Bulk-Hinge Correspondence and Three-Dimensional Quantum Anomalous Hall Effect in Second-Order Topological Insulators

PHYSICAL REVIEW RESEARCH 3, 033177 (2021)

Bulk-hinge correspondence and three-dimensional quantum anomalous Hall effect in second-order topological insulators

Bo Fu ,Zi-AngHu , and Shun-Qing Shen * Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China

(Received 3 February 2021; revised 29 April 2021; accepted 7 July 2021; published 20 August 2021)

The chiral hinge modes are the key feature of a second-order topological insulator in three dimensions. Here we propose a quadrupole index in combination of a slab Chern number in the bulk to characterize the ﬂowing pattern of chiral hinge modes along the hinges at the intersection of the surfaces of a sample. We further utilize the topological ﬁeld theory to present a picture of three-dimensional quantum anomalous Hall effect as a consequence of chiral hinge modes. The two bulk topological invariants can be measured in electric transport and magneto-optical experiments. In this way we establish the bulk-hinge correspondence in a three-dimensional second-order topological insulator.

DOI: 10.1103/PhysRevResearch.3.033177

I. INTRODUCTION In the present work, we address the bulk-hinge correspondence and three-dimensional (3D) QAHE as a physical The bulk-boundary correspondence lies at the heart of consequence of the CHMs in a second-order topological topological states of matter and topological materials [1–4]. It insulator. We start with a minimal four-band model to re- bridges the topology of bulk band structures and the physical veal different flowing patterns of CHMs. It is found that a observables near the boundary. In the quantum Hall effect quadrupole index is associated with the flowing direction of and quantum anomalous Hall effect (QAHE), the quantized four hinge modes of the system along one direction and a slab Hall conductance is associated with the TKNN number of the quantized Hall conductance reveals the formation of a closed band structure and the number of the edge modes of electrons loop of the CHMs. We further demonstrate the correspondent around the boundary [5–8]. In a topological insulator, a Z 2 connection of the CHMs to the quadrupole index and the slab index in the bulk is associated with the number of the gap- Chern number by means of topological field theory. Finally less Dirac cones of the surface electrons [9–11]. This reflects we propose to utilize magneto-optical Faraday and Kerr ef- intrinsic attributes of the topological phenomena. A recent fects to detect these topological invariants. advance in the field of topological materials is the discovery of higher-order topological insulators [12–20]. A second-order topological insulator in three dimensions refers to an insu- II. MODEL HAMILTONIAN AND SYMMETRY ANALYSIS lator with one-dimensional the chiral hinge modes (CHMs) We start with a minimal four-band Hamiltonian, localized on the hinges at the intersection of adjacent side 3 surfaces [15–27]. Over the past few years, a great of efforts H = H + V , (1) have been made to explore the possible relation of the bulk 0 i = bands and existence of hinge modes as an extension of the i 1 bulk-boundary correspondence, such as effective mass anal- which consists of four parts (Appendix A). The primary part ysis [18–23], the symmetry indicator [28–36], and spectral is flow analysis [37]. All the approaches have their own merits. H0 =h¯σx[v⊥(kxsx + kysy ) + vzkzsz] However, CHMs in a second-order topological insulator may + + 2 + 2 + 2 σ , display various flowing patterns as illustrated in Fig. 1. It lacks m0 m⊥ kx ky mzkz zs0 (2) a systematic method to provide a comprehensive description where k , k , k are the wave vectors, and m and v are the of diverse flowing patterns. Also it is desirable to learn which x y z i i model parameters. We consider the basis functions (|+ ↑, observable in the bulk is associated with the CHMs. |+ ↓, |− ↑, |− ↓) with ± being the parity eigenvalues and ↑ (↓) denoting the spin-up (-down) state, thus si and σi (i = x, y, z) are the Pauli matrices and s0 and σ0 the identity matrices acting in spin and orbital space, respectively. The H *[email protected] point group of 0 is D4h, which can be generated by a fourfold Rz Rx rotation 4 around z axis, a twofold rotation 2 around x axis Published by the American Physical Society under the terms of the and the spatial inversion I. H0 possesses the time reversal 2 Creative Commons Attribution 4.0 International license. Further symmetry T = isyK(T =−1, and K is complex conjugate) distribution of this work must maintain attribution to the author(s) and belongs to the symplectic symmetry class AII. Here we and the published article’s title, journal citation, and DOI. focus on the case of both m0m⊥ < 0 and m0mz < 0 such that

2643-1564/2021/3(3)/033177(14) 033177-1 Published by the American Physical Society BO FU, ZI-ANG HU, AND SHUN-QING SHEN PHYSICAL REVIEW RESEARCH 3, 033177 (2021)

surface states parallel with the z axis induced by V1,itonly modifies its value and has no influence on the four hinge states along the z direction. When all the surface states are gapped out and the Fermi level is located in the surface band gap, the electrons can only propagate unidirectionally along the hinges shared by adjacent side surfaces due to time reversal symmetry breaking. The presence of the antiunitary symmetry IT imposes the strong restriction that there must be another FIG. 1. Illustration of selected patterns of chiral hinge modes hinge state propagating in the same direction on its spatial and their projection in a second-order topological insulator in three inversion if one CHMs propagates along any hinge. Thus, the dimensions. (a) 3D antiferromagnetic quantum anomalous Hall in- CHMs may form two closed loops on the surfaces (100) and ¯ V sulator: A double-loop pattern with the quadrupole indices xy = (100) as shown in Fig. 1(a). The relative sign between c in 1 V −zx = 1andyz = 0 and the slab Chern number nx = ny = nz = and d in 2 will determine which surface the two hinge mode 0. (b) 3D rotoinversion quantum anomalous Hall insulator: A single- loops locate around. loop pattern with xy = 1andyz = zx = 0andnx = ny = 0 (iii) 3D rotoinversion QAHI: HRQAHI = H0 + V1 + V3 and nz =−1. (c) 3D inversion quantum anomalous Hall insulator: with V3 = bzsz. The inclusion of V3 with magnetic field in z = = = = = A single-loop pattern with xy yz zx 0andnx ny direction reduces the magnetic group to G2 = S4 ⊕ T (D2d − − = nz 1. S4 ). The ferromagnetic term bzsz behaves similar to the antiferromagnetic term V2 but open gaps with different signs for the surface Dirac electrons of the 3D CHOTI on the (001) H0 describes a 3D strong topological insulator with gapless and (001)¯ surfaces. As a consequence, the chiral hinge modes Dirac cone of the surface states at all surfaces [4,38]. H0 also of 3D rotoinversion QAHI will exhibit a distinctly different respects the global chiral symmetry C = σys0, i.e., {C, H0}= pattern compared with 3D antiferromagnetic QAHI. The pres- S = IRz 0. Including the crystalline symmetries, the total point sym- ence of the improper rotation symmetry 4 4 protects metry group is G0 = D4h ×{1, T , P, C} with the particle-hole a single-loop CHMs wriggling around the bulk as shown in symmetry P ≡ CT −1 [39]. As shown below all the terms in H Fig. 1(b). The relative sign between b and c determines the preserve P, it is more convenient to rewrite G0 as G0 = G0 × wriggling way of the single-loop CHMs. { , P} G = ×{ , T }= ⊕ H = H + V V = 1 with the magnetic group 0 D4h 1 D4h (iv) 3D inversion QAHI: IQAHI 0 3 with 3 T / V D4h (or 4 mm1 ). The three additional terms breaks different b i=x,y,z si. Only in the presence of 3 that magnetic field V = 2 − 2 σ T = = = symmetries, respectively. 1 c(kx ky ) ys0 breaks both points to (111) direction, i.e., bx by bz b, the magnetic Rz Rz T and 4 individually, but respects their combination 4 . point group is G3 = Ci ⊕ T (C2h − Ci ). Due to the presence of V = σ 2 d ys0 is an antiferromagnetic term which breaks both the inversion symmetry I, the CHMs at the inversion symmet- T I IT V = and , but respects the combination . 3 i=x,y,z bisi ric hinges are propagating in the opposite directions, and form is the magnetic Zeeman interaction or ferromagnetic term a closed loop as shown in Fig. 1(c). which breaks T but preserves I. The group G0 is a Heesch- Here, we only take several higher-order topological phases Schubnikov magnetic group of type II; the inclusion of any the with distinct flowing patterns of the hinge currents as ex- term in Vi breaks time reversal symmetry and several spatial amples to illustrate the bulk-hinge correspondence. All the symmetries and reduces it to a magnetic group of type III. In possible patterns of hinge currents can be constructed by the general, the reduced magnetic point group can be expressed as strategy in Appendix C and the relationship between different [39] Gi = N ⊕ T (G − N), where G is subgroup of D4h and N patterns and the topological invariants are also checked. is a halving subgroup of G. Combination of the three terms Vi may generate higher-order topological phases with different III. THE QUADRUPOLE INDEX symmetries (Appendix B). (i) 3D chiral higher-order topological insulator (CHOTI): To characterize the topological hinge modes, we introduce HCHOTI = H0 + V1. The presence of V1 reduces the magnetic a quadrupole index as topological invariant. There are the group to G = D2d ⊕ T (D4h − D2d ). The term opens an gap CHMs along four hinges in the z direction in the case of with opposite sign for the surface states on the neighboring Figs. 1(a) and 1(b). The energy dispersions of the four hinge surfaces parallel to z axis and the CHMs may be localized modes connect the conduction and valence bands, and cross at their intersections. The CHMs are protected by the com- at kz = 0(seeFig.4 in Appendix A). For a specific kz, H(kz ) bination of fourfold rotational symmetry and time-reversal can be viewed as a 2D system in the x-y plane and there Rz T ¯ symmetry 4 . The surface states on the bottom (001) and are four corner states. The existence of corner states can be top (001) surface remain gapless. characterized by the quadrupole moment [40–43], (ii) 3D antiferromagnetic quantum anomalous Hall insu- 1 † † lator (QAHI): HAQAHI = H0 + V1 + V2. The inclusion of V2 qxy(kz ) = Im log det[U QxyUk ] det Qxy , (3) π kz z in the Hamiltonian of CHOTI further reduces the magnetic 2 G = ⊕ T − V group to 1 D2 (D2h D2 ). The 2 term anticommutes where the matrix Ukz is constructed by the occupied lowest 2πirˆx rˆy/Lx Ly with the linear termsh ¯v⊥(kxσxsx + kyσxsy ) along x and y di- energy states, Qxy = e ,ˆrα are the position operators, rections. Thus, it acts as the mass terms and gap out the surface and Lα are the lengths of the system in the α direction. ¯ H = states on (001) and (001) while being projected onto the x-y For the occupied states, we have (kz )Ukz Ukz Eocc(kz ), V σ s with E (k ) = diag[ (k ), (k ), ... (k )]. and |un surface. Since 2 commutes with the mass term y 0 for the occ z 1 z 2 z Nocc z n kz

033177-2 BULK-HINGE CORRESPONDENCE AND THREE- … PHYSICAL REVIEW RESEARCH 3, 033177 (2021)

H | n = are the energy eigenvalue and eigenstate, (kz ) uk with the periodic boundary condition along the x and y di- n z ε (k )|u . U is constructed by the occupied states U = rection. Denote the Bloch eigenstates by |u (k⊥, z) are the n z kz kz kz n N H , | , =ε | , (|u1 , ...|u occ ), which has dimension of N × N . N = Bloch eigenstates, (k⊥ z) un(k⊥ z) n(k⊥) un(k⊥ z) kz kz tot occ tot with k⊥ = (k , k ) and the index n for the bands. The Lx × Ly × Norb denotes the dimension of total Hilbert space x y space-resolved Berry connection is given by Aα;n,n (k⊥, z) = with Norb being the “orbitals” per site and Nocc denotes the − , |∂ | , , number of occupied bands. We hence build the the projection i un(k⊥ z) α un (k⊥ z) for the two occupied bands n n . † In this way we define the slab Hall conductance and its rela- operator into the occupied energy bands Pocc = UU and we † also have relation U U = 1 × . The determinant can be tion to a slab Chern number nz [36], Nocc Nocc reformed as Lz e2 σ slab = σ = , † † xy dz xy(z) nz (8) = × + − det Uk QUkz det 1Nocc Nocc Uk (Q 1)Ukz (4) 0 h z z e2 2 Since the Hilbert space of the Hamiltonian (occupied and σ = ⊥ F ⊥, where xy(z) 2πh d k Tr[ xy(k z)] and unoccupied energy bands included) at each kz is complete, we † † F = ∂ A − ∂ A + A , A have the relation V V + U U = 1 where we denote V as αβ (k) α β (k) β α (k) i[ α (k) β (k)] (9) kz kz kz kz kz the eigenstates for unoccupied band. By using the Sylvester’s is the non-Abelian Berry curvature in terms of Aα n,n (k⊥, z). determinant identity, we have ; Because of the periodicity of the Berry connection in the first † † † det U QUk = det V Q Vk det[Q]. (5) Brillouin zone, it can be proved that the slab Chern number nz kz z kz z is quantized if the filled bands has a band gap to the excited We then focus on the anti-symmetries (e.g., the chiral symme- states for a band insulator. According to the bulk-boundary try C or particle-hole symmetry P) because these symmetries correspondence [8], each nonzero Chern number is associated will relate the occupied states to the unoccupied states which with the closed loop of chiral edge state. In Fig. 1(a),for is essential in the quantization of the quadrupole moment [42]. antiferromagnetic QAHI we have n = n = n = 0, while O x y z Under the antisymmetry a, the Hamiltonian obeys two quadrupole indices are not vanishing xy =−zx = 1. O H(k )O−1 =−H(χk ), (6) The system in a slab geometry (the open boundary condition a z a z is imposed in the y direction) is analog to the quantum spin with χ =−1 denotes the operation Oa will inverse the mo- Hall insulator except the the two counter-propagating hinges mentum kz and χ =+1 means that Oa keeps kz unchanged. modes are localized on the opposite sides. Experimentally, the Following the same procedure in Ref. [42], it is found that an quantized anomalous Hall effect can be measured by using the O O H O−1 = antisymmetry a leaves xy plane invariant a (kz ) a surface-sensitive method [44]. In Fig. 1(b), for rotoinversion −H(−kz ) which will impose a constraint on the quadrupole QAHI we have nz =−1 and nx = ny = 0. There is a closed moment qxy(kz ): qxy(kz ) + qxy(−kz ) = 0 or 1. At two high loop of chiral edge mode around the z axis. Combined with symmetry points z = 0orπ, the symmetry is restored, the nonzero quadrupole index xy = 1. there are four CHMs O H O−1 =−H a ( z ) a ( z ), and qxy( z ) must be quantized along the four hinges along the z axis, the two indices can 1 determine that a single-loop of CHMs that wriggles around the to0or 2 . Nonzero quantized qxy( z ) indicates the system topologically nontrivial and the existence of four zero-energy bulk. QAHE can be detected through a global quantum Hall corner states in the reduced 2D subspace. For example, if measurement probing the whole sample due to the nonzero nz. = =− = qxy(kz = 0) = 1/2, then qxy(±π ) = 0 or 1. In this case, there In Fig. 1(c), for inversion QAHI we have nx ny nz 1. exist CHMs which compensate for the difference of the corner There is a single loop of chiral edge mode around each axis. charges. Thus, we can introduce a quadrupole index, Because of the zero quadrupole indices around the three axes, there is no four CHMs along one direction. It exhibits a single 2π = ∂ , CHM traversing half of its hinges, which can be projected out xy dkz kz qxy(kz ) (7) 0 a single closed loop in the direction of x, y, and z. The QAHE to characterize the existence and the flowing direction of four can be observed for three directions due to the nonvanishing CHMs. For antiferromagnetic QAHI in Fig. 1(a),wehave slab Chern numbers. xy =−zx = 1 and yz = 0, which are protected by the combination of chiral symmetry and the mirror symmetry V. 3D QUANTUM ANOMALOUS HALL EFFECT CMα and the combination of chiral symmetry and the time The CHMs can be further understood in the framework of reversal symmetry CT . For rotoinversion QAHI in Fig. 1(b), = = = topological field theory with an effective action [45], we have xy 1 and yz zx 0. The quadrupole index along the z direction is protected only by CT and along the x 1 1 θ(r, t )e2 CM CT S = d3rdt E2 − B2 + E · B , (10) (y) is protected by both x(y) and . For inversion QAHI 8π μ 4π 2hc¯ in Fig. 1(c), xy = yz = zx = 0. where E and B are the electromagnetic fields, and and μ are the dielectric constant and magnetic permeability, respec- IV. THE SLAB CHERN NUMBER tively. θ(r, t ) is known as the axion angle [46]. The product The slab Chern number is another topological invariant as E · B is odd under the time reversal or spatial inversion, θ has the quadrupole index alone are not enough to characterize to be 0 (modulo 2π) for a trivial insulator and the vacuum and the diversity of the flowing pattern of the CHMs. Consider π for a topological insulator with respect to the symmetries a slab geometry of the sample with a finite thickness Lz [47,48]. In the quadratic order of electric and magnetic fields,

033177-3 BO FU, ZI-ANG HU, AND SHUN-QING SHEN PHYSICAL REVIEW RESEARCH 3, 033177 (2021) besides the Maxwell term, the θ term may give rise to the The first term depends on the temporal gradient of the θ-field topologically magnetoelectric effect that an electric field can and is proportional to magnetic field, i.e., the so-called chiral induce a magnetic field and vice verse [45,49–53]. By taking magnetic field, and vanishes in a static limit. The second term the functional derivative of θ term with respect to a gauge depends spatial gradient of the θ field and is perpendicular to field, the induced electric current density depends on the spa- the electric field, i.e., the anomalous Hall effect. Thus, there tial and temporal gradients of the θ-field [45,46], will be surface anomalous Hall effect at the interface between two regions with different θ values and no Hall response will e2 jθ (r, t ) = [∂ θ(r, t )B − ∇θ(r, t ) × E]. (11) exist in the bulk as θ takes a constant value θb [54,55]. 2πh t

The value of θb is given by the three-dimensional integration of the Chern-Simons 3-form over momentum space [56,57], 3 θb d k i = αβγ tr Fαβ (k)Aγ (k) − [Aα (k), Aβ (k)]Aγ (k) . (12) 2π 16π 2 3

Consider a Bloch Hamiltonian H(k) which is invariant under O, the eigenstates of H(k)atk and DOk must be related by a gauge transformation, where DO is an operator transforming k to DOk in momentum space with the Jacobian matrix as Jab = ∂(DOk)b/∂ka. We can prove that only those symmetries with Jacobian determinant det J =−1 will quantize the axion angle. There are two types symmetries satisfying this condition: the improper rotational symmetries and the combined symmetries from time reversal symmetry and the proper rotation symmetry, which will put the following constraints on axion angle θb, respectively, θ 3 b d k ∗ T ∗ T ∗ T 2 = αβγ tr[(U (k)∂αU (k))(U (k)∂βU (k))(U (k)∂γ U (k))], (13) 2π 24π 2 and θ 3 b d k † † † 2 = αβγ tr[(U (k)∂αU (k))(U (k)∂βU (k))(U (k)∂γ U (k))]. (14) 2π 24π 2 ∗ and T represents the complex conjugate and transpose, respectively, and U (k) is the unitary transformation matrix acting on O| = | the space of occupied bands, un(k) m Unm(k) um(DOk) with n as the occupied band index. U (k) is periodic in kx, ky, and kz, and thus defines a map from a 3-torus to the space of Nocc × Nocc unitary matrices. Such maps are classified by an integer θ b ∈ ∈ topological invariant or winding number which is given by the righthand side of the above equations, 2 2π Z.IfU (k) U (1), then the terms in the brackets can be interchanged, considering the antisymmetric form of the winding number, the axion angle θb always trivially vanishes. Therefore to yield a nontrivial axion angle, the occupied bands need to be degenerate such that the gauge transformation U (k) becomes non-Abelian. (i) For 3D antiferromagnetic QAHI (b = 0, c = 0, and d = 0), the all five 4 × 4 matrices in H anticommute. The eigenvalues for two valance bands are easily obtained as εv =− 2 + 2 + + 2 − 2 2, ±(k) p(k) M(k) d c kx ky (15) = + 2 + 2 + 2 = , , where M(k) m0 m⊥(kx ky ) mzkz and p(k) (v⊥kx v⊥ky vzkz ). The two bands are doubly degenerate for the whole Brillouin zone which is guaranteed by the combined symmetry TI (with (IT )2 =−1). (ii) For the3D rotoinversion QAHI (b = (0, 0, bz ), c = 0 and d = 0), the eigenvalues for two valance bands are εv =− 2 + 2 + 2 ± 2 2 2 + 2 + 2 2 − 2 2 . ±(k) bz p(k) M(k) 2 bz vz kz M(k) c kx ky (16) The band degeneracy occurs only for topological nontrivial case and at the four momenta (± − m0 , ± − m0 ). (iii) For the3D 2m⊥ 2m⊥ inversion QAHI (b = (bx, by, bz ) and c = d = 0), the eigenvalues for two valance bands are explicitly written as v 2 2 2 2 2 2 ε±(k) =− b + p(k) + M(k) ± 2 [b · p(k)] + b M(k) . (17)

The band degeneracy occurs at the momenta which satisfy tized and nontrivial. While for topological trivial insulators both the conditions M(k) = 0 and b · p(k) = 0. The ﬁrst con- (m0m⊥, m0mz > 0), the band degeneracy is removed for all dition requires band topology is nontrivial (m0m⊥, m0mz < the Brillouin zone the axion angle must trivially vanish 0), and restricts the momenta to a 2D ellipsoidal momentum in this situation. We want to emphasize the band degen- surface. The second condition further restrict the momenta eracy for these two cases is not due to the symmetry but to the line nodes perpendicular to the vector b.Forthe the band topology. Therefore, the band degeneracy is nec- case (i), the axion angle is not quantized and can take any essary but not sufﬁcient for the quantized nontrivial axion value. For the cases (ii) and (iii), the axion angle are quan- angle.

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In addition to the inversion or time-reversal symmetry, θb will be quantized with improper rotation symmetries or a combination of time-reversal symmetry and proper rotation symmetries [58]. From Eq. (11), the layer-resolved Hall conductivities in the xy plane is associated with the gradient of θ, σ = e2 ∂ θ xy(z) 2πh z (z). Thus, the slab Hall conductance Eq. (8) is given by the difference of the θ values of bottom and top vacuum, ∞ e2 θ − θ σ slab = σ = T B , xy dz xy(z) (18) −∞ h 2π which is integerquantized independent of the θ value of the bulk. The gradient ∇θ(r) is no vanishing at the interface between the two homogeneous materials with different θ values. Thus, the surface Hall conductances for the top and bottom interfaces are given by e2 θ − θ σ T = T b , xy h 2π e2 θ − θ σ B = b B . (19) xy h 2π Apparently, the total slab Hall conductance is given by the σ slab = σ T + σ B . summation of surface Hall conductances xy xy xy Since the θ value for the top and bottom vacuum can only T/B T/B FIG. 2. (a) Schematic view of the hinge current. The planar take the values θT/B = 2πn with n being integer, for θ surfaces of the topological insulator are characterized by integers later convenience, we introduce the difference of the values n and n , describing the integer change of the θ value nearby the / 1 2 between the top bottom vacuum and the bulk of the material surfaces. (b) Schematic view of θ-term as a function of the angle as ϕ for topologically nontrivial and trivial cases. (c, d) Plots of the T/B T/B layer-resolved Hall response σαβ (rγ ) and panels (e, f) are plots of 2πn˜ = 2πn − θb. (20) z z the θ-angle as function of the layer index for antiferromagnetic and As a result, the surface Hall conductance for the top and rotoinversion quantum anomalous Hall insulator, respectively, from σ T = e2 T σ B = a layer-resolved Kubo formula in a slab geometry for 10 layers. bottom interfaces can be expressed as xy h n˜z and xy − e2 B h n˜z , respectively. not quantized, the above argument for the gapless chiral hinge θ VI. RELATION BETWEEN THE TERM AND THE channel is still valid. CHIRAL HINGE MODES As show in Figs. 2(c)–2(f), we plot the layer-resolved Hall The current carried by the CHMs can be evaluated from responses σαβ (rγ )(αβγ = 1) and the integrated value for the spatial dependent θ, and each chiral hinge channel carries θ(rγ ) as a function of the layer index for three directions. In one conductance quantum (e2/h). We calculate the current numerical evaluation, we consider a slab geometry with the αβ through a 2D section disk (D) encircling a hinge normal periodic boundary in the plane and open boundary con- = · dition in rγ direction. The layer resolved Hall response only to the plane as illustrated in Fig. 2(a), I D dS jθ .The electric ﬁeld is determined by the gradient of a scalar po- distributes near the slab surfaces where θ changes and quickly tential, E =−∇(r), and we choose the boundary of the drops to zero as the position moves into the bulk where θ takes constant value. For antiferromagnetic QAHI in Figs. 2(c) and disk as an equipotential line e. By utilizing Stokes theorem, 2 = e · ∇θ 2(e), the magnetic point group G1 will put a constraint on the I 2πh C ds (r) (r). Thus, there is no current or equiv- alently gapless conducting channel on the hinge when θ(r)in Hall response that the layer-resolved Hall conductivity takes the opposite values for the slab center. Thus, the slab Chern the two vacuum areas takes the same value n1 = n2.Iftheyare numbers vanish for three directions. Due to the presence of different n1 = n2, then there will be a branch cut separating the two vacuums where θ(r) is singular. In this situation, the the mass term d, the axion angle will deviate form the quan- π θ / π − . contour integral gives the number of the conducting channels, tized value , for example, b 2 0 59 in Fig. 2(e).It is also consistent with the symmetry analysis that there is no I θ G = n − n = n˜ − n˜ , (21) such symmetry to guarantee the quantization b in 1.Asa 2/ 2 1 2 1 σ B e h consequence, the surface Hall conductance xy for the bottom σ T which is the winding number of the ﬁeld θ(r). = e − interface and xy for the top interface are not half quantized in in denotes the potential difference between the outer contour sharp contrast to the axion insulators. However, the summa- Cout and the inner contour Cint. In other words, the gapless tion of the surface Hall conductance of the adjacent surface θ σ i + σ j = e2 i − j , = hinge mode tracks the singularity of the term and vice versa. must be quantized since zx zy h (ny nx ) with i j We also want to emphasize that, even when θ in the bulk is T, B, indicates whether the hinge mode at the intersection

033177-5 BO FU, ZI-ANG HU, AND SHUN-QING SHEN PHYSICAL REVIEW RESEARCH 3, 033177 (2021)

reflection and transmission tensors, and can be solved by matching the electrodynamic boundary conditions. The Kerr θ =− y/ x and Faraday angles are defined by the tan K Er Er and θ = y/ x tan F Et Et , respectively [49,50]. When the chemical potential is located within the surface gap Eg andh ¯ω Eg, the magnetic fields at the interface of the two materials are discontinuous due to the presence of surface Hall current. The reflection and transmission tensors for a slab can be obtained FIG. 3. (a) Schematic illustration of the measurement of Kerr and by composing the single-interface scattering matrices for top Faraday angle. Incident linearly polarized light becomes elliptically and bottom surfaces. For simplicity we only consider a free- polarized after transmission (Faraday effect) and reflection (Kerr standing sample, the influence of a substrate do not change our conclusion qualitatively. The reflectivity R ≡|E |2/|E |2 effect), with polarization angles as θF and θK , respectively. (b) The r t reflectivity R as a function of the slab thickness Lz along z direction will depend on the relative magnitude of the slab thickness λ = 2√πc in the units of half of photon wavelength λb/2 for suspended single and the wavelength ( b ω μ ) inside the bulk. As shown in loop case with = 10 and μ = 1. Fig. 3(b), when the slab thickness contains an integer multiple of half wavelength Lz = Nλb/2 with an integer N (the reso- of two surfaces exists or not. For rotoinversion QAHI in nance condition), R reaches the minima. At the resonance, the Fig. 2(d), the symmetry G constrains that the layer-resolved reflectivity can be obtained as 2 2 Hall conductivities for z direction are symmetric about the α n˜T − n˜B slab center, while for x and y directions are antisymmetric R = z z . (24) + α T − B 2 about the slab center. The layer-resolved Hall conductivities 1 n˜z n˜z in xz plane and yz plane are also related to each other by the If the slab Chern number vanishesn ˜T = n˜B = 0, then the light S symmetry. Furthermore, as shown in Fig. 2(f), θ will be z z 4 b can propagate through the sample without reflection and the quantized due to the presence of improper rotation symmetry S transmitted light is still linearly polarized in x direction. In this 4 and a combination of time-reversal and the diagonal mirror θ = TMx+y situation, the Faraday angle F 0. If the slab Chern number symmetry . As a result, the surface Hall conductance θ does not vanish, then the Faraday angle F will be quantized are half-quantized for three directions. More detailed discus- 2 in integer multiples of the fine structure constant α ≡ 1 e sion symmetry constraints on Hall response can be found in 4π0 hc¯ [51], Appendix D. In this way, we establish the relation between the the CHMs and the two physical invariants, θ = α T − B , tan F n˜z n˜z (25) 2 slab e T B which is universal and independent of the specific value of θ . σ = αβγ n˜ − n˜ , (22) b αβ h γ γ As α ≈ 1/137, θ ≈ α(ñT − n˜B ). The slab Hall conductance F z z T T Eq. (22) can thus be expressed in terms of the fundamental αβ = δ T − B, δ T − B, n˜α − n˜β , (23) n˜β n˜β 0 n˜α n˜α 0 constants and the experimentally measurable quantity = with αβγ 1. e2 e2 σ slab = tan θ = n˜T − n˜B . (26) xy αh F h z z VII. MAGNETO-OPTICAL EFFECT AS A DETECTION For the Kerr angle, we have OF TOPOLOGICAL INVARIANTS cot θ = α n˜T − n˜B , (27) We propose an optical experiment to directly measure K z z θ = this phenomenon independent of material details as shown in For the slab Chern number that is not so large, K −1 /α T − B ≈ π − α T − B Fig. 3(a). Consider a normally incident linearly x-polarized tan [1 (ñz n˜z )] 2 (ñz n˜z ), the polarization of light with frequency ω propagating along the z direction the reflected light will exhibit a full-quarter rotation relative = − ω = T B through the sample Ein Ein exp[i(k0z t )]xˆ with k0 to the incident light [51]. To determinen ˜z andn ˜z for top and ω/c. Er and Et are the reflected and transmitted electric bottom surface, we also need to use the results at reflectivity = + 1 λ / field, respectively. Their values at the interface between two maxima when Lz (N 2 ) b 2. materials are related to the incident field Ein by the 2 × 2 The corresponding reflectivity is

˜ 4 μ R = 1 − ˜ , (28) ˜ 2 + ˜ − α 2 T B + + α T 2 + α B 2 μ˜ 2 μ˜ 1 (2 ) n˜z n˜z 1 2 n˜z 1 2 n˜z with ˜ andμ ˜ being the relative permittivity and permeability ≈ ˜ − 2/ ˜ + 2 number R ( μ˜ 1) ( μ˜ 1) is only determined by the for the material, respectively. For relative small slab Chern θ material constants. The corresponding Kerr K and Faraday

033177-6 BULK-HINGE CORRESPONDENCE AND THREE- … PHYSICAL REVIEW RESEARCH 3, 033177 (2021)

θ angles F can be obtained as APPENDIX A: MODEL HAMILTONIAN, BAND STRUCTURE, AND THE WINDING θ OF QUADRUPOLE MOMENTS tan K ˜ α − B − α T + α B 2 To facilitate numerical calculations, we map the continuous 2 μ 2 n˜z 2 n˜z 1 2 n˜z = ˜ , Hamiltonian model in the main text on a square lattice in the − ˜ 2 − ˜ α T α B+ − α T 2 + α B 2 → 1 2 → 2 − μ˜ 2 μ˜ 2 n˜z 2 n˜z 1 2 n˜z 1 2 n˜z tight binding approximation, ki a sin kia and ki a2 (1 cos kia)fori = x, y, z. The lattice constant a = 1 is chosen as (29) 1 the same for three directions. (iv⊥σ s − 2m⊥σ s ∓ 2cσ s ) 2 x i z 0 i 0 α T − B are the hopping matrices connecting the nearest neighbor sites 2 n˜z n˜z 1 tan θ = . (30) along x, y direction, respectively, and (ivzσxsz − 2mzσzs0 )is F + ˜ + α T α B 2 1 μ˜ 2 n˜z 2 n˜z the matrix connecting the next nearest neighbor sites along two directions in the x-y plane, which encode the hopping + + By canceling the explicit dependence on the materials between degrees of freedom between two sites. (m0 4m⊥ μ θ properties ˜ and ˜ , the measured Faraday angle F and Kerr 2mz )σzs0 + dσys0 + = , , bisi are the on-site energies. In θ i x y z angle K give a relation (see Appendix E) this representation, the Hamiltonian in momentum space is tan θ H(k) = H (k) + V (k), (A1) tan(θ + θ ) 1 − F = α n˜T + n˜B . (31) 0 i K F θ z z i=1,2,3 tan F with Using the two relations in Eqs. (25) and (31), we can determine the values ofn ˜T andn ˜B as z z H0(k) =h¯σx[v⊥(sin kxsx + sin kysy ) + vz sin kzsz] + + − − 1 tan θ [m0 2m⊥(2 cos kx cos ky ) nT = θ + θ − F + θ , ˜z α tan( K F ) 1 θ tan F (32) + − σ , 2 tan F 2mz(1 cos kz )] zs0 1 tan θ V =− − σ , nB = θ + θ − F − θ . 1(k) 2c(cos kx cos ky ) ys0 ˜z α tan( K F ) 1 θ tan F (33) 2 tan F V2(k) =dσys0, V3(k) = bisi. T/B T/B i=x,y,z Similarly,n ˜x andn ˜y can be determined by the magneto- optical Faraday and Kerr measurements in the samples of thin In the Hamiltonian Eq. (A1), H0 describes a 3D quantum spin ﬁlms normal to other two crystallographic axes. Armed with Hall insulator. Throughout this work, we choose the parame- T/B α = , , alln ˜α ( x y z), one can determine the values of all the ters quadrupole indices [Eq. (23)] and the slab Chern numbers

[Eq. (22)], and figure out the flowing pattern of chiral hinge vz = v⊥ = 1, mz = m⊥ =−0.5, m0 = 0.5, currents [Eq. (21)]. to realize the topological nontrivial phase. To illustrate the differences of the hinge states between distinct higher-order VIII. CONCLUSION topological phases, we calculate the band structure in a In a 3D second-order topological insulator, the CHMs quasi-1D geometry along three different directions for the circulate along the hinges of a sample and lead to the 3D antiferromagnetic QAHI and the rotoinversion QAHI. To for- quantum anomalous Hall effect. The quadrupole index and mulate these two cases, we need to add additional terms to the H the slab Chern numbers are introduced to characterize the model of 3D quantum spin Hall system 0. V flowing pattern of the CHMs. The nonzero slab Chern num- (i) To realize the rotoinversion QAHI, we include both 1 V bers determine the formation of the close path of the CHMs and 3 and use the parameters while the quadrupole index indicates the existence of CHMs c = 1, b = 0.3, b = 0, b = 0. along the four parallel hinges of a sample. The two topo- z x y logical invariants can be determined by the magneto-optical In Figs. 4(a)–4(c), we show the band structures in the bar Faraday and Kerr measurements. The relation between the geometry along three directions. We can find two 1D chiral two topological invariants and the flowing pattern of the chiral modes in x and y directions in the energy gap of the surface hinge states reflects the bulk-boundary correspondence in 3D states which are singly degenerated with oppositely chiral second-order topological insulators. modes localized on the neighboring two hinges. The rod bands along z direction exhibit four chiral modes which are doubly degenerated and localized at four hinges with the same chiral ACKNOWLEDGMENTS modes on opposing sides. In Figs. 5(a)–5(c), we show the B.F. and Z.A.H. contributed equally. This work was sup- quadrupole moments for the two-dimensional subspace as ported by the Research Grants Council, University Grants functions of the momentum in the third direction. Only the Committee, Hong Kong under Grant No. 17301220, and quadrupole moment along the z direction exhibits a nontrivial the National Key R&D Program of China under Grant No. winding number which is consistent with the band calcula- 2019YFA0308603. tions in Figs. 4(a)–4(c).

033177-7 BO FU, ZI-ANG HU, AND SHUN-QING SHEN PHYSICAL REVIEW RESEARCH 3, 033177 (2021)

3 (a) 3 (b) 1.5 (c)

2 2 1

1 1 0.5

0 0 0

-1 -1 -0.5

-2 -2 -1

-3 -3 -1.5 - 0 - 0 - 0

3 3 1.5 (d) (e) (f)

2 2 1

1 1 0.5

0 0 0

-1 -1 -0.5

-2 -2 -1

-3 -3 -1.5 - 0 - 0 - 0 kx ky kz

FIG. 4. The band structure of the rotoinversion QAHI and the antiferromagnetic QAHI in three directions. Panels (a–c) are the band structures of the 40 lowest bands along three directions of the rotoinversion QAHI. We use the parameters c = 1andbz = 0.3, bx = 0, by = 0. Panels (d–f) are band structures of the antiferromagnetic QAHI along three directions. We use the parameters d = 0.2, c = 1. The parameters in H0 are same for two cases: vz = v⊥ = 1, mz = m⊥ =−0.5, m0 = 0.5.

(ii) To realize the antiferromagnetic QAHI, we include magnetic point group can be expressed as [39] both V1 and V2 and use the parameters Gi = N ⊕ T (G − N), d = 0.2, c = 1. where G is a subgroup of D4h and N is a halving subgroup of In Figs. 4(d)–4(f), we show the band structures in the bar G. The subgroups of D4h has the following relations: ⎧ geometry along three directions. We can only find the chiral ⎨D2d ⊃ S4 ⇒ G , G2, modes along y and z directions in the band gap. The in-gap ⊃ ⊃ ⇒ G , D4h ⎩D2h D2 1 chiral modes for two directions are all doubly degenerated ⊃ ⇒ G , and localized at four hinges. It is further confirmed by the C2h Ci 3 winding of the quadrupole moments in Figs. 5(d)–5(f) that from which the reduced magnetic point group can be gen- the quadrupole moments along both the y and z directions erated. The group G = D4h ⊕ T D4h are given explicitly in exhibit nontrivial winding. Only directions with four chiral Table I and the symmetry breaking due to the inclusion of hinge modes exhibit nontrivial winding number in quadrupole Vi=1,2,3 is also presented. moments. In order to fully characterize the pattern of the hinge states, we also need to introduce the slab Chern number as APPENDIX C: HINGE STATE PATTERN discussed in the main text. AS COLORING THE FACE Identifying all possible chiral hinge patterns on a cuboid APPENDIX B: SYMMETRY OF THE HAMILTONIAN geometry is equivalent to find all the distinct ways that a The group G0 of Eq. (1) in the main text is a Heesch- six-sided cuboid can be painted by using two different colors Schubnikov magnetic group of type II; the inclusion of any (each face in one color). The hinge currents can be viewed term in Vi=1,2,3 breaks time reversal symmetry and several to flow along the boundary of different colored faces. Two spatial symmetries and reduces it to a magnetic group of colored cubics are not distinct if they are only up to a rotation. type III (G , Gi=1,2,3 in the main text). In general, the reduced We list all distinct colored cuboid patterns in the Fig. 6.The

033177-8 BULK-HINGE CORRESPONDENCE AND THREE- … PHYSICAL REVIEW RESEARCH 3, 033177 (2021)

0.5 0.5 1

(a) (b) (c) 0.8

0.6

0 0

0.4

0.2

-0.5 -0.5 0 - 0 - 0 - 0

0.5 1 1

(d) (e) (f) 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

-0.5 0 0 - 0 - 0 - 0 kx ky kz

FIG. 5. The winding of quadrupole moments for the rotoinversion QAHI (a–c) and the antiferromagnetic QAHI (d–f) along three directions. The parameters are the same as described in the caption of Fig. 4. O| ξ = ξ | ξ ξ ﬁrst line in Fig. 6 lists all the distinct patterns that one of the un( ) m Bn,m( ) um(DO ) with Bn,m( )arethema- faces is painted gray and all others white. The second line lists trix elements of a unitary transformation acting on the space of the patterns that two faces are gray and all others white. And the occupied bands. From which one can verify, for an antiu- the third line lists the patterns that three faces are gray. nitary symmetry operator Oa and unitary symmetry operator All this results can be uniquely determined by the quadru- Ou, the Berry curvature satisﬁes ple index and slab Chern number in Table II. O O F ξ =− a a ξ F T ξ † ξ , αβ ( ) Jαδ Jβζ B( ) δζ DOa B ( ) APPENDIX D: SYMMETRY CONSTRAINTS O O ∗ F ξ = u u ξ F ξ T ξ , ON HALL RESPONSE αβ ( ) Jαδ Jβζ B ( ) δζ DOu B ( ) The Bloch Hamiltonian H(ξ) respects the symmetry −1 O, i.e., OH(ξ)O = H(DOξ), where ξ = (kα, kβ , rγ ) with O respectively, with J = ∂(DOk)β /∂kα.FromEq.(8)inthe α,β, γ representing three orthogonal directions. As shown αβ main text, the layered-resolved Hall responses have the fol- in the second column of Table I, the unitary symmetry O = u lowing relations: U is a unitary matrix that acts on the internal degrees of O freedom of the unit cell. The antiunitary symmetry a can O O be written as Oa = UK, where U is a unitary operator and σ =− a a σ ξ , αβ (rγ ) Jαδ Jβζ δζ DOa χ K is the complex conjugate operator. Due to the symmetry, O O H ξ ξ Oξ σ = u u σ ξ , the eigenstates of ( )at and D can be related by a αβ (rγ ) Jαδ Jβζ δζ DOu χ (D1) gauge transformation. Explicitly, we can write the relations, H(DOξ)O|un(ξ)=OH(ξ)|un(ξ)=εn(ξ)O|un(ξ), for any eigenstate |un(ξ) of H(ξ) with eigenvalue εn(ξ) (the eigen- where δ, ζ , χ are another three orthogonal directions. The value only depends on the momentum components). Thus, constraints on the Hall response due to the certain symmetry O|un(ξ) is an eigenstate of H(DOξ) with the same en- for three different higher-order topological phases are pre- ergy. We can thus expand O|un(ξ) by the states at DOξ, sented in Table III.

033177-9 BO FU, ZI-ANG HU, AND SHUN-QING SHEN PHYSICAL REVIEW RESEARCH 3, 033177 (2021)

TABLE I. Symmetry of the Hamiltonian H0(kx, ky, kz ), on lines: time-reversal antiunitary symmetry (T ), chiral antisymmetry (C = PT ), P Rz R j particle-hole antiunitary antisymmetry ( ), fourfold rotation (around z axis, 4), twofold rotation (around j axis, 2), inversion symmetry (I), fourfold improper rotation (around z axis, S4), mirror reﬂection (M j the mirror plane is normal to j axis), and the antiunitary symmetries (the combination of time reversal symmetry T with the above spatial symmetries). For each symmetry, the operator of transformation is given in the second column and the momentum is transformed according to the third column. The forth column: the Jacobian of the transformation. The last ﬁve columns indicate which symmetry is further broken with the inclusion of Vi=1,2,3,.

, , σ 2 − 2 σ Symmetries Operators DO (kx ky kz )detJdys0 c(kx ky ) ys0 bzsz bxsx bysy

T isyK (−kx, −ky, −kz ) −1 χχχχχ C σ , , χχχχχ y (kx ky kz )1√√√√√ P = CT −1 −iσ s K (−k , −k , −k ) −1 √ y y x y z √ √ Rz (Rz−1 ) 2 (s + is )(k , −k , k )1 χ χχ 4 4 2 0 z y x z √√√ Rz is (−k , −k , k )1 χχ 2 z x y z √√ √ √ Rx (Ry ) is (k , −k , −k )1 χ (χ) χ( ) 2 2 √ x x y z + − √ Rx y (Rx y ) 2 i(s + s )(k , k , −k )1 χχχχ 2 2 2 x y y x z √√ √ I σ (−k , −k , −k ) −1 χχ √ z x y z √√ S (S−1 ) 2 σ (s + is )(−k , k , −k ) −1 χ χχ 4 4 2 z 0 z y x z √ M σ , , − − χχ χχ z i zsz (kx ky kz ) 1 √ √ M (M ) iσ s (−k , k , k ) −1 χχχ(χ) χ( ) x y √ z x x y z √ 2 M + (M − ) iσ (s + s )(−k , −k , k ) −1 χ χχ χ x y x y √2 z x y y x z √ √√ TRz (Rz−1 ) 2 i(s + s )K (−k , k , −k ) −1 χ χ 4 4 2 y x y x z √√ TRz is K (k , k , −k ) −1 χχχ 2 x x y z √ √ √ TRx (Ry ) −is K (−k , k , k ) −1 χχ χ( ) (χ) 2 2 √ z x y z + − √√√√ TRx y (Rx y ) − 2 (s + is )K (−k , −k , k ) −1 χ 2 2 2 0 z y x z √√ TI σ is K (k , k , k )1 χχ χ √ z y x y z √ √√ TS (S−1 ) 2 iσ (s + s )K (k , −k , k )1 χχ 4 4 2 z y x y x z √√ √√ TM σ K − , − , χ z i zsx ( kx ky kz )1 √√√√ √ TM (M ) −iσ s K (k , −k , −k )1 χ( ) (χ) x y √ z z x y z √ √√ √ 2 TM+ M − − + σ K , , − χ x y ( x y ) 2 (s0 isz ) z (ky kx kz )1

APPENDIX E: MAGNETO-OPTICAL FARADAY AND KERR EFFECTS In the main text we propose an optical experiment to measure the topological invariants. We consider a suspended sample with a slab geometry of thickness Lz. A similar ap- proach was considered in Ref. [50], but for the sake of completeness, we present the detailed derivations here. We first solve the reflection and transmission coefficients for the interface between two materials. Consider a light wave propagating along the z direction through two materials, labeled by i and j with dielectric constant and magnetic permeability as i, μi and j, μ j, respectively. The interface is

TABLE II. The corresponding quadrupole index and slab Chern number of all pattern in Fig. 6.

12345

nx 0000−1 ny 00111 nz 10101 Δx 0 −100 0 Δ FIG. 6. All possible patterns of chiral hinge modes as the bound- y 01010 Δ ary of colored face on a cubic. z 00000

033177-10 BULK-HINGE CORRESPONDENCE AND THREE- … PHYSICAL REVIEW RESEARCH 3, 033177 (2021)

TABLE III. The constraints on the Hall response due to the certain symmetry for three cases discussed in the main text according to Eqs. (D1). The symmetry constraints on the Hall response for each reduced magnetic point group are listed in the last line for each topological phase.

D2 ⊕ T (D2h − D2 ) Antiferromagnetic QAHI Rz σ =−σ − ,σ =−σ − 2 xz(y) xz( y) yz(x) yz ( x) Rx σ =−σ − ,σ =−σ − 2 xy(z) xy( z) xz(y) xz( y) TI σxy(z) =−σxy(−z),σxz(y) =−σxz(−y),σyz(x) =−σyz (−x) TMz σxy(z) =−σxy(−z) TMx σyz(x) =−σyz (−x) G1 σxy(z) =−σxy(−z),σxz(y) =−σxz(−y),σyz(x) =−σyz (−x)

S4 ⊕ T (D2d − S4 ) Rotoinversion QAHI Rz σ =−σ − ,σ =−σ − 2 xz(y) xz( y) yz(x) yz ( x) S4 σxy(z) =−σyx (−z),σxz(y) = σyz(x),σyz(x) =−σxz(−y) TRx+y σ =−σ − ,σ = σ ,σ = σ 2 xy(z) yx ( z) xz(y) yz (x) yz(x) xz(y) TMx σyz(x) =−σyz (−x) G2 σxz(y) =−σxz(−y) = σyz(x) =−σyz (−x),σxy(z) = σxy(−z)

Ci ⊕ T (C2h − Ci ) Inversion QAHI

I σxy(z) = σxy(−z),σzx (y) = σzx (−y),σyz (x) = σyz(−x) TRx+y σ =−σ − ,σ = σ ,σ = σ 2 xy(z) yx ( z) yz (x) xz(y) zx (y) zy(x) x+y TM σxy(z) =−σyx (z),σyz(x) = σxz(−y), σzx (y) = σzy(−x) G3 σxy(z) = σxy(−z),σzx (y) = σzx (−y) = σzy(x) = σzy(−x) located at z = z0. The electric field in the medium i can be in the medium i written in terms of the reflected and transmit- written as ted components of electric field is ikiz t −ikiz r ik z −ik z Ei = e E + e E , (E1) e i e i t i i Ei Ei = − . i ikiz − τ − i ikiz − τ t (E4) r(t ) = r(t ), r(t ) T Hi μ e ( i y ) μ e ( i y ) E j where Ei [Eix Eiy ] denote reflected (transmitted) i i components of the electric fields in the medium i, k = √ i Now consider the interface conditions for the electromagnetic ω μ is the wave vector. The magnetic field is given by c i i fields, the electric field is continuous across the interface Faraday’s law, − | = = , (Ei E j ) z z0 0 (E5) i − = − τ ikiz t − ikiz r , Hi ( i y ) e Ei e Ei (E2) μi and the magnetic field will be discontinuous across the interface due to the presence of the surface current τ = 0 −i where y [i 0 ] is the Pauli matrix acting on xy space. The −iτ (H − H )| = = 4πJ | = , (E6) incoming and outgoing fields at the interface are related by the y j i z z0 i z z0 S matrix: with the surface current density i i Er rt Et Ji|z=z = σiEi|z=z , = , (E3) 0 0 E j trE j t r where σ is the 2 × 2 conductivity tensor. The electromagnetic where r, r and t, t are all 2 × 2 reflection and transmission boundary conditions Eqs. (E5) and (E6) can be recast into the tensors. The electric field Eq. (E1) and magnetic field Eq. (E2) matrix form,

t Ei E j 00Ei − =−4π − . τ σ ikiz0 τ σ ikiz0 t Hi H j i y ie i y ie E j z=z0 z=z0

Using Eq. (E4) and comparing with Eq. (E3), we directly obtain i j 2 2 i − + 4πσxx − 4πσxy τ0 − 8πσxyiτy μi μ j μi r = e2ikiz0 , 2 j i 2 + + 4πσxx + (4πσxy) μ j μi i j + i + πσ τ − i πσ τ 2 μ μ μ 4 xx 0 μ 8 xyi y − i j i i t = ei(ki k j )z0 , 2 j i 2 + + 4πσxx + (4πσxy) μ j μi

033177-11 BO FU, ZI-ANG HU, AND SHUN-QING SHEN PHYSICAL REVIEW RESEARCH 3, 033177 (2021)

2 2 j − i + πσ − πσ τ − j πσ τ μ μ 4 xx 4 xy 0 μ 8 xyi y − j i j r = e 2ikj z0 , 2 j i 2 + + 4πσxx + (4πσxy) μ j μi j j i j 2 + + 4πσxx τ0 − iτy 8πσxy μ j μ j μi μ j t = . (E7) 2 j i 2 + + 4πσxx + (4πσxy) μ j μi

Thus, r is related to r by making the replacement ki →−k j and the Kerr and Faraday angles are defined by and interchanging i/μi and j/μ j and t can be obtained from /μ /μ () y t by interchanging i i and j j. The reflection (rS ) and Er ryxEin ryx rxy θ =− =− =− = , ( ) tan K x transmission tensor (tS ) for a slab can be composed from the Er rxxEin rxx rxx () () single-interface scattering matrices rT,B and tT,B for the top and bottom surfaces. We assume the two interfaces between and the vacuum and the sample are at z = 0 and z = Lz, respec- = μ = y tively. The vacuum outside the sample has 0 0 1. The E t E t t tan θ = t = yx in = yx =− xy , electric fields at the top and bottom interface have the follow- F E x t E t t ing relations: t xx in xx xx ET r t ET r = T T t , respectively. The reflectivity R depends on the relative mag- I I E tT r E nitude of the slab thickness and the wavelength inside the t T r = π bulk. For the√ cavity resonance condition kLz N is satisfied EI r t EI = μω/ r = B B t , where k c is the wave number in the slab film and B B Et tB rB Er N is an integer, we have where EI and EI are the transmitted and reflected components ⎧ ⎫ t r ⎨ 2 ⎬ of the electric fields in the sample, respectively. We also have min − 4π σ T + σ B rS,xx 1 xy xy = min 2 ⎩ 0 T B ⎭ T T rS,xy 0 + π σ T + σ B − 8π σ + σ E rS t E 4 μ 4 μ0 xy xy r = S t . 0 xy xy B B Et tS rS Er Then we can obtain the reflection and transmission tensor for and the slab − ik0Lz 0 −1 min e μ − 0 rS =rT + t rB(1 − r rB ) tT , t , 0 4 μ T T S xx = 0 , min 2 −1 t , 0 T B π σ T + σ B = − , S xy 4 + 4π σ + σ 8 xy xy tS tB(1 rT rB ) tT μ0 xy xy = − −1 , tS tT (1 rBrT ) tB with k0 = ω/c is the vacuum wave vector. The corresponding = + − −1 . θ θ rS rB tBrT (1 rBrT ) tB Kerr K and Faraday angles F can thus be obtained, For an incident light linearly polarized in the x direction 0 2 μ E = E xˆ. 0 in in tan θ = cot θ = . (E8) K F π σ T + σ B The reflectivity is defined as 4 xy xy |E |2 |r E |2 +|r E |2 R ≡ r = xx in xy in =|r |2 +|r |2, At the reflectivity maxima kL = (N + 1 )π,wehave 2 2 xx xy z 2 |Ein| |Ein| max r , 1 S xx = max 2 2 2 rS,xy + 2 0 − 4πσT 4πσB + 0 + 4πσT 0 + 4πσB μ μ μ0 xy xy μ0 xy μ0 xy 2 2 2 − + πσT πσB + 0 − πσT 0 + πσB μ 2 μ 4 xy4 xy μ 4 xy μ 4 xy 0 0 × 2 −2 0 − 4πσB + 4πσT 0 + 4πσB μ0 μ xy xy μ0 xy and ⎧ ⎫ max −ik L ⎨ 0 + 0 − πσT πσB ⎬ t , e 0 z 2i μ μ μ μ 4 xy4 xy S xx = 0 0 . max 2 2 2 t 0 T B 0 T 0 B ⎩ − 0 π σ T + σ B ⎭ S,xy + 2 − 4πσ 4πσ + + 4πσ + 4πσ 2i μ μ 4 xy xy μ μ μ0 xy xy μ0 xy μ0 xy 0

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θ θ The corresponding Kerr K and Faraday angles F can be obtained as 2 2 0 4πσB − 4πσT 0 + 4πσB μ0 μ xy xy μ0 xy tan θ = , K 2 2 2 − + 2 4πσT 4πσB + 0 − 4πσT 0 + 4πσB μ μ xy xy μ0 xy μ0 xy 0 π σ T + σ B μ 4 0 xy xy tan θ = . (E9) F + 0 − 4πσT 4πσB μ μ0 xy xy From the results Eqs. (E8) and (E9), we can obtain Eqs. (6) and (7) in the main text.

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