The Bulk-Hinge Correspondence and Three-Dimensional Quantum Anomalous Hall Effect in Second Order Topological Insulators
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b The Bulk-Hinge Correspondence and Three-Dimensional Quantum Anomalous HallEffect in Second Order Topological Insulators
Bo Fu, Zi-Ang Hu, and Shun-Qing Shen ∗ Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China
The chiral hinge modes are the key feature of a second order topological insulator in three di-mensions. Here we propose a quadrupole index in combination of a slab Chern number in thebulk to characterize the flowing pattern of chiral hinge modes along the hinges at the intersectionof the surfaces of a sample. We further utilize the topological field theory to demonstrate thecorrespondent connection of the chiral hinge modes to the quadrupole index and the slab Chernnumber, and present a picture of three-dimensional quantum anomalous Hall effect as a consequenceof chiral hinge modes. The two bulk topological invariants can be measured in electric transportand magneto-optical experiments. In this way we establish the bulk-hinge correspondence in athree-dimensional second order topological insulator.
Introduction
The bulk-boundary correspondence liesat the heart of topological states of matter and topologi-cal materials [1–4]. It bridges the topology of bulk bandstructures and the physical observables near the bound-ary. In the quantum Hall effect and quantum anomalousHall effect (QAHE), the quantized Hall conductance isassociated with the TKNN number of the band structureand the number of the edge modes of electrons aroundthe boundary [5–8]. In a topological insulator, a Z indexin the bulk is associated with the number of the gaplessDirac cones of the surface electrons [9–11]. This reflectsintrinsic attributes of the topological phenomena. A re-cent advance in the field of topological materials is thediscovery of higher-order topological insulators [12–20].A second-order topological insulator in three dimensionsrefers to an insulator with one-dimensional the chiralhinge modes (CHMs) localized on the hinges at the inter-section of adjacent side surfaces [15–26]. Over the pastfew years, a great of efforts have been made to explorethe possible relation of the bulk bands and existence ofhinge modes as an extension of the bulk-boundary cor-respondence, such as effective mass analysis [18–23], thesymmetry indicator [27–35], and spectral flow analysis[36]. All the approaches have their own merits. How-ever, CHMs in a second order topological insulator maydisplay various flowing patterns as illustrated in Fig. 1.It lacks a systematic method to provide a comprehensivedescription of diverse flowing patterns. Also it is desir-able to learn which observable in the bulk is associatedwith the CHMs.In this Letter, we address the bulk-hinge correspon-dence and three-dimensional (3D) QAHE as a physicalconsequence of the CHMs in a second-order topologicalinsulator. We start with a minimal four-band model toreveal different flowing patterns of CHMs. It is foundthat a quadrupole index is associated with the flowingdirection of four hinge modes of the system along onedirection and a slab quantized Hall conductance revealsthe formation of a closed loop of the CHMs. We further ∗ [email protected] Figure 1. Illustration of selected patterns of chiral hingemodes and their projection in a second order topological in-sulator in three dimensions. (a) A double-loop pattern withthe quadrupole indices ∆ xy = − ∆ zx = 1 and ∆ yz = 0 and the slab Chern number n x = n y = n z = 0 . (b) Asingle-loop pattern with ∆ xy = 1 and ∆ yz = ∆ zx = 0 and n x = n y = 0 and n z = − . (c) A single-loop pattern with ∆ xy = ∆ yz = ∆ zx = 0 and n x = n y = − n z = 1 . demonstrate the correspondent connection of the CHMsto the quadrupole index and the slab Chern number bymeans of topological field theory. Finally we propose toutilize magneto-optical Faraday and Kerr effects to de-tect these topological invariants. Model Hamiltonian and symmetry analysis
We startwith a minimal four-band Hamiltonian, H = H + P i =1 V i , which consists of four parts. The first part is H = ~ σ x [ v ⊥ ( k x s x + k y s y ) + v z k z s z ]+ [ m + m ⊥ ( k x + k y ) + m z k z ] σ z s (1)where k x , k y , k z are the wave vectors, m i and v i are modelparameters. s and σ are the Pauli matrices acting inspin and orbital space, respectively. H possesses thetime reversal symmetry T ( T = − and belongs tothe symplectic symmetry class AII. Here we focus on thecase of both m m ⊥ < and m m z < such that H describes a 3D strong topological insulator with gaplessDirac cone of the surface states at all surfaces [4, 38]. H also respects the global chiral symmetry C = σ y s , {C , H } = 0 . Including the crystalline symmetries, thetotal point symmetry group is G = D h × { , T , P , C} with the particle-hole symmetry P ≡ CT − [37]. Asshown below all the terms in H preserve P , it is moreconvenient to rewrite G as G = e G × { , P} with themagnetic group e G = D h × { , T } = D h ⊕ T D h (or /mm ′ ). V = c ( k x − k y ) σ y s breaks the time rever-sal symmetry T . The presence of V reduces magneticgroup to e G ′ = D d ⊕ T ( D h − D d ) . The term pro-portional to c opens an gap with opposite sign for thesurface states on the neighboring surfaces parallel to z axis and the CHMs may be localized at their intersec-tions. The CHMs are protected by the combination offour fold rotational symmetry and time-reversal sym-metry R z T . The surface states on the bottom (00¯1) and top (001) surface remain gapless. V = dσ y s and V = P i = x,y,z b i s i is the magnetic Zeeman interaction.The two terms dσ y s and b z σ s z anticommute with thelinear terms ~ v ⊥ ( k x σ x s x + k y σ x s y ) along x and y direc-tions. Thus they act as the mass terms and gap out thesurface states on (001) and (00¯1) while being projectedonto the x-y surface. Since both of them commute withthe mass term σ y s , so they only modifies the mass termfor the surface states parallel with the z-axis and have noinfluences on the four hinge states along the z direction.When all the surface states are gapped out and the Fermilevel is located in the surface band gap, the electrons canonly propagate unidirectionally along the hinges sharedby adjacent side surfaces due to time reversal symmetrybreaking. However, with different parameters, the chi-ral hinge modes can exhibit distinctly different patterns.The presence of both V and V reduces the magneticgroup to e G = D ⊕ T ( D h − D ) . The term propor-tional to d breaks both the time reversal T and inversionsymmetry I , respectively, but respects the antiunitarycombination IT which means the fact that if one CHMspropagates along any hinge there must be another hingestate propagating in the same direction on its spatial in-version. Thus the CHMs may form two closed loops onthe surfaces (100) and (¯100) as shown in Fig. 1(a). Therelative sign between c in V and d in V will determinewhich surface the two hinge mode loops locate around.The presence of both V and V = b z s z with mag-netic field in z direction reduces the magnetic group to e G = S ⊕ T ( D d − S ) . The term breaks the IT sym-metry while preserving the S symmetry which protectsa single-loop CHMs wriggling around the bulk as shownin Fig. 1(b). The relative sign between b and c deter-mines the wriggling way of the single-loop CHMs. Onlyin the presence of V = b P i = x,y,z s i that magnetic fieldpoints to (111) direction, the magnetic point group is e G = C i ⊕ T ( C h − C i ) . Due to the presence of the inver-sion symmetry I , the CHMs at the inversion symmetrichinges are propagating in the opposite directions, andform a closed loop as shown in Fig. 1(c). A detailedsymmetry analysis can be found in Ref. ([39]). Quadrupole index and slab Chern number
In order tocharacterize the topological hinge modes, we introducetwo topological invariants: the quadrupole index and slabChern number. There are the CHMs along four hingesin the z direction in the case of Figs. 1(a) and (b). The energy dispersions of the four hinge modes connect theconduction and valence bands, and cross at k z = 0 (seeFig.S1 in [39]). For a specific k z , H ( k z ) can be viewed asa 2D system in the x-y plane and there are four cornerstates. The existence of corner states can be character-ized by the quadrupole moment [40–42], q xy ( k z ) = 12 π Im log (cid:20)
Det[ U † k z Q xy U k z ] q Det Q † xy (cid:21) where the matrix U k z is constructed by the occupied low-est energy states, Q xy = e πi ˆ r x ˆ r y /L x L y , ˆ r α are the posi-tion operators, and L α are the lengths of the system inthe α direction. Any anti-symmetry O a leaves xy planeinvariant O a H ( k z ) O − a = −H ( − k z ) will put a constrainton the quadrupole moment q xy ( k z ) : q xy ( k z )+ q xy ( − k z ) =0 or 1. At two high symmetry points Λ z = 0 or π ,the symmetry is restored, O a H (Λ z ) O − a = −H (Λ z ) , and q xy (Λ z ) must be quantized to or (see Ref. [39]).Non-zero quantized q xy (Λ z ) indicates the system topo-logically nontrivial and the existence of four zero-energycorner states in the reduced 2D subspace. For example,if q xy ( k z = 0) = 1 / , then q xy ( ± π ) = 0 or 1. In this case,there exist CHMs which compensate for the difference ofthe corner charges. Thus we can introduce a quadrupoleindex, ∆ xy = ˆ π dk z ∂ k z q xy ( k z ) (2)to characterize the existence and the flowing direction offour CHMs. For the double-loop case in Fig. 1(a), wehave ∆ xy = − ∆ zx = 1 and ∆ yz = 0 , which are protectedby the combination of chiral symmetry and the mirrorsymmetry CM α and the combination of chiral symmetryand the time reversal symmetry CT . For the single-loopcase in Fig. 1(b), we have ∆ xy = 1 and ∆ yz = ∆ zx = 0 .The quadrupole index along the z direction is protectedonly by CT and along the x ( y ) is protected by both CM x ( y ) and CT . For the case in Fig. 1(c), ∆ xy = ∆ yz =∆ zx = 0 .The slab Chern number is another topological in-variant as the quadrupole index alone are not enoughto characterize the diversity of the flowing pattern ofthe CHMs. Consider a slab geometry of the samplewith a finite thickness L z with the periodic boundarycondition along the x and y direction. Denote theBloch eigenstates by | u n ( k ⊥ , z ) i are the Bloch eigen-states, H ( k ⊥ , z ) | u n ( k ⊥ , z ) i = ε n ( k ⊥ ) | u n ( k ⊥ , z ) i with k ⊥ = ( k x , k y ) and the index n for the bands. The space-resolved Berry connection is given by A α ; n,n ′ ( k ⊥ , z ) = − i h u n ( k ⊥ , z ) | ∂ α | u n ′ ( k ⊥ , z ) i for the two occupied bands n, n ′ . In this way we define the slab Hall conductanceand its relation to a slab Chern number n z [35] σ slabxy = L z ˆ dzσ xy ( z ) = n z e h (3)where σ xy ( z ) = e πh ´ d k ⊥ Tr[ F xy ( k ⊥ , z )] and F xy ( k ⊥ , z ) is the non-Abelian Berry curvature interms of A α ; n,n ′ ( k ⊥ , z ) . Because of the periodicity ofthe Berry connection in the first Brillouin zone, it canbe proved that the slab Chern number n z is quantizedif the filled bands has a band gap to the excited statesfor a band insulator. According to the bulk-boundarycorrespondence [8], each non-zero Chern number is asso-ciated with the closed loop of chiral edge state. In Fig.1(a), n x = n y = n z = 0 , while two quadrupole indicesare not vanishing ∆ xy = − ∆ zx = 1 . The system in aslab geometry (the open boundary condition is imposedin the y direction) is analogue to the quantum spinHall insulator except the the two counter-propagatinghinges modes are localized on the opposite sides. Exper-imentally, the quantized anomalous Hall effect can bemeasured by using the surface-sensitive method [43]. InFig. 1(b), n z = − and n x = n y = 0 . There is a closedloop of chiral edge mode around the z axis. Combinedwith the non-zero quadrupole index ∆ xy = 1 . there arefour CHMs along the four hinges along the z axis, thetwo indices can determine that a single-loop of CHMsthat wriggles around the bulk. QAHE can be detectedthrough a global quantum Hall measurement probingthe whole sample due to the nonzero n z . In Fig. 1(c), n x = n y = − n z = 1 . There is a single loop of chiral edgemode around each axis. Because of the zero quadrupoleindices around the three axis, there is no four CHMsalong one direction. It exhibits a single CHM traversinghalf of its hinges, which can be projected out a singleclosed loop in the direction of x, y and z. The QAHEcan be observed for three directions due to the nonvanishing slab Chern numbers.
3D QAHE
The CHMs can be further understood inthe framework of topological field theory with an effectiveaction[44], S = ˆ d rdt (cid:20) π (cid:18) ǫ E − µ B (cid:19) + θ ( r , t ) e π ~ c E · B (cid:21) , (4)where E and B are the electromagnetic fields, ǫ and µ are the dielectric constant and magnetic permeability, re-spectively. θ ( r , t ) is known as the axion angle[45]. Theproduct E · B is odd under the time reversal or spatialinversion, θ has to be (modulo π ) for a trivial in-sulator and the vacuum and π for a topological insula-tor with respect to the symmetries[39, 46, 47]. In thequadratic order of electric and magnetic fields, besidesthe Maxwell term, the θ term may give rise to the topo-logically magneto-electric effect that an electric field caninduce a magnetic field and vice verse[44, 48, 50–52]. Bytaking the functional derivative of θ term with respectto a gauge field, the induced electric current density de-pends on the spatial and temporal gradients of the θ -field[44, 45], j θ ( r , t ) = e πh [ ∂ t θ ( r , t ) B − ∇ θ ( r , t ) × E ] . (5) (a) (b)(d)(c) (d)(c) (f)(e) Figure 2. (a) Schematic view of the hinge current. The pla-nar surfaces of the topological insulator are characterized byintegers n and n , describing the integer change of the θ value nearby the surfaces. (b) Schematic view of θ -term as afunction of the angle ϕ for topologically nontrivial and trivialcases. (c) and (d) Plots of the layer-resolved Hall response σ αβ ( r γ ) and (e) and (f) plots of the θ -angle as function of thelayer index for two cases phases from a layer-resolved Kuboformula in a slab geometry for layers. The first term depends on the temporal gradient of the θ -field and is proportional to magnetic field, i.e., the so-called chiral magnetic field, and vanishes in a static limit.The second term depends spatial gradient of the θ -fieldand is perpendicular to the electric field, i.e., the anoma-lous Hall effect. Thus there will be surface anomalousHall effect at the interface between two regions with dif-ferent θ values and no Hall response will exist in thebulk as θ takes a constant value θ b [53, 54]. The valueof θ b is given by the three-dimensional integration of theChern-Simons 3-form over momentum space[55, 56]. Inaddition to the inversion or time-reversal symmetry, θ b will be quantized with improper rotation symmetries or acombination of time-reversal symmetry and proper rota-tion symmetries[39, 57]. From Eq. (5), the layer-resolvedHall conductivities in the xy plane is associated with thegradient of θ , σ xy ( z ) = e πh ∂ z θ ( z ) . Thus the slab Hallconductance (3) is given by the difference of the θ valuesof bottom and top vacuum σ slabxy = e h θ T − θ B π , which isinteger-quantized independent of the θ value of the bulk. Relation between the θ term and the chiral hinge modes The current carried by the CHMs can be evaluated fromthe spatial dependent θ , and each chiral hinge channelcarries one conductance quantum ( e /h ). We calculatethe current through a 2D section disk ( D ) encircling ahinge normal to the plane as illustrated in Fig. 2(a), I = ˜ D d S · j θ . The electric field is determined by the gra-dient of a scalar potential, E = −∇ Φ( r ) , and we choosethe boundary of the disk as an equipotential line Φ e .By utilizing Stokes theorem, I = e πh ı C d s · ∇ θ ( r )Φ( r ) .Thus there is no current or equivalently gapless conduct-ing channel on the hinge when θ ( r ) in the two vacuumareas takes the same value n = n . If they are differ-ent n = n , there will be a branch cut separating thetwo vacuums where θ ( r ) is singular. In this situation,the contour integral gives the number of the conductingchannels I/ (∆Φ e /h ) = n − n which is the windingnumber of the field θ ( r ) . ∆Φ = Φ e − Φ in denotes thepotential difference between the outer contour C out andthe inner contour C int . In other words, the gapless hingemode tracks the singularity of the θ term and vice versa.We also want to emphasize that, even when θ in the bulkis not quantized, the above argument for the gapless chi-ral hinge channel is still valid.As show in Fig. 2 (c-f), we plot the layer-resolvedHall responses σ αβ ( r γ ) ( ǫ αβγ = 1 ) and the integratedvalue for θ ( r γ ) as a function of the layer index for threedirections. In numerical evaluation, we consider a slabgeometry with the periodic boundary in the αβ planeand open boundary condition in r γ direction. The layerresolved Hall response only distributes near the slab sur-faces where θ changes and quickly drops to zero as the po-sition moves into the bulk where θ takes constant value.For the double-loop case in Fig. 2 (c) and (e), the mag-netic point group e G will put a constraint on the Hallresponse that the layer-resolved Hall conductivity takesthe opposite values for the slab center. Thus the slabChern numbers vanish for three directions. Due to thepresence of the mass term d , the axion angle will deviateform the quantized value π , for example, θ b / π ≃ − . in Fig. 2 (e). It is also consistent with the symmetryanalysis that there is no such symmetry to guaranteethe quantization θ b in e G . As a consequence, the sur-face Hall conductance σ Bxy = e h ( θ b π − n Bz ) for the bottominterface and σ Txy = e h ( n Tz − θ b π ) for the top interfaceare not half quantized in sharp contrast to the axion in-sulators. However, the summation of the surface Hallconductance of the adjacent surface must be quantizedsince σ izx + σ jzy = e h ( n iy − n jx ) with i, j = T, B , indi-cates whether the hinge mode at the intersection of twosurfaces exists or not. For the single-loop case, the sym-metry e G constrains that the layer-resolved Hall conduc-tivities for z direction are symmetric about the slab cen-ter, while for x and y directions are antisymmetric aboutthe slab center. The layer-resolved Hall conductivities in xz plane and yz plane are also related to each other bythe S symmetry. Furthermore, θ b will be quantized dueto the presence of improper rotation symmetry S anda combination of time-reversal and the diagonal mirrorsymmetry T M x + y . As a result, the surface Hall con-ductance are half-quantized for three directions. In thisway, we establish the relation between the the CHMs (a) (b) Figure 3. (a) Schematic illustration of the measurement ofKerr and Faraday angle. Incident linearly polarized light be-comes elliptically polarized after transmission (Faraday effect)and reflection (Kerr effect), with polarization angles as θ F and θ K respectively. (b) The reflectivity R as a function of the slabthickness L z along z direction in the units of half of photonwavelength λ b / for suspended single loop case with ǫ = 10 and µ = 1 . and the two physical invariants, σ slabαβ = e h ( n Tγ − n Bγ ) and ∆ αβ = δ n Tβ n Bβ δ n Tα n Bα ( n Tα − n Tβ ) with ǫ αβγ = 1 . Magneto-optical effect as a detection of topologicalinvariants
Consider a normally incident linearly x-polarized light with frequency ω propagating along the z direction through the sample E in = E in exp[ i ( k z − ωt )]ˆ x with k = ω/c . E r and E t are the reflected and trans-mitted electric field, respectively. Their values at the in-terface between two materials are related to the incidentfield E in by the × reflection and transmission ten-sors, and can be solved by matching the electrodynamicboundary conditions. The Kerr and Faraday angles aredefined by the tan θ K = − E yr /E xr and tan θ F = E yt /E xt ,respectively[48, 49]. When the chemical potential is lo-cated within the surface gap E g and ~ ω ≪ E g , the mag-netic fields at the interface of the two materials are dis-continuous due to the presence of surface Hall current.The reflection and transmission tensors for a slab canbe obtained by composing the single-interface scatter-ing matrices for top and bottom surfaces. For simplic-ity we only consider a free-standing sample, the influ-ence of a substrate do not change our conclusion qual-itatively. The reflectivity R ≡ | E r | / | E t | will dependon the relative magnitude of the slab thickness and thewavelength ( λ b = πcω √ ǫµ ) inside the bulk. When the slabthickness contains an integer multiple of half wavelength L z = N λ b / with an integer N (the resonance condition), R reaches the minima. At the resonance, the Faraday θ ′ F and Kerr θ ′ K rotations have the same universal quan-tized value [39, 50], tan θ ′ F = cot θ ′ K = α ( n Tz − n Bz ) , where α ≡ πǫ e ~ c the fine structure constant. At the resonance,the difference of θ values between the top and bottomvacuum can be obtained irrespective of the specific valueof θ b . In order to determine n Tz − θ b π and n Bz − θ b π for topand bottom surface, we also need to use the results atreflectivity maxima when L z = ( N + ) λ b / . The mea-sured Faraday angle θ ′′ F and Kerr angle θ ′′ K give a relation[39] tan( θ ′′ K + θ ′′ F ) (cid:18) − tan θ ′ F tan θ ′′ F (cid:19) = α ( n Tz + n Bz − θ b π ) . Using the two relations, we can determine the values ofthe quadrupole indices and the slab Chern numbers.In short, the quadrupole index in combination with theslab Chern number can determine the flowing pattern ofthe CHMs, which gives rise to a 3D QAHE in a secondorder topological insulator.
ACKNOWLEDGMENTS
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