Spectral signatures of the surface anomalous Hall effect in magnetic axion insulators

The topological surface states of magnetic topological systems, such as Weyl semimetals and axion insulators, are associated with unconventional transport properties such as nonzero or half-quantized surface anomalous Hall effect. Here we study the surface anomalous Hall effect and its spectral signatures in different magnetic topological phases using both model Hamiltonian and first-principles calculations. We demonstrate that by tailoring the magnetization and interlayer electron hopping, a rich three-dimensional topological phase diagram can be established, including three types of topologically distinct insulating phases bridged by Weyl semimetals, and can be directly mapped to realistic materials such as MnBi2Te4/(Bi2Te3)n systems. Among them, we find that the surface anomalous Hall conductivity in the axion-insulator phase is a well-localized quantity either saturated at or oscillating around e2/2h, depending on the magnetic homogeneity. We also discuss the resultant chiral hinge modes embedded inside the side surface bands as the potential experimental signatures for transport measurements. Our study is a significant step forward towards the direct realization of the long-sought axion insulators in realistic material systems.


Supplementary Note 1. k·p model of the multilayer topological heterostructure
In the absence of inter-bilayer hopping and exchange, the multilayer model Eq. (1) in the main text reduces to a -independent form where = Δ − ∥ 2 . Since the inter-bilayer hopping terms couple the first and last block in the present space, only the effective interactions between these two blocks are essential while considering the inter-bilayer hopping. We take the eigenvector of intra as = ( 1 , 2 , 3 , 4 ) . The Schrodinger's equation of intra reads The eigen energies are solved as ±, = ±� 2 − ℏ 2 2 where Λ = �( 0 ) 2 + 4 2 − 8ℏ 2 2 ∥ 2 and = 1,2. We choose the energy pair with = 1 to capture the property of phase transition since the other pair with = 2 never close the gap. Therefore, the effective Hamiltonian in the basis of ( 1 , 4 ) is reached with the effective mass term ,eff = sgn( 0 )(| 0 | − Λ )/2. Expanding ,eff up to the square of ∥ and adding magnetic exchange, we find that those two bands of the bilayer take the form of Bernevig-Hughes-Zhang model Here Δ eff ( 0 ) = sgn( 0 )(| 0 | − Λ =0 )/2 and eff ( 0 ) = −2sgn( 0 )�ℏ 2 2 + Δ �/ Λ =0 , in which Δ eff / eff > 0 demarcates the no trivial phase in this quasi-2D bilayer even if Δ/ < 0.
Afterwards we turn on the inter-bilayer hopping with periodic boundary on z direction. One directly finds the dispersions Critical boundaries of distinct phases in Fig. 1

Supplementary Note 2. The fragile topology of the 3D -broken QSH phase
The parity distribution at inversion-invariant k-points of the 3D -broken QSH insulator can be viewed as nearly isolated layer stacking of 2D T-broken QSH insulators. Similar to the other phases (axion insulator, Weyl semimetal, and 3D Chern insulator), the 3D -broken QSH phase is unable to decompose into the symmetric Wannier functions. However, such a Wannier obstruction for the QSH phase can be removed by adding a set of trivial elementary band representations (EBRs).
To check the fragile nature of the 3D -broken QSH phase, we construct the EBRs, i.e.,

Supplementary Note 3. Electronic structures of different topological phases in MnBi2Te4
The

Supplementary Note 4. Locality of the half-quantized surface AHC
To further demonstrate that the half-quantized AHC of an axion insulator is a local property at the gapped surface, we consider a 16-layer slab of FM MnBi4Te7, which is insulating for MnBi2Te4 termination but metallic for Bi2Te3 termination [see Supplementary In addition, we also build symmetric slabs with (4 + 1) VdW layers which both surfaces terminated by the Bi2Te3 to check whether half quantization can be realized in the Bi2Te3-termination. Since there is no net magnetization of these slabs, the total Chern number should be zero. DFT calculation proves that such a slab has a global gap, thus the layered Chern numbers for the two surfaces are well defined and should be symmetrically canceled out. Supplementary Fig. 3(c,d) shows the geometry of these slabs, electronic band structure and the resulting layered Chern number.
The above results show the locality of the half-quantized surface AHC.

Supplementary Note 5. Calculations of the surface state and chiral hinge state
The surface state is computed by iterative Green's function implemented in WannierTools package 4 . The spectral function, i.e. imaginary part of the Green's function is then plotted in Fig. 3 and 4 in the main text.
For the hinge states, two methods have been developed and the results are cross-checked.
The first method directly computes the spectral function for the hinge along a particular direction (in our case, along y). x and z directions are semi-infinite. The bi-semi-infinite geometry of the system is plotted in Supplementary Fig. 5.
An approximation is then made that The first method described above gives the spectral function right at the hinge. In order to investigate the spectral function away from the hinge, a slab that is finite in the x-direction, semi-infinite in the z-direction and infinite in the y-direction is used, plotted in Supplementary   Fig. 5(a). In this sense, is a well-defined quantum number. The thickness convergence along x has been tested and we find that 20-unit-cell is thick enough to decouple the hinge state and the top surface state. The spectral functions at the two hinges and the center of the top surface, denoted as B, C and D in Supplementary Fig. 5(a) In addition, the opposite Berry curvature Ω ( ) = ∇ × ⟨ |∇ | ⟩ of bands above and below the gap shows the topological property of both the magnetization-and hybridizationinduced gap, as displayed in Supplementary Fig. 9.