ordered phases (CDW). Reprinted with permission from (Raghu et al. , 2008). Copyright (2008) by the American Physical Society. (Right) Density-matrix renormalization group phase diagram for the same model containing a variety of charge-ordered and modulated phases but not a topological phase. Reprinted with permission from (Motruk et al. , 2015). Copyright (2015) by the American Physical Society.  \nA standard mean-field decoupling leads to order parameters 〈 χ 0 〉 (Chern insulator phase) and 〈 χ i 〉 , i = 1 , 2 , 3 (QSH phase). Raghu et al. (Raghu et al. , 2008) showed using meanfield and functional RG methods that for sufficiently large V 2 a topological phase is stabilized with a slight tendency towards the QSH phase.  \nFor the spinless case, a similar procedure led to the proposal that a Chern insulator phase should be realized when V 2 is sufficiently large (Raghu et al. , 2008), see the original phase diagram in Fig. 12 (left). While the spinful problem has been too challenging to be attacked by numerical methods so far, many authors studied the spinless case (Capponi and Läuchli, 2015; Ðuri'c et al. , 2014; Daghofer and Hohenadler, 2014; Dauphin et al. , 2012; García-Martínez et al. , 2013; Grushin et al. , 2013; Liu et al. , 2016; Motruk et al. , 2015; Venderbos and Fu, 2016; Weeks and Franz, 2010a; Wen et al. , 2010). In most of these works a topological Chern insulator phase has not been found. Instead, charge density wave orders were stabilized. For instance, the phase diagram obtained within the density matrix renormalization group is shown in Fig. 12 (right); a topological phase is absent. For details and references see the recent review (Capponi, 2017). An analogous study for spinless Dirac fermions on the π -flux (square) lattice led to comparable results: while a topological phase was present within the mean-field treatment, exact diagonalization found charge order instead (Jia et al. , 2013). The Lieb lattice represents a third setup where the Raghu-idea can be tested; it might have the advantage that a cold-atom realization seems to be feasible (Dauphin et al. , 2016).  \nA somewhat analogous idea to generate topological phases by opening the gap at a 'Dirac crossing' has been proposed by Sun et al. (Sun et al. , 2009): a quadratic band crossing (QBC) point instead has the major advantage of a drastically increased density of states. Short-range repulsive interactions turn out to be marginally relevant in the renormalization group sense, i.e. , that the QBC point will acquire a gap even for arbitrarily weak interactions. Such weak interactions will be unable to generate charge order as it happens for the case of Dirac fermions. In the spinless case, a quantum anomalous Hall phase was found; in the spinful case, both quantum  \nwith a semimetal state; the logarithmic divergence of en-  \ntanglement of a metallic state requires a matrix product  \nstate with χ →∞ . The bond asymmetry reduces the entanglement by shifting the Dirac cones away from the K and K ' points. 38 Together, the previous numerical evidence are consistent with the semimetallic state. We note that numerically the semimetal state extends beyond { V 1 ,V 2 } /lessorsimilar t towards higher interaction strengths through a narrow semimetal trench in the phase diagram. The larger size of this region at L y = 6 (not shown) suggest that it is likely to shrink as the circumference of the cylinder is increased but a definitive statement requires going beyond the numerically accessible sizes. B. Charge density wave I (CDW I) Upon increasing V 1 we identify a charge density wave state labeled CDW I in the phase diagram of Fig. 3. This state has a two site unit cell and is characterized by a finite order parameter of symmetry B 2 under the symmetry group C '' 6 v 39 B 2 = 〈 n A 〉-〈 n B 〉 , (4) that can be calculated using (2), where n A and n B are the fermion densities in the A and B sublattice sites of the two site unit cell respectively. The resulting charge order is depicted schematically in Fig. 2(b). For instance, anomalous Hall and QSH phases can be realized. In addition, also nematic phases might emerge for stronger interactions (Sun et al. , 2009). All these predictions are in agreement with subsequent analytical and numerical work (Uebelacker and Honerkamp, 2011; Vafek, 2010). While the Chern insulator phase could never been detected within exact diagonalization or density matrix renormalization group for the case of Dirac crossings ( e.g. on the honeycomb lattice) (Capponi and Läuchli, 2015; Motruk et al. , 2015), recently this has been accomplished for the case of QBC points on the kagome (Zhu et al. , 2016) and on the checkerboard lattice (Wu et al. , 2016; Zeng et al. , 2018). For the latter, also an intermediate bondordered phase sandwiched between the Chern insulator and nematic charge ordered phase has been found (Zeng et al. , 2018).  \nOne should not conclude, however, that the Raghu proposal is wrong; it rather seems that the honeycomb lattice with its linear density of states is a difficult terrain to search for this kind of physics. The success in finding the interactioninduced phase for the case of QBC points demonstrates that the intuition of Raghu et al. (Raghu et al. , 2008) was correct, only the energetics on the honeycomb lattice were unfavorable to stabilize the topological phase by virtue of interactions. As a side remark, it might be interesting to consider the Raghu proposal for spinless fermions on an anisotropic honeycomb lattice: lattice anisotropies could be chosen such that the charge density wave order becomes energetically less favorable. It would be interesting to test whether this is sufficient to eventually stabilize the topological phase.  \nUltimately, it would be desirable to probe the physical mechanism elucidated in this subsection experimentally. Fermionic optical lattices might provide the proper environment to test these predictions experimentally (Sun et al. , 2012). Motivated by such proposals, it was realized that QBC points can also lead to interaction-driven topological phases of bosons: the bands which form the QBC point exhibit both non-negative curvature. Onsite-interactions induce a quantum anomalous Hall phase for the Bogoluibov quasiparticle spec-  \nat  \nV  \n1  \n/t  \n= 4  \nmagnitude of  \ncuracy this p  \nFor  \nl a r g e  \nV  \nbility since  \nbalance bet  \nbeen found  \nact  \ndiagon  \nsimulations.  \nC.  \nSubl  \nFor small  \ngenerate. W  \nas  \nV  \n2  \n>V  \n1  \nwith charge  \nthe CMs ph  \nparameters f  \ncorrespondin  \ntion values of  \nwhere A and  \nvenient to a  \nthat has well  \nsymmetry of  \ntry group  \nC  \nfour dimensi  \n6  \nThe CMs sta  \nof both the  \nthat obey  \nK  \nb  \ni  \n1  \ntrum characterized by a finite Chern number (Di Liberto et al. , 2016).  \nOther systems where interaction-induced topological phases were proposed are certain heterostructures, so-called digital transition metal oxide heterostructures (Rüegg and Fiete, 2011; Rüegg et al. , 2012; Xiao et al. , 2011). The involved transition metal oxides have a cubic environment but are grown in the (111) direction which results in a 'buckled' honeycomb net. The interplay of the complex orbital order and electron-electron interactions leads to either Chern insulator or topological insulator phases if interactions are not too strong. Otherwise Mott-type physics dominates (Okamoto et al. , 2014) which might even lead to spin liquid groundstates (Okamoto, 2013).", '3. Topological Kondo Insulators': "The topological Kondo insulator (Alexandrov et al. , 2013; Dzero et al. , 2010, 2016) has been proposed to be realized in SmB 6 and is the only interaction-induced topological insulator phase present in a real material so far. Other material proposals such as YbB 12 have been suggested (K. Hagiwara et al. , 2016; Weng et al. , 2014). For a detailed review about topological Kondo insulators we refer the reader to (Dzero et al. , 2016; Sun and Wu, 2017). Note that the Kondo effect (Kondo, 1964) is an interaction-driven low temperature phenomenon. The so-called Kondo insulators or heavy fermion semiconductors are materials with a narrow band gap which is caused by the Kondo effect, the hybridization between conduction electrons and (localized) f electrons.  \nThe main idea of topological Kondo insulators is explained in the following. Kondo insulators such as SmB 6 exist for more than 50 years and some of them are well-studied. The difference between a conventional and topological Kondo insulator can be summarized in two main points: (i) the relevance of strong spin-orbit coupling of f electrons (much larger than the typical Kondo gap) and (ii) the different parity of f and d electrons, where the latter exhibits odd and the former even parity. That is, each time there is a band crossing between f and d electrons the Z 2 index changes, which leads under the right circumstances to a TI phase.  \nThe experimental situation is far more complicated; alternatively, one could say that the topological insulator proposal for SmB 6 has proven to be sufficiently exciting that it has stimulated a variety of experiments performed by many groups. While several experiments are compatible with or proof for the topological insulator picture, others are not. In the following, we only mention a few relevant experiments; a detailed discussion is given in the reviews by Dzero et al. (Dzero et al. , 2016) and by Sun and Wu (Sun and Wu, 2017); note that the latter proposes the scenario of an 'accompany-type valence fluctuation state' which possibly coexists with the Kondo ground state of SmB 6 . Large single crystals of SmB 6 are available (Hatnean et al. , 2013) and many experiments find evidence of a surface state at low temperatures: com-  \nbined ARPES and ab initio studies (Frantzeskakis et al. , 2013; Jiang et al. , 2013; Kim et al. , 2013; Neupane et al. , 2013; Wolgast et al. , 2013; Zhu et al. , 2013), point-contact spectroscopy (Zhang et al. , 2013), thickness-dependent transport measurements (Kim et al. , 2014), magnetoresistance measurements (Chen et al. , 2015), and - possibly most convincingly - quantum oscillation measurements (Tan et al. , 2015). 2D Fermi surfaces have also been observed using torque magnetometry (Li et al. , 2014; Xiang et al. , 2017). The spinpolarized structure of the surface states was detected (N. Xu et al. , 2014) being in good agreement with topological surface states. One of the obstacles are the low temperatures T < 4 K which are required to observe the surface states (Kim et al. , 2013). In a recent strain experiments it has been reported that with 0.7% tensile strain the surface-dominated conduction can be observed up to a temperature of 240 K, persisting even after the strain has been removed. Theoretically, the surface quasiparticle interference has been calculated (Baruselli and Vojta, 2014). Also, it has been shown that the cubic topological Kondo insulator models possess distinct topological crystalline insulating phases (Baruselli and Vojta, 2015).  \nIn terms of topological toy models one can understand topological Kondo insulators in a simplified picture: while the d electrons form a conduction band, the f electrons are typically localized and can be described as Ising or Heisenberg spins ( i.e. , the f band is (almost) flat). The Kondo hybridization which is present due to interactions features in certain crystals a symmetry such that it mimics the form of a gapless Dirac-theory. The aforementioned normal-state dispersions act as a mass-term for the gapless Dirac-theory associated with the Kondo hybridization. The result is a QSH-type insulator in 2D (Werner and Assaad, 2013) or a strong/weak TI in 3D(Dzero et al. , 2016). Similar findings were reported using the Gutzwiller wave function technique thus showing that also non-local correlation effects lead to topological Kondo insulating phases on the square lattice (Wysoki'nski and Fabrizio, 2016). Also 1D versions of a topological Kondo insulator have been discussed (Alexandrov and Coleman, 2014; Pillay and McCulloch, 2018; Zhong et al. , 2017). Most works concentrate, however, on 3D and the SmB 6 compound. As previously mentioned, whether or not SmB 6 is indeed a topological Kondo insulator has been debated in the past years (Dzero et al. , 2016).  \nLet us emphasize again that all the interaction induced topological phases discussed in Sec. III.C possess a (noninteracting) band-structure analogue. That is, one could adiabatically transform the interaction-induced phase into a topological band structure without closing of the bulk gap. That would even apply to the strongly interacting topological Kondo insulators.", 'D. Exotic Strongly Correlated Topological Phases': "In this section, we consider exotic quantum states of matter which are also 'interaction-induced topological phases'.  \nThe major difference to the models discussed in the previous section is, however, that these systems do not have a bandstructure analog. Moreover, these states of matter merely exist due to the interplay of non-trivial band topology and strong electron-electron interactions. Examples discussed in the following are topological Mott insulators (Sec. III.D.1), fractional Chern insulators and fractional topological insulators (Sec. III.D.2); topological spin liquids (not further discussed here) should also be mentioned (Balents, 2010).", '1. Topological Mott Insulators': "The Topological Mott insulator (TMI) it is the prototype of an interacting topological insulator . Originally proposed by Pesin and Balents (Pesin and Balents, 2010) for Pr 2 Ir 2 O 7 , it realizes a three-dimensional U(1) spin liquid (Wang and Senthil, 2016). One might intuitively think about this state of matter in the following way: due to the strong correlations the charge degrees of freedom have been stripped from the original electrons and are frozen in a Mott insulating phase; the spinons ( i.e. , emergent quasiparticles carrying only spin degree of freedom) inherit the non-trivial band topology of the underlying TI bandstructure. The TMI exhibits spin-only Dirac cone surface states. Today, more than seven years after the proposal, there are no experimental signs or clues of the TMI phase; even the theoretical understanding is rather limited (Cho et al. , 2012; Kargarian et al. , 2011; Pesin and Balents, 2010; Rachel and Le Hur, 2010; Scheurer et al. , 2015; Wang and Senthil, 2016; Witczak-Krempa et al. , 2010; Yoshida et al. , 2014; Young et al. , 2008). The relationship between interacting topological insulators and quantum spin liquids as well as their classification was discussed in Ref. (Guo et al. , 2018).  \nThe reader may be warned that the term 'Topological Mott insulator' has also been introduced and widely used for the interaction-induced Chern or QSH insulators as discussed in Sec. III.C.2. But these systems have nothing in common with the TMI phase of Pesin and Balents: the former are adiabatically connected to non-interacting band insulators while the latter is a strongly correlated state of matter not having any bandstructure analog.  \nHere we will first briefly explain why the TMI phase does not exist in the KMH model or any other 2D Z 2 TI. Then we will discuss the pyrochlore model of Pesin and Balents and its extensions. Eventually we will introduce a 3D generalization of the Hofstadter-Hubbard model (Scheurer et al. , 2015) discussed in Sec. III.B.6 which provides a scenario to realize the TMI phase using ultracold quantum gases in an optical lattice setup (Scheurer et al. , 2015).  \nWithin slave-rotor theory (Florens and Georges, 2002, 2004; Zhao and Paramekanti, 2007), one introduces phase variables θ j conjugate to the total number of charges or fermions on lattice site j and fermionic auxiliary ('spinon')  \noperators f jσ by rewriting the original fermion operators:  \nc jσ = e iθ j f jσ , c † jσ = e -iθ j f † jσ . (34)  \nThe motivation is that the electrons c jσ are represented by a collective phase degree of freedom θ (conjugate to charge) and spinons f jσ (describing spin degree of freedom). Introducing an additional quantity, the angular momentum L j ≡ -i∂ θ j associated with a quantum O(2) rotor θ , simplifies the original Hubbard term: the quartic fermionic Hubbard interaction term reduces to a bilinear L 2 j in the angular momentum operators, a simple kinetic term. This simplification comes for the price that (i) bandstructure terms of the bare electrons are now quartic in the rotor and spinon variables and (ii) the Hilbert space has been enlarged, unphysical states are present. The first point can be resolved by performing all kinds of meanfield decouplings (Florens and Georges, 2002, 2004); the second point is more fundamental as it requires the introduction of a constraint,  \n∑ σ f † iσ f iσ + L i = 1 . (35)  \nImposing this constraint guarantees to involve only physical states.  \nAs a side remark, let us emphasize that already on this operative level one can claim that any Z 2 topological band insulator will be stable towards weak electron-electron interactions at least up to U ∼ t . This can be best seen by considering the Hubbard term at half filling,  \nU ∑ i n i ↑ n i ↓ = U 2 ∑ i ( ∑ σ n ( f ) iσ -1 ) 2 = U 2 ∑ i L 2 i . (36)  \nApparently HubbardU only affects the rotor sector, and by readily approximating the hopping terms in a mean-field fashion the resulting model is an XY model with its well-known phase transition from a Bose-condensed or superfluid phase ( U < t ) to a Mott insulating phase ( U > t ) . Condensation of the rotor means that a uniform ansatz θ i -θ j = 0 is favored. Since exp( ± i [ θ i -θ j ]) = 1 , the auxiliary fermions are proportional to the original electrons. With other words, as long as U < t the bandstructures of spinons and electrons are (up to some renormalization factors) identical - the TI phase persists (Rachel and Le Hur, 2010).  \nWorking out the slave-rotor theory quantitatively for the KMHmodel suggests that at U c ( λ ) a phase transition into the TMI phase occurs,  \nU c ( λ ) = [ 1 2 N Λ ∑ k ' 1 √ ξ k -min( ξ k ) ] -2 . (37)  \nFurther analysis including the effect of gauge fluctuations reveals that the presence of a dynamical gauge field is inevitable. Following Polyakov the fractionalized TMI phase is not stable against gauge fluctuations ('instanton proliferation') (Polyakov, 1975) causing an XY instability, i.e. ,  \nNATURE PHYSICS  \nDOI: 10.1038/NPHYS1606  \n<!-- image -->  \n/  \nt  \nλ  \nPhase diagram based on the slave-rotor approximation and  \nstrong coupling limit, as a function of Hubbard repulsion  \n(relative to hopping  \noccurring for moderately strong electron-electron repulsion are the  \nmetallic, the TBI, the TMI and the gapless Mott insulator (GMI) phases.  \nThe dashed line denotes a zero-gap semiconductor state resulting from an  \n'accidental' gap closing. The dotted line schematically demarcates the  \nregion, where magnetic ordering is expected. As discussed in the  \nmain text, long-range Coulomb interactions are expected to induce an  \nexcitonic region in the vicinity of the metal-TBI boundary shown here.  \nNow consider the SOI, denoting its strength by  \nt  \nmomentum operator is equal to minus the usual spin-one angular  \nmomentum operator  \nBand structure  \n<!-- image -->  \n<!-- image -->  \nL  \n2  \ng  \n(ref. 16). Thus, we may effectively consider  \nthe pseudo-total angular momentum J = L + S , which commutes with the SOI. The multiplet structure for the usual /lscript = 1 states is inverted, and the t 2 g manifold splits into a j = 1 / 2 doublet with energy λ , and a j = 3 / 2 quadruplet with energy -λ / 2. We will work in the local diagonal basis of the j eigenstates, and introduce a single label α such that α = 1 , 2 and α = 3 ··· 6 denote the doublet and quadruplet, respectively, with orbital energy εα = λ for the doublet and εα = -λ / 2 for the quadruplet. The specific forms of the wavefunctions of these states are given in the Supplementary Information. We assume that hopping between nearest-neighbour Ir ions is accomplished through the oxygen atoms nearest to a given pair, Fig. 2b. In reality, this is not necessarily the case 17 . However, our assumption minimizes the number of free parameters, and is resilient to perturbations that are not too strong. The model Now consider large limit SOI is weak only if 'bandwidth' for large Figure 13 (a) Interacting phase diagram of the topological pyrochlore model (Pesin and Balents, 2010) where the topological Mott insulator was originally proposed. Reprinted with permission from (Pesin and Balents, 2010). Copyright (2010) by Nature Springer (Nature Physics). (b) Surface state of the STI phase within the TR invariant 3D Hofstadter-Hubbard model shown in the surface BZ with k surf = ( k y , k z ) . (c) Interacting phase diagram of the TR invariant 3D Hofstadter-Hubbard U vs. t z including various topological Mott phases as explained in the text. Panels (b), (c) are taken from (Scheurer et al. , 2015).  \nARTICLES  \nThe Hamiltonian (1) contains two dimensionless parameters: λ / t and U / t , which define the phase diagram in Fig. 1. It is instructive first to consider various simple limits. For U = 0, we have a free-electron model, which is of course exactly soluble. Owing to inversion symmetry, one in general obtains 12 doubly degenerate bands. For small λ / t , these overlap at the Fermi energy and one obtains a metal. For large λ / t , the upper four bands originating from the j = 1 / 2 doublet become well-separated from the lower eight bands. As there are four holes per unit cell, the upper four bands are half-filled in total. On inspection, we can see (Fig. 3) that they show a bandgap, indicating the formation of a band insulator at large λ . As shown by Fu and Kane 19 , one can determine the band topology of an insulator with inversion symmetry either from the parity of the Hamiltonian eigenstates at time-reversal-invariant momenta, or from the number of Dirac points on the surface of the insulator. Applying the first criterion (see Supplementary Information), we The previously discussed TR invariant Hofstadter-Hubbard model (see Sec. III.B.6) can be extended to three spatial dimensions by introducing hopping in the z direction such that it picks up different Peierls phases for ↑ - and ↓ -spins (Scheurer et al. , 2015). Starting at t z = 0 with the 2D system, we can investigate the dimensional crossover to 3D. The surface state of the STI phase at t z = t is shown in Fig. 13 (b) proving the topological character of the STI phase. Similar to the behavior of edge states of the Hofstadter problem in 2D (see for instance Fig. 1 (b)), also in 3D the diabolic point ( i.e. , the Dirac nodal point) is buried in the bulk.  \nfind that the large λ / t state is a pure 'strong' TBI of the electrons (the weak Z 2 invariants vanish, consistent with cubic symmetry). Wealso calculated the surface-state spectrum (Fig. 4), which shows the required odd number of intersections with the Fermi level on passing between time-reversal-invariant surface momenta. The behaviour for general λ / t is as follows. For λ / t < 2 . 8, one obtains a metallic state, whereas for λ / t > 2 . 8 the bands separate at the Fermi energy. For almost all of this range of large λ / t , the system is a (strong) TBI. However, an 'accidental' closing of the bandgap occurs at λ / t ≈ 3 . 3, at which point it is a zero-gap semiconductor with eight Dirac points located along the 〈 111 〉 directions in reciprocal space. As the number of these Dirac points is even, there is no change in band topology resulting from the gap closure. Strong coupling limit U / t . In this limit, one has a Mott insulator, and the Hamiltonian is effectively projected into the space of one hole per Ir site, and the system is described by a spin-orbit The corresponding interacting phase diagram is shown in Fig. 13 (c). It contains correlated WTI and STI phases. At sufficiently large interaction strength U c ( t z ) in the range 2 . 25 < U c ( t z ) < 4 . 4 , one finds the transition into the fractionalized phase. The semimetallic phase boundary between WTI and STI phases becomes a gapless Mott insulator (GMI) with a semimetallic spinon spectrum (Scheurer et al. , 2015) being very similar to nodal spin liquid states. The correlated WTI phases become weak TMI (WTMI) (Kargarian et al. , 2011) phases for U > U c . Most importantly, the rather large correlated STI phase undergoes a phase transition in the strong TMI (STMI) regime for U > U c , see Fig. 13 (c).  \nDii  \n'  \nunique is that  \ni  \n'  \n>  \ni  \n. As  \nwith oxygen-mediated hopping is preferable, as it contains a single parameter determining the hopping strength: the hopping integral between Ir t 2 g states and O p orbitals ( Vpd π in the terminology of ref. 18). Integrating over the oxygens, and taking the simplest on-site Coulomb interaction involving only the total charge, we arrive at the Hubbard Hamiltonian, H = ∑ Ri α ( εα -µ ) d † Ri α dRi α + t ∑ 〈 Ri , R ' i ' 〉 αα ' T ii ' αα ' d † Ri α d R ' i ' α ' + U 2 ∑ Ri ( ∑ α d † Ri α dRi α -nd ) 2 (1) where R and i label the sites of the Bravais lattice and the tetrahedral basis and nd = 5 is the number of 5 d electrons on each Ir 4 + ion. The strength of hopping is parameterized by a single energy scale t ∝ V 2 pd π / ∆ , where ∆ is the energy difference between the Ir d and O p states. The procedure to obtain the dimensionless hopping matrices T ii ' αα ' , arising from the Ir-O-Ir hopping path, taking into account the rotation between the local cubic axes of each Ir ion, is given in the Supplementary Information. (Kugel-Khomskii type) model. Superexchange leads to spin (and orbital) exchange of order J ∼ t 2 / U /lessmuch t . We see that in this λ /lessmuch J /lessmuch t . Thus, the strong SOI regime is greatly enhanced with increasing correlations, as the relevant U is exchange rather than hopping. As the general spin-orbital Hamiltonian is cumbersome, and relevant only for very weak SOI, we will focus only on the strong SOI regime. Here, only the half-filled doublet at each site is relevant, and the effective Hamiltonian is of Heisenberg spin-exchange type, with an effective spin 1/2 at each site. The derivation of the low-energy Hamiltonian by second-order degenerate perturbation theory is standard. It is customary to write the resulting Hamiltonian as a sum of isotropic exchange, Dzyaloshinskii-Moriya and anisotropic exchange parts (suppressing the Bravais lattice index R for brevity): H spin = 4 t 2 U ∑ ii ' [ -19 243 + J S i S i ' + D ii ' · S i × S i ' + S i · ↔ /Gamma1 ii ' · S i ' ] (2) The convention to make the choice of all D and Γ terms are related by symmetry, it is enough to specify them for only one bond: J = 49,132 177,147 D 01 = 7,280 59,049 (0 , 1 , -1) Γ 01 = 1,568 59,049   -2 3 0 0 0 1 3 -1 0 -1 1 3   NATURE PHYSICS | VOL 6 | MAY 2010 | www.nature.com/naturephysics 377 an easy-plane antiferromagnetically ordered phase (Hermele et al. , 2008; Rachel and Le Hur, 2010). Since these arguments are rather generic for 2D TR invariant systems, we conclude that the TMI phase cannot exist in 2D TIs. The original TMI proposal of Pesin and Balents is formulated as a multi-band Hubbard model on the pyrochlore lattice which should be suitable for certain Ir-based pyrochlore oxides A 2 Ir 2 O 7 (Pesin and Balents, 2010). The model is sufficiently simplified and contains as free parameters only the spin-orbit coupling λ and local Coulomb interactions U ; the phase diagram is shown in Fig. 13 (a). The noninteracting system features a metallic regime for weak and a strong TI phase for strong spin-orbit coupling. While very strong electron correlations cause magnetic ordering, here the intermediateU regime is of interest. Within slave-rotor meanfield theory, a stable TMI phase can be found in addition to a gapless Mott insulating phase. The TMI phase is an exotic Mott insulator with spin-charge separation. In contrast to conventional U(1) spin liquids, here the spinons (the fractionalized excitations of the spin liquid or TMI state) exhibit a topologically non-trivial behavior; in particular, in the TMI phase the system features a spinon surface state (Pesin and Balents, 2010). When lattice distortions are incorporated, also weak TMI phases were suggested (Kargarian et al. , 2011). This analysis demonstrates that the TMI phase of Pesin and Balents is not limited to the pyrochlore model (Pesin and Balents, 2010) and can be found on other lattices and for other models. Moreover, the cold-atom setup proposed in (Scheurer et al. , 2015) opens the possibility to investigate TMI phases in a controlled way: as most bandstructure parameter as well as HubbardU can be tuned with high precision it promises to be a successful route to the observation of topological Mott insulators. Another interesting system involving electron fractionalization in topological band structures was proposed to exist in heavy transition metal oxides (Maciejko et al. , 2014). 2. Fractional Chern and Topological Insulators The fractional quantum Hall effect is realized in a twodimensional electron gas - nevertheless it represents a strongly correlated electron system. Due to the heavily quenched kinetic energy of perfectly flat continuum Landau levels, the ratio of interactions and kinetic energy is effectively large. For a surprisingly long time there was no proposal to stabilize the fractional quantum Hall effect in a lattice system. When it was realized that (topological) bandstructures could be tuned to become sufficiently flat without the need of unphysically large hopping ranges (Neupert et al. , 2011a; Sun et al. , 2011; Tang et al. , 2011), fractional Chern insulators (FCIs) became popular. Here we will briefly discuss the main ideas behind FCIs and also sketch the TR invariant extension, fractional topological insulators (FTIs). The interested reader should consult the review articles about FCIs and FTIS (Bergholtz and Liu, 2013; Neupert et al. , 2015a). Also a brief but more general review about Fractional Topological Insulators is available (Maciejko and Fiete, 2015).  \nThe main reason why FCIs were not proposed earlier might be due to the insight that most fractional quantum Hall states require longer-ranged Coulomb interactions to be stabilized (Haldane, 1983a). Including longer-ranged Coulomb interactions on a lattice typically leads to a ground state with conventional order such as charge order ( cf. discussion in Sec. III.C.2). In fact, FCI ground states do compete with charge ordered states and fine-tuning is indeed required.  \nThe first ingredient for FCIs is a sufficiently flat band with a non-zero Chern number or finite Hall conductivity. For the continuum Landau levels the non-zero Chern number stems from the external magnetic field. The lattice Chern insulators are more elegant as they do not require any magnetic field - the bandstructure itself leads to the non-trivial topology. Including longer-ranged hoppings to the minimal Chern insulator models allows to tune the 'band flatness' (Neupert et al. , 2011a; Sun et al. , 2011; Tang et al. , 2011). The second ingredient is an appropriate band filling in analogy to the fractional filling of Landau levels. For instance, in order to stabilize the lattice version of the ν = 1 / 3 Laughlin state the band filling also needs to be adjusted to 1/3. The third and final ingredient are electron-electron interactions (Regnault and Bernevig, 2011; Sheng et al. , 2011). In case of spinless particles, first- and second-neighbor Coulomb repulsion are already sufficient to stabilize several fractional quantum Hall states (Regnault and Bernevig, 2011). For hardcore bosons, the ν = 1 / 2 Laughlin state forms even without the need of any non-local interactions. Flat bands on arbitrary lattices (Hu et al. , 2011; Wu et al. , 2012b) and with higher or arbitrary Chern number have been discussed (Liu et al. , 2012b; Trescher and Bergholtz, 2012; Wang et al. , 2012c; Yang et al. , 2012). FCI ground states with Abelian and nonAbelian anyon excitations have been identified (Sterdyniak et al. , 2013) including Moore-Read type states (Regnault and Bernevig, 2011). Also a hierarchy of FCIs states has been shown to exist (Läuchli et al. , 2013) as well as multicomponent fractional states (Zeng et al. , 2017a). Pseudo potential descriptions in analogy to the fractional quantum Hall effect have been dervied (Lee et al. , 2013; Wu et al. , 2013). Fractional many-body states have also been proposed and investigated for the Hofstadter problem (see Sec. II.A). In contrast to FCIs, the fractional phases in Hofstadter bands realize a lattice version of the fractional quantum Hall effect in the presence of an external magnetic field. For both fermions and bosons fractional ground states have been shown to exist (Möller and Cooper, 2015).  \nAll these works about FCIs have in common that they consider non-interacting bandstructures which are already topologically non-trivial. In Sec. III.C we have seen that interactions can also 'make' topologically trivial bandstructures non-trivial. In a recent work, it was shown that it is even possible to directly induce an FCI phase in topologically trivial bands (Kourtis, 2018): in solid-liquid composites, the fractional phase can be induced by the formation of charge order, based on the idea of symmetry-breaking topological order (Kourtis and Daghofer, 2014).  \nIn the following, let us consider the spinful extension of FCIs. Bernevig and Zhang proposed such a state of matter already in 2006 (Bernevig and Zhang, 2006). Levin and Stern pointed out that it is not sufficient to 'glue' together arbitrary fractional Chern insulator ground states; they derived a criteria for spin-conserving systems which involves the spin-Hall conductance σ sH and the elementary charge e glyph[star] in units of e : if σ sH /e glyph[star] is odd, the resulting state is a fractional topological insulator; if it is even, one obtains a trivial insulator instead (Levin and Stern, 2009). Microsocpic models and studies based on exact diagonalization started after FCIs had been established (see above). The construction is usually straightforward: the FCI states originating from a spinless bandstructure with Chern number C = 1 and C = -1 , respectively, will be the same but possess opposite chirality. For a topologically non-trivial Z 2 bandstructure (in the spirit of the KaneMele construction) it must, hence, be possible to obtain a TRinvariant fractionalized phase where the ↑ -spins display the chiral FCI state while the ↓ -spins display the anti-chiral FCI state. Both combined lead to a TR invariant ground state just like the TR invariant TIs: the resulting state is a fractional topological insulator (Neupert et al. , 2011b; Repellin et al. , 2014; Wu et al. , 2012c) given it fulfills the Levin-Stern criteria (Levin and Stern, 2009). Using exact diagonalization, such constructions have explicitly been shown to exist (Neupert et al. , 2011b; Repellin et al. , 2014).  \nTo date there is no experimental evidence for the realization of a fractional Chern or topological insulator phase. This has inspired physicists to think about schemes which allow for the engineering of such phases: coupled-wire constructions. They can be thought of as arrays of one-dimensional systems ('wires') which can be treated within Luttinger liquid formalism. By coupling them in a smart way, they can reproduce topological states of matter. Originally pioneered for the integer (Lee, 1994; Yakovenko, 1991) and fractional quantum Hall effects (Kane et al. , 2002; Teo and Kane, 2014) in two dimensions, these ideas have been generalized and used to engineer essentially all types of topological states of matter. Given that the wires are modelled as Luttinger liquids, electron-electron interactions are intrinsically embedded into these systems, thus allowing to analytically study the interacting versions of these topological systems including fractional topological insulators (Iadecola et al. , 2016; Klinovaja and Tserkovnyak, 2014; Klinovaja et al. , 2015; Meng, 2015; Meng et al. , 2015; Meng and Sela, 2014; Neupert et al. , 2014; Sagi and Oreg, 2015; Volpez et al. , 2017). In principle, it seems possible to experimentally build such wire networks thus underlining the importance of coupled-wire constructions.", 'E. Interacting Surface States of Topological Insulators': "So far we always considered electron-electron interactions affecting the total system, i.e. , both bulk and edges or surfaces, respectively. 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Unfortunately, our finite size studies cannot  \npreclude the appearance of new phases in larger systems, a  \nFIG. 3 (color online). Numerical evidence for the emergence of (a) a chiral fermionic mode and (b) a chiral Majorana mode at the boundary between two ferromagnetic domains and a ferromagnetsuperfluid domain wall, respectively. Shown are finite size energy spectra for a system restricted to the n ≤ 2 . Magnetic domains on Figure 14 (a) Phase diagram of the STI surface in the presence of contact interactions with amplitude U vs. filling fraction ν . (b) Level distribution of the finite-size spectra for the chiral Majorana mode. Figures are taken from (Neupert et al. , 2015b).  \n<!-- image -->  \n0  \n0  \nthe northern or southern hemisphere are enforced by Hamiltonian  \n,  \nU(1) mode that consists of six states with m ¼ -5 = 2 ; … ; 5 = 2 assuming that the levels with m> 0 are occupied in the half-filled ground state. In (b), the level counting in the fermion parity sectors The surface states of 3D STIs are very special with respect to different aspects:  \nEq. (2) with a mean-field magnetization m z ð θ Þ σ z with j m z j ¼ Λ while the superfluid domain is interaction induced with U ¼ -4 . 3 Λ . In (a), the level counting is the one expected for a surface of a strong TI in 3D, ignore the bulk since it is gapped, and ask how interactions might influence the surface states.  \n- are as expected if one assumes that the chiral Majorana mode at the boundary consists of three operators with m ¼ -5 = 2 ; -3 = 2 ; -1 = 2 which do not annihilate the ground state. 1. They are described by a two-dimensional Dirac theory and there must be an odd number of such Dirac cones; in some cases there is only a single Dirac cone per surface. It is interesting to ask whether or not the surface of an STI violates the fermion doubling theorem (stating that Dirac cones on the lattice always need to come in pairs) (Nielsen and Ninomiya, 1981). A possible solution to this riddle is to recall that the slab geometry features on both top and bottom surface a Dirac cone (Hasan and Kane, 2010). In total, surface Dirac cones still come in pairs.\n- 2. Fu and Kane pointed out that breaking of the protecting U(1) charge symmetry, for instance by a superconducting proximity effect, leads on the 3D STI surface to an unconventional gapped s -wave superconductor with Majorana zero modes bound to the vortex cores (Fu and Kane, 2008).\n- 3. When TR symmetry is broken, such as by magnetic coating, the Dirac cone surface state acquires a gap, resulting in a ν = 1 / 2 quantum Hall effect (Qi et al. , 2008b; Sitte et al. , 2013) without fractionalized excitations, as expected from a single massive Dirac theory.  \nU=  \nΛ  \n→  \n-∞  \nis a direct manifestation of the localization  \nproperties of the single-particle states. If the single particle  \nstates were fully localizable in real space, pairs of electrons  \ncould bind into pointlike particles and the ground state  \nwould be exponentially degenerate. Conclusions. -We have developed a formalism to study interaction effects on fermionic 3DTI surface states numerically. From the analysis of a two-body contact interaction, we found both ferromagnetic and topologically nontrivial superconducting phases, as well as chiral fermion and chiral Majorana fermion boundary modes between different emerging on the surface of a strong 3D TI to be of NonAbelian type. In Ref. (Fidkowski et al. , 2013) a similar idea was pushed forward for the surface of 3D topological superconductors. A non-perturbative lattice construction for SPT and for symmetry-enriched topological phases and their gapped surfaces is provided in Ref. (Wang et al. , 2017).  \nphases. Several branches of future investigation can be anticipated, such as the application to bosons and studies of more sophisticated interaction profiles. The formalism establishes an ideal testing ground for topologically ordered TI surface state scenarios. T. N. acknowledges several fruitful discussions with F. D. M. Haldane and thanks him for sharing the results of Ref. [28] prior to publication. R. T. thanks T. Dumitrescu and M. Metlitski for discussions. This work was supported by DARPA SPAWARSYSCEN Pacific N66001-11-14110, by the Helmholtz association through VI-521, by the DFG through FOR 960, SPP 1666, and SFB 1143, and by the European Research Council through the grant TOPOLECTRICS (ERC-StG-Thomale-336012). These findings are most remarkable in the following sense: as discussed previously, a 3D TI is an SPT phase which has a gapped bulk but metallic surface states which are protected as long as the protecting symmetries are intact. Apparently the presence of electron-electron interactions circumvents these strict requirements, leading to a gapped surface which fully respects the protecting symmetries; the stronlgy correlated groundstate is accompanied by topological order. Most of these proposals are based on field-theoretical considerations; microscopic studies of interaction effects on the STI surface are limited to an exact diagonlization study (Neupert et al. , 2015b). Therein the STI surface has been studied in a microscopic setup by employing a spherical geometry. The Dirac cone surface states are described by a two-dimensional Dirac Hamiltonian, which is given in the limit of long wavelengths as  \n[1] R. Roy, Phys. Rev. B  \n79  \n, 195322 (2009).  \n[2] J. E. Moore and L. Balents, Phys. Rev. B 75 , 121306 (2007). H = v ˆ n ( -i ∇× σ ) (38)  \n017001-4 where σ is the spin operator, ˆ n is the surface normal, and v the Dirac velocity. For a spherical TI with radius R , (38) becomes(Imura et al. , 2012) H 0 = v/R ( σ x Λ θ + σ y Λ φ ) where Λ is the dynamical electron angular momentum when a magnetic monopole with strength 2 πσ z is present. Similar to Haldane's elegant idea to map the QHE for spinless fermions onto a sphere with a monopole in its center, here we end up with a setup where spinful electrons move on a sphere with a monopole in its center which possesses opposite sign for opposite spins to ensure TR invariance. Eigenstates in this setup are readily labeled, because Landau levels on the sphere are spanned by two mutually commuting SU(2) algebras. One is associated with the guiding center momentum L and one with the cyclotron momentum S . Another advantage is the absence of an edge; keep in mind that a surface state cannot have an edge in a normal sense. The spherical geometry provides a natural way of avoiding an edge.  \nRecently, it was proposed that there is yet a third possibility to gap the single Dirac cone on the 3D STI surface, transcending mean-field scenarios but involving topological order (see Sec. II.D). Motivated by the study of bosonic topological insulators in 3D (Burnell et al. , 2014; Metlitski et al. , 2013; Vishwanath and Senthil, 2013; Wang and Senthil, 2013), it was shown that such gapped surface states without breaking any symmetries are indeed possible. These works directly stimulated the analogous investigation of the fermionic 3D TIs (Bonderson et al. , 2013; Metlitski et al. , 2015; Wang et al. , 2013) and showed the topological order  \nA straightforward exact diagonalization study with Hubbard-like density-density interactions leads at half filling to a ferromagnetically polarized half-QHE (repulsive interactions) or to an s -wave superconductor (attractive interactions), see Fig. 14 (a). These calculations show that local density-density interactions are not sufficient to stabilize the predicted topologically ordered phases (Bonderson et al. , 2013; Metlitski et al. , 2015; Wang et al. , 2013), longer-ranged and potentially more-complex interaction terms need to be added. This remains an open problem suited for future studies because any interaction term can be implemented within the discussed framework.  \nAn s -wave superconducting termination of a 3D STI is a topological superconductor in the following sense: (i) it supports Majorana zero modes in its vortex cores and (ii) a chiral  \nture contains the honeycomb lattice as shown in Fig. 1(a).  \n[For the three-dimensional structure, see Fig. 2(d).] Each Ir  \natom is surrounded by an octahedron of six O atoms, which  \nleads to the energy level scheme similar to  \nSr  \nIrO  \n, i.e.,  \nconnecting a NNN pair.  \natom in the center of the he  \nlattice, the transfer integral  \ncan be  \nthe 180  \ncanting  \nis  \ndistortions.  \n/C11  \nin  \nSr  \n2  \nAs in  \nmodel  \ngered  \ntan  \n2  \n/C30  \nanisotropy.  \nð  \n2  \n1.5  \n~  \n~  \nS  \nKitaev-Heisenberg (or Heisenberg-Kitaev) model in the literature, given by the Hamiltonian 2 4 states. Therefore we can confer integral t 0 0 and breaks the  \nÞ  \nMajorana mode at the boundary with a ferromagnetic region of the surface. Using exact diagonalization, this can be conveniently shown by considering the following modified setups: (i) a superconductor-ferromagnet domain wall and (ii) the region between two ferromagnetic domains. The corresponding level counting for the chiral Majorana mode is shown in Fig. 14 (b). H KH ( α ) = α ∑ 〈 ij 〉 S i S j -2 α ∑ 〈 ij 〉 γ S γ i S γ j (39) tional statistics, topological degeneracy, and, in particular, playing PRL 102, 017205 (2009) PHYSICAL REVIEW one electron in j eff ¼ 1 = 2 struct the effective single-band model on the honeycomb lattice. Since the O p -level /C15 p are around 3 eV lower than the Ir d -level /C15 d [17], we can integrate out p orbitals to obtain the following effective Hamiltonian: t 0 0 ¼ 0 for the moment the topological properties To summarize these  \na  \n'  \n/C14  \nAn interesting aspect of interacting surface states of STIs is the emergence of supersymmetry (Grover et al. , 2014; Ponte and Lee, 2014). It has been predicted to occur at the quantum critical point at the end of the semimetallic (SM) line (with one Dirac fermion) where the superconducting (SC) phase evolves for U < 0 [see Fig. 14 (a)]. Indeed a recent Monto Carlo study reported the observation of supersymmetric behavior (Li et al. , 2017). An extension of this line of research led to the prediction of emergent supersymmetric electrodynamics on the surface of the topological Kondo insulator SmB 6 (see Sec. III.C.3) at the critical point between the semimetal (with three Dirac fermions) and the nematic pair density wave phase (Jian et al. , 2017). IV. RELATION TO MAGNETISM IN IRIDATES The honeycomb iridates A 2 IrO 3 (A=Na, Li) together with α -RuCl 3 are today known to play a major role in the active field of Kitaev materials (Hermanns et al. , 2018; Trebst, 2017; Winter et al. , 2017). Here we are not reviewing the experimental and theoretical developments of the past years but focus instead on the relationship of Kitaev spin exchange and topological insulating bandstructures, following the original papers which initiated the excitement about Na 2 IrO 3 (Jackeli and Khaliullin, 2009; Shitade et al. , 2009). where 0 ≤ α ≤ 1 and γ = x, y, z depending on the direction of the bond 〈 ij 〉 . The second term in (39) represents the previously mentioned compass or Kitaev interactions. The phase diagram of the KH model hosts for dominating Heisenberg exchange antiferromagnetic Neel order ( 0 ≤ α ≤ 0 . 4 ) and for dominating Kitaev exchange the spin liquid phase ( 0 . 8 ≤ α ≤ 1 ). In between another collinear antiferromagnet, dubbed stripy antiferromagnet, is realized. Both the paper by Shitade et al. and the one by Jackeli and Khaliullin have been very influential. Early experiments showed that magnetic long-range order occurs below T N = 15 K in Na 2 IrO 3 (Singh and Gegenwart, 2010; Singh et al. , 2012). As if the competing proposals of Shitade et al. and Jackeli and Khaliullin are not controversial enough, subsequent inelastic neutron scattering experiments revealed that the magnetic order is of zigzag type (Choi et al. , 2012) - not present in the Kitaev-Heisenberg (KH) model. In the meantime, it is also known that the sister compound Li 2 IrO 3 is likewise magnetically ordered with T N = 15 Kbut the order is of incommensurate spiral type (Williams et al. , 2016). Apparently both materials are not correlated TIs - but their magnetic groundstate is not contained in the original KH model (39). In another work, Na 2 IrO 3 was suggested to realize a 3D strong TI phase (Kim et al. , 2012); most research is, however, motivated by the idea that these compounds feature Kitaev spin exchange. As emphasized before, here we are not reviewing the manifold experimental and theoretical efforts of the past years; an overview of the current status can be found here (Hermanns et al. , 2018; Trebst, 2017; Winter it is relevant for quantum computation [18]. This generated an enormous interest in a possible realization of this model in real systems, with current proposals based on optical lattices [27]. Here, we outline how to ''engineer'' the Kitaev model in Mott insulators. Shown in Fig. 3(a) is a triangular unit formed by 90 /C14 bonds together with ''compass'' interactions that follow from Eq. (3). Such a structure is common for a number of oxides, e.g., layered compounds AB O 2 (where A and B are alkali and TM ions, respectively). The triangular lattice of magnetic ions in an AB O 2 structure can be depleted down to a honeycomb lattice (by periodic replacements of TM ions with nonmagnetic ones). One then obtains an A 2 B O 3 compound, which has a hexagonal unit shown in Fig. 3(b). There are three nonequivalent bonds, each being perpendicular to one of the cubic axes x , y , z . Then, according to Eq. (3), the spin coupling, e.g., on a ( x )-bond, is of S x i S x j type, precisely as in the Kitaev model. The honeycomb lattice provides a particularly striking example of new physics introduced by strong SO coupling: the Heisenberg model is converted into the Kitaev model with a spin-liquid ground state. The compound Li 2 RuO 3 [28] represents a physical example of the A 2 B O 3 structure. By replacement of spin-one Ru 4 þ with spin-one-half Ir 4 þ ions, one may realize a strongly spin-orbit-coupled Mott insulator with low-energy physics described by the Kitaev model. ''Weak'' ferromagnetism of Sr 2 IrO 4 .-As an example of a spin-orbit-coupled Mott insulator, we discuss the layered compound Sr 2 IrO 4 ,a t 2 g analog of the undoped highT c cuprate La 2 CuO 4 . In Sr 2 IrO 4 , a square lattice of Ir 4 þ ions is formed by corner-shared IrO 6 octahedra, elongated along the c -axis and rotated about it by /C11 11 /C14 [19] (see Fig. 4). fact is ' 0 : 14 /C22 B than that angle /C30 rigidly we show natural e We first Sr 2 IrO 4 Here, the sin 2 /C18 þ 1 2 and third pies, with [For /C11 ¼ H 0 ¼/C0 t X h ij i ½ d y i d j þ H : c : /C138þ X hh ij ii ½ d y i ^ t 0 ij d j þ H : c : /C138 ; (2) where h ij i and hh ij ii denote the nearest-neighbor (NN) and next-nearest-neighbor (NNN) pairs, respectively. The transfer integral t between a NN pair is real and spin independent as given by t ¼ 1 3 ð pd /C25 Þ 2 /C15 d /C0 /C15 p ð pp /C27 Þþ 3 ð pp /C25 Þ /C15 d /C0 /C15 p ; (3) where ( pd /C25 ), ( pp /C27 ), and ( pp /C25 ) are Slater-Koster parameters between pd and pp , respectively [20]. Note that the contributions of the order of ð pd /C25 Þ 2 = ð /C15 d /C0 /C15 p Þ cancel out in the honeycomb lattice, in sharp contrast to Sr 2 IrO 4 with the square lattice. The transfer integral between a NNN pair depends on spin, leading to a topological insulator. The local x , y , and z axes at an Ir atom are chosen to point in the direction of neighboring O atoms as shown in Fig. 1. Therefore Z ¼ð x þ y þ = ffiffiffi 3 p is perpendicular to the honeycomb plane. With this convention, the transfer integral is a 2 /C2 2 matrix in the spin space, and is written as ^ t 0 ij it 0 /C27 a þ t 0 0 ; (4) is the direction whose projection onto the and spin independent for spin dependent for a NNN is related to the Haldane  \nSr  \n2  \nIrO  \n4  \nundergoes a magnetic transition at  \nS  \nz  \nz  \nFIG. 2 (color online).  \n<!-- image -->  \nFIG. 3 (color online).  \nExamples of the structural units formed  \n/C24  \n240 K  \ndis-  \nzz  \n(a)  \nα  \n/  \nφ  \n1  \ning the SOI in two different  \nred thin solid, green dashed,  \nj  \n'  \n<!-- image -->  \neff  \n1  \n=  \n2  \n, Ir  \nj  \neff  \n3  \n=  \n2  \n, Ir  \ne  \n0.5  \n¼  \n¼  \nFermi energy is set to zero.  \nby 90 /C14 TM-O-TM bonds and corresponding spin-coupling patterns. Gray circles stand for magnetic ions, and small open circles denote oxygen sites. (a) Triangular unit cell of AB O 2 -type layered compounds, periodic sequence of this unit forms a triangular lattice of magnetic ions. The model (3) on this structure is a realization of a quantum compass model on a triangular lattice: e.g., on a bond 1-2, laying perpendicular to x -axis, the interaction is S x 1 S x 2 . (b) Hexagonal unit cell of A 2 B O 3 -type layered compound, in which magnetic ions ( B -sites) form a honeycomb lattice. (Black dot: nonmagnetic A -site). On an xx -bond, the interaction is S x i S x j , etc. For this structure, the model (3) is identical to the Kitaev model. 0 FIG. 4. of the an IrO 2 about z -axis /C18 ' /C25 = 5 , the in-plane z -axis (a) The honeycomb lattice of Ir atoms in axis. A large black circle shows an Ir atom surrounded by six O atoms (red small circles). (b) The transfer integrals on the honeycomb lattice. A black solid line , while blue short-dashed, red dash-dotted, and green it 0 /C27 x , it 0 /C27 y , it 0 /C27 z , respectively. (thin lines) and the extended parameters t ¼ 310 K , t 0 ¼ 60 K (thick lines). (d) The by black dashed lines, shown in Fig. 1(b). Because layers are not stacked in graphene. 256403-2 Figure 15 (a) Topological insulator picture. The transfer integrals on the honeycomb lattice: a black solid line shows -t , while blue short-dashed, red dash-dotted, and green long-dashed arrows indicate it ' σ x , it ' σ y , it ' σ z , respectively. Reprinted with permission from (Shitade et al. , 2009). Copyright (2009) by the American Physical Society. (b) Frustrated magnet picture. Hexagonal unit cell of A 2 BO 3 -type layered compound, in which magnetic ions (B-sites) form a honeycomb lattice. (Black dot: nonmagnetic A-site). On an xx -bond, the interaction is S x i S x j , etc. For this structure, the spin Hamiltonian contains Kitaev spin exchange. Reprinted with permission from (Jackeli and Khaliullin, 2009). Copyright (2009) by the American Physical Society.  \n017205-3  \ng  \nS  \nz  \n¼  \nThe Kane-Mele proposal (Kane and Mele, 2005a) stimulated the search for other honeycomb lattice compounds aiming to find a successful material realization of the Kane-Mele model. As pointed out in Sec. III.B.4, the layered honeycomb lattice iridate Na 2 IrO 3 was amongst the first proposals along this line. Ab initio calculation revealed (Shitade et al. , 2009) that the relevant nearest and next-nearest neighbor hoppings integrals are similar to the Kane-Mele scenario, see Fig. 15 (a). Shitade et al. claimed that-if Coulomb interactions are not too strong-the groundstate should be a correlated QSH insulator. About the same time, a competing proposal by Jackeli and Khaliullin was put forward: they claimed that Na 2 IrO 3 is deep in a Mott phase such that charge fluctuations are frozen out and the system is well described by a spin model (Chaloupka et al. , 2010; Jackeli and Khaliullin, 2009). Moreover, they proposed that due to large SOC and 90 degree bonding of the edge-sharing oxygen tetrahedra, which surround the Ir atoms, the interactions are dominated by Kitaev spin exchange (Kitaev, 2006) which is known to realize a spin liquid ground state. In Fig. 15 (b) the hexagonal unit cell in Na 2 IrO 3 is shown with the nearest-neighbor Kitaev spin exchange. Together with additional generic Heisenberg interactions, the resulting spin Hamiltonian is referred to as where a ¼ x , y , z FIG. 1 (color online). Na 2 IrO 3 viewed from the c shows /C0 t long-dashed arrows indicate  \nS  \ny  \ny  \n2 3  \nS  \ni  \n/C1  \net al. , 2017).  \nIn the following, we will discuss the connection between the TI physics of Shitade et al. and the KH model of Jackeli and Khaliullin. We note that the strong coupling expansion of the topological Hubbard model (the SIH model in Sec. III.B.4) results in the spin Hamiltonian Eq. (27) (Reuther et al. , 2012). The topological Hubbard model generates, hence, in the strong-coupling limit Kitaev spin exchange on next-nearest neighbor bonds instead on nearest neighbor bonds as suggested by Jackeli and Khaliullin. Partially motivated by this discrepancy, models were proposed containing both nearest and next-nearest neighbor Kitaev exchange (dubbed K 1 -K 2 models) and rich groundstate phase diagrams derived (Reuther et al. , 2014; Rousochatzakis et al. , 2015).  \nThe considerations in the previous paragraph might raise the question which bandstructure could result in nearestneighbor Kitaev spin exchange. Defining the spin-dependent, imaginary spin-orbit term of the SIH model on nearestneighbor bonds in addition to the standard real nearest neighbor hopping, indeed results in Kitaev and Heisenberg terms. The strong coupling expansion produces, however, another term which results from processes where an electron hops from site i to j with spin-orbit amplitude ˜ λ and hops back from j to i with normal hopping amplitude t (Laubach et al. , 2017; Yu et al. , 2013). The corresponding spin exchange in the strong coupling is the symmetric of-diagonal Γ exchange. In order to avoid the cross term, one could replace the spinorbit term by a spin-dependent real hopping term - then the mixed terms cancel and the resulting spin Hamiltonian is the KH model (39). Such a one-band Hubbard model was indeed discussed and shown to have several interesting properties (Hassan et al. , 2013; Liang et al. , 2014). The underlying bandstructure explicitly breaks, however, TR symmetry. That immediately implies that the weak-coupling regime cannot contain any TR invariant Z 2 TI phase as proposed by Shitade et al.  \nThe simplest model which captures all these aspects consists of three spinful orbitals, which one might identify with the t 2 g manifold consisting of d yz , d xz , and d xy orbitals (Laubach et al. , 2017; Rau et al. , 2014). The Hamiltonian is given by 6  \nH = H kin + H SOC + H UH , (40)  \nconsisting of kinetic, spin-orbit, and interaction part, respectively. H kin reads  \nH kin = ∑ 〈 ij 〉 γ ∑ n,n ' ,α d † inα T γ nn ' d jn ' α +H . c . (41)  \nwhere d inα denotes a fermionic annihilation operator on site i with orbital n ∈ { yz, xz, xy } and spin α ∈ {↑ , ↓} . For a z  \nbond the hopping matrix T z nn ' is gien by  \nT z nn ' =   t 1 t 2 0 t 2 t 1 0 0 0 t ' 1   , (42)  \nand the other matrices T y nn ' and T z nn ' follow by cyclic permutations of the rows and columns. The spin-orbit term H SOC is given by  \nH SOC = λ 2 ∑ i ∑ n,n ' ,α,α ' d † inα l nn ' · s αα ' d in ' α ' , (43)  \nwhere l is an angular momentum operator with l 2 = l ( l +1) = 2 and σ is the vector of Pauli matrices. For positive λ , (43) generates a low-energy J = 1 / 2 doublet, representing the subspace for the KH model.  \nThe simplest SU(2) symmetric extension of the Hubbard repulsion, which generates anisotropic spin exchange in the J = 1 / 2 subspace, has the form  \nH int = U -3 J H 2 ∑ i ( N i -1) 2 -2 J H ∑ i S 2 i (44)  \nglyph[negationslash]  \nwith Hund's coupling J H , with N i = ∑ n,α d † inα d inα and S µ i = 1 2 ∑ nαα ' d † inα s µ αα ' d inα ' . Projecting the result of the standard strong coupling expansion ( U glyph[greatermuch] t 1 / 2 and λ glyph[greatermuch] t 2 1 / 2 /U ) on the low-energy J = 1 / 2 doublet leads to the pure KH model (39) for t 1 = t ' 1 (Laubach et al. , 2017). Due to the orbital structure of Na 2 IrO 3 , t 1 = t ' 1 is more realistic leading to additional symmetric off-diagonal Γ spin exchange, S x i S y j + S y i S x j .  \nAnalysis of the 3-band model at U = J H = 0 yields a phase diagram containing several metallic and insulating phases (Laubach et al. , 2017). As a central result, at the relevant filling 1 / 6 a large region of the non-interacting phase diagram is in a topological insulator phase. This topological phase is clearly adiabatically connected to the one found in Ref. (Shitade et al. , 2009) based on the lesson learned from Kane and Mele that there are in 2D only two types of TR invariant insulators: trivial insulators and topological insulators.  \nThe interacting phase diagram is parametrized by Hubbard U (we assume that J H is much smaller than U ) and some bandstructure parameters. Thus the phase diagram is framed by the weak-coupling regime of Shitade et al. (Shitade et al. , 2009) and the strong-coupling regime of the KH spin model (Chaloupka et al. , 2010; Jackeli and Khaliullin, 2009). Attempts in solving the intermediateU regime agree that a magnetic phase with zigzag-order is present (Igarashi and Nagao, 2016; Laubach et al. , 2017). It seems as if there was a natural tendency to form zigzag magnetic order when driving the topological insulator within the Na 2 IrO 3 environement into the Mott phase. Whether or not this zigzag phase (being in the vicinity to the weak-coupling TI phase) is the same as the one which has been observed experimentally (Choi et al. , 2012; Singh and Gegenwart, 2010; Singh et al. , 2012) remains unclear.  \nThe reader may note that there have been several attempts to explain the experimentally found zigzag phase in Na 2 IrO 3 by extending the original KH spin Hamiltonian (39): longerranged Heisenberg exchange (Katukuri et al. , 2014; Kimchi and You, 2011; Singh et al. , 2012), second neighbor Kitaev exchange (Reuther et al. , 2014; Rousochatzakis et al. , 2015), additional Ising terms (Kimchi et al. , 2015), a negative-sign version of the original KH model (Chaloupka et al. , 2013), significant Γ exchange (Catuneanu et al. , 2017; LampenKelley et al. , 2018; Rau et al. , 2014) and a combination of Γ and longer-ranged isotropic exchange (Winter et al. , 2016). Ab initio based works also suggested a more itinerant explanation for the phenomenology of Na 2 IrO 3 such as the formation of quasimolecular orbitals (Foyevtsova et al. , 2013; Mazin et al. , 2012); in particular, the magnitude of Coulomb interactions, spin-orbit coupling and hopping strength was found to be of similar size which would support the above discussed scenario of the three-band Hubbard model (Laubach et al. , 2017) where charge fluctuations are responsible for the magnetic zigzag order.  \nAs a final remark, we emphasize that a magnetically ordered zigzag phase which is close to the weak-coupling TI phase might display remnants of the TI physics. In Ref. (Laubach et al. , 2017) topologically trivial edge states were observed in the single-particle spectral function. Indeed, recently the observation of metallic surface states were reported in an angle-resolved photoemission spectroscopy (ARPES) measurement (Alidoust et al. , 2016). In a more recent ARPES and x-ray absorption study the relevance of the surface termination was pointed out as well as the role of charge transfer from Na atoms to Ir-derived states (Moreschini et al. , 2017).", 'V. CONCLUSION AND OUTLOOK': "In this review various aspects of electron-electron interactions applied to topological insulators are discussed. The consequences are manifold and include many different fields of contemporary condensed matter physics: from quantum Hall effects to quantum magnets and spin liquids, from onedimensional Luttinger liquids to three-dimensional Hubbard models, from homotopy groups to renormalization group methods, and from SmB 6 and its long history to the most recent developments of ultracold quantum gases. Each of this topics clearly deserves its own review article; here only their important aspects for interacting topological insulators are considered.  \nThe central theme is the effect of electron-electron interactions in topological bandstructures. TR invariant topological insulators, prototypes of symmetry protected topological phases, are typically stable up to moderate interaction strength. Also the metallic edge states, which are protected against single-particle backscattering are stable towards interactions as long as the bulk phase remains intact. Stronger interactions then often induce phase transitions into more  \nconventional phases such as antiferromagnetically or chargeordered states. In certain situations, the competition between strong interactions and non-trivial topology at half filling leads to exotic states of matter such as the topological Mott insulator phase and fractional Chern insulators. In addition, it is shown that there are several interesting proposals where topologically non-trivial bandstructures can be driven into a topological insulator phase by virtue of interactions. For the class of the honeycomb iridates A 2 IrO 3 it is shown that the frustrated magnetism of these materials is intertwined with the underlying topological insulating bandstructures.  \nWe wish to conclude with an outlook about possible future directions. Currently a lot of interest has been attracted by topological Weyl semimetals which feature nodal points in the bulk spectrum (Burkov and Balents, 2011; Murakami, 2007; Volovik, 2003; Wan et al. , 2011). Since they possess topologically protected Fermi arc surface states these systems are sometimes regarded as 'gapless topological insulators'. Several of these materials are known to be superconducting (Qi et al. , 2016) or magnetic (Wang et al. , 2016) implying that electron-electron interactions play a substantial role. Consequently, the process of systematically understanding the effect of interactions in Weyl semimetals has begun (Jian et al. , 2015; Laubach et al. , 2016; Maciejko and Nandkishore, 2014; Wang and Ye, 2016; Wang and Zhang, 2013; Wei et al. , 2012; Witczak-Krempa et al. , 2014). Even more interesting, in the recently predicted and discovered 'type-II Weyl semimetals' (Soluyanov et al. , 2015; Xu et al. , 2015b) where the Weyl nodes are overtilted, instabilities are expected to play a more dominant role due to the drastically enhanced density of states at the Fermi level.  \nAnother exciting idea is to search for magnon modes in quantum antiferromagnets which exhibit topological properties. For instance, the presence of topological linear band crossings of magnon modes in antiferromagnets provide the analog of Weyl fermions in electron systems (thus dubbed 'Weyl magnons') (Li et al. , 2016). In another recent example, magnon bands with finite Chern number have been predicted (Laurell and Fiete, 2016). One might wonder to what extent other topological phases exist in the excitation spectra of quantum magnets.", 'VI. ACKNOWLEDGMENTS': "I am particularly grateful to Karyn Le Hur and Matthias Vojta for countless discussions and for their continued support in the past years. I would like to thank my friends, colleagues, and co-authors I had the pleasure to work with in the context of interacting topological insulators: M. Buchhold, D. Cocks, M. Ezawa, L. Glazman, M. Greiter, W. Hofstetter, M. Laubach, W.-M. Liu, T. Neupert, F. v. Oppen, P. P. Orth, J. Reuther, A. Rod, M. Scheurer, T. L. Schmidt, R. Thomale, and W. Wu. For interesting and fruitful discussions I further want to acknowledge D. Abanin, D. P. Arovas, F. F. Assaad, B. A. Bernevig, P. Brydon, R. Claessen, M. Daghofer, L. Fritz, P.  \nGegenwart, M. Hohenadler, G. Jackeli, J. Knolle, S. Kourtis, T. C. Lang, Y. 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