Perspective: Floquet engineering topological states from effective models towards realistic materials
Fangyang Zhan Rui Chen Zhen Ning Da-Shuai Ma Da-Shuai Ma Dong-Hui Xu Rui Wang


Article Title]Perspective: Floquet engineering topological states from effective models towards realistic materials









1,2]Fangyang Zhan
These authors contributed equally to this work.

3]Rui Chen
These authors contributed equally to this work.

1,2]Zhen Ning


1,2]Da-Shuai Ma

1,2]Ziming Wang



[1,2]Dong-Hui Xudonghuixu@cqu.edu.cn

[1,2]Rui Wangrcwang@cqu.edu.cn

[1]Institute for Structure and Function & Department of Physics & Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing, 400044, P. R. China

[2]Center of Quantum Materials and Devices, Chongqing University, Chongqing, 400044, P. R. China

[3]Department of Physics, Hubei University, Wuhan, 430062, P. R. China







With significant advances in classifying and cataloguing topological matter, the focus of topological physics has shifted towards quantum control, particularly the creation and manipulation of topological phases of matter. Floquet engineering, the concept of tailoring a system by periodic fields, offers a powerful tool to manipulate electronic properties of condensed systems, and even to create exotic non-equilibrium topological states that are impossibly present in equilibrium scenarios. In this perspective, we give a brief review of recent progress in theoretical investigations of Floquet engineering topological states from effective models towards realistic materials. We show that light irradiation can realize various desired topological states through the introduction of symmetry breaking, such as first- and higher-order Weyl fermions, quadrupole topological insulator with periodic driving and disorder, quantum anomalous Hall effects with a tunable Chern number, as well as beyond. Moreover, based on first-principles calculations and Floquet theorem, we show several realistic material candidates proposed as potential hosts for promising Floquet topological states, facilitating their verification in experiments. We believe that our perspective on Floquet engineering of topological states will advance further studies of rich exotic light-induced phenomena in condensed matter physics.











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    September 9, 2024
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§ INTRODUCTION

In the past decades, the study of topological phases of matter has been a significant subject <cit.>. Due to their profound significance and extensive potential for next-generation devices with ultralow power dissipation, topological materials have emerged as a frontier in fields of condensed matter physics as well as material science. Recent research has seen a surge in classifying and realizing topological states in crystalline materials, thanks to progress in symmetry-based theoretical frameworks and computational methods <cit.>. Various catalogues of topological materials, such as nonmagnetic and magnetic topological electronic materials <cit.>, topological phononic materials <cit.>, and topological superconductors <cit.>, have been successively established. More recently, by further utilizing symmetry arguments, researchers have comprehensively constructed the effective models of all magnetic space groups <cit.> and established the encyclopedia of emergent quasiparticles in three-dimensional crystals <cit.>, strongly facilitating the development of topological physics. However, the current focus mainly lies in investigating equilibrium states of topological physics, while studies on topological physics in non-equilibrium scenarios are still in the infancy. With the increasing achievements of topological states and topological materials, performing investigations of quantum control such as creation and manipulation of topological states is an imperative task.

Light-matter interaction is an important approach to dynamically modulate material properties on ultrafast timescales, enabling the creation of exotic non-equilibrium topological states that are otherwise not possible in the equilibrium cases <cit.>. Among various mechanisms of light-matter interaction, of particular interest is the concept of Floquet engineering. Within the framework of Floquet theory <cit.>, periodic light field transfers Bloch energy bands of crystalline solids to periodic Floquet-Bloch sidebands through multiphoton absorption or emission. Thus, light driving offers a mean to manipulate electronic structures, enabling a great potential to control electronic topology out of equilibrium in materials. By appropriately selecting incident light that matches the target system, Floquet engineering can provide a wide range of pathways for dynamically manipulating topological states and even inducing topological phase transitions. For instance, light can gap out the Dirac cone in graphene or drive a band inversion in semiconductor quantum wells, thereby developing the notion of a Floquet topological insulator (FTI) <cit.>. The  topologically nontrivial band gap can also be induced from avoided crossings of photon-dressed Floquet sidebands via the optical Stark effect <cit.>.


The Floquet topological phases are strongly dependent on the drive frequency, amplitude, and polarization of incident light, exhibiting high tunability. It has been demonstrated that irradiation of circularly polarized light (CPL) can break the time-reversal (𝒯) symmetry, and thus quantum anomalous Hall (QAH) insulators with a non-zero Chern number can be obtained under irradiation of CPL <cit.>. Unlike the QAH effect that is present in magnetic materials, the Floquet QAH effect does not require initial magnetism and can generally be present in light-irradiated magnetic and nonmagnetic materials <cit.>. Besides, Floquet engineering can give rise to Chern flat bands with tunable large Chern numbers in twisted systems irradiated by CPL <cit.>. Inspired by the development of conventional FTI with the dipole polarization, it is found that periodic driving can further lead to Floquet higher-order topological insulators with multipole polarization <cit.>. Beyond gapped topological phases, tailoring topological semimetallic phases with first-order or higher-order topology through light irradiation have also been intensely studied <cit.>. Through changing the propagation or polarization direction of incident light, light control of symmetry breaking can be accordingly achieved, and periodic driving via light irradiation offers a fascinating avenue to realize desired gapless topological fermions. The typical examples are the Floquet Weyl semimetals (WSMs) with highly controllable Weyl nodes, which can be generated in light-irradiated topological insulators <cit.>, Dirac semimetals (DSMs) <cit.>, nodal-line semimetals (NLSMs) <cit.>.

On the other hand, disorder plays an important role on the observation of topological edge or surface transport in a realistic topological system <cit.>, and even can induce a phase transition from a topologically trivial insulator to a topological Anderson insulator (TAI) phase <cit.>. In periodically driven systems, it has been demonstrated that disorder can induce topological phases that go beyond the well established paradigm of static disorder-induced topological phases <cit.>. The topological phase arises from the interplay of disorder and periodic driving, dubbed a Floquet TAI. Experimentally, the Floquet topological sates as well as their emerged exotic properties can be captured by time-resolved and angle-resolved photoemission spectroscopy (TrARPES) or time-resolved transport measurements <cit.>. For instance, the Floquet sidebands and CPL-induced gap of topological surface states were observed in three-dimensional topological insulator Bi_2Se_3 by using TrARPES <cit.>. Through the application of ultrafast time-resolved transport measurements using a laser-triggered ultrafast photoconductive switch, the light-induced anomalous Hall effect was confirmed in CPL-driven graphene <cit.>. Recently, the Floquet band engineering in a semiconductor has achieved important progress in experiments, such as pseudospin-selective Floquet sidebands in black phosphorus <cit.>, and optical control of valley polarization in semiconductors MoS_2 <cit.> and BN <cit.>.

Overall, design and control of topological states via Floquet engineering has gradually become an attractive focus in topological physics, with rich exotic phenomena and promising application prospects. To date, numerous representative works have been proposed theoretically, suggesting rich light-induced topological phases based on effective modes, thereby significantly advancing studies of Floquet engineering in condensed-matter community <cit.>. However, only a few experimental evidences have confirmed light-induced topological sates and phase transitions in periodically driven systems <cit.>. The exploration of realistic material candidates that can realize these theoretical effective models is relatively slow. Therefore, to drive Floquet engineering forward by laying down a foundation for experiments, the combination of first-principles calculations and Floquet theorem is an effective mean to predict novel topological properties in realistic material candidates. In comparison with effective model calculations, the first-principles based approach can map momentum- and spin- resolved Floquet-Bloch bands in whole Brillouin zone of solids, and further depict complex and entangled band manifolds on ultrafast timescales <cit.>. Therefore, this perspective will provide a brief review of recent progress in theoretical investigations of Floquet engineering topological states from effective models towards realistic materials. In particular, most of material candidates are well studied in literature, facilitating the realization of Floquet engineering topological states and their device design in experiments.




§ BASIC METHOD OF FLOQUET ENGINEERING

In this section, we review theoretical formalisms and computational methods for Floquet engineering electronic states in crystalline materials under irradiation of time-periodic light fields.



 §.§ Floquet-Bloch Hamiltonian

Firstly, we consider a system driven by a time-periodic and space-homogeneous light field with a Bloch Hamiltonian H(𝐤) with crystal momentum 𝐤. The light field can be described as a vector potential as 𝐀(t)=𝐀(t+T) with period T=2π/ω and the light frequency ω, and then the polarized electric field is 𝐄(t)=-∂_t𝐀(t). In the presence of periodic drive, we obtain a time-dependent Hamiltonian H(𝐤,t)=H[𝐤+𝐀(t)]. When the Hamiltonian is time periodic, the Floquet theorem <cit.> allows us to map it to a time-independent Hamiltonian.
Specially, the Bloch wavefunction |Ψ(𝐤)⟩ develops into the time-dependent formalism as

    |Ψ(𝐤,t)⟩=exp[-iε(𝐤)t]|Φ(𝐤,t)⟩

with time-periodic auxiliary function |Φ(𝐤,t)⟩=|Φ(𝐤,t+T)⟩, which can be expanded in discrete Fourier series as

    |Φ(𝐤,t)⟩=Σ_α e^-αω t|u^α(𝐤)⟩,

where α∈(-∞,+∞) is an integer and termed as the Floquet index. Besides, the electronic wavefunction |Ψ(𝐤,t)⟩ in light-driven
crystals is determined through the time-dependent Schrödinger equation as

    i∂/∂ t|Ψ(𝐤,t)⟩=H(𝐤,t)|Ψ(𝐤,t)⟩.

Combining with Eqs. (<ref>) and (<ref>), the time-dependent Schrödinger equation can be transformed into a series of time-independent equations as

    Σ_αH^mn(𝐤)|u^α(𝐤)⟩=[ε(𝐤)+βħω]|u^β(𝐤)⟩,

with

    H^α-β(𝐤)=1/T∫_0^TH(𝐤,t)e^-i(α-β)ω tdt,

Here, the time-independent square matrix H^(α-β)(𝐤) is dubbed as Floquet-Bloch Hamiltonian. With this representation, the time-dependent
Schrödinger equation is mapped to an eigenvalue problem in an extended Hilbert space. The eigenvalue ε(𝐤) is the energy of Floquet-Bloch states in a system with periodic driving, dubbed Floquet-Bloch band structures. In fact, eigenvalues of ε(𝐤) and ε(𝐤)+nħω represent the same Floquet state, and thus we could define the first Floquet Brillouin zone in (-ħω/2, +ħω/2). The states beyond the first Floquet Brillouin zone can be obtained through multiphoton absorption or emission from states inside the first Floquet Brillouin zone. Overall, the Floquet-Bloch bands form a series of photo-dressed replica bands, which can be deformed by light irradiation and induce hybridization with the folded bands at the first Floquet Brillouin zone boundary, resulting in the electronic and topological properties changed by coupling with light fields.



 §.§ Floquet Hamiltonian in tight-binding Wannier function

To reveal the light-induced modification of crystalline materials, it is required to carry out first-principles calculations to obtain the basis of plane waves. By projecting plane waves of
Bloch states onto localized Wannier basis using
the WANNIER90 package <cit.>, we constructed real-space tight-binding Wannier Hamiltonian as

    H^W=∑_m,n,𝐑,𝐑't_mn(𝐑-𝐑')C_m^†(𝐑)C_n(𝐑')+h.c.,

where 𝐑 and 𝐑' are lattice vectors, (m,n) is the index of Wannier orbitals, t_mn(𝐑-𝐑') are the hopping integrals between Wannier orbital m at site 𝐑 and Wannier orbital n at site 𝐑', and C_m^†(𝐑) or C_m (𝐑) creates or annihilates an electron of Wannier orbital m on site 𝐑.
When a time-periodic and space-homogeneous monochromatic light field is applied to a material, the time-dependent hopping is obtained by using the Peierls substitution <cit.>,

    t_mn(𝐑-𝐑', τ)=t_mn(𝐑-𝐑')e^ie/ħ𝐀(τ)·𝐝_mn,

where 𝐀(τ) is the time-dependent vector potential of an applied light-field, and 𝐝_mn is the related position vector between Wannier orbital m at site 𝐑 and Wannier orbital n at site 𝐑'. The corresponding light-driven operator is C_m(𝐑, τ)= ∑_α=-∞^∞ C_α m(𝐑)e^iαωτ with the Floquet operator C_α m(𝐑) <cit.>.  In this case, the time-dependent H^W(τ) hosts both lattice and time translational symmetries, so we can map it onto a time-independent Hamiltonian according to the Floquet theory <cit.>. By carrying out a dual Fourier transformation, the static Floquet Hamiltonian can be expressed as


    H^F(𝐤, ω)=∑_m, n∑_α, β[H_mn^α-β(𝐤, ω)+αħωδ_mnδ_αβ]C_α m^†(𝐤)C_β n(𝐤)+h.c.,


where ω is the frequency of an incident light and thus ħω represents the energy of photon, and the matrix H_mn^α-β(𝐤, ω) can be obtained by Wannier Hamiltonian as


    H_mn^α-β(𝐤, ω)=∑_𝐑∑_𝐑'e^i𝐤· (𝐑-𝐑')(1/T∫_0^Tt_mn(𝐑-𝐑')e^ie/ħ𝐀(τ)·𝐝_mne^i(α-β)ωτdτ).


According to Eq. (<ref>), one can use first-principles calculations based on density functional theory to quantitatively simulate evolution of electronic properties in light-matter coupled materials. Beyond effective models, the Floquet-Wannier Hamiltonian obtained from density functional theory calculations with Floquet theorem can predict the specific light-driven electronic topology of crystalline materials in nonequilibrium.




§ LIGHT-DRIVEN TOPOLOGICAL SEMIMETALLIC STATES FROM EFFECTIVE MODELS

In this section, we review the recent developments in light-driven topological phases from different kinds of topological semimetallic phases based on effective models.



 §.§ Light-driven type-I, type-II, and hybrid NLSMs

The general model Hamiltonian of the undriven NLSM with a single nodal ring has the form <cit.>,

    H_0=c_ik_i^2σ_0+( m_0-m_ik_i^2) σ
    _z+v_yk_yσ_y,

where m_0, m_i (i=x,y,z) and c_i are model parameters, v_y is the
velocity along the y-axis, k_i are the crystal momenta, σ_i are
Pauli matrices and σ_0 is the identity matrix. Here, Einstein's
summation convention is used, where the repeated indices imply the summation.

Depending on the parameters, the NLSM can be categorized into three types, as illustrated in Figs. <ref>(a)-<ref>(c). In Fig. <ref>(a), where the tilt is weak, the band touching exhibits a type-I nodal ring. Conversely,
Fig. <ref>(c) shows that with strong tilt, both bands align in the same direction, forming a type-II nodal ring at their intersection.
Figure <ref>(b) presents the band spectrum for a hybrid NLSM, revealing that the tilt ratio is smaller near the k_z-axis and larger near the k_x-axis.


To study the interaction of NLSMs with light,  a time-dependent
vector potential 𝐀( t) =𝐀( t+T) is considered,
which is a periodic function with a period of T=2π/ω. Applying Floquet theory <cit.>
in the high-frequency limit, the periodically driven system can be
described by a static effective Hamiltonian given by <cit.>

    H_eff=H_0,0+[ H_0,-1,H_0,1]/ħω+O( A_L^4),

where ω and A_L describe the frequency and amplitude of light, H_m,n=1/T∫_0^TH (t) e^i(m-n)ω tdt is the discrete Fourier components of the Hamiltonian.

When the light propagates along the x-axis, 𝐀(t) is
given by 𝐀=A_L(0,cosω t,ηsinω t),
where η=± 1 indicates the chiralities of CPL. From Eq. (<ref>), the Floquet correction is

    Δ H^x=-A_L^2/2(m_y+m_z)σ_z-Lm_zk_zσ_x,

with L=2η A_L^2v_y/(ħω). In the presence of the light, the coupling term gaps out the nodal ring except at two Weyl points ±𝐤_0=( ±√(m̃_0/m_z),0,0) with m̃_0=m_0-A_L^2( m_y+m_z)/2.
The results indicate that a light traveling along the x-axis gaps out
the nodal ring, leaving a pair of Weyl nodes and causing the system enters into a WSM
phase <cit.>. However, the type of the Weyl nodes is independent of the intensity
and the frequency of the incident light, suggesting that a type-II
Floquet WSM state arises by driving the type-II NLSM with a light along the x-axis.





The band spectrum of the driven type-I NLSM [Fig. <ref>(d)] shows the type-I Weyl nodes. The bulk band spectra of the driven hybrid NLSM and the driven type-II NLSM are depicted in Figs. <ref>(e) and <ref>(f), respectively. 



Besides, it has been shown that the type of Floquet WSM phases depends on the orientation of the incident light <cit.>. When the incident light propagates along the x-axis or along the z-axis, a type-II NLSM is converted into a type-II WSM 
, while for a driven hybrid NLSM, depending on the tilt direction, the photoinduced Floquet WSM could be of type-I [Fig. <ref>(b)] or type-II [Fig. <ref>(d)]. When the applied light propagates along the y-axis, only the positions of nodal rings change Fig. <ref>(c) 
. Surprisingly, by rotating the incident light on the x-z plane, both type-I and type-II WSMs can be realized by tuning the driving angle and amplitude Fig. <ref>(e) 
. For the sake of comparison, the Figs. <ref>(a)-<ref>(e) also give the Floquet states of driven type-I NLSMs by a CPL, which show different features from those of type-II and hybrid NLSMs. Furthermore, the anomalous Hall effects of these photoinduced Floquet WSM phases are also investigated by use of the Kubo formula <cit.>.






 §.§ Light-induced higher-order WSM phases

In this section, we show that CPL can also induce higher-order WSM phases that support both surface Fermi arcs and hinge Fermi arcs in a higher-order NLSM <cit.> or a higher-order DSMs <cit.>.

The Hamiltonian for the NLSM has the form <cit.>,

    H(𝐤) =    im(Γ_1 Γ_4 + Γ_2 Γ_4) + tsink_xΓ_1 + tsink_yΓ_2 
        + [M-t(cosk_x+cosk_y+cosk_z)] Γ_3,

where the Dirac matrices are defined as Γ_1=σ_0 τ_3, Γ_2=σ_2 τ_2, Γ_3=σ_0 τ_1, Γ_4=σ_1 τ_2, where σ_j and τ_j (j=1,2,3) are Pauli matrices labeling the sublattice and layer degrees of freedom, σ_0 and τ_0 are identity matrices. t is the amplitudes of hoppings, M is the Dirac mass, and m is the additional mass that generates higher-order topology. The system supports two bulk nodal rings, two-dimensional (2D) drumhead surface states, and one-dimensional hinge Fermi arc states. 

This model can describe the higher-order NLSM materials as XTe_2 (X=Mo, W) <cit.> and 3D ABC stacked graphdiyne <cit.>.
Figure <ref>(a) illustrates two mirror-protected bulk nodal rings on the k_n-k_z mirror plane with 𝐤_𝐧 along the k_x=-k_y axis in the first Brillouin zone. The drumhead surface states depicted in Fig. <ref>(b) are the projection of bulk nodal rings on the k_y-k_z plane. Figure <ref>(c) demonstrates the hinge Fermi arc states, which are located on two mirror-symmetric off-diagonal hinges.









When the CPL propagates along the z-axis, 𝐀(t) is given by 𝐀=A_L(ηsinω t, cosω t,0). The effective Hamiltonian of the driven higher-order NLSM can be found in Ref. <cit.>.








The light irradiation breaks both the 𝒯-symmetry and chiral symmetry, gapping out the nodal rings and leaving a pair of Weyl nodes, as shown in Fig. <ref>(d). Figure <ref>(e) demonstrates that the surface drumhead states are replaced by surface Fermi arc. However, the mirror symmetry and the product of time-reversal and chiral symmetries are still preserved, which protect the higher-order hinge Fermi arcs shown in Fig. <ref>(f). The above results indicate that the driven system turns into a higher-order WSM, which supports both first-order surface Fermi arc and second-order hinge Fermi arc states.
Moreover, when light propagates along the other axis, CPL can always drive a higher-order NLSM to a higher-order WSM. More importantly, it is found that the propagation axis of CPL can control the location of the Weyl nodes.

A similar conclusion is found in the CPL-driven higher-order DSM <cit.>. The model describing an undriven higher-order DSM model is given by <cit.>:

    H(𝐤) =   ϵ_0(𝐤) + λsink_xΓ_1^' + λsink_yΓ_2^'
        + M(𝐤)Γ_3^' + G(𝐤)Γ_4^',

where ϵ_0(𝐤)=t_1 (cosk_z-cosK_z^0)+t_2(cosk_x+cosk_y-2), M(𝐤)=t_z (cosk_z-cosK_z^0)+t(cosk_x+cosk_y-2), Γ_1^'=s_3 ρ_1, Γ_2^'=s_0 ρ_2, Γ_3^'=s_0 ρ_3, Γ_4^'=s_1 ρ_1, Γ_5^'=s_2 ρ_1, s_j and ρ_j (j=1,2,3) are Pauli matrices denoting the spin and orbital degrees of freedom, and s_0 and ρ_0 are identity matrices. This model can describe the higher-order DSM materials including but not limited to Cd_3As_2 and KMgBi <cit.>.  Here, t, λ, and t_1,2,z  are the amplitudes of hoppings. The two Dirac cones are located at 𝐤=(0,0,± K_z^0), as shown at the boundary of Fig. <ref>(a). 
G(𝐤)=g(cosk_x-cosk_y)sink_z represents the higher-order topological term, which gives rise to second-order hinge Fermi arc states as displayed in Fig. <ref>(b)-<ref>(c).
This system has different surface states, as demonstrated in Fig. <ref>(a). The closed Fermi ring, instead of helical Fermi arc states, emerges in the surface Brillouin zone <cit.> when the Fermi energy cuts through the surface Dirac cone.













The CPL also drives the higher-order DSM into a Floquet WSM, separating each Dirac point into a pair of Weyl points. These Weyl points host surface Fermi arc states that connect the projections of each pair of Weyl nodes, as depicted in Fig. <ref>(d)-<ref>(e). Moreover, Fig. <ref>(f) shows that the Floquet WSM also hosts hinge Fermi arc states, terminated by the projection of two adjacent Weyl nodes from two different pairs.
Moreover, CPL can drive the tilted Weyl cones in Floquet WSMs, when the axis of light propagation is changed. On the other hand, the higher-order term also can be written as G(𝐤)=g(cosk_x-cosk_y); this term also breaks the four-fold rotational symmetry, but preserves the effective parity-time reversal (𝒫𝒯) symmetry. In this case, the CPL plays a similar role in inducing the topological phase in the higher-order DSM.

The above content focuses on the higher-order topological semimetals with the four-fold rotational symmetry. In the higher-order DSM with the six-fold rotational symmetry, CPL irradiation can also produce Floquet WSM that supports both first-order surface Fermi arc and higher-order hinge Fermi arc states, and the location of Weyl nodes depends on the propagation direction of angle of incidence of CPL <cit.>. Furthermore, CPL can also control the degree of tilt of the
resulting Weyl cones by adjusting the incident direction of the CPL, enabling the realization of different types of WSMs.




§ TOPOLOGICAL STATES INDUCED BY THE INTERPLAY OF PERIODIC DRIVING AND DISORDER

The investigation of the condensed matter systems in the presence of disorder is a longstanding research area of fundamental importance. For instance, the disorder is crucial for the observation of integer quantized Hall plateau <cit.>. Over the past two decades, significant progress has been made in the study of disorder effects in topological systems  <cit.>. Surprisingly, despite the presence of disorder-induced localization effects, the disorder may also endow a system with non-trivial topological states i.e., the TAI <cit.>. 

The interplay of disorder and periodic driving can give rise to rich topological phenomena <cit.>. In this section, we review the recent works about topological states induced by the interplay of periodic driving and disorder. 




 §.§ Light-induced QAH effects in disordered systems

In the original version of TAI, 𝒯-symmetry is preserved and the phenomenology is similar to the quantum spin Hall effect. Under the breaking of 𝒯-symmetry, the QAH state can be realized <cit.>. A natural analogy is as follows: Can we induce the QAH effect in disordered systems? In the following, we briefly review that the QAH state arises from the interplay of light and nonmagnetic disorder. Here, a strategy was proposed to induce QAH in disordered systems <cit.>. The idea is to explore systems in which bulk topology is driven by the nonmagnetic disorder, and the spin degeneracy is lifted under irradiation of light fields, as shown in Fig. <ref>. As the light intensity increases beyond a critical value k_A^c, the energy gap of one spin sector first closes and then reopens, while the other spin sector still possesses the nontrivial topology. In this case, the system evolves into the QAH phase with one gapless chiral edge channel [Figs. <ref>(b)-(d)].



The four-band Bernevig-Hughes-Zhang (BHZ) effective Hamiltonian <cit.> can be used to reveal the light-modulated topological phases mentioned above.

    h_s(𝐤) = d_0(𝐤)τ_0+𝐝_s(𝐤) ·σ,

where the index s=± denotes spin, or the inequivalent valleys ± K due to spin-valley locking in morié superlattices. For the low-energy effective minimal model, we have d_0(𝐤) = -Dk^2 and 𝐝_s(𝐤) = (sAk_x,Ak_y,M-Bk^2), where M is the Dirac mass term depicting the band inversion, the other parameters A, B, D can be obtained from experiments. The sign of Dirac mass term is crucial to characterize the topological phase transition from normal insulator (M>0) to topological insulator (M<0).

In the presence of CPL and disorder, the Hamiltonian becomes time-dependent and the translational symmetry is broken by lattice disorder. However, a simple picture can still be obtained by the following approximations. In off-resonant regime (i.e., high frequency), the time-independent effective Hamiltonian can be approximately obtained through the Magnus expansion <cit.>. On the other hand, the effects of disorder can be taken into account by effective medium theory, such as the self-consistent Born approximation (SCBA) <cit.>. After considering these approximations, the renormalized Dirac mass can be expressed as:

    M^eff_s = M - Δ^cpl_s(k_A) - Σ^dis_s(γ),

where the second term on the right hand side of Eq. (<ref>) Δ^cpl_s(k_A) is a function of light intensity k_A induced by CPL. The third term Σ^dis_s(γ) is disorder induced self-energy which can be calculated by SCBA for a given disorder strength γ. It is noted that both corrections Δ^cpl_s, Σ^dis_s are spin-dependent, leading to two spin sectors with different responses to CPL.
The effective mass M^eff_s can be extracted from the width of the quasi-band band gap, as shown in Fig. <ref>(e). If the disorder is too weak to induce the TAI phase, the increasing of the light-intensity k_A cannot induce any topological phase transitions [Fig. <ref>(f)]. Therefore, one need a moderate strength of disorder to drive the system into TAI. Specifically, as shown in Fig. <ref>(g), the effective mass is negative in the absence CPL, M^eff_s(k_A=0)<0. Once irradiation of CPL is applied, two spin sectors exhibit different responses, and the system shows 𝒯-symmetry broken TAI when M^eff_↑,↓<0. As the light intensity exceeds a critical value k_A^c, the energy gap of one spin sector first closes and then reopens M^eff_↓>0, while the other spin sector still possesses the nontrivial topology M^eff_↑<0 [Fig. <ref> (g)]. In this case, the system evolves into the QAH phase. The spin-polarized topological phases can also be characterized by the spin Hall conductivity σ^spin_xy and charge Hall conductivity σ^c_xy [Fig. <ref>(h)]. These results conceptually demonstrate the possibility of realizing QAH effect in TAI using CPL.








































 §.§ Quadrupole topological insulator with periodic driving and disorder

Recently, the concept of topological phase of matter has been extended to higher-order topological phases <cit.>. 

Among various higher-order topological phases, the quadrupole topological insulator (QTI) <cit.> associated with a quantized quadrupole moment is of particular interest, which accommodates topologically protected corner states. It has been believed that spatial symmetries (such as mirror symmetries and/or four-fold rotation symmetry) and internal symmetries (such as chiral symmetry, 𝒯-symmetry and particle-hole symmetry) are crucial ingredients to design and realize QTIs. The presence of disorder explicitly breaks crystal symmetries. Therefore, the new phenomena induced by disorder in the higher-order topological states have attracted wide attention  <cit.>.  


In the following, we review that an exotic QTI created by the intertwined periodic driving and disorder emerges from a topologically trivial band structure. This intriguing QTI possesses a quantized quadrupole moment only protected by particle-hole symmetry.

Starting with the Benalcazar-Bernevig-Hughes (BBH) model, a paradigmatic model of QTIs, the Hamiltonian can be written as

    H_q(𝐤) =  λsin(k_y)τ_2σ_1 +λsin(k_x)τ_2σ_3 
       [γ + λcos(k_x)]τ_1σ_0 + [γ + λcos(k_y)]τ_2σ_2.

The schematic illustration of the lattice structure is shown in Fig. <ref> (a).
The topological phase transition is determined by the ratio of γ / λ. In the static and clean limit, the BBH model Eq. (<ref>) describes a trivial insulator when γ / λ>1 or a QTI when γ / λ<1 . The phase boundary is at γ / λ = 1. Under the irradiation of CPL, electronic structures of the system are effectively modified by the virtual photon absorption processes, which can be expressed as an effective Floquet-Bloch Hamiltonian:

    H^eff(𝐤)    = H_q(𝐤) + H'(𝐤),

where the second term H'(𝐤)  induced by CPL gives an important modification to the original BBH model <cit.>. It is noted that chiral symmetry and 𝒯-symmetry preserved by the static BBH Hamiltonian H_q(𝐤) are both broken under the CPL. Nevertheless, the combination of these two symmetries (i.e., particle-hole symmetry) is preserved.
Particle-hole symmetry is critical to the quantization of quadrupole moment in the even presence of both disorder and periodic-driving.
The quadrupole moment defined in real space Q_xy <cit.> can characterize the QTI phase in disordered systems. When the periodic driving and disorder are simultaneously present, all crystalline symmetries and chiral symmetry are destroyed. However, the preserved particle-hole symmetry can protect the quantization of quadrupole moment <cit.>. The established topological invariant Q_xy allows one to investigate topological phase transitions in the presence of periodic driving and disorder.



Figure <ref>(b) depicts the phase diagram of the quadrupole moment Q_xy on the W-k_A parameter plane. Significantly, there is an explicit area of the QTI phase, which is created by the joint effort of periodic driving and disorder. This QTI phase can neither be created by the driving field without disorder, nor disorder in the absence of driving field. For example, in Fig. <ref>(b), the QTI phase cannot be induced by tuning the field strength k_A in the weak disorder regime (W∼1 ) or by increasing the disorder strength in the presence of weak driving field (k_A ∼1). 
The energy eigenstates by directly diagonalizing the tight-binding Hamiltonian with open boundaries can fully illustrate the characteristic features of the emergent QTI phase. As evident in Fig. <ref>(c), four in-gap modes at E=0 emerge as a function of k_A, implying that the presence of topologically nontrivial QTI phase. The zero-energy modes in bulk gap correspond to corner states. Furthermore, the topological phase transitions induced by the interplay of periodic driving and disorder can be understood by a simple picture based on the effective medium theory. The hopping amplitudes renormalized by periodic driving and disorder (γ_x,y→ t^d_x,y) are displayed in Fig. <ref>(d). When t^d_x<1 and t^d_y <1 simultaneously, the disorder-induced Floquet QTI phase is created. This picture agrees with the numerical computations. The intriguing QTI phase, protected only by particle-hole symmetry, which necessitates the simultaneous presence of disorder and periodic driving further enriches the symmetry-protected mechanism of higher-order topology.



§ FLOQUET ENGINEERING OF TOPOLOGICAL STATES IN REALISTIC MATERIALS

In this section, we review the progress of light-driven QAH states and controllable Weyl fermions proposed using first-principles calculations combined with Floquet theory.





 §.§ The realization of light-driven QAH and VQAH states




The light-driven QAH states can be realized in 2D nonmagnetic MX_2/WTe_2 (M=Mo, W; X= S, Se) transitional metal dichalcogenides (TMDs) heterobilayers under the irradiation of CPL. Considering the locking of spin to valley in TMDs, the valley-polarized quantum anomalous Hall (VQAH) state with one spin- and valley-resolved chiral edge channel in TMDs heterobilayers under light irradiation behaves as a perfect topological spin-valley filter [Fig. <ref>(a)]. Figures. <ref>(b)-<ref>(e) illustrate the evolution of spin-resolved band structures around the K and K' valleys under CPL irradiation. We can find that the bands of spin-up states around the K valley have been more drastically modified than those of spin-down states around the K' valley. More importantly, with increasing light intensity, the band gap of spin-up states first closes and then reopens; that is, only spin-down states preserve the inverted band topology, resulting in a valley quantum spin Hall (VQSH) to VQAH topological phase transition. 


By integrating the Berry curvature as shown in Figs. <ref>(f) and <ref>(g), one can obtain 𝒞_K=1 and 𝒞_K'=-1, and the valley Chern number is 𝒞_v= 2. For the VQAH state, the Berry curvature Ω_z only distributes and diverges near K' valleys, giving 𝒞_K=0 and 𝒞_K'=-1, and then the Chern number is 𝒞=-1. The topological phase with a specific topological invariant gives rise to uniquely nontrivial edge states. 

Without light irradiation, the 𝒯-invariant VQSH state shows that two opposite chiral edge states with Kramers degeneracy are visible at the K and K' valleys [Fig. <ref>(h)]. With increasing light intensity to 0.045 Å^-1, the 𝒯-broken VQSH state removes the Kramers degeneracy and exhibits different inverted band gaps at the K and K' valleys, but the chiral edge states are also visibly present [Fig. <ref>(i)], confirming its nontrivial feature. As shown in Fig. <ref>(j), the VQAH state possesses one chiral edge state connecting the valence and conduction bands around the K' valley, while bands around the K valley exhibit the topologically trivial NI state. Furthermore, depending on light helicities, this CPL can selectively switch the states between two valleys and spin, providing a reliable scheme to realize an optically switchable topological spin-valley filter <cit.>.





The Floquet QAH states can be further obtained from topologically trivial semiconductors. As illustrated in Fig. <ref>(a), under irradiation of CPL, periodic driving gives rise to Floquet-Bloch bands, and then two certain bands [i.e., labeled as E_c^F and E_v^F in Fig. <ref>(a)] move close to the Fermi level via the optical Stark effect <cit.>. 
This proposal can be implemented in the 2D semiconductors MSi_2Z_4 (M = Mo, W, V; Z = N, P, As) family materials. 
This family of materials, including magnetic and nonmagnetic members, host excellent stability, and especially MoSi_2N_4 and WSi_2N_4 were successfully synthesized in experiments <cit.>. As a representative example, the band structures of VSi_2N_4, with the valence band and conduction band respectively contributed by d_xy & d_x^2-y^2 and d_z^2 orbitals of V atoms, depict a trivial semiconducting feature with the valley degeneracy in Fig. <ref>(b). Under light irradiation, in addition to the equilibrium bands (black solid lines), one can find that the Floquet-Bloch bands in Fig. <ref>(c), that are created by absorption (red dashed lines) or emission (blue dashed lines) of a photon, are present. 
Interestingly, the band gaps near the K and K' points indicate that band gap at the K point closes and reopens twice [Fig. <ref>(d)]. Due to the threefold rotational (C_3) symmetry, a triangular distortion of the Fermi surface around the K points would be present [Fig. <ref>(e)], which is known as trigonal warping <cit.>. 
The presence of trigonal warping would like to strongly enrich topological phases. As shown in Fig. <ref>(f), the phase diagram characterized by 𝒞_K (K') as functions of ħω and eA/ħ indicates that there are five distinct topological phases, such as regime I: 𝒞_K = 0 and 𝒞_K'=0, regime II: 𝒞_K = +1 and 𝒞_K'=0, regime III: 𝒞_K = -3 and 𝒞_K'=0, regime IV: 𝒞_K = -3 and 𝒞_K'=-1, and regime V: 𝒞_K = 0 and 𝒞_K'=-1. Except the topologically trivial regime I, other four regimes are all related to the topologically nontrivial VQAH state. The Berry curvature distributions for regimes III and IV are plotted in Figs. <ref>(g) and <ref>(h). The nonzero Berry curvature Ω_z(𝐤) diverges near the K and/or K' points. In particular, the Ω_z(𝐤) near the K point exhibits the C_3-symmetry as shown in the insets of Figs. <ref>(g) and <ref>(h), further confirming nontrivial band topology associated with trigonal warping. The Floquet VQAH states with specific first Chern number 𝒞 and valley-resolved Chern number 𝒞_K (K') correspond to the valley-dependent chiral edge channels [Fig. <ref>(i)] and quantized Hall conductance σ_xy [Fig. <ref>(j)], characterizing the global band topology of VQAH states.



Besides, the photoinduced high-Chern-number QAH states can also occur in a higher-order topological insulators. As earlier mentioned, the light-driven BBH model can capture directly the influence of light field on the QTI. Under the irradiation of CPL, the energy spectrum of BBH model shows that both one-dimensional edge states (colored blue) and zero-dimensional corner states (colored red) inside the gap [Fig. <ref>(a)], indicating the coexistence of QTI and Chern insulator phases. As shown in Fig. <ref>(b), the quadruple moment Q_xy can always be quantized to 1/2 with the increase of light intensity, indicating that the irradiation of CPL does not destroy the higher-order topology; meanwhile, the Chern number 𝒞 accompanies with a transition from 0 to 1. Further, the phase with 𝒞=2 or 𝒞=3 and  Q_xy=1/2 are present at higher frequencies <cit.>. 

These photoinduced high-Chern-number QAH states can be realized in experimentally synthesized 2D graphdiyne <cit.>. The crystal structure of graphdiyne constructed from sp- and sp^2-hybridized carbon atoms is shown in Fig. <ref>(c). The calculated band structures around Γ under various light intensity and photon energy of CPL are shown in Fig. <ref>(d). 

Similar to the case of BBH model, the multiple band inversion occurs as expected, leading to the Chern number manipulated continuously from trivial state 𝒞=0 to nontrivial QAH state with 𝒞=3. 
Figure <ref>(e) illustrates the edge states of graphdiyne under the irradiation of CPL along zigzag direction. One can see that the number of the chiral edge states with different light intensity and  photon energy, further supporting the fact that one can obtain the QAH states by irradiating CPL on a 2D higher-order topological insulators. Moreover, by manipulating the light intensity and  photon energy, one can realize the high-Chern-number QAH states up to 𝒞=4 for graphdiyne. The phase diagram can completely summarize the parameter regimes of different phases and gain a deep insight into the topological phase transition. As shown in Fig. <ref>(f), one can find that there are five distinct phase regimes corresponding to the continuously changed Chern number (ranging from 0 to 4). 





 §.§ Light-manipulated Weyl nodes in topological semimetallic materials







Distinct from gapped topological phases, such as Chern insulators, topological semimetals possessing the gapless nodal points (or nodal lines) near the Fermi level are particularly relevant to symmetries.


The nontrivial band topology of topological semimetals often leads to attractive phenomena, such as ultrahigh carrier mobility <cit.>, half-integer quantum Hall effects <cit.>, large diamagnetism <cit.>, and electromagnetic duality <cit.>, and thus considered to have a wide range of applications in future devices and technologies.
Benefiting the efforts donated to uncover mappings between symmetries and band topology <cit.>, it is now possible to actively manipulate the transition between different topological states and thereby design desired topological semimetals under the irradiation of light fields.
For instance, it has been shown that Floquet WSMs can be generated in materials with QSH state, Dirac fermion, triple fermion, and nodal-line fermion subjected to periodic driving light fields <cit.>.
Here, we briefly review the material proposal of light-induced WSM from triple fermion in TiO, and nodal-line fermion in carbon allotrope bct-C_16.

The TiO crystallizing in tungsten carbide type (WC-type) structure with a space group P6m2 (D_3h, No. 187) was demonstrated to be an ideal candidate with triple fermions near the Fermi level.
As schematically illustrated in Fig. <ref>(a), the triply degenerate nodal points in WC-type TiO without light irradiation are protected by the C_3-symmetry, vertical mirror symmetry plus 𝒯-symmetry σ_v⊕𝒯. Under time-periodic and space-homogeneous CPL, the light irradiation can break the symmetry of σ_v⊕𝒯 and thereby lead to the triply degenerate nodal points splitting to twofold degenerate Weyl nodes.
The orbital-resolved band structures along high-symmetry paths are shown in Fig. <ref>(b).
There is a band crossing point along the high-symmetry Γ-A path, which is mainly contributed by the e_g orbital (d_xy and d_x^2-y^2) and the d_z^2 orbital.
In fact, the enlarged view of the bands along Γ-A exhibits three sets of bands with distinct band degeneracy [Fig. <ref>(c)], i.e., the non-degenerate Λ _4 and Λ _5 bands, and the doubly degenerate Λ _6 band. The Λ _4(Λ _5) band crosses with the doubly degenerate Λ _6 band, forming the triply degenerate nodal points that are protected by C_3 and σ_v⊕𝒯.
The light irradiation can lift this doubly degenerate band, as shown in Figs. <ref>(d) and <ref>(e), forming two non-degenerate bands that are respectively featured by two C_3 eigenvalues of e^iπ/3 and e^-iπ/3. Consequently, the triply degenerate nodal points along Γ-A are absent. These four Weyl points located along the rotation-invariant high-symmetry line Γ-A are constrained by the rotation symmetry C_3, which indicates that CPL irradiation breaks σ_v⊕𝒯 but preserve C_3. The obtained LDOS projected on the semi-infinite (010) surface are plotted in Figs. <ref>(f) and  <ref>(g), and one can find the characteristic surface states terminated at projections of bulk Weyl nodes.
The application of lattice strain and its coupling of CPL would offer an effective way to manipulate the electronic and topological properties of WC-type TiO. Figure <ref>(h) shows the surface states connecting the projections of bulk Weyl nodes are clearly visible under 4% tensile strain. With various intensities of CPL, the strained WC-type TiO exhibits distinct Weyl semimetallic phases with different numbers of Weyl nodes and Fermi arcs. To be specific, as the light intensity increases, the photon-dressed band structures of strained WC-type TiO show that the two lowest conduction bands and the two highest valence bands simultaneously move away from the Fermi level. Consequently, the Weyl nodes W_1 and W_2 come together and then annihilate at a specific light intensity as described in Fig. <ref>(i).





The carbon allotrope bct-C_16 crystallizes in a body-centered tetragonal (bct) structure with space group I4_1/amd [Fig. <ref>(a)], which can be obtained from the famous T-carbon through a temperature-driven structural transition <cit.>. Due to the extremely tiny SOC of the carbon element, the interplay between the SOC and light irradiation can be ignored. Therefore, it can be considered as an ideal platform to study the photon-dressed topological states. As shown Fig. <ref>(b), the bct-C_16 is a NLSMs protected by the 𝒫𝒯 symmetry.
The Dirac nodal-line is located at the the mirror reflection invariant k_x-k_y plane with k_z = 0. Under a periodic field of a linearly polarized laser (LPL), the NLSM phase is transitioned to a mixed-WSM phase with two pairs of tunable Weyl nodes.
As shown in Fig. <ref>(c), one can see that the light irradiation obviously influences the electronic band structures of bct-C_16. The previous band crossings in the Γ-X and Γ-M directions are both gapped, indicating that the nodal-line fermions in bct-C_16 disappear. With increasing the light intensity, the band gaps are further enlarged [the inset of Fig. <ref>(c)]. The band profiles around one pair of Weyl nodes (i.e., W_1^+ and W_1^-) evolve with increasing the amplitude of LPL, as shown in Figs. <ref>(d)-(l). The other pair of Weyl nodes shows the same behaviors with respect to the 𝒯-symmetry. The band dispersion around the W_1^- with a light intensity eA_z /ħ of 0.03, 0.059, and 0.066 Å^-1 are illustrated in Figs. <ref>(d), <ref>(e), and <ref>(f), respectively. It is found that the right-handed Weyl node W_1^- keeps to be type-I though it becomes more tilted with increasing the light intensity. On the contrary, light-dependent change around W_1^+ is more remarkable. When the light intensity eA_z /ħ increases from 0.03 to 0.066 Å^-1, the left-handed Weyl node W_1^+ undergoes a transition from type-I [Fig. <ref>(g)] to type-II [Fig. <ref>(i)]. In this transition process, the critical type-III Weyl node is present between type-I and type-II states with eA_z /ħ = 0.059 Å^-1 [Fig. <ref>(h)]. The 3D plots of band profiles around W_1^+ with eA_z /ħ = 0.03, 0.059, and 0.066 Å^-1 are respectively shown in Figs. <ref>(j), <ref>(k), and <ref>(l), which are consistent with topologically nontrivial features of type-I, type-II, and type-III Weyl fermions. The results demonstrate that the unconventional Weyl pairs composed of distinct types of Weyl nodes are realized in bct-C_16.

In addition to the material mentioned above, other realistic material systems with controllable Floquet topological states have also been shown to form the analogous proposals, such as the graphene <cit.>, FeSe <cit.>, black phosphorus <cit.>, Na_3Bi <cit.>, MnBi_2Te_4 films <cit.> and so on. These experimentally synthesized materials offer excellent platforms to control the electronic properties for achieving the new and desired topological states.





§ SUMMARY AND PERSPECTIVE

Last but not least, we have provided a brief review of recent progress in theoretical investigations of Floquet engineering topological states in condensed-matter systems under light irradiation. The study of various Floquet topological states, such as the light control of type-I, type-II, and type-III Weyl fermions and their topological phase transitions, photoinduced QAH effects with a tunable Chern number, higher-order topological insulators arising from periodic driving and disorder, paves a fascinating path for realizing desired topological states with high tunability. The strong dependence of Floquet topological phases on light-induced symmetry breaking provides a valuable platform for investigating the interplay between symmetry and topology, as well as for generating exotic non-equilibrium topological phases unattainable in equilibrium conditions. Moreover, we show that the combination of first-principles calculations and Floquet theorem offers a reliable avenue to map momentum- and spin- resolved Floquet-Bloch bands in whole Brillouin zone of solids, and most of predicted material candidates are synthesized or well studied in experimental level, facilitating the measurements of Floquet engineering topological states and their device design in experiments. 

Besides, although we point out that the Floquet theorem combining with  effective model as well as first-principles calculations can effectively map  theoretical investigations of Floquet engineering topological states to experiments, but the complex and entangled band manifolds on ultrafast timescales are still not clear. Especially, the thorough understanding of Floquet topological states in periodically driven systems, such as interplay between non-equilibrium topological states and electron–electron corrections, is a long-term challenge.  We believe that further study of Floquet engineering topological states will uncover even more exotic phenomena yet to be imagined, establishing this field as a promising area of study with attractive physical phenomena in the condensed-matter community.





  

Acknowledgements

Not applicable.

 

Funding

This work was supported by the National Natural Science Foundation of China (NSFC, Grants No. 12204074, No. 12222402, No. 92365101, No. 12347101, No. 12304195, No. 12304191, and No. 12074108) and the Natural Science Foundation of Chongqing (Grants No. 2023NSCQ-JQX0024, and No. CSTB2022NSCQ-MSX0568).


  

Availability of data and materials

Not applicable.

  

  

Declarations

  

Ethics approval and consent to participate

Not applicable.

  

Consent for publication

Not applicable

  

Competing interests

The authors declare no competing interests.

 

Author contributions

RW and DHX supervised the work. FZ and RC contribute to the work equally. All authors read and edited the full manuscript. Introduction (RW and DHX); basic method of Floquet engineering (RW); light-driven topological semimetallic states from effective models (RC and ZW); Topological states induced by the interplay of periodic driving and disorder (ZN); Floquet engineering of topological states in realistic materials (FZ and DSM); summary (RW). Overview of the review (RW and DHX). All authors have approved the manuscript.





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Maebashi, H.,
Tajima, N.,
Tsumuraya, T.,
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Ogata, M.,
Kato, R.:
Large diamagnetism and electromagnetic duality in two-dimensional Dirac electron system.
Phys. Rev. Lett.
128,
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(2022)
10.1103/PhysRevLett.128.027201




[Ding et al.2020]Ding_2020

Ding, X.-Y.,
Zhang, C.,
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Wang, R.:
Topological phase transition from T-carbon to bct-C16.
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10.1088/1367-2630/ab990b







