Geometric Response and Disclination-Induced Skin Effects in Non-Hermitian Systems
GGeometric response and disclination-induced skin effects in non-Hermitian systems
Xiao-Qi Sun, ∗ Penghao Zhu, ∗ and Taylor L. Hughes Department of Physics and Institute for Condensed Matter Theory,University of Illinois at Urbana-Champaign, Illinois 61801, USA (Dated: February 12, 2021)We study the geometric response of three-dimensional non-Hermitian crystalline systems withnon-trivial point gap topology. For systems with four-fold rotation symmetry, we show that inpresence of disclination lines with a total Frank angle which is an integer multiple of 2 π , there can benon-trivial, one-dimensional point gap topology along the direction of disclination lines. This resultsin disclination-induced non-Hermitian skin effects. We extend the recently proposed non-Hermitianfield theory approach to describe this phenomenon as a Euclidean Wen-Zee term. Furthermore, bydoubling a non-Hermitian Hamiltonian to a Hermitian 3D chiral topological insulator, we show thatthe disclination-induced skin modes are zero modes of the surface Dirac fermion(s) in the presenceof a pseudo-magnetic flux induced by disclinations. Introduction.–
Non-Hermitian Hamiltonians provide anatural formalism to describe wave phenomena in thepresence of loss and gain, which are ubiquitous in a vari-ety of physical contexts including condensed matter [1–4],photonic [5–17] and cold atom [18, 19] systems. Recently,there has been a growing interest in the interplay be-tween non-Hermiticity and topological phases. The syn-ergy of these two concepts has resulted in fruitful resultsthat are distinct from Hermitian topological physics, suchas new transport and dynamical features [20–27], newforms of bulk-boundary correspondence [28–35], and non-Hermitian analogy of topological insulators [36–44] andsemimetals [45–60].One of the most remarkable consequences of non-Hermiticity is new types of topological phases withoutHermitian analogs. These intrinsically non-Hermitiantopological phases are defined in the presence of a pointgap with respect to a reference energy in the complexenergy plane [61–68]. In one spatial dimension, the non-trivial point gap topology produces the celebrated non-Hermitian skin effect (NHSE) [31, 69–76], which gener-ates an extensive number of states localized at the bound-aries of a system. Recently, a topological field theoryhas been proposed which describes the response of non-Hermitian systems having non-trivial point gap topologyto external gauge fields [77]. This approach led to a fieldtheoretic description of the magnetic field induced NHSEin three-dimensional (3D) non-Hermitian Weyl semimet-als with non-trivial point-gap topology [78]. In additionto electromagnetic response, there have been extensivestudies of the geometric response of Hermitian topolog-ical systems both in the continuum limit [79–88] and atlattice level [89–97]. However, the understanding of theinterplay between geometry and non-Hermitian point-gap topology is still preliminary.In this letter, we consider the geometric response of3D non-Hermitian crystalline systems having non-trivialpoint gap topology. We show that disclination lines inrotationally invariant systems can support 1D point gaptopology along the direction of the disclination lines, hence leading to a corresponding NHSE. In addition toour microscopic calculations, we show that we can de-scribe this phenomenon with the inclusion of a Wen-Zeeterm [80] in the non-Hermitian field theory approach.Furthermore, by mapping the non-Hermitian problem toa 3D chiral topological insulator, we show that the discli-nation skin modes are zero modes of the surface Diracfermions subjected to a pseudo-magnetic field inducedby the disclinations.
Topological response.–
3D non-Hermitian crystallinesystems can have a non-trivial point gap topology char-acterized by the point gap winding number W ( E ) thatis defined with respect to a reference energy E as [77, 78]: W ( E ) = − (cid:90) BZ d k π (cid:15) ijk tr (cid:104) ( (cid:101) H − ∂ k i (cid:101) H ) × ( (cid:101) H − ∂ k j (cid:101) H )( (cid:101) H − ∂ k k (cid:101) H ) (cid:105) , (1)where (cid:101) H ( k ) ≡ H ( k ) − E, and H ( k ) is the non-HermitianBloch Hamiltonian. Non-Hermitian Hamiltonians withnon-vanishing W ( E ) exhibit a chiral magnetic skin ef-fect, i.e., a NHSE along the direction of an applied mag-netic field. This response is captured by a Euclideantopological field theory [77]: S E = W ( E )4 π (cid:90) d x (cid:15) ijk A i ∂ j A k + ..., (2)where we have written down the topological Chern-Simons term in the gradient expansion of the spatialvector potential A ( x ). From this action, the current re-sponse is j = W ( E )2 π B , where the external magnetic field B = ∇ × A .To illustrate this effect consider B = B ˆ z . We canchoose a gauge that preserves translation symmetry along z, and then define a 1D point gap winding number W ( E )that captures the NHSE along the z -direction [63, 65, 77,78]: W ( E ) = − (cid:90) π dk z π ∂∂k z arg [det ( H ( A , k z ) − E )] . (3) a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b It has been shown [77, 78] that the chiral current re-sponse is related to this 1D winding number, and hencethe NHSE in the z -direction, through W ( E ) = I z = (cid:90) S j z dxdy = W π (cid:90) S B z dxdy, (4)where S is a surface at a fixed value of z . We note that thequantization of W ( E ) is guaranteed for compact mani-folds where the total magnetic flux along the z -directionis quantized.Inspired by this formalism, we study the response ofnon-Hermitian systems with non-trivial W to defects ofthe background geometry, which, as we will see, can effec-tively produce a pseudo-magnetic flux. From field-theoryintuition of Hermitian systems, we can tentatively con-sider the effects predicted by a Euclidean Wen-Zee geo-metric response action [80, 98]: S WZ = L z W ( E )2 π (cid:90) d x (cid:15) ijk A i ∂ j ω k , (5)where ω is the spin connection that we treat as an ef-fective gauge field, and L z is an orbital angular momen-tum. In exact analogy to Eq. (2) we can calculate the1D point gap winding number induced by a flux of thespin-connection along the z -direction as W ( E ) = I z = (cid:90) S j z dxdy = L z W ( E )2 π (cid:90) S Ω z dxdy, (6)where Ω z ≡ ∂ x ω y − ∂ y ω x , and W > W <
0) indicatesa NHSE with skin modes on the top (bottom) surface. Toobserve this effect we will generate the flux Ω z by insert-ing disclinations in a lattice, which are represented by theFrank-angle vector Θ i ≡ (cid:82) Ω i dS i [90, 91, 94, 99]. We notethat we have not derived Eq. (5), but instead proposedit to use as a guide to understand disclination-inducedphenomena in concrete lattice models with C z rotationsymmetry. This is because though treating disclinationsas a pseudo-magnetic field has been studied extensivelyin the literature of disclinations [89, 93, 99–101], it canhave subtleties due to the lattice regularization. We willsee below that in certain cases the response of a latticeHamiltonian to disclinations can be captured by Eq. (5),and we will clarify the deviation for the other cases. Hermitian description.–
In order to bridge the low en-ergy effective theory discussed above with the NHSE ob-served in lattice models, it is convenient to introduce adoubled and Hermitianized Hamiltonian with chiral sym-metry [63, 66]: H ( k ) = (cid:18) H ( k ) † − E ∗ H ( k ) − E (cid:19) , (7)where H ( k ) is the non-Hermitian Hamiltonian we wantto study, and E is the reference energy on which we focus.Two main features of this approach are: (i) the topolog-ical winding number W ( E ) of H ( k ) equals the chiral winding number of H ( k ). Hence, the bulk-boundary cor-respondence of chiral symmetric insulators indicates thata nonzero W ( E ) implies the existence of | W ( E ) | pro-tected surface Dirac cones (SDCs), and (ii) the existenceof an exact zero mode of H implies the existence of aneigenstate at energy E ( E ∗ ) for H ( H † ) depending on itschirality. X YZ (+ , +) ( , ) ( , )( , ) ( , ) ( , +) ( , +)( , +) HermitianrBZ(a) (b) k x k y k z U R S T k x k y C z MX Y FIG. 1. (a) A tabulation of (sgn [ γ − Im E ] , χ i ) at the eightWeyl points of Eq. (8) when 1 < Im E <
3. (b) the locationsof Dirac cone(s) on the surface of the Hermitian Hamiltonianin Eq. (7) when 1 < Im E < − < Im E < − < Im E < − For an explicit illustration, let us focus on a con-crete non-Hermitian Weyl semimetal model with a BlochHamiltonian: H ( k ) = t sin k x σ x + t sin k y σ y + t sin k z σ z + iγ ( k ) , (8)where γ ( k ) = cos k x + cos k y + cos k z . This model has athe C z rotational symmetry represented by: C z = e − i π σ z e − iπ L z / ,C z H ( k x , k y , k z ) C † z = H ( k y , − k x , k z ) , (9)where the Pauli matrices represent the spin degree of free-dom, and L z is the orbital angular momentum that takesvalues − , , , , for C z symmetric systems. Wewill see below that the orbital rotation phase in Eq. (9) isimportant for the geometric response. For this model, the3D winding number W ( E ) has a simplified formula [78]: W ( E ) = (cid:88) i
12 sgn [ γ ( Q i ) − Im E ] χ i , (10)where Q i are the momenta of the eight Weyl points inthe Hermitian limit, and χ i = ± γ − Im E ] , χ i ). This model therefore hasa nontrivial 3D point gap winding number W ( E ) = 1for 1 < | Im E | <
3, and W ( E ) = − | Im E | < H in a semi-infinitebulk geometry terminated with a surface normal ˆ z andits 2D reduced Brillouin zone (rBZ) (see Fig. 1(b)). Westart with the simple case where 1 < Im E < W ( E ) = 1. The low-energy effective Hamiltonian onthe top C z -symmetric surface termination is a rotation-ally symmetric SDC at Γ as shown in Fig. 1(b) : H eff = v ( k x τ x + k y τ y ) , (11)where the Pauli matrices τ i are the effective degrees offreedom, and the chiral symmetry is represented by τ z .Now we introduce disclination lines parallel to the z -direction. In addition to the coupling to τ z , the spin con-nection also couples to L z , which adds an effective gaugeflux L z Θ z to a Dirac fermion affected by the disclina-tion curvature sources [98]. According to the index theo-rem [102], this flux L z Θ z , will lead to robust zero modeson the top surface with total number ν = |L z Θ z / (2 π ) | ,assuming a continuous ω -field. Furthermore, the zeromodes are eigenmodes of τ z with eigenvalue (chirality) τ = sgn L z Θ z [98]. For L z Θ z >
0, the eigenmodeswill have τ = +1 and correspond to skin modes of H on the top surface and at energy E . By a similar ar-gument for the bottom surface, one can derive that for L z Θ z <
0, there are zero modes of H corresponding toskin modes of H on the bottom surface and at energy E .It is known [63] that the number of zero modes of H with τ = +1 equals | W ( E ) | , and that sgn[ W ( E )] = + / − in-dicates that the zero modes with τ = +1 are on thetop/bottom surface. Hence, we can conclude that when1 < Im E < W ( E ) = 1, the disclination inducesa winding number W ( E ) = L z Θ z / (2 π ), which is fullycaptured by the Wen-Zee term in Eq. (5). We note thatthe effective flux is contributed only by the orbital rota-tion generator, which commutes with τ i . From here on,we focus only on this part to compute the effective fluxfor a Dirac fermion in order to determine W ( E ).Let us proceed to discuss cases where the SDCs are atmomenta away from Γ. Since disclinations are classifiedby a Frank angle Θ z and a Burgers vector (equivalenceclass) b = ( b x , b y ) [90, 91, 94], for a SDC with non-zeromomentum, the Burgers vector will also contribute aneffective flux of − Q ⊥ · b , which cannot be captured by theWen-Zee action [103]. With this consideration in mind,let us analyze other topologically non-trivial regimes ofour model. First, when − < Im E < −
1, there is asingle SDC at M [see Fig.1 (b)], and the effective flux itfeels is [98] Φ eff ( M ) = L z Θ z − ( π, π ) · b . (12)From the index theorem for 2D Dirac fermions, and therelation between W and the number of surface zeromodes, it is straightforward to see that the effectiveflux in Eq.(12) leads to a winding number W ( E ) =Φ eff ( M ) / (2 π ). Second, when − < Im E <
1, there isa pair of SDCs at X and Y [see Fig.1 (b)]. Since a C z rotation takes one SDC to another, and the translationphase obtained by each SDC is different, these two SDCstogether form an irreducible representation of the space group. A subtlety arises that we cannot individually de-fine the effective flux for each SDC, and need to considerthe (possibly non-Abelian ) effective flux of both. In do-ing so, we write the C z rotation operator for these twoSDCs acting on orbital degrees of freedom: C z = σ vx e − i π L z = exp (cid:16) − i π L z σ vx (cid:17) , (13)where the superscript v indicates the operator is in thevalley space, and σ vx exchanges the two valleys X and Y . From Eq.(13), we can see L z σ vx is the orbital rotationgenerator, which leads to a non-Abelian flux for SDCsat the two valleys. The effective flux can be written asΦ r = L z Θ z σ vx , where Θ z is a multiple of π/ C z -symmetric case. In addition, the translation phase ofSDCs at the two valleys contributes another matrix flux:Φ t = − π (cid:18) b x b y (cid:19) . (14)Combining these two contributions, we define the totalnon-Abelian effective flux felt by SDCs at two valleys tobe e i Φ eff ( XY ) = e i Φ t e i Φ r . (15)Now applying the index theorem again [98], we find awinding number W ( E ) = − tr[Φ eff ( XY )] / (2 π ), wherethe minus sign comes from the chiral winding number − (a) (b) FIG. 2. Schematic illustration of a lattice construction withfour disclinations, each of which has Frank angle − π/
2. (a) isthe gluing procedure [98], and (b) is a completed lattice withdisclinations.
Numerical results.–
To find a quantized W ( E ) and thecorresponding NHSE, we need a total pseudo-magnetic(time-reversal odd) flux that is an integer multiple of 2 π .However, there is a subtlety on the lattice that a fluxof nπ passing through a single lattice plaquette is time-reversal invariant and incompatible with a chiral current I z response and the corresponding NHSE [c.f. Eq.(6)].Thus, for our lattice calculations this motivates us to con-sider orbital angular momentum L z = ±
1, and introducefour disclinations each with Frank angle Θ sz = ± π/ s is to distinguish the Frank angleof a single disclination and the total Frank angle Θ z inthe system. For illustration purposes, let us fix L z = 1 , and focus on the case where each of the four disclina-tions has a Frank angle Θ sz = − π/ sz = − π/ , L z = − sz = π/ , L z = +1), etc., in the Supplemental Mate-rial [98].For our lattice calculations we construct four disclina-tions with Frank angle Θ sz = − π/ H dis ( k z ). We note thatwe are using disclinations with “plaquette-type” cores[90] which implies a Burgers vector class b s = a ˆ x, where a is the lattice constant and the superscript s indicatesthe Burgers vector is for a single disclination. Note thatbecause of the C z symmetry we could equivalently saythat b s = a ˆ y, hence a Burgers vector class. Let us firstfocus on the simplest regime when 1 < Im E < W ( E ) = 1. To show the nontrivial 1D point gap wind-ing number W ( E ) = − H dis ( k z ), we plotarg det ( H dis ( k z ) − E ) at E = 2 i in Fig. 3 (a). Fig. 3(b) shows the spectrum of H dis under periodic boundaryconditions along the z -direction, where a loop circlingthe 1 < Im E < bottom surface, which are qualitatively capturedby W ( E ) = − , are indicated by blue dots in Fig. 3 (c)which is the spectrum with open boundary conditions inthe z -direction. We find ten skin modes in the region1 < Im E < , which is consistent with the extensiveNHSE for a 1D line in a 3D system where N z = 10 . In Fig. 3 (d) we show an exponentially decaying wave-function of a representative skin mode (circled in Fig. 3(c)). Hence, when 1 < Im E < L z = +1, and thereis pseudo-magnetic flux − π , we have shown numeri-cally that W ( E ) = − − < Im E < − b s = a ˆ x , the effective flux [c.f. Eq. (12)] con-tributed from each disclination is − π/ , which is equiv-alent to π/
2. In total, there is a Φ eff ( M ) = 2 π pseudo-magnetic flux, and we expect a nontrivial 1D point gapwinding, W ( E ) = +1, which is confirmed by our numer-ical calculation shown in Fig. 4 (a). Finally, consider theregime when − < Im E <
1. By substituting b s = a ˆ x and Ω z = − π/ t and Φ r defined above, one cancalculate e i Φ eff ( XY ) = σ vy , of which the two eigenmodesfeel time-reversal invariant flux (Φ = 0 and Φ = π ) oneach disclination, and thus we expect a trivial 1D pointgap winding W ( E ) = 0. This is confirmed by our nu-merical calculation shown in Fig. 4(b). Conclusions.–
We studied the geometric response of - - - - E=2i k z Re E I m E (a) (b) - - - Re E I m E z | | (c) - - - - (d) a r g d e t( H d i s ( k z ) E ) FIG. 3. Numerical calculations for the non-Hermitian Weylsemimetal described by Eq. (8) on a lattice with four discli-nations as shown in Fig. 2, and t = 1 / z -direction, and200 unit cells in the x - y plane at each z . (a) shows the non-trivial one-dimensional point gap winding number W ( E ) at E = 2 i . (b) and (c) show the energy spectrum under periodicand open boundary condition along z , where the blue dotsin (c) are skin modes on the bottom surface. (d) shows thewave-function along z direction for the state indicated by ared circle in (c). k z (a) (b)E=-2i E=0i k z - - - - - - a r g d e t( H d i s ( k z ) E ) FIG. 4. The 1D point gap winding number W ( E ) on a latticewith four disclinations as shown in Fig. 2 for (a) E = − i and(b) E = 0 i . a C z -symmetric 3D non-Hermitian lattice model withnon-trivial point-gap topology characterized by W . Wefind that disclinations can induce NHSEs along the discli-nation lines. We compared our explicit lattice calcula-tions with a proposed Euclidean Wen-Zee term in aneffective non-Hermitian response field theory. Interest-ingly, we found that for the case where the SDC of theHermitianized Hamiltonian is located at the Γ point, theWen-Zee action predicted the correct 1D winding number W , while in other cases one must consider a combinationof rotation and translation lattice information for the lowenergy theory to properly explain the resulting 1D wind-ing number and existence/non-existence of the NHSE. Inthe future, we expect that our model can be realized invarious platforms, including photonic, mechanical, andcircuit systems [70–75]. Furthermore, our model studyserves as a first concrete example to understand geomet-ric response in non-Hermitian topological phases and willmotivate future theoretical endeavors in this direction. Acknowledgments.–
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In this section, we justify the index theorem for SDCs used in the main text. In our case, we have rotation (around z -direction) generator L z + τ z / ω µ with µ = 1 , /D SDC = e µa γ a [ ∂ µ − iω µ ( L z + τ z / e µa γ a ( ∂ µ − i L z ω µ − iω µ τ z / , (S1)where e µa is the frame field and γ , = τ x,y . If we compare Eq.(S1) with the most general Dirac operator coupled to aU(1) gauge field and to the background geometry: /D = e µa γ a ( ∂ µ − iA µ − i ω abµ γ ab ) , (S2)where ω abµ is the spin connection, we can see ω µ in Eq.(S1) is just a short form for ω µ = − ω µ , and correspondingly γ = − γ = τ z . If we further consider L z ω µ as an effective Abelian gauge field A eff µ , it is straightforward to see thatEq. (S1) is just a special case of Eq. (S2) on a manifold M described by the fixed ω abµ . Then we can apply the indextheorem for the general Dirac operator [see Chapter 12 of Ref. 102] to Eq. (S1): ν + − ν − = 12 π (cid:90) M ∂ x A eff y − ∂ y A eff x = L z π (cid:90) M ∂ x ω y − ∂ y ω x = L z Θ z π , (S3)where ν + ( ν − ) is the number of zero modes with chirality τ = +1 ( − ν ≡ | ν + − ν − | number ofrobust zero modes. In the case of ν + − ν − >
0, the ν robust zero modes have chirality τ = +1, which corresponds toskin modes. We note that the RHS of Eq.(S3) only has contribution from the flux of the (effective) abelian gauge field,and has no contribution from the coupling of spin connection to γ ab , which is true for Dirac operators in 2D [102].We also note that the above discussion is for a SDC with chiral winding number +1, for SDCs with chiral windingnumber − W ( E )] and the bottom SDC have the oppositechiral winding number in our model. With similar analysis, we can also find the effective Abelian/non-Abelian gaugeflux and use the correct index theorem in other cases.When the SDC at M , we add an extra flux related to translation, − ( π, π ) · b , to the effective flux. This is becausefor SDCs with non-zero momenta Q ⊥ = ( Q x , Q y ), there will be another effective gauge field − Q ⊥ ,a e aµ besides ω µ L z and the extra effective gauge flux can be computed as − Q ⊥ ,a (cid:73) dx µ e aµ = − Q ⊥ ,a b a = − Q ⊥ · b . (S4)Like the effective flux from ω µ L z can be captured by a bulk Wen-Zee term in Eq. (5) of the main text, we tentativelypropose a term at low energy to capture the translation phase: S t = 12 π (cid:90) d x (cid:15) ijk A i Q ⊥ ,a ∂ j e ak , (S5)the details of which will be left to future research.When there are a pair of SDCs at X and Y of the same winding number, we have a 2D Dirac operator under anon-Abelian effective gauge field related to orbital angular momentum, L z σ vx ⊗ , and a non-Abelian effective gaugefield related to translation, (cid:18) ( π, · e µ
00 (0 , π ) · e µ (cid:19) ⊗ . Then, we need to consider the non-Abelian effective fluxΦ eff ( XY ) contributed by both and we have ν + − ν − = ± sgn[ W ( E )] tr (cid:2) Φ eff ( XY ) (cid:3) π , (S6)for the top/bottom surface as used in the main text. SUPPLEMENTAL INFORMATION FOR NUMERICAL CALCULATIONS
Here, we first describe more details about how we perform the numerical calculations. When gluing the open edgesat φ and φ + π/ − i L z π/ π/ − i L z π/ σ x to σ y , and rotate σ y to − σ x when gluing. This part is given by the rotation of theinternal degrees of freedom (or spin), which is generated by σ z .Next, we present numerical results for L z = ± sz = ± π/ < Im E < − < Im E < − − < Im E < - - - - - - - - - - - - - - - - - - E=2iE=2i k z k z (a) (b) E=-2i(c) k z (e) k z E=-2i(f) (g) E=0(d) E=0 k z a r g d e t( H d i s ( k z ) E ) FIG. S1. Numerical calculations for a non-Hermtian Weyl semimetal described by Eq. (8) in a lattice where there are fourdisclinations with Frank angle − π/ L z = 1. (e), (f) and (g) are for L z = − - - - - - - E=2iE=2i k z k z (a) (b) - - - E=-2i(c) k z (e) - - - k z E=-2i(f) - - - (g) E=0 - - - (d) E=0 k z k z a r g d e t( H d i s ( k z ) E ) FIG. S2. Numerical calculations for a non-Hermtian Weyl semimetal described by Eq. (8) in a lattice where there are fourdisclinations with Frank angle π/ L z = 1. (e), (f) and (g) are for L z = −−