Anomalous Hall effects in magnetic weak topological insulator films
Rui Chen Xiao-Xia Yi Bin Zhou Dong-Hui Xu







Department of Physics, Hubei University, Wuhan 430062, China

Department of Physics, Hubei University, Wuhan 430062, China

Department of Physics, Hubei University, Wuhan 430062, China
Key Laboratory of Intelligent Sensing System and Security of Ministry of Education,

Hubei University, Wuhan 430062, China
[]donghuixu@cqu.edu.cn
Department of Physics and Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing 400044, China
Center of Quantum Materials and Devices, Chongqing University, Chongqing 400044, China

§ ABSTRACT 
The interplay between magnetism and strong topological insulator gives rise to distinct new topological phases and various intriguing phenomena, attracting significant attention in recent years. However, magnetic effects in weak topological insulators remain largely unexplored. In this work, we systematically investigate the magnetic effect on thin films of weak topological insulators. We focus on ferromagnetic and antiferromagnetic effects, which have been extensively studied in strong topological insulators, as well as the recently highlighted altermagnetic effect. We reveal that the interplay between magnetism and weak topological insulators leads to a variety of Hall effects in the absence of an external magnetic field, including the metallic quantum anomalous Hall effect without chiral edge states, the quantum anomalous Hall effect with a higher Hall conductance plateau, the quantized layer Hall effect, the metallic half-quantized valley-like Hall effect, and a quantized valley-like Hall effect.  This work provides valuable insights for exploring magnetic effect on weak topological insulators.

Anomalous Hall effects in magnetic weak topological insulator films
    Dong-Hui Xu
    September 9, 2024
===================================================================



§ INTRODUCTION

Over the past two decades, topological insulators have attracted considerable attention due to their fundamental novelty and potential applications <cit.>. Unlike conventional insulators, topological insulators exhibit hallmark gapless surface Dirac states with a linear dispersion, resulting from the symmetry-protected nontrivial bulk topology. Within the topological band theory <cit.>, three-dimensional topological insulators are
classified into strong topological insulators (STIs) and weak topological insulators (WTIs), distinguished by their odd or even number of surface Dirac cones, respectively. STIs have been experimentally realized in materials like Bi_1-xSb_x <cit.>, Bi_2Se_3 <cit.> family, and TiBiSe_2 family <cit.>, while WTIs have emerged in ZrTe_5 <cit.>, Bi_4I_4 <cit.>, and Bi_4Br_2I_2 <cit.>.

The interplay between magnetism and STIs leads to the discovery of novel topological phases, such as the quantum anomalous Hall insulator <cit.>, axion insulator <cit.>, semi-magnetic topological insulator <cit.>, metallic quantum anomalous Hall insulator <cit.>, and half-quantum mirror Hall insulator <cit.>.
These phases have been experimentally observed in ferromagnetic topological
insulators like (Bi, Sb)_2Te_3 and the intrinsic antiferromagnetic topological insulator MnBi_2Te_4 <cit.>. However, the magnetic effect on WTIs remains an unexplored territory. Concurrently, the emergence of altermagnetism, a unique form of collinear antiferromagnetism, has attracted significant attention. This magnetic state possesses the ability to induce a spontaneous Hall effect <cit.>. Moreover, the band topology induced by altermagnetism has been found in different works <cit.>.

In this work, we investigate the effect of ferromagnetism, antiferromagnetism, and altermagnetism on a WTI film. Our research uncovers an array of intriguing Hall effects facilitated by the interplay between magnetism and WTIs.  Moreover, in the magnetic WTI film, we find that the half-quantized Hall conductance is revealed through the integration of the Berry curvature over half of the Brillouin zone. This is different from the previous studies on the magnetic STI film, that the half-quantization is manifested through the integration of the Berry curvature over the whole Brillouin zone <cit.>. Below, with close reference to Fig. <ref>, we provide a summary of these closely-related topological phases in the ferromagnetic and altermagnetic systems. It is noticed that the total Hall conductance is related to the number of magnetism-induced gapped surface Dirac cones.


(i) Figure <ref>(a): a WTI film without magnetization.  Each of the top and bottom surfaces hosts two gapless Dirac cones. Time-reversal symmetry ensures that the Hall conductance of this system is zero.

(ii) Figure <ref>(b): a metallic quantum anomalous Hall effect, a phenomenon that challenges conventional expectations and highlights the interplay of magnetism and the topology in WTIs.  The ferromagnetism is selectively introduced to the top surface of the WTI film.  The two Dirac cones on the top surface are gapped due to local time-reversal symmetry breaking caused by the magnetization. In contrast, the two Dirac cones on the bottom surface remain gapless, as this layer is not subjected to magnetization.



(iii) Figure <ref>(c): a quantum anomalous Hall effect with a higher Hall conductance plateau σ_xy=2e^2/h, resulting from contributions of e^2/h from both top and bottom surfaces. Introducing ferromagnetism to both surfaces in the WTI film gaps all four Dirac cones. This scenario can be regarded as a double version of Case (ii), but it features two pairs of gapless chiral edge states within the insulating gap.

(iv) Figure <ref>(d): a quantized layer Hall effect with the top and bottom surfaces possessing quantized Hall conductances of opposite sign. Introducing ferromagnetism to both surfaces, with oppositely oriented magnetic moments, gaps all four Dirac cones. However, the two Dirac cones on the bottom surface acquire an opposite Dirac mass compared to those on the top.

(v) Figure <ref>(e): a metallic half-quantized valley-like Hall effect with a half-quantized Hall conductance e^2/2h or -e^2/2h distributed across different halves of the Brillouin zone. This effect is protected by the C_4T symmetry, where C_4 is the 4-fold rotational symmetry and T is time-reversal symmetry. Due to altermagnetism, the two Dirac cones at X and Y points on the top surface gain opposite masses, while those on the bottom surface remain gapless due to the lack of magnetization.

(vi) Figure <ref>(f): a quantized valley-like Hall effect with a quantized Hall conductance e^2/2h or -e^2/2h distributed across different halves of the Brillouin zone. This effect is protected by the C_4T symmetry. The two Dirac cones at X and X' points, as well as those at Y and Y' points gain opposite masses due to the altermagnetism. This scenario can be regarded as a double version of Case (v),  but with two pairs of counter-propagating gapless chiral edge states within the insulating gap.


Additionally, we explore two distinct scenarios showcasing the antiferromagnetic effect depicted in Figs. <ref>(a) and <ref>(b). We discover that these configurations lead to two distinct phenomena: a quantum anomalous Hall effect characterized by a higher Hall conductance plateau and a quantized layer Hall effect.





§ MODEL AND METHOD



 §.§ Model

We start with a generic four-band topological insulator model defined on a cubic lattice <cit.>

    H=H_0(k)+H_fm(z)+H_am(k_x,k_y,z)+H_afm(z),


    H_0(k)=ℳ(k) Γ_4+A (Γ_1 sin k_x+Γ_2 sin k_y+Γ_3 sin k_z),

where ℳ(k)=M_0+6 B-2 B ∑_i cos k_i, Γ_1=σ_1  τ_1, Γ_2=σ_2  τ_1, Γ_3=σ_3 τ_1,  and Γ_4= σ_0  τ_3. Here M_0, B and A are model dependent parameters. σ and τ are Pauli matrices representing spin and orbital degrees of freedom, respectively.

In the absence of magnetization [i.e., H_fm=0, H_am=0, and H_afm=0], H_0(k) represents a trivial insulator when M_0>0. When M_0<-12 B, it depicts an STI phase with the Z_2 index (1 ; 000) or a STI phase with (1 ; 111) when 0>M_0>-4 B or -8 B>M_0>-12 B.  It corresponds a WTI phase with (0 ; 111) when -4 B>M_0>-8 B <cit.>. This four-band model, originally proposed by Zhang et al., has been successfully used to describe the STI of Bi_2 Se_3 family <cit.>. As our primary focus is on the WTI phase, we concentrate on the parameter regime -4 B>M_0>-8 B.

The last three terms of the Hamiltonian in Eq. <ref>, H_fm(z), H_am(k_x,k_y,z), and H_afm(z) depict the layer-dependent effect of ferromagnetism, altermagnetism, and antiferromagnetism, respectively.
Specifically, the three types of magnetization have the following form

    H_fm(z)   =m(z)σ_3 τ_0,
    
    H_am(k_x,k_y,z)   =m'(z)(cos k_x-cos k_y)σ_3 τ_0,
    
    
    H_afm(z)   =m”(-1)^zσ_3 τ_0.

Ferromagnetism can be induced through magnetic doping <cit.>.  Equation (<ref>) represents a d-wave altermagnetic ordering  <cit.>, which can be achieved via the proximity effect. Equation (<ref>) depicts the antiferromagnetic ordering, where neighboring layers possess opposite magnetizations. The amplitude of the magnetization within each layer is m”.

The ferromagnetic term m( z )  and altermagnetic term m^'( z )  are given by

    m( z ) =
    	m_t
    
    	0
    
    	m_b
    [           z=1,2,3,;         elsewhere,; z=n_z-2,n_z-1,n_z,;                    ]

and

    m^'( z ) =
    	m_t^'
    
    	0
    
    	m_b^'
    [           z=1,2,3,;         elsewhere,; z=n_z-2,n_z-1,n_z.;                    ]

This is  more clearly illustrated in Fig. <ref>, where ferromagnetic and altermagnetic effects are confined solely to the top and bottom surface layers.

In the following content, we first introduce the nonmagnetic system in Sec. <ref>, then focus on the ferromagnetic and altermagnetic effects in Sec. <ref>, and finally discuss the antiferromagnetic effect in Sec. <ref>.



 §.§ Method


The ferromagnetic and antiferromagnetic systems are characterized by the total Hall conductance
 
    σ_xy=∑_k_x,k_y,zσ_xy(k_x,k_y,z),

and the layer Hall conductance

    σ_xy(z)=∑_k_x,k_yσ_xy(k_x,k_y,z).

In Appendix <ref>, we provide more details for calculating the layer- and momentum-resolved Hall conductance σ_xy(k_x,k_y,z) <cit.>. Here, the layer Hall conductance σ_xy(z) counts the contributions to the total Hall conductance from the z-th layer. The total Hall conductance σ_xy counts the contribution to the Hall conductance from all layers, i.e., σ_xy=∑_zσ_xy(z). Moreover, we define

    σ_xy^t=∑_z=n_z-3^z=n_zσ_xy(z),
    σ_xy^b=∑_z=1^z=4σ_xy(z),

which count the contributions to the Hall conductance from the top and bottom surface layers, respectively.

On the other hand, we adopt a momentum-resolved Hall conductance to characterize the altermagnetic system, with

    σ_xy^'(θ)=∑_k_x,k_y∈ P(θ),zσ_xy(k_x,k_y,z),

where P(θ) is the area enclosed by the green lines with a θ angle in Fig. <ref>(a).


§ NONMAGNETIC SYSTEM


As depicted schematically in Fig. <ref>(a), the nonmagnetic system corresponds to a WTI film with four gapless Dirac cones. Two of these Dirac cones are situated at X and Y points of the top surface Brillouin zone, while the other two are located at X^' and Y^' points of the bottom surface Brillouin zone. This is further clarified in Figs. <ref>(a) and <ref>(b), which show the local density of state (LDOS) of the nonmagnetic surface. The Hall conductance of the system is zero due to the existence of time-reversal symmetry T=I_n_zσ_y τ_0 K, where I_n_z is the identity matrix and K denotes the complex conjugation.


§ FERROMAGNETIC AND ALTERMAGNETIC SYSTEMS





In this section, we consider five different scenarios for the ferromagnetic system [Figs. <ref>(b) and <ref>(c)] and altermagnetic systems [Figs. <ref>(e) and <ref>(f)]. In the numerical calculations, the coefficients are given by the following tabular,

    c]c|ccccccc   (i)   (ii)   (iii)   (iv)   (v)   (vi)
    m_t   	          0    V    V    V    0    0
    m_b   	          0    0    V    -V    0    0
    m_t^'    0    0    0    0    V    V
    m_b^'    0    0    0    0    0    V

The other parameters are taken as A=1, B=1, M=-6, and V=0.6.






 §.§ Metallic quantum anomalous Hall
effect without chiral edge states


Let us first consider the simplest case where the ferromagnetization is introduced only to the top surface layers of the WTI film [Fig. <ref>(b)].  The Dirac cones on the top surface are gapped out due to the breaking of local time-reversal symmetry induced by the magnetization [Figs. <ref>(b) and <ref>(c)]. The gapless Dirac cones on the bottom surface remain due to the absence of local magnetization [Figs. <ref>(b) and <ref>(b)].

Figure <ref>(a) shows the numerically calculated Hall conductance σ_xy and the DOS as functions of the Fermi energy E_F. The system exhibits a quantized anomalous Hall conductance (the blue circle line) when the Fermi energy is inside the magnetic gap of the top surface (|E_F|<V). However, it is important to note that the system differs from conventional quantum anomalous Hall insulators, where the quantized Hall conductance only emerges when the Fermi energy resides within the insulating gap. In contrast, the present system lacks an insulating energy gap, as revealed by the numerical results of the DOS (the red circle line). Moreover, the DOS exhibits a linear increase as the Fermi energy increases, indicating its origin from the 2D gapless Dirac cones on the bottom surface.

Figure <ref>(c) shows the longitudinal conductance σ_xx as a function of the Fermi energy E_F,  numerically obtained using the Kubo-Greenwood formula expressed in terms of the Chebyshev polynomials <cit.> (see Appendix <ref>). The system manifests a metallic quantum anomalous Hall effect, distinguished by a quantized Hall conductance and a nonvanishing longitudinal conductance. Figure <ref>(e) displays the layer Hall conductance as a function of layer index z with E_F=0.  We find that σ_xy^t=0.9998e^2/h and σ_xy^b=0.0001e^2/h [see Eq. (<ref>)]. The slight deviation from perfect quantization arises from the quantum confinement effect. These results further confirm that the quantized Hall conductance originates from the ferromagnetic top surface.

Very recently, the metallic quantum anomalous Hall effect has been proposed in an STI film with a ferromagnetic layer at interior layers <cit.>. Our work presents an alternative approach to realize the metallic quantum anomalous Hall effect. Moreover, the metallic quantum anomalous Hall effect can be regarded as a double version of the semi-magnetic topological insulator, which has been experimentally detected using the traditional six-terminal Hall-bar measurement and is characterized by a half-quantized Hall conductance and a nonvanishing longitudinal conductance <cit.>. Thus, we believe that the metallic quantum anomalous Hall effect in our work could be observed using a similar experimental setup.



 §.§ Quantum anomalous Hall
effect with a higher plateau


Here we consider the case shown in Fig. <ref>(c), with parallel magnetization alignment on the top and bottom surfaces.  All the surface Dirac cones are gapped out due to the magnetic effect, and the top and bottom surfaces share the same LDOS shown in Fig. <ref>(c). The system hosts a global energy gap characterized by a vanishing DOS when |E_F|<V  [see the red circle line in Fig. <ref>(b)]. This energy gap features a higher quantized Hall conductance σ_xy=2e^2/h [the blue circle line in Fig. <ref>(b)]. Consequently, the system is identified as a quantum anomalous Hall insulator. The insulating nature of the system is further confirmed by calculating the longitudinal conductance σ_xx [Fig. <ref>(d)], which remains zero as long as the chemical potential resides within the energy gap. Analysis of the layer Hall conductance reveals that both the top and bottom surfaces contribute a nearly quantized Hall conductance [Fig. <ref>(f)], with σ_xy^t=σ_xy^b=0.9998e^2/h. In experiments, the quantum anomalous Hall effect can be detected using the traditional six-terminal Hall-bar measurement.



 §.§ Quantized layer Hall effect






Now we consider the case shown in Fig. <ref>(d), where the magnetizations on the top and bottom surfaces are antiparallel. All the surface Dirac cones are gapped due to the magnetic effect, and the top and bottom surfaces share the same LDOS as shown in Fig. <ref>(c). It is noticed that the top and bottom surfaces gain an opposite Dirac mass due to the antiparallel magnetization alignments on the top and bottom surfaces [labeled by the red and blue gapped Dirac cones in Fig. <ref>(d)].

The breaking of local time-reversal symmetry results in a nonzero layer Hall conductance as shown in Fig. <ref>(a). Moreover, the layers connected by the
PT symmetry are exactly compensated, leading to a zero net Hall conductance. Here, P=Mσ_0τ_z depicts the inversion symmetry, and M is the orthogonal matrix that permutes the layers of the entire system perpendicularly. Figure <ref>(b) shows σ_xy^t/b and the total Hall conductance σ_xy as functions of the chemical potential E_F [see the dashed lines]. Both the top and bottom surfaces exhibit a quantized surface Hall conductance but with an opposite sign, establishing the quantized layer Hall effect.

The quantized layer Hall effect can be viewed as a double version of the axion insulator, which has been proposed for experimental detection by various approaches <cit.>. One of the most direct approaches involves using the traditional transport measurement by applying an external magnetic field [Figs. <ref>(a)-<ref>(c)] <cit.>. By varying the strength and direction of the external magnetic field, the Hall conductance of the system should fluctuate between -2e^2/h, 0, and 2e^2/h. The quantized layer Hall effect is characterized by the intermediate zero plateau, while the quantum anomalous Hall effect mentioned in Sec. <ref> is characterized by the quantized Hall conductance ± 2e^2/h.

Another method to detect the quantized layer Hall effect is by applying an external perpendicular electric field [Figs. <ref>(d)-<ref>(f)] <cit.>, which breaks the PT symmetry and induces an energy offset between the plateaus of the surface Hall conductance [Fig. <ref>(b)]. The electric-field-induced Hall conductance has opposite signs for opposite Fermi energies and its amplitude increases with the increasing strength of the field [Fig. <ref>(c)].  For moderate film thickness and strength of the external electric field, we find that the emergent Hall conductance can approximate the quantized value of e^2/h [Fig. <ref>(d)]. These signatures provide evidence for observing the quantized layer Hall effect in the ferromagnetic WTI films.





 §.§ Metallic half-quantized valley-like Hall effect


We consider the altermagnetic case, which can be achieved by approximating the WTI to an altermagnetic material. The system breaks time-reversal symmetry but preserves the combined C_4 T symmetry, where C_4=I_n_zexp(-iσ_zτ_0π/4)R_4 and R_4 is the rotational matrix with R_4 (k_x,k_y)=(k_y,-k_x). The C_4 T symmetry requires that 
    Ω(k)=-Ω(R_4k),

which guarantees a zero total Hall conductance.

Figure <ref>(a) shows the Berry curvature of the altermagnetic system as a function of k_x and k_y. Our numerical results align with  Eq. (<ref>), where the Berry curvature Ω(k) is opposite in sign to Ω(R_4k).
The blue curve in Fig. <ref>(c) shows the Hall conductance σ_xy^'(θ) as a function of θ, where σ_xy^'(θ) is the total Hall conductance contributed by the k points within the green cone area shown in Fig. <ref>(a). We observe that half of the Brillouin zone contributes to a half-quantized Hall conductance. Consequently, the other half of the Brillouin zone must contribute another half-quantized Hall conductance with the opposite sign due to the C_4 T symmetry. Moreover, similar to the situation discussed in Sec. <ref>, the bottom surface is gapless, making the system a metal. Therefore, we identify this system as a metallic half-quantized valley-like Hall effect.

For a comparative study, we also examine the Berry curvature distribution of the ferromagnetic case [see Sec. <ref> and Fig <ref>(b)]. The ferromagnetic system is protected by the C_4 symmetry and requires that Ω(k)=Ω(R_4k). Thus, each half of the Brillouin zone will contribute a half-quantized Hall conductance, resulting in a quantized Hall conductance overall. This implies that each gapped Dirac cone is associated with a half-quantized Hall conductance, and the half-quantization is manifested through half of the Brillouin zone in the magnetic WTI film. This scenario differs from previous studies on magnetic STI films, where the half-quantization in the semimagnetic topological insulator phase is manifested through the integration of the Berry curvature over the entire Brillouin zone <cit.>.

Furthermore, Figure <ref>(d) shows σ_xy^'(θ) as a function of θ, in a system where both the ferromagnetic and altermagnetic effects coexist. We find that the relationship σ_xy(π/2)= 0.5 is stable by varying the strength of ferromagnetic ordering.





 §.§ Quantized valley-like Hall effect

Now we consider the case shown in Fig. <ref>(f), where the altermagnetic effect is introduced to both the top and bottom surface layers. The situation in Fig. <ref>(f) closely resembles that in Fig. <ref>(e). However, there are three key distinctions: (1) The Berry curvature and the corresponding Hall conductance double. In other words, half of the Brillouin zone contributes a quantized Hall conductance, and the other half contributes another quantized Hall conductance with the opposite sign; (2) the system is an insulator because both the top and bottom surfaces are gapped; and (3) the system hosts a chiral edge state inside the insulating surface gap [see Fig. <ref> and Appendix <ref>]. Consequently, the system is identified as a quantized valley-like Hall effect.



§ ANTIFERROMAGNETIC SYSTEM




In this section, we investigate the antiferromagnetic effect on WTI films,  focusing on the two scenarios depicted in Figs. <ref>(a) and <ref>(b). In the presence of antiferromagnetism, the Dirac cones on the top and bottom surfaces acquire local topological masses with  the same sign for odd layers [Fig. <ref>(a)] and the opposite signs for even layers [Fig. <ref>(b)]. Figures <ref>(c) and <ref>(d)  illustrate the layer Hall conductance σ_xy(z)  as a function of the layer index z. Each magnetic layer exhibits a nonzero layer Hall conductance with its sign depending on the direction of its magnetic moment. Additionally, the layer Hall conductance is enhanced at the surface layers due to the surface Dirac cones. In the odd-layer system, the total Hall conductance is 2e^2/h due to the breaking of time-reversal symmetry. In the even-layer system, the total Hall conductance is always zero protected by the PT symmetry.

To mitigate the oscillatory behavior of the bulk layers, we define the renormalized Hall conductance <cit.> as

    σ̃_xy(z)=σ_xy(z)+σ_xy(z+1)/2,

where z=2,⋯,n_z, σ̃_xy(1)=σ_xy(1)/2, and σ̃_xy(n_z+1)=σ_xy(n_z)/2. This approach effectively counts each layer once, except for the surface layers, which are counted with a weight of 1/2. It can be considered an application of the sliding window averaging method <cit.>.

Figures <ref>(e) and <ref>(f) show the renormalized layer Hall conductance σ̃_xy(z) as a function of layer index z. The Hall conductance contributed by the bulk layers is compensated to zero due to the local PT symmetry at the bulk. Moreover, we obtain ∑_z=1^z=3σ̃_xy(z)=0.9980 e^2/h for the top surface layers of both systems, and ∑_z=n_z-2^z=n_zσ̃_xy(z)=(-) 0.9980 e^2/h for the odd-layer (even-layer) systems. By adopting the renormalized layer Hall conductance, we extract the quantized surface Hall conductance from the bulk layers.  This indicates that the system in Fig. <ref>(a) realizes a quantum anomalous Hall effect with a higher Hall conductance plateau, while the system in Fig. <ref>(b) realizes a quantized layer Hall effect.






§ CONCLUSION

In this study, we investigate the magnetic effects on films of WTIs. Our findings demonstrate that the interplay between magnetism and WTIs gives rise to a diverse range of anomalous Hall effects, including the metallic quantum anomalous Hall effect without chiral edge states, the quantum anomalous Hall effect with a higher Hall conductance plateau, the quantized layer Hall effect, the metallic half-quantized valley-like Hall effect, and the quantized valley-like Hall effect. Notably, the quantized layer Hall effect, metallic half-quantized valley-like Hall effect, and quantized valley-like Hall effect are not observed in magnetic STIs.


D.-H.X. was supported by the NSFC (under Grant Nos. 12074108, 12474151 and 12347101), the Natural Science Foundation of Chongqing (Grant No. CSTB2022NSCQ-MSX0568). R.C. acknowledges the support of the NSFC (under Grant No. 12304195) and the Chutian Scholars Program in Hubei Province.  B.Z. was supported by the NSFC (under Grant No. 12074107), the program of outstanding young and middle-aged scientific and technological innovation team of colleges and universities in Hubei Province (under Grant No. T2020001) and the innovation group project of the natural science foundation of Hubei Province of China (under Grant No. 2022CFA012).



Note-added.—
Recently, we became aware of a complementary study, which focus on a two-band two-dimensional topological semimetal, where the Brillouin zone is divided into two patches characterized by half-quantized Berry curvature fluxes with opposite signs due to the presence of the C_4z T magnetic symmetry <cit.>.






§ HALL CONDUCTANCE


In the calculations, the Hall conductances in Eqs. (<ref>)-(<ref>) are calculated from the following expression <cit.>:

    σ_xy(k_x,k_y,z)=-4 π e^2/N_kShIm∑_v v^' c  X_v c  Y_v^' c ^†ρ_v v^'(z),

which corresponds to the contribution to the total Hall conductance σ_xy from the z-th layer of a certain (k_x,k_y) point. The matrix element for the position operator along the x or y directions, is denoted as X(Y)_v c =⟨ψ_v |x(y)| ψ_c ⟩= ⟨ψ_v |i ħ v_x(v_y)| ψ_c ⟩/E_c -E_v, which is related to the energy difference between the conduction and valence bands E_c -E_v. The indices v and c represent the valence and conduction bands. ρ_v v^'(z) is the projection matrix on to the corresponding z-th layer, which implies a summation over all orbitals v, v^', c belonging to that layer. N_k represents the number of k-points and S represents the unit cell area.










§ LONGITUDINAL CONDUCTANCE


The longitudinal conductance is obtained by using the Kubo-Greenwood formula expressed in terms of the Chebyshev Polynomials <cit.>. In zero temperature, it can be expressed as <cit.>

    σ_xx(Ẽ_F)=4 e^2 ħ/π S∑_m, n ≤ mμ_n m^xx T_n(Ẽ_F) T_m(Ẽ_F),

where

    μ_m n^xx≡g_m g_n/(1+δ_n 0)(1+δ_m 0)Tr[v_x T_m(H̃) v_x T_n(H̃)],

and

    g_m^J=(M-m+1) cosπ m/M+1+sinπ m/M+1π/M+1/M+1

is the Jackson Kernel, T_m(x)=cos(marccos x) is the Chebyshev polynomials of the first kind. H̃ is the rescaled Hamiltonian so that its eigenvalues Ẽ_F is contained in the interval [-1,1].

In the numerical calculation for the longitudinal conductance, the system size is taken as 400× 400, the moment M=1000, the disorder strength W=0.1 and the results are obtained after averaging on 20 independent disorder configurations.






§ SPECTRUM WITH OPEN BOUNDARY CONDITIONS ALONG THE Y- AND Z-DIRECTIONS


In addition, we plot the energy spectrum of the six systems, with periodic boundary condition along the x-direction and open boundary conditions along the y- and z-directions. For Case (i) shown in Fig. <ref>(a), there is no chiral current due to the absence of magnetism. The systems in Cases (ii) and (iii) break the T symmetry, which allow chiral current propagating along the same direction near k_x=0 and k_x=π points. In Case (iv), each band is double degenerate and there is no net chiral current for the degenerate bands due to the PT symmetry. In Cases (v) and (vi), there appear chiral currents propagating along the opposite directions k_x=0 and k_x=π points, which results in a vanishing net chiral current.

Furthermore, it is noticed that the energy gaps between the edge states in Figs. <ref>(b) and <ref>(e) are attributed to the quantum confinement effect. For a slab system with a large enough n_y, the system corresponds to a metal due to the gapless Dirac cone on the bottom surface layers. The situation is different from the case in Figs. <ref>(c) and <ref>(f), with an insulating bulk even in the large n_y limit.

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