Multifractaility, topology and anomalous Hall conductivity on a 30 degrees twisted bilayer honeycomb lattice
Grigory Bednik

arabic







 
Department of Physics, University of Nebraska Omaha








§ ABSTRACT 








We consider 30^∘ twisted bilayer formed by two copies of Haldane model and explore evolution of its properties with varying interlayer coupling strength. Specifically, we compute the system's energy spectrum, its fractal dimensions, topological entanglement entropy, local Chern markers and anomalous Hall conductivity. We find that at weak interlayer coupling, the system remains gapped and retains topological properties of the isolated layers, but at strong interlayer coupling, the system forms a gapless multifractal state.  We also establish that anomalous Hall conductivity can be used to characterize the system's topological properties in the same way as a local Chern marker.





Multifractaility, topology and anomalous Hall conductivity on a 30 degrees twisted bilayer honeycomb lattice
    Grigory Bednik
    September 4, 2024
============================================================================================================





§ INTRODUCTION




Quasicrystals are a novel class of materials discovered in 1980-s, whose lattice is non-periodic and yet is ordered according to certain rules. 
One possible way to obtain a quasicrystal is to consider a periodic lattice in higher dimension and project it into a plane with a normal vector incommensurate to the original lattice <cit.>. Other specific models of quasicrystals include Fibbonacci lattice <cit.>, Penrose tiling, Rauzy tiling <cit.>, Aubry-André model <cit.> etc. It was established that quasicrystals generally host multifractal electronic states, which are neither extended, nor localized, but instead characterized by non-integer fractal dimension <cit.>. 

It is widely known that periodic crystals can host multiple topological phases. 
On the other hand, topological properties of quasicrystals are still not well-understood. In a recent work <cit.>, it  was suggested that topological quasicrystals can be built from crystals in higher dimensions, and their topology can be deduced from crystal topology. In a few other works, empirical models of topological quasicrystals were proposed  <cit.>, 
which host gapless  edge states and whose topology was characterized directly using empirical 'local Chern markers' <cit.>. However, there is still no systematic understanding of how non-trivial topology in quasicrystals may arise.

 Motivated by this, we are interested whether it is possible to obtain a topological quasicrystal by smoothly deforming a crystal with known non-trivial topology. Specifically, we choose to consider 30 degrees twisted bilayer honeycomb lattice - a model which is currently actively studied in the context of twisted bilayer graphene <cit.>. However we consider a 
 twisted bilayer, in which each monolayer is described by gapped Haldane model <cit.>.  We explore our model's evolution once the interlayer coupling is increased. We obtain that at weak coupling, the model retains all properties of Haldane model: bulk states remain extended and edge states still exists. 
But as the interlayer coupling increases, the model undergoes a transition into a strongly coupled state, which is gapless, and whose eigenstates are multifractal. 

Characterizing topological properties of our model turns out to be challenging because most well-established topological concepts (e.g. Chern numbers) were developed in momentum space, i.e. only for crystalline systems. Nevertheless, we are still able to find out if the system is topological by computing its topological entanglement entropy <cit.>. Furthermore, we study its topological properties by computing 'local Chern marker' <cit.>. Finally we numerically compute its anomalous Hall conductivity by using Kubo formula and find that it can be used to characterize topological properties of the quasicrystal in the same way as local Chern marker.

This paper is organized as follows. In Sec. <ref> we introduce our model, and in Sec. <ref> we describe its phase diagram, which consists of three phases: topological weakly-coupled, non-topological weakly coupled and strongly coupled multifractal. In Sec. <ref> we describe its topological properties. Specifically, in Sec.  <ref> we present our results for its topological entanglement entropy and in Sec. <ref> we describe its topological properties using the 'local Chern markers'. In Sec. <ref> we discuss anomalous Hall conductivity. We summarize our findings in Sec. <ref> .






§ MODEL




We consider a tight-binding model of electrons on a  bilayer of two honeycomb lattices twisted relative to each other by 30^∘ angle (see Fig. <ref>). Each of the lattices is described by Haldane Hamiltonian, and in addition, electrons from the different layers interact via exponentially decaying potential. The total Hamiltonian has the form

    H = ∑_α H_α + V_1-2,

where α = 1, 2 is the layer index, H_α is Haldane Hamiltonian of each layer, and V_1-2 describes interaction between the layers. Specifically,

    H_α   =    t_intra∑_<ij> c_i, α^† c_j, α  
    + t_2 ∑_≪ i, j≫ e^-i ν_ijϕ c_i, α^† c_j, α
           + 3 √(3)m ∑_i ϵ_i c_i, α^† c_i, α,
    
    
    V_1-2   =    t_inter∑_i, j v_i,j c_i, 1^† c_j, 2 + h.c.

Here i, j numerate the lattice sites within each layer, < i, j > is a sum over the nearest neighbors, ≪ i, j ≫ is a sum over the next-nearest neighbors. t_intra is the nearest neghbors hopping (between A, B sites), t_2 is the next-nearest neighbors hopping (between A-A or B-B sites), ν = ± 1 depending on the relative direction between i, j, ϕ = π/2, and ϵ = ± 1 for A(B) sites (see e.g. ref. <cit.> for more details). The interlayer potential has the form

    v_i, j = 
    {
    e^- r_ij/r_0,
       r < r_max
    0,      r > r_max.
    .

Here r_ij is a distance between the sites i, j, and to simplify the model we neglect interlayer spacing. Also we introduced long-distance cutoff r_max to decrease the complexity of our numerical calculations. We fix the numerical values of our parameters as t_intra=1, t_2=1, ϕ=π/2,  r_0 =1, r_max = 2 and explore its evolution over  m, t_inter.







 §.§ Phase diagram




To study the properties of our model, we compute its energy spectrum and wavefunctions numerically using exact diagonalization. Further, we characterize their spatial properties by computing fractal dimensions D_q = log(P_q)/(q-1) log(N), where P_q is a participation ratio defined as P_q = ∑_i (ψ^†ψ )^q, and i=1… N runs over all  lattice sites.

First, let us recall our model's properties in the known case when the interlayer coupling t_inter is set to zero. This model is a Chern insulator at |m| < 1 (we assume that both layers have the same signs of Chern numbers). Its bulk states form two bands separated by an energy gap, which in turn, is filled by edge states. At m=1, the bulk gap closes, and at m>1, it reopens again, and the system becomes a trivial insulator. 

Now let us look at the properties of our model once the interlayer coupling  is turned on. We find that the phase diagram of the system in m ,t_inter space has three phases, which we label as 'topological weakly coupled', 'non-topological weakly coupled' and 'strongly coupled' respectively (see Figs. <ref>). To be more specific, if we start from the topological phase of the Haldane model, we observe (see Figs. <ref>) that at small coupling, the bulk states are still separated by an energy gap and the edge states persist, but as t_inter is increasing, the bulk energy gap shrinks. Once t_inter reaches a critical value, the bulk gap closes, and the edge states disappear. As we increase t_inter further, the system remains gapless. Similarly, if we start from the non-topological phase (see Fig. <ref>), at small t_inter, the system remains gapped, and as t_inter is increasing, the gap becomes smaller and eventually disappears. 









To explore multifractal properties of our model, we plot average fractal dimension and its standard deviation for all states against the system size and a few values of q (Fig. <ref>). At zero, as well as at small t_inter, we observe that as the system size increases, the average fractal dimension of all states in the system approaches 1 and its standard deviation approaches 0. In other words, we confirm that in both topological and non-topological weakly coupled phases, the bulk eigenstates are fully extended in the same way as in a fully periodic system. 




Characterizing multifractal properties at large t_inter turns out to be challenging because finite size corrections to fractal dimensions scale as 1/log(L), and because of that it is impossible to obtain reliable results for any realistic system size, which can be handled numerically. Nevertheless, 
when we plot average fractal dimensions and their deviations as a function of the system size in the 'strongly coupled' phase, we observe that the former converge to values smaller than 1, and the latter converge to values greater than 0. Furthermore, we can see that as the system size grows, average fractal dimensions converge to different values for various q. Thus we confirm that eigenstates in the strongly coupled phase are multifractal - they are neither extended, nor localized.

To confirm that the system indeed undergoes a phase transition between the weakly and strongly coupled phases, we have to plot its derivatives of free energy and confirm that it diverges at the transition point. To be more specific, we consider a many body system at half filling 
 and assume zero temperature. Simply speaking, we consider a many particle system, in which all states with negative energies are filled, but all states with positive energies are empty. Its free energy is just equal to the total energy, i.e. sum of energies of all filled states. First, we confirm that the topological phase transition is indeed a phase transition by plotting its second derivative as a function of m (see Fig. <ref>) and observing that it 
is discontinuous.

Next, we plot the second derivative of free energy as a function of t_inter  (see Fig. <ref>) and confirm that it has a peak at the transition point between the weakly and strongly coupled phases, whose magnitude grows with the system size.












 §.§ Topological properties






  §.§.§ Topological entanglement entropy




To study entanglement properties of a many body system, one has to partition it into two subsystems in coordinate space. For a subsystem with density matrix ρ_A, the entanglement entropy is defined as S = - trρ_A logρ_A. If a 2D system has smooth boundary, its entanglement entropy scales linearly with system size L as

    S = α L - γ,

where the constant γ is a known as a topological entanglement entropy - a term present only in topological systems. We note that in this definition, it is important that the system's boundary is smooth, otherwise its entanglement entropy might get additional contributions e.g. due to corners (<cit.>).



To extract the topological contribution γ, we compute the difference between entanglement entropies of several systems in such a way that their leading term, which is linear over L, as well as subleading contributions due to corners exactly cancel out. To do this, we choose a configuration of four systems shown on the Fig. <ref>. We compute entanglement entropies of each subsystem using the method suggested in Refs. <cit.>, which we briefly describe in Sec. <ref>.








We plot the results for γ as functions of m, t_inter on the Figs. <ref> (we assume E_F = 0). We can see that in the 'weakly coupled topological' phase, γ is a non-zero constant, which in turn confirms that it is a topological quasicrystal. In a similar way, in the 'weakly coupled non-topological' phase, γ is equal to zero, which means that the system is non-topological. In the strongly coupled phase, the concept of topological entanglement entropy becomes ill-defined because the system is gapless. 








  §.§.§ Local Chern marker




It was argued in Ref <cit.>, that Chern number is not well-defined for a system of finite size because 
if one defines it in the same way as the corresponding infinite-size expression, the answer becomes exactly zero.  
 Instead it was suggested that topological properites of a system can be characterized empirically using 'local Chern marker' defined in the following way. First, one can define a projector to the subspace of filled 
 states as P(r_i, r_j) = ∑_E_λ < E_Fψ_λ(r_i) ψ^†_λ(r_j).

Next, one may project 
the coordinate operators as 
X̃(r_i , r_j) = ∑_r_k P(r_i, r_k) x_k P(r_k, r_j)
and Ỹ(r_j , r_i) = ∑_r_k' P(r_j, r_k') y_k' P(r_k', r_i). The local Chern marker is defined as


    C(r_i) = 2 π i/S∑_r_j[ X̃ (r_i, r_j) , Ỹ (r_j, r_i)  ],

where S is an average area per each lattice site on the lattice. One may check that for conventional Haldane model on a finite lattice, this local Chern marker inside the bulk is equal to the value of Chern invariant in the corresponding continuum model, whereas near the edges, its value is very large and has opposite magnitude in such a way that its sum over all lattice sites is zero: ∑_i C(r_i) = 0.

We plot the local Chern marker of 30^∘ twisted bilayer Haldane model for various values of t_inter on Fig. <ref>. One may see that in the 'weakly coupled' phases, it behaves in the same way as in the Haldane model on a monolayer. Specifically, in the 'weakly coupled topological phase', its bulk value remains the same and equal to the value in the case of zero interlayer coupling. Similarly, in the 'weakly coupled non-topological' phase, its bulk value is equal to zero. On the other hand, at larger values of t_inter, the local Chern marker starts behaving differently from the case of periodic lattice. 
Specifically, the local Chern marker no longer has a form of constant value in the bulk and large, but opposite value at the edge. Instead, regions with C(r_i) values of opposite signs start emerging within the bulk of the system, and sufficiently far away from the transition point, these regions become indistinguishable from the edges of the system. In other words, at large t_inter, the local Chern marker C(r_i) strongly fluctuates within the bulk. so that there is no distinction between the bulk and the edges as in a crystalline topological insulator.

Finally, we point out that the local Chern marker's behavior starts deviating from the limit of non-interacting layers at smaller value of t_inter than the gap closing phase transition. We interpret this result in a way that the local Chern marker can be used as a tool to discover a novel 'hidden' phase transition which does not manifest itself through the behavior of the eigenenergies.











  §.§.§ Anomalous Hall conductivity



We start our analysis from introducing electric current of our model. Specifically, if a tight-binding Hamiltonian for a given system has hoppings H_ij between sites i, j, then in the presence of electric field they change according to Peierls substitution and thus become H_ij e^ie/ħ∫_r_i^r_jA⃗d⃗r⃗, where A⃗ is a vector potential. If we expand these hoppings to the first order in A⃗, we obtain that the electric current operator has the following expression 

    J⃗_ij = ie/ħ  H_ij (r⃗_j - r⃗_i).


Here J⃗_ij is a vector directed along the bond between the sites i, j. In fact, the last expression could be immediately obtained by writing Heisenberg equation and assuming that the current is proportional to the velocity operator. 

Further, one can compute a response to applied electric field by using an expression for the electric current (<ref>)  and applying Kubo formula. Anomalous Hall conductivity would be its antisymmetric part, and thus it would 
be given by the Eq. (<ref>).













After lengthy, but straightforward calculations (specifically writing electrons Green functions and performing Matsubaba summation), the expression for anomalous Hall conductivity can be brought to the following form

    σ^(xy)_ij   =    - e^2/2 ħ^2 ∑_f,e∑_k, l1/(E_f - E_e)^2
    
          ×{ψ^†_e, i H_ij (x_j - x_i) ψ_f, j·ψ^†_f, k H_kl (y_l - y_k) ψ_e, l. 
    
          .
     - ψ^†_f, i H_ij (x_j - x_i) ψ_e, j·ψ^†_e, k H_kl (y_l - y_k) ψ_f, l. 
    
           -
    .
    ψ^†_e, i H_ij (y_j - y_i) ψ_f, j·ψ^†_f, k H_kl (x_l - x_k) ψ_e, l. 
    
          .
     + ψ^†_f, i H_ij (y_j - y_i) ψ_e, j·ψ^†_e, k H_kl (x_l - x_k) ψ_f, l}.

In this equation, i, j are lattice sites between which the electric current is computed, k, l are other pairs of sites over which summation is performed, E, ψ are eigenenergies and eigenfunctions of the model, indices f, e numerate filled/empty states respectively. 

Let us emphasize once again, that in the Eq. (<ref>), σ_ij describes electric current through a bond between the lattice sites i, j (we assumed that the electric field is uniform). However, one can sum electric currents through all bonds next to a given lattice site i and consider σ_i  = ∑_j σ_ij. For a crystalline system, one can take a step further and split the lattice site index i into two indices i', α numerating unit cells and sublattice degrees of freedom respectively. The well-known integer value of the anomalous Hall conductivity is obtained after an additional summation over α. A simple explanation of the latter is as follows: for an infinite crystalline system one may transform any dependency over unit cell index i' into momentum representation, and the well-known integer AHE occurs at zero momentum. However, its expression in terms of Berry curvature still contains summation over sublattice degrees of freedom. Nevertheless, on a honeycomb lattice, the two sublattice degrees of freedom are equivalent to each other, and therefore the AHE for each of them would be the same, i.e. half-integer. 

More interestingly, properties of AHE obtained from the Eq. (<ref>) on a finite lattice 
are fundamentally different from the case of an infinite crystal. Namely AHE on a finite lattice behaves in a similar way to a local Chern marker: its sum over all lattice sites i, j is equal to zero, its bulk value is constant for a crystalline lattice and equal to the value it would have in a corresponding infinite lattice, and its value at the edges has opposite sign and large magnitude so that its sum over all sites near the eges exactly cancels out the sum over all sites in the bulk.  At first sight, this may seem paradoxal, but there is a simple argument for it. Indeed it is well-known that on an infinite lattice, AHE is proportional to an integer topological invariant, and phases with different topological invariants cannot be smoothly transformed into each other, but are separated by phase transitions. However, it is also well-known that phase transitions do not exist on a finite lattice, but appear only in thermodynamic limit. This means that topologically distinct phases cannot exist on a finite lattice! In other words, any phase on a finite lattice is topologically equivalent to trivial. For this reason, if one sums the local Chern marker (<ref>) over all lattice sites r_i, or one sums AHE (<ref>) over all sites i, j, one obtains exactly zero. 



 


We present our numerical calculations of  σ_i for our model of twisted bilayer on the Fig. <ref>. One can see that in the case of zero interlayer coupling, i.e. when the model is just a superposition of two periodic layers, σ_i behaves just like a local Chern marker. In the 'weakly coupled topological phase', it has a constant value inside the bulk equal to the value it would have in the limit of infinite size, whereas at the boundary its value has an opposite sign and large magnitude so that the total sum ∑_i σ_i is zero. Once the interlayer coupling t_inter is turned on, the behavior remains the same while t_inter is small. However, once t_inter is increased, σ_i is no longer uniform inside the bulk. The behavior of σ_i shows a phase transition at the same value of t_inter as the local Chern marker, and as t_inter increases further, the distinction between the bulk and the edge behavior smoothly disappears. This fact supports our argument that for a quasicrystal there is no distinction between bulk and edge as in a crystal, but instead there are 'bulk-like' and 'edge-like' regions.

Finally let us explain the meaning of zero total ∑_i σ_i from the experimental perspective. Indeed, introducing the vector potential A⃗ into our Hamiltonian with open boundary conditions is equivalent to placing our sample under external electric field. However, if one places a finite and isolated sample of a material with non-trivial AHE under electric field, there would be no electric currents flowing into or out of the sample because it does not have any contacts, through which the current might flow. In other words, our calculations 
physically mean that if a finite and isolated sample is placed under electric field, there will appear electric currents inside the sample, but no currents into or out of it. In order to obtain a physically measurable anomalous Hall conductivity, one has either to consider explicitly the flow of electric current through contacts, or just to assume that the sample is infinite in the direction of electric current, as it was done in multiple past works  (e.g. <cit.>). Since within our model, the case of an infinite sample is fundamentally different from a finite one, our results can be applied only to the latter. Nevertheless, we anticipate that if our model is realized experimentally, then at zero as well as small interlayer coupling, while the system is gapped, the total Hall conuctivity would be just an integer, i.e. the sum of two Hall conductivities of the isolated samples. On the other hand, in the gapless phase, the Hall conductivity will become non-universal, and possibly it may not even have a well-defined limit at infinite system size. The latter fact can be interpreted in a way that 'strong quasicrystallinity' is qualitatively similar to strong disorder: in its presence, all observable properties become sample-dependent. 









§ DISCUSSION




In this work, we have studied topological properties of quasicrystalline 30 degrees twisted bilayer honeycomb lattice. We started from the case of two uncoupled layers each of which forms Haldane model, and tracked its evolution once the interlayer coupling is increased. We found that at small interlayer coupling, two layers of crystalline topological insulators become a quasicrystalline topological insulator. The latter fact is not obvious because topological properties of Haldane model are characterized by Chern invariant defined in momentum space, which is in turn well-defined only in the presence of crystalline translational symmetry. In this regard, one could be concerned whether translational symmetry breaking might break topological protection, but we established that 
at small, but non-zero interlayer coupling, this is not the case: our bilayer has exactly the same topological properties as a superposition of two monolayers. Specifically, it still has edge states, has exactly the same value of topological entanglement entropy and the same behavior of local Chern marker and anomalous Hall conductivity. 


 
As the interlayer coupling reaches a critical value, the system undergoes a 'hidden' phase transition which manifests itself only through the behavior of local Chern markers and anomalous Hall conductivity. As the interlayer coupling increases further, our system undergoes a phase transition into a gapless phase, whose eigenstates are multifractal. 
Rigorous characterization of such a gapless phase's topological properties is challenging, but can just qualitatively say that its topology starts 'deteriorating'. Specifically, in the weakly coupled phases, the local Chern marker and anomalous Hall conductivity is constant in the bulk, but have a value of opposite sign in the boundary, whereas in the strongly coupled phase, these quantities significantly fluctuate within the bulk, but do not have any special features at the boundary. In fact, one may consider an analogy between our model and a disordered topological insulator. At weak disorder, the system has exactly the same topological properties as the topological insulator without disorder. However, once the disorder becomes strong, fluctuations of the disorder potential effectively behave like edges themselves, and this fact qualitatively explains that the bulk becomes indistinguishable from the physical edges of the system. 

It would be of interest to realize our model in experiments, but it is challenging because typically (e.g. in twisted bilayer graphene), at large twist angles, interlayer coupling is very weak. However, one may still try to make it larger by applying high pressure between the layer. More plausibly, one might try to realize our model using ultracold atoms (see e.g. <cit.>), which have been proved to be a powerful tool to realize various physical models.


In summary, we proposed a way to realize topological quasicrystals by starting from topological crystalline materials and breaking their translational symmetries. Moreover we proposed a model of a quasicrystal, which explicitly hosts a topological phase transition. We demonstrated that non-trivial topological properties of quasicrystals can be characterized not only by local Chern markers, but also by topological entanglement entropy and anomalous Hall conductivity. Most surprisingly, we established that the anomalous Hall conductivity in an open system, in our case in a quasicrystal behaves qualitatively in the same way as a local Chern marker. We hope that in the future, our results may be used to obtain rigorous topological classification of quasicrystals. We are also interested in studying their unusual transport properties and possibilities for future applications.










§ ACKNOWLEDGEMENTS


The author would like to thank Institute of Basic Science (Daejeon, S. Korea), in which this project started and Profs. Moon Jip Park, Kyong Min Kim, Sergej Flach, Sergey Syzranov, Predrag Nikolic, Renat Sabirianov, Luis Santos, Ivan Khaymovich for helpful discussions. 
Financial support by the National Science Foundation through EPSCoR RII Track-1: Emergent Quantum Materials and Technologies (EQUATE), Award OIA-2044049 is acknowledged.







§ CALCULATION OF THE ENTANGLEMENT ENTROPY



Here we briefly describe the method we use to compute entanglement entropy, which was proposed in Refs. <cit.> and used in a number of works later on, e.g. in <cit.>. We consider the model of twisted bilayer at half filling given by the Hamiltonian (<ref>) and are interested in entanglement entropy of any subsystem from the Fig. <ref>. First, we compute eigenvectors ψ of the Hamiltonian (<ref>) and select only the ones corresponding to its filled states. Next, we compute 'correlation matrix', i.e. sum of their outer products 

    C_ij = ∑_filledψ (r⃗_i) ψ^† (r⃗_j).

After that we select the components whose coordinates r⃗_i, j are inside the subsystem whose entanglement entropy we are interested in and thus obtain 'reduced correlation matrix' C_A, ij. The entanglement entropy can be expressed in terms of its eigenvalues ζ as

    S = - ∑_m(
    ζ_m logζ_m
    + (1 - ζ_m)  log (1 - ζ_m).
    )

Finally, to substract the leading 'area law' terms and extract the topological contribution we compute sum of the entanglement entropies over several figures as shown on the Fig. <ref>.







§ KUBO FORMULA FOR ANOMALOUS HALL EFFECT


In this section, we derive an expression for the anomalous Hall conductivity on a finite lattice using Kubo formula. Importantly, we do not use momentum representation, which in turn makes our derivation applicable to the case of non-periodic lattice. 

We start from partition function of the system, which we write as

    𝒵 = ∫ D ψ D ψ̅
    e^∑_w, i i w c^†_i, w c_i, w - ∑_w, i, j c^†_i, w H_ij c_j, w .

In this expression, we have transformed imaginary time into Matsubara frequencies representation, but left the coordinate representation. H_ij is the lattice Hamiltonian, and i, j numerate its sites. Electric field is introduced via Peierls substitution, which in turn may be approximated as

    H_ij→ H_ij e^i ∫_r_i^r_jA⃗d⃗r⃗≈ H_ij + i H_ijA⃗ (r⃗_j - r⃗_i).

From the last equation we can see that the current components have the form

    J⃗_ij, Ω = i c^†_i, w+Ω H_ij (r⃗_j - r⃗_i) c_j, w

Note, that the current is defined here as  a vector for each bond connecting the lattice sites i, j.

Now we apply Kubo formula and write that the expectation current has a form

    J_x, ij, Ω = 
    ⟨ 
    J_x, ij, Ω∑_k. l J_y, kl, -Ω⟩ A_y, kl, Ω.

Now we assume that the electric field is constant in space, and the vector potential is written in the gauge A⃗ = E⃗/(iΩ). The anomalous Hall conductivity is given by the antisymmetric part of the current-current correlator, namely

    σ^(Hall)_ij = 1/2iΩ∑_kl(
    ⟨ J_x, ij, Ω J_y, kl, -Ω⟩
    - ⟨ J_y, ij, Ω J_x, kl, -Ω⟩).

Note that in our notations, the conductivity has just two indices i, j, which physically describe electric current at the bond connecting lattice sites i, j as a response to constant electric field (if the electric field was not constant, we would have to include 4 components).

We evaluate the current-current correlator in a conventional way by substituting fermionic Green's functions and summing over Matsubara frequencies. The Green's function can be written in terms of eigenstates ψ_n, i and energies E_n (here n numerates the states) as 

    G_ij, w = ∑_n ψ_i ψ^†_j/i w - E_n.

After we sum over Matsubara frequencies, assume zero temperature and take the limit of zero external frequency, we can obtain an answer for σ^(Hall)_ij, which is given by the Eq. (<ref>).

Finally, let us look at the sum  ∑_ijσ^(Hall)_ij over all lattice sites i, j. We can simplify the Eq. (<ref>) by using explicitly the fact that ψ are eigenvectors, namely by applying an identity

    ∑_k, lψ^†_f, k H_k, l(x_l - x_k) ψ_e, l
    = (E_f - E_e) ∑_kψ^†_f, k  x_k ψ_e, k.

In this way we can obtain that the total sum has the form

    ∑_i, j σ^(Hall)_ij
    = 1/2∑_f,e∑_i, k
          ×{ψ^†_e, i x_i ψ_f, i·ψ^†_f, k y_k ψ_e, k 
    - ψ^†_f, i H_ij x_i ψ_e, i·ψ^†_e, k y_k ψ_f, k.  
    
          . 
    - ψ^†_e, i y_i ψ_f, i·ψ^†_f, k x_k ψ_e, k
    + 
    ψ^†_f, i y_i ψ_e, i·ψ^†_e, k x_k ψ_f, k}.

After applying the completeness relation

    ∑_e ψ_e, i·ψ^†_e, k
    = 1 - ∑_f ψ_f, i·ψ^†_f, k

one can derive that the sum σ_ij is equal to 

    ∑_i, j σ^(Hall)_ij
    = 1/2∑_f∑_i{ψ^†_f, i( y_i x_i  - x_i y_i) ψ_f, i},

i.e. it is indeed zero for any finite lattice. From the last expression, one can also see how this argument may break down for an infinite lattice: we found that anomalous Hall conductivity is proportional to a trace of a commutator [X, Y]. The latter is always zero in a finite-dimensional space, but does not have to be zero in an infinite-dimensional space. 






§ ANOMALOUS HALL CONDUCTIVITY ON AN INFINITE CRYSTALLINE LATTICE


The goal of this section is to show that the expression for Hall conductivity (<ref>) from the main text in the case of an infinite crystalline lattice is indeed equivalent to a well-known expression in terms of Berry curvature. Here, for clarity we separate lattice indices i, j. ... into pairs (i, α), (j, β), ..., which refer to unit cells and sublattice degrees of freedom respectively.  

The key feature of a crystalline lattice is that its eigenvectors are plane waves

    ψ_iα = 1/√(N) e^i k⃗x⃗_i ϕ_α, k.

Here the wavefunctions ϕ_α, k are eigenfunctions of the Schrodinger equation in momentum representation 

    E ϕ_α, k = ∑_β H_αβ (k) ϕ_β, k

and the corresponding momentum-space Hamiltonian is just a Fourier transformation of the original Hamiltonian

    H_αβ(k) = 
    ∑_j, β
    e^i k⃗(x⃗_j - x⃗_i) 
     H_i,j α, β.

Due to translational symmetry, we can reverse the above expression and thus write it as 

    H_i,j α, β 
    = 1/N∑_p H_αβ(p) e^-i p⃗ (x⃗_j - x⃗_i) .

After substituting the above expression, as well as expressions for the eigenstates in the form of plane waves (<ref>) into the main expression for anomalous Hall conductivity (<ref>), one can obtain the lattice version of TKNN formula

    ∑_jσ^(Hall)_i,j α, β   =   1/2∑_γ, δ1/N∑_k⃗1/(E_f - E_E)^2
          ×{ϕ^†_e, α∂ H_αβ/∂ k_xϕ_f, β·ϕ^†_f, γ∂ H_γδ (k)/∂ k_yϕ_e, δ.
    
               -
    ϕ^†_f, α∂ H_αβ/∂ k_xϕ_e, β·ϕ^†_e, γ∂ H_γδ (k)/∂ k_yϕ_f, δ
          
    .
    ∂ H_γδ (k)/∂ k_y
    - ( k_x ↔ k_y )
    }.

Notice that in the left side of this equation we have sum ∑_j σ_ij of conductivities through all bonds adjacent to a site i - the quantity we discuss in the main text. In addition, to obtain the physical answer we have to sum over all sublattice degrees of freedom, i.e. over the indices α, β. We also note that in the Eq. (<ref>) we still assume that the lattice is discreet, but the momentum is continuum because the lattice is infinite. Hence it is straightforward to replace the momentum summation with an integration 

    1/N∑_k → S ∫d^2 k/(2π)^2,

 where S is an area of the unit cell. 

Finally, by making use of identities

    ∑_αβϕ^†_e, α∂ H_αβ/∂ k_xϕ_f, β
    = 
    ∑_α (E_f - E_e ) ϕ^†_e, α∂ϕ_f, α/∂ k_x

combined with the completeness relation (<ref>) we obtain a familiar expression for anomalous Hall conductivity in terms of Berry curvature
 
    ∑_j αβσ_i, j α, β
     = S ∫d^2 k/(2π)^2∑_α{∂ϕ^†_f, α/∂ k_y∂ϕ_f, α/∂ k_x
     -
     ∂ϕ^†_f, α/∂ k_x∂ϕ_f, α/∂ k_y}.






 














apsrev4-2




