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Scattering theory of topological insulators and superconductors I. C. Fulga, F. Hassler, and A. R. Akhmerov Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands (Dated: July 2011) Thetopological invariantofatopologicalinsulator(orsuperconductor)isgivenbythenumberof symmetry-protectededgestatespresentattheFermilevel. Despitethisfact,establishedexpressions forthetopologicalinvariantrequireknowledgeofallstatesbelowtheFermienergy. Here,wepropose 2 awaytocalculatethetopological invariantemployingsolelyitsscatteringmatrixattheFermilevel 1 without knowledge of the full spectrum. Since the approach based on scattering matrices requires 0 muchlessinformationthantheHamiltonian-basedapproaches(surfaceversusbulk),itisnumerically 2 more efficient. In particular, is better-suited for studyingdisordered systems. Moreover, it directly connectsthetopological invarianttotransportpropertiespotentiallyprovidinganewwaytoprobe y topological phases. a M PACSnumbers: 72.20.Dp,73.43.-f,74.20.Rp 7 I. INTRODUCTION relatethescatteringmatrixinddimensionstoaHamilto- ] l nian in d 1 dimensions. Our scheme of dimensional re- l − a GivenaHamiltonianH(k)ofabandinsulatororasu- duction does not preserve the symmetry, unlike the field h theorybasedscheme of Ref. 10. Insteadour dimensional perconductoranditssymmetriesasafunctionofthemo- - s mentum k in d-spatial dimensions, a topological invari- reductionpreservesthetopologicalinvariant,similarlyto e the dimensional reduction of clean Dirac-like Hamiltoni- ant (H)canbedefined. Itcountsthenumberofsurface m Q ans of Ref. 11. states insensitive to disorder which are present at an in- t. terfacebetweenthesystemandthevacuum. Ineachspa- In the remainder of the introduction we first illustrate a our approach by revisiting the familiar example of the tialdimensionexactly5outof10Altland-Zirnbauersym- m metry classes (distinguished by time-reversal symmetry integer quantum Hall effect. Subsequently, we present a - ,particle-holesymmetry ,andchiral/sub-latticesym- brief outline of the paper. nd mTetry )1 allow for a nontPrivial topological invariant.2,3 C o The evaluation of the topological invariant conven- c tionally involves an integral over a d-dimensional Bril- A. Dimensional reduction in the quantum Hall [ louin zone of some function of the Hamiltonian. Re- effect 3 cently, various approximations to the topological invari- v ant have been developed which require only the knowl- A 2D system exhibiting the integer quantum Hall ef- 1 edge of eigenvalues and eigenvectors of the Hamiltonian fect is a topological insulator in the symmetry class A 5 at one point in momentum space (rather than in the en- (all symmetries broken). It is characterized by a quan- 3 tire Brillouin zone).4–6 tized transverse conductance σ = ng with n Z .6 Despite the fact that these approximations are more and g0 = e2/h. The quantumxynumber0 n is a t∈opo- 6 efficient, we argue that they do not use one important logical invariant (the so-called Chern number) of the 0 property of a topological invariant. By definition, the Hamiltonian.12 It equals the number of protected chiral 1 topological invariant describes the properties of the sys- edge states at the Fermi level, each of which contributes 1 temattheFermilevel,namelythenumberofedgestates. e2/h to the transverse conductance.13,14 : v This observation suggests that it should be possible to Charge pumping provides an alternative way to relate i X calculate the topological invariant without knowing the the topological invariant to a quantized transport prop- full spectrum of the Hamiltonian, but rather calculating erty: inserting a flux quantum inside a quantum Hall r a only properties of the system at its Fermi energy. For samplerolled-uptoacylinderadiabaticallypumpsnelec- one-dimensional(1D) systems, this was demonstratedin trons across the sample.15 There exists a scattering ma- Ref. 7. Here, we show that for any dimensionality the trix formulation of charge pumping,16,17 which allows to topologicalinvariantcanbeobtainedfromthescattering express pumped charge per cycle (in units of e), matrix of the system at the Fermi level. Our results offer two benefits. Firstly, since the scat- 1 2π d = dϕ logdetr(ϕ), (1.1) tering matrix contains less degrees of freedom than the Q 2πi dϕ Z0 Hamiltonian, the computation of the topological invariant is much more efficient. Secondly, the scattering ma- through the flux dependence of the reflection block r(ϕ) trix relates the topological invariant to transport prop- ofthescatteringmatrixofonelead.18 Hereϕdenotesthe erties,suggestingwaystoprobethe topologicalphaseby dimensionless flux Φ = ~ϕ/e and the system is assumed electrical or thermal conduction measurements.8,9 to be insulating such that the reflection matrix r(ϕ) is The approach is based on dimensional reduction: We unitary. Equation(1.1)is nothing but the winding num- 2 ber of detr(ϕ) when ϕ is varied from 0 to 2π, which is a Weconsiderthenumericalefficiencyofourmethodand topological invariant. showexamplesofitsapplicationinSec.V. Wealsocom- The winding number occurs in a different context in pare the finite size effects of different approximations to the theory of topological insulators. The topological the topological invariant, and introduce the ‘fingerprint’ invariant of a one-dimensional Hamiltonian H(k) with of phase transitions between different topological phases chiral/sub-lattice symmetry in 2D. Finally, we conclude in Sec. VI. 0 h(k) H(k)= , (1.2) h†(k) 0 II. SCATTERING MATRIX FROM A (cid:18) (cid:19) HAMILTONIAN is expressed via the winding number given by19,20 This section contains the necessary preliminaries: the 1 2π d (H)= dk logdeth(k). (1.3) definitionofscatteringmatrixandaproofthattheshape Q 2πi dk Z0 oftheFermisurfacecanbecalculatedfromthescattering Here momentum k is measured in units of ~/a, with a matrix. While the formulas in this section are needed for the the lattice constant. We see that upon the identification actual implementation of our method of dimensional re- h r and k ϕ we are able to express the topological ≡ ≡ duction, the method itself can be understood without invariant in a 2D system without any symmetries as the them. This section can thus be skipped at first reading. topological invariant of an effective Hamiltonian in 1D Any Hamiltonian H(k) of a translationally invariant with chiral symmetry. We will show that a similar di- systemwithafiniterangehoppingcanbebroughttothe mensional reduction applies to all topological invariants tight-binding form by choosing a sufficiently large unit in all dimensions. cell d d B. Outline of the paper H(k)=H + tieiki + t†ie−iki. (2.1) i=0 i=0 X X As a prerequisite for the dimensional reduction, we Herek isad-dimensionalvectorofBlochmomenta, H is have to open up the system to obtain a scattering ma- the on-site Hamiltonian,and ti arethe hoppings in posi- trix from a given Hamiltonian. Section II explains how tivei-direction. Westartourconsiderationfromopening this can be done. This section may be skipped on first the system and attaching 2d fictitious leads to it. First reading. The dimensional reduction proceeds along the we attach d sites to the original system without on-site following lines: First we form out of a scattering ma- Hamiltonian, and connect them with hoppings ti to the trix S a reflection block r(k) from one surface of the system. The Hamiltonian of this ‘unfolded’ system be- system, when all the dimensions except one are closed comes by twisted periodic boundary conditions. Then, the ef- H t† H˜ = , (2.2) fective Hamiltonian Hd−1(k) in one dimension lower is t 0 defined according to the simple rule (cid:18) (cid:19) t=(t ,t ,...t )T. (2.3) 1 2 d H (k) r(k), with chiral symmetry, (1.4a) d−1 In the next step we attach the fictitious leads to the ≡ 0 r(k) unfolded system, as illustrated in Fig. 1 for the case of H (k) , without chiral symmetry. d−1 ≡ r†(k) 0 two dimensions. The hopping to the leads in positive i- (cid:18) (cid:19) (1.4b) directionischosentobeequalto+1,andinthenegative i-direction to be equal to 1. − In Sec. III we show how to evaluate r(k) given the scat- We are now ready to construct the scattering matrix tering matrix S of the initial system and prove that the of the open system by using the Mahaux-Weidenmu¨ller reduced Hamiltonian H has the same topological in- formula21 (see also Appendix A) d−1 variant as the original H, i.e. (H )= (H). Q d−1 Q S =1+2πiW†(H˜ iπWW†)−1W. (2.4) After the generalproof we turn to the particular ways − to evaluate the topological invariant in 1–3 dimensions ThecouplingW betweentheleadandthesystemisequal in Sec. IV. In 1D we show that our expressions coincide tow√ρ,withw thehoppingfromtheleadtothesystem, with the ones derived in Ref. 7 in a different way, with- and ρ the density of states in the lead. We choose ρ = out using dimensional reduction. For 2D we formulate 1/wπ, such that the evaluation of the topological invariant as a general- 1 0 1 0 1 0 izedeigenvalueproblem. For3Dtopologicalinsulatorsin ··· 0 1 0 0 0 0 class AII the topological invariant reduces to a product 1 0 −0 0 1 0 0  of 2D invariants, while the other symmetry classes re- W = − ; (2.5) quire usage of a Bott index.6 We alsomention how weak √π ... ... ...    topological invariants fit into our approach. 0 0 0 0 0 1  ··· −    3 with the twist matrix Z(k) given by 0 eik1 0 0 ··· . e−ik1 0 ..  Z(k)  0 ... 0 . (2.8) ≡   ... 0 eikd    0 0 e−ikd 0   ···    We show that Eqs. (2.6) and (2.7) have a solution for a given k if and only if the equation H(k)ψ = 0 has a nontrivial solution. The condition for the nontrivial solution of Eqs. (2.6) and (2.7) to exist is det[S Z(k)]=0. (2.9) − Performing block-wise inversion of H˜ iπWW† yields FIG. 1: Sketch of the tight binding model used to attach − leads in order to open-up the Hamiltonian H(k) of (2.1). In J iJt† 2Dweintroducefourleadsshownascircleslabeledby1,¯1,2, S =1+2iW† itJ i− tJt† W, (2.10) and ¯2. The on-site terms (boxes) are connected by hoppings (cid:18)− − (cid:19) (lines). The additional trivial hoppings 1 and −1 are intro- J =(H0 id it†t)−1. (2.11) − − duced such that the lead properties drop out when twisted We simplify this expression further by noting that periodic boundary conditions are applied. For the Mahaux- Weidenmu¨llerformula(2.4),thethreenodesform theon-site S =γ +2iγ U†JU, U = 1 it† W, (2.12) z z Hamiltonian H˜ which is then connected via the trivial hop- − pings to ideal leads. withγz thethirdPaulimatrixinth(cid:0)edirect(cid:1)ionspace. We now write det[S Z(k)]=det 1+γ Z(k)+2iU†JU (2.13) z − =detJdet[1+γ Z(k)] here, we have set w = 1 for convenience. The values of (cid:2) z (cid:3) hopping and the lead density of states are chosen such det J−1+2iU[1+γ Z(k)]U† z × thatin the processofrolling-up,the fictitious leadsdrop =detJdet[1+γ Z(k)]detH(k). (cid:0) z (cid:1) out. Since both J and 1+γ Z(k) are nonsingular, the last Thescatteringmatrix(2.4)relatestheincomingstates z identitymeansthatdet[S Z(k)]anddetH(k)canonly in the leads to the outgoing ones: − bezerosimultaneously,whichiswhatwesetouttoprove. This proof shows that the Fermi surfaces as defined ψ ψ by the original Hamiltonian and the scattering matrix 1 1 ψ¯1 ψ¯1 are identical. This is the reason why it is at all possible ψ  ψ  to determine the topological invariant using solely the 2 2 ψ¯2 =Sψ¯2 . (2.6) scattering matrix S. Even though the scattering matrix  .   .  only describes scattering at the Fermi level, it contains  ..   ..  informationabout the complete Brillouinzone, andthus ψd ψd cannotbeobtainedfromalongwavelengthorlowenergy ψd¯out ψd¯in expansionof the Hamiltonian, but requires the complete     Hamiltonian. Note however that the scattering matrix at a single energy contains less information about the To prove that the scattering matrix contains all of the system than the Hamiltonian: in order to determine the information about the Fermi level at energy E = 0, F Hamiltonianfromthescatteringmatrix,theinversescat- we impose twisted periodic boundary conditions on the teringproblemhastobesolvedwhichrequiresknowledge scattering states: of the scattering matrix at all the energies. The size of the scattering matrix (2.4) is 2d-times larger than the size of Hamiltonian. However, if the ψ ψ 1 1 Hamiltonianislocalonalarged-dimensionallatticewith ψψ¯1 ψψ¯1 size Ld, the hoppings ti are very sparse. This allows to 2 2 efficientlyeliminateallofthemodesexcepttheonesthat ψ¯2 =Z(k)ψ¯2 , (2.7) arecoupledtothehoppings. Theresultingscatteringma-  .   .   ..   ..  trix is of size 2dLd−1, and accordingly for large systems     ψ  ψ  itis adense matrix ofmuchsmallerdimensions thanthe  d  d ψd¯in ψd¯out Hamiltonian.     4 III. DIMENSIONAL REDUCTION The aim of this section is to provide a route to the topological classification of scattering matrices by elim- ination of one spatial dimensions. This approach of dimensional reduction is inspired by the transport properties of topological systems. When applied to 1D systems it reproduces the results of Ref. 7, and in quantum Hallsystemsitreproducestherelationbetweenadiabatic pumping and the Chern number of Refs. 15,18. We begin from substituting the first 2(d 1) equa- − tionsfrom(2.7)into(2.6). Thisisequivalenttoapplying twistedperiodicboundaryconditionstoallofthedimen- sions except the last one, which is left open. Then we study the reflectionfromthe d-directionbackontoitself. The reflection is given by ψ =r(k)ψ , (3.1) d,out d,in TABLE I: (Color online) Topological classification of theten r(k)=D C[A Z (k)]−1B, (3.2) − − d−1 symmetry classes in different dimensions. Combinations of symmetryclassanddimensionalitywhichsupportnon-trivial with Z given by Eq. (2.8) in d 1 dimensions. The d−1 − topological invariants are indicated by the type of the topo- matrices A, B, C, and D are sub-blocks of S given by logical invariant (Z or Z2). Classes which support only trivial insulators are denoted by ‘-’. The arrows indicate the S1,1 ··· S1,d−1 S1,d changeofsymmetryclass upondimensionalreductionasdis- A= ... ... ... , B = ... , cussedinthemaintext. ThetopmostsymmetriesAandAIII (which do not have any anti-unitary symmetries) transform S S S  d−1,1 ··· d−1,d−1  d−1,d into each other, whereas the remaining 8 classes (with anti-     C = Sd,1 ··· Sd,d−1 , D = Sd,d . (3.3) unitarysymmetries)exchangecyclically. Thedimensionalre- duction changes the symmetry class, but preserves the topo- To st(cid:0)udy topological (cid:1)properties of r(k)(cid:0)we c(cid:1)onstruct logical invariant (‘-’, Z, or Z2). an effective Hamiltonian H (k) which has band gap d−1 closings whenever r(k) has zero eigenvalues. In classes possessing chiral symmetry one may choose a basis such Hd−1(k) is straightforwardinall ofthe cases,except the that r(k)=r†(k). If chiral symmetry is absent, there is time-reversalsymmetryinsymmetry classesAII andAI. no Hermiticity condition on r, so we double the degrees There we have r(k)= rT( k), and hence ± − of freedom to construct a single Hermitian matrix out of a complex one. The effective Hamiltonian is then given 0 r(k) H (k) = by d−1 ≡ r†(k) 0 (cid:18) (cid:19) Hd−1(k)≡r(k), with chiral symmetry, (3.4a) [rT(0 k)]† ±r(−0k)T =±τxHd∗−1(−k)τx. (3.6) 0 r(k) (cid:18)± − (cid:19) H (k) , without chiral symmetry. d−1 ≡ r†(k) 0 (cid:18) (cid:19) The details of the symmetry properties of r and H, as (3.4b) well as the relations between these symmetries are given in App. A. It is straightforward to verify that in both cases the The way the symmetry class of the d-dimensional Hamiltonian H (k) has band gap closings simultane- d−1 Hamiltonian transforms into the symmetry class of ously with the appearance of vanishing eigenvalues of H (k) expresses the Bott periodicity of the topolog- r(k). d−1 ical classificationof symmetry classes.2 Namely, symme- Ifr(k)haschiralsymmetry,H (k)doesnothaveit. d−1 tryclassesAandAIIItransformintoeachother,andthe On the other hand, if r(k) has no chiral symmetry, then other 8 classes with anti-unitary symmetries are shifted H (k)= τ H (k)τ , (3.5) by one, as shown in Table I. This reproduces the nat- d−1 z d−1 z − ural succession of symmetry classes that appears in the with τ the third Pauli matrix in the space of the dou- context of symmetry breaking22 (see also Appendix A). z bled degrees of freedom. This means that in that case Thecombinedeffectsofthechangeindimensionalityand H (k) acquires chiral symmetry. in symmetry class is that the Hamiltonians H(k) and d−1 The way in which the dimensional reduction changes Hd−1(k) have the same topological classification. the symmetry class is summarized in Fig. 2. The trans- We now turn to provethat for localized systems topo- formation of symmetries of r(k) into symmetries of logicalinvariants (H)and (H ) areidentical. This d−1 Q Q 5 particle-hole symmetry CII AII DIII y r t e m m C A – no symmetry D y s l a s r e v e AIII r - e m CI AI BDI i t FIG.2: Symmetrypropertiesofr(k)andH(k)inthetensymmetryclasses. Time-reversalsymmetryisdenotedbyT,particle- hole symmetry by P. The signs at the top and left of the table denote either the absence (×) of a corresponding symmetry, or the value of the squared symmetry operator. The entries of the table with a gray background have an additional chiral symmetry C, which always has theform shown in theAIII entry of thetable. In particular, we always chose a basis such that r(k) = r†(k) in the chiral symmetry classes. The way symmetry classes transform under our definition of Hd−1, cf. (3.4), is denoted by the arrows; the double arrow implies a doubling of degrees of freedom as in Eq. (3.4b). Going along an arrow, the symmetry of the reflection block r(k) (marked by a solid box) transforms into the symmetry of the reduced Hamiltonian (markedbyadashedbox). Inthechiralclasses, thereisanadditionalsymmetry(notmarkedbyabox)whichcanbeobtained from the other bycombining it with thechiral symmetry,H(k)=−τzH(k)τz and r(k)=r†(k), respectively. correspondence was proven in 1D in Ref. 7, so here we invariants. accomplish the proof in higher dimensions. Conversely, if H and H′ have different topological in- First of all, we observe that a topologically trivial variants,thereexistsatransmittingmodeattheinterface Hamiltoniancanbedeformedintoabunchofcompletely betweentwopartsofthesystem,whichappearsirrespec- decoupledlocalizedorbitalswithout closingits gap. In a tive of system size and microscopic details of the inter- sufficiently large system, this also means that the gap of face. This means that it is not possible to construct an H (k)doesnotcloseduringthisprocess. Forasystem interface between H and H′ which would be com- d−1 d−1 d−1 of decoupled orbitals, r(k) and accordingly H (k) are pletely gapped. d−1 momentum-independent(andhenceH (k)istopologi- Finally, the edge states in d 1 dimension have to d−1 − callytrivial). Thismeansthatasufficientlylargesystem have the same group properties as the surface states with trivialH(k) maps ontoa trivial H (k) under the in d dimensions, leading us to the conclusion that d−1 scheme of dimensional reduction outlined above. (H)= (H ), aswesetouttoprove. The topology- d−1 Q Q Let us now consider an interface between two systems preserving property of our dimensional reduction proce- with different bulk Hamiltonians H and H′, shown in dure is the same as that of the mapping from a general Fig. 3. If the Hamiltonians H and H′ constructed d-dimensionalHamiltoniantoad+1-dimensionalHamil- d−1 d−1 out of reflection blocks of the two systems have differ- tonian presented in Ref. 23. ent topological invariants, a topologically protected zero At this point one might wonder why we apply the energy edge state in d 1 dimensions must appear at dimensional reduction only once. Indeed, the reduced − theinterfacebetweenthem. Recallingthatazeroenergy Hamiltonian H can be straightforwardly approxi- d−1 edge state in d 1 dimension corresponds to a perfectly mated by a tight-binding Hamiltonian on a d 1 dimen- − − transmitting mode of the original d-dimensional system, sional lattice using a Fourier transform. This allows to we conclude that H and H′ have different topological repeat the procedure of dimensional reduction until we 6 yields =ν(r), for AIII, BDI, and CII (4.2a) Q =Pf ir, for DIII, (4.2b) Q 0 ir =Pf Q irT 0 (cid:18)− (cid:19) =detr, for D. (4.2c) We confirm that the Eqs. (4.2) are in agreement with Ref. 7. B. Topological invariant in 2D Starting from2D, the dimensionalreductionbrings us to a 1D Hamiltonian. In this subsection we first review FIG.3: Asysteminddimensionsconsistingoutoftwoparts the known expressions for the topological invariants of with different Hamiltonians H and H′. Reflection blocks of 1D Hamiltonians, and then describe how to efficiently the scattering matrix r and r′ are used to define the lower evaluate it for the effective Hamiltonian (3.4). The Z dimensional Hamiltonians Hd−1 and Hd′−1. We prove the topologicalinsulators in 1D (classes AIII, BDI, and CII) correspondencebetween topological invariantsin dand d−1 are characterized by a winding number19,20 dimensions using the relation between the surface state at the interface between H and H′ and the edge state at the interface between Hd−1 and Hd′−1. H(k)≡ h†0(k) h(0k) , (4.3) (cid:18) (cid:19) 1 2π d (H)= dk logdeth(k), arriveatazerodimensionalHamiltonian. Westopatthe Q 2πi dk Z0 first dimensional reduction for practical purposes, since for AIII, BDI, and CII. (4.4) the advantage of considering only Fermi level properties is achieved already at the first step. The topological invariant for the Hamiltonian in class D is given by Kitaev’s formula24 Pf H(0) IV. RESULTS FOR ONE–THREE DIMENSIONS (H)=sign , for D. (4.5) Q Pf H(π) (cid:20) (cid:21) A. Topological invariant in 1D Finally, in class DIII the expression for the topological invariant was derived in Ref. 25: Webeginbyverifyingthatwerecoverthe1Dresultsof Ref.7,wherethetopologicalinvariantwasrelatedtothe (H)= Pf[UTh(π)] exp 1 πdk d logdeth(k) scattering matrix without going through the procedure Q Pf[UTh(0)] "−2Z0 dk # of dimensional reduction. Dimensional reduction in this case brings us to a zero-dimensional Hamiltonian. The Pf[UTh(π)] deth(0) = , for DIII, (4.6) topological invariant of a zero-dimensional Hamiltonian Pf[UTh(0)] pdeth(π) without symmetry between positive and negative energies (symmetry classes A, AI, and AII) is given just by where the square root pis defined through analytic con- the number of states below the Fermi level. In class AII tinuation over the first half of the Brillouin zone, h is Kramers’degeneracymakesthisnumberalwayseven. In defined by Eq. (4.3), and UT is the unitary part of the addition, in 0D there exist two Z2 topological insulators time reversaloperator T =UTK. in symmetry classes D and BDI. The topological num- SubstitutingEq.(3.4)intotheexpressionsfortopolog- ber is in that case the ground state fermion parity, or ical charge we get the Pfaffian of the Hamiltonian in the basis where it is antisymmetric. To summarize, 1 2π d = dk logdetr(k), for A, C, D (4.7a) Q 2πi dk Z0 (H)=ν(H), for A, AI, and AII, (4.1a) Pf[U r(π)] detr(0) Q T = , for AII, (4.7b) Q(H)=Pf iH, for D and BDI, (4.1b) Q Pf[UTr(0)] pdetr(π) Pf r(0) whereν(A)denotesthenumberofnegativeeigenvaluesof =sign p , for DIII. (4.7c) the Hermitianmatrix A. Substituting H fromEqs.(3.4) Q Pf r(π) (cid:20) (cid:21) 7 In order to efficiently evaluate the integral given in onto the negative real axis. In symmetry class DIII the Eq. (4.4), and the analytic continuation in Eq. (4.7b) evaluation of the topological invariant is most straight- using Eq. (3.2), we define a new variable z = eik. Then forward, and yields we perform an analytic continuation of det r(z) to the Pf r(0) complex plane from the unit circle z =1. To find zeros =sign for DIII. (4.14) | | Q Pf r(π) and poles of det r(z) we use (cid:20) (cid:21) The physical meaning of the topological invariant in detr(z)=det A−CZ1(k) DB det[A−Z1(k)], cmlaasgsnAetiicsflquuxa.ntiIznedthpeumqpuianngtuomf chspairngeHaasllairnessuploantosre tino (cid:18) (cid:19)(cid:30) (4.8) class AII the invariantcanbe interpreted either as time- where reversal polarization pumping26, or as pumping of spin which is quantized along an unknown axis.27,28 In the 0 eik 0 z Z (k)= = ; superconductingclassesC,D,andDIIIitisananalogous 1 e−ik 0 z−1 0 thermal or gravitationalresponse.29,30 (cid:18) (cid:19) (cid:18) (cid:19) Equation (4.8) follows from Eq. (3.2) and the determi- nant identity C. Topological invariant in 3D M B det(D CM−1B)=det detM. (4.9) Turning now to 3D, we need to consider topological − C D (cid:18) (cid:19)(cid:30) invariantsof2DHamiltonians. Thesymmetryclasswith Since both the numerator and the denominator of the simplest expression for the topological invariant in Eq. (4.8) are finite at any finite value of z, the roots of termsofthescatteringmatrixin3DisAII.The2Dtopo- the numeratorz are the zeros of detr(z), and the roots logicalinvariantofasysteminclassDIII(intowhichAII n ofthe denominatorw arethe poles. InApp.Bweshow transformsupondimensionalreduction)isaproduct25 of n that due to unitarity of the scattering matrix, the poles the topological invariants (4.6) of 1D Hamiltonians ob- of detr(z) never cross the unit circle. By multiplying tained by setting one of the momenta to 0 or π, the secondcolumnofthe numeratorofEq.(4.8)by z we [H(k ,k )]= [H(k ,0)] [H(k ,π)], (4.15) 1 2 1 1 bring the problem of finding roots z of this numerator Q Q Q n to the generalized eigenvalue problem, with [H(k1)]givenbyEq.(4.6). Substituting Eq.(3.4) Q into this expression we obtain S1,1 1 S1,2 0 S1,¯1 0 − − Pf[U r(π,0)] detr(0,0) S¯1,1 0 S¯1,2 ψn =zn 1 S¯1,¯1 0 ψn, (4.10) = T    −  S2,1 0 S2,2 0 S2,¯1 0 Q Pf[UTr(0,0)] pdetr(π,0)× −     which can be efficiently evaluated. The roots wn of the Pf[UTr(π,pπ)] detr(0,π), for AII. (4.16) denominatorcanalsobefoundbysolvingthegeneralized Pf[UTr(0,π)] pdetr(π,π) eigenvalue problem, Direct evaluation of thpe Hamiltonian topological invariant in 2D in classes with nontrivial Chern number (cid:18)SS¯11,,11 −01(cid:19)ψn =wn(cid:18)10 −−SS¯11,,¯1¯1(cid:19)ψn. (4.11) (tAo,fixC,thDe),gaaungdeitnhrcolausgshAouIIt tishehaBrrdillboeucianuzsoenoef12t,h26e.nIeteids usuallymoreefficienttouseamethodwhichreliesonthe Since the poles of detr(z) never cross the unit circle, real space structure of H evaluated in a single point in in classes A, C, and D the topological invariant is given momentum space.4,5,31,32 These methods using the Bott by indexorasimilarexpressionforthetopologicalinvariant require the so-called band-projected position operators: =# z : z <1 N , for A, C, and D, (4.12) n n 1 Q { | | }− x = Pexp(2πix)P and y = Pexp(2πiy)P. Here P is P P i.e., the number of z ’s inside the unit circle minus the the projector on the states of the Hamiltonian with neg- n number of modes N in the direction 1. In class AII ative energies, and x and y are the coordinate operators 1 (quantum spin Hall insulator) the topological invariant in the unit cell of the system. In order to evaluate these is given by operatorsin our case we note that the eigenvalues of the effective Hamiltonian in the symmetry classes of interest i1+zn approach 1whentheoriginalsystembecomeslocalized. = n 1−zn Pf UTr(π) for AII, (4.13) In that ca±se P =(1 r)/2 [with r r(0,0)], and we can Q Q q( i)1+wn × Pf UTr(0) avoid the need to ca−lculate the pro≡jector explicitly if we n − 1−wn approximate x and y by q P P Q withthebranchcutofthesquarerootalongthenegative x (1+r)/2+(1 r)e2πix(1 r)/4, (4.17) P real axis. Note that the linear fractional transformation ≈ − − y (1+r)/2+(1 r)e2πiy(1 r)/4. (4.18) z i(1+z)/(1 z)mapstheupperhalfoftheunitcircle P ≈ − − 7→ − 8 Using the 2D Hamiltonian expressions from Ref. 33 we arrive at a scattering formula for the 3D topological invariant, 1 = Imtrlog[x y x† y†], for AIII, CI, DIII. Q 2π P P P P (4.19) ThesymmetryclassCIIin3Dtransformsupondimen- sional reduction to class AII in 2D. The expressions for the Pfaffian-Bott index required to calculate the topological invariant for a 2D Hamiltonian in class AII are quite involved. We do not give them here, but refer the interested reader to Eqs. (7), (9), and (10) of Ref. 4. FIG. 4: The value of the chemical potential µc where the D. Weak invariants ensemble averaged topological invariant equals to 0.5, as a function of system size L. Red: topological invariant defined Allofthealgorithmsdescribedaboveapplydirectlyto in terms of the scattering matrix, from Eq. (4.12). Green: the weak topological invariants.10,34,35 In order to eval- topological invariant obtained from the Hamiltonian expres- uate a weak invariant one just needs to eliminate one of sion of Ref. 4. Lines represent fitsas described in thetext. the dimensions by setting the momentum along that di- mensiontoeither0orπ,andtoevaluatetheappropriate In addition to tight-binding models, our method ap- topological invariant for the resulting lower dimensional plies very naturally to various network models,43–45 system. The only caveat is that since weak topologi- which are favorite models for the phase transitions. cal indices do not survive doubling of the unit cell, the Hamiltonian-based approaches are not applicable to the thicknessofthesysteminthetransversedirectionshould network models, since those only have a scattering ma- be equal to the minimal unit cell. In the same fashion trix, and no lattice Hamiltonian. We have checked (eliminating one momentum or more) one can calculate thepresenceofsurfacestates36 inchiralsuperconductors that calculating a topological invariant of the Chalker- and Fermi arcs37 in 3D systems. Coddingtonnetworkmodelofsize1000 1000onlytakes × several minutes on modern hardware. V. APPLICATIONS AND PERFORMANCE B. Finite size effects A. Performance The expressions for the topological invariant given in The complexity of the Hamiltonian expressions scales terms ofthe scattering matrix inSec. IV do not coincide with linear system size L as L2 in 1D, and as L3d in with (H) very close to the transition. This is a finite Q higher dimensions. In contrast, the complexity of the size effect. In order to estimate the importance of finite scattering matrix expressions scales proportionally to L size effects we have computed the shift of the transition in 1D and to L3d−3 in higher dimensions.38,39 All the point between the n = 0 and n = 1 plateaus of a disor- subsequentoperationshavethesameoramorefavorable deredquantumHallsystemasafunctionofsize. Wehave scaling. We use the algorithmof Ref. 40 to calculate the usedasquarelatticediscretization(latticeconstanta)of Pfaffian of an arbitrary skew-symmetric matrix. a single band tight binding model with nearest neigh- We have verified that using the scattering matrix bor hopping t = 1. The magnetic flux per unit cell of method allows to efficiently calculate the topological the lattice was fixed at 0.4~/e. We used on-site disorder invariant of a quantum Hall system and of the BHZ homogeneously distributed on an interval [ 0.05,0.05]. − model41 discretized on a square lattice with a size of Thetransitionpointisdefinedasthevalueofthechem- 1000 1000. This improves considerably on previously icalpotential µ atwhich the disorder-averagedtopolog- c repor×ted4,42 results of up to 50 50 lattice sites for the ical invariant equals 0.5. We have compared two expres- × BHZ model. sions for the topological invariant: the scattering matrix In 3D the improvement in performance is not as large expression (4.12) and the Hamiltonian expression from because the values of L that we can reach are smaller. Ref. 4. The results are shown in Fig. 4. We fit the data Nevertheless,wehaveconfirmedthatitispossibletocal- obtainedvia the scattering matrix approachto the func- culate the topological invariant of 3D systems in classes tion f(L) = c +c /L obtaining a value c 0.026. In 1 2 2 ≈ AIIandDIIIusinga4-bandmodelonacubiclatticewith the case of the expression of Ref. 4, the finite size effect systemsize50 50 50. Thisisasignificantimprovement arebestfittothefunctiong(L)=c′ +c′ sin(c L+c )/L, overthe12 1×2 1×2size,reportedforHamiltonian-based with c′ 0.116. We conclude that1the2finite3size e4ffects methods.6 × × of our2al≈gorithm are significantly lower. 9 C. Applications systemaveragedover100disorderrealizationsareshown in Fig. 7 as a function of M. We observe that, anal- In2Dweillustrateourapproachbyapplyingittonet- ogously to the two-dimensional case, the presence of a work models in classes A, AII, and DIII. In class A we metallic phase is accompanied by a plateau in the topo- use the Chalker-Coddingtonnetwork model.43 In classes logical charge at a value of zero. AII and DIII we have used the quantum spin Hall net- workmodelof Ref. 45. Inclass DIII we haveset the link phases to zero in order to ensure particle-hole symme- VI. CONCLUSION try. In each of these cases the parameter which tunes throughthe transitionis theangleαrelatedtoreflection In conclusion, we have introduced a procedure of di- probability at a node of the network by R=cos2α. mensional reduction which relates a scattering matrix of OurresultsaresummarizedinFig.5. Toppanelsshow a d-dimensional system to a Hamiltonian in d 1 di- the evolution of zeros and poles of detr(z) across the − mensions with a different symmetry class, but with the phasetransition—the‘fingerprint’ofatopologicalphase same topological invariant as the originalsystem. When transition.46 There are no symmetry constraints on this applied repeatedly this dimensional reduction procedure fingerprint in class A. The time-reversal symmetry en- servesasanalternativederivationoftheBottperiodicity suresthatforeveryzeroorpoleatz thereisanotherone 0 of topological insulators and superconductors. at 1/z . The particle-hole symmetry translates into the 0 Since our approach uses only Fermi surface properties mirror symmetry with respect to the real axis: for every itis muchmoreefficientthan existing alternativeswhich zero or pole at z there is one at z∗. The bottom panels 0 0 require the analysis of the full spectrum. We have de- showthe behaviorofthe topologicalinvariantandofthe scribed how to implement our method efficiently in all conductance G = trt†t, with t the transmission matrix the symmetry classes in 1–3 dimensions. We have veri- through the system. The simulations were performed fied that it allows to analyze much larger systems than on systems of size 300 300 in each of the symmetry × previously possible. classes and averaged over 1000 samples. The presence This paper focused on the description of the method of plateaus around zero in the curves for the topological and we only touched on a few applications at the end. invariant coincides with the presence of a metallic phase More applications can be envisaged and we believe that in the phase diagram of symmetry classes AII and DIII. the scattering approachwill leadto the discoveryof new Although we introduced the topological invariant observable physics at topological phase transitions. throughtransportproperties,itdoesnotalwayshavethe same features as the conductance. The topological invariant characterizes winding of scattering modes in the transverse direction. Accordingly, in a system with a Acknowledgments large ratio of width W to the length L, the width of the transition of the topological invariant is reduced. The This research was supported by the Dutch Science width of the peak in the conductance, on the contrary, Foundation NWO/FOM. We thank B. Béri, L. Fu, is reduced if W/L becomes small. This is in agreement and J. Tworzydl o for useful discussions. We are es- withwhatweobserveinnumericalsimulations. We have pecially grateful to M. Wimmer for explaining the ef- calculatedthe topologicalinvariantandconductance av- ficient method to calculate transport properties and to eraged over 1000 disorder realizations in the Chalker- C. W. J. Beenakker for expert advice. Coddington network model in systems with W = 300 and L = 60 and vice versa. The results are shown in Fig. 6 and they agree with our expectations. Appendix A: Introduction to discrete symmetries We have also studied a 3D topological system in class AII on a cubic lattice. We haveused a simplified version Herewedefinethethreecorediscretesymmetries,and of the Hamiltonian of Ref. 47: the corresponding symmetry constraints on the Hamil- (k) vkz 0 vk− tonians and on the scattering matrices. We also specify H =Mv0kz −Mvk+(k) vk(−k) v0kz −µ (5.1) hFoigw. 2t.o choose the symmetry representation we used in vk 0 Mvk − (k)  + − z −M    discretizedoncubiclatticewithlatticeconstanta,where k =k ik , and (k)=M αk2. The Hamiltonian Definitions and properties of discrete symmetries ± x y paramete±rs were choMsen to be α−= a2, v = a. We chose µ = µ +δµ with µ = 0.4, and δµ being a random un- The discrete symmetries are defined as follows: The 0 0 correlated variable uniformly distributed in the interval timereversalsymmetryoperator isananti-unitaryop- T [ 2,2]. The topological invariant defined by Eq. (4.16) erator. When it is applied to an arbitrary eigenstate ψ − aswellasthe longitudinalconductancefora20 20 20 of the Hamiltonian H at energy ε, returns an eigenstate × × 10 A AII DIII FIG. 5: Top panel: Evolution of the poles (green dots) and the zeros (red dots) of detr(z) as a function of a parameter α which tunes through the topological phase transition in classes A, AII, and DIII in 2D. Shown is the complex plane with the unit circle |z| = 1 indicated in blue. Time-reversal symmetry in AII and DIII implies that for every zero/pole at z0 there is additionally one at 1/z0. In DIII,thereis additional particle-hole symmetry which additionally dictates zeros/poles at z0∗ and 1/z0∗.Thephasetransition happenswhen atleast oneof thezeroscrosses theunitcircle. Thiseventcoincideswith achangeof thetopological invariant Q (green) defined byEqs. (4.12 – 4.14) , as shown in thebottom panels. FIG. 7: Conductance and topological invariant (4.16) for a disordered 3D topological insulator in class AII. Ontheotherhand,theanti-unitaryparticle-holesymme- tryoperator returnsaneigenstatewithoppositeenergy P when applied to any eigenstate of the Hamiltonian: FIG. 6: Average topological invariant Q (4.12) and longitu- Hψ =εψ H ψ= ε ψ (A1b) ⇒ P − P dinalconductanceGofadisorderedquantumHallsamplefor Chiral symmetry also reverses energy, but unlike the different aspect ratios as a function of themixing angle α. C other two has a unitary operator. Allthreesymmetries , , areZ symmetries,sothe 2 T P C symmetry operators must square to a phase factor. of the Hamiltonian at the same energy: In an arbitrary basis the symmetry operators are rep- resented by Hψ =εψ H ψ =ε ψ (A1a) =U , =U , =U , (A2) ⇒ T T T TK P PK C C