Topological Defects and Gapless Modes in Insulators and Superconductors

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9-22-2010

Topological Defects and Gapless Modes in Insulators and Superconductors

Jeffrey C.Y. Teo University of Pennsylvania, [email protected]

Charles L. Kane University of Pennsylvania, [email protected]

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Recommended Citation Teo, J. C., & Kane, C. L. (2010). Topological Defects and Gapless Modes in Insulators and Superconductors. Retrieved from https://repository.upenn.edu/physics_papers/35

Suggested Citation: Teo, J.C.Y. and C.L. Kane. (2010). "Topological defects and gapless modes in insulators and superconductors." Physical Review B. 82, 115120.

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/35 For more information, please contact [email protected]. Topological Defects and Gapless Modes in Insulators and Superconductors

Abstract We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians.We consider Hamiltonians H(k,r) that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined yb time-reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified and explicit formulas for the opologicalt invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.

Disciplines Physical Sciences and Mathematics | Physics

Comments Suggested Citation: Teo, J.C.Y. and C.L. Kane. (2010). "Topological defects and gapless modes in insulators and superconductors." Physical Review B. 82, 115120.

This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/35 PHYSICAL REVIEW B 82, 115120 ͑2010͒

Topological defects and gapless modes in insulators and superconductors

Jeffrey C. Y. Teo and C. L. Kane Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA ͑Received 3 June 2010; published 22 September 2010͒ We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians H͑k,r͒ that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined by time-reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majo- rana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.

DOI: 10.1103/PhysRevB.82.115120 PACS number͑s͒: 73.20.Ϫr, 73.43.Ϫf, 71.10.Pm, 74.45.ϩc

I. INTRODUCTION a structure similar to an ordinary Bloch Hamiltonian, except that it has an exact particle-hole symmetry that reflects the The classification of electronic phases according to topo- particle-hole redundancy inherent to the BdG theory. Topo- logical invariants is a powerful tool for understanding and logical superconductors are also characterized by gapless predicting the behavior of matter. This approach was pio- boundary modes. However, due to the particle-hole redun- 1 ͑ ͒ neered by Thouless, et al. TKNN , who identified the inte- dancy, the boundary excitations are Majorana fermions. The ger topological invariant characterizing the two-dimensional simplest model topological superconductor is a weakly ͑2D͒ integer quantum-Hall state. The TKNN invariant n paired spinless p wave superconductor in one-dimensional ␴ 2 / 38 gives the Hall conductivity xy =ne h and characterizes the ͑1D͒, which has zero-energy Majorana bound states at its H͑ ͒ Bloch Hamiltonian k , defined as a function of k in the ends. In 2D, a weakly paired px +ipy superconductor has a 39 magnetic Brillouin zone. It may be expressed as the first chiral Majorana edge state. Sr2RuO4 is believed to exhibit a 40 Chern number associated with the Bloch wave functions of triplet px +ipy state. The spin degeneracy, however, leads to the occupied states. A fundamental consequence of this topo- a doubling of the Majorana edge states. Though undoubled logical classification is the bulk-boundary correspondence, topological superconductors remain to be discovered experi- which relates the topological class of the bulk system to the mentally, superfluid 3He B is a related topological number of gapless chiral fermion edge states on the sample phase34,35,37,41,42 and is predicted to exhibit 2D gapless Ma- boundary. jorana modes on its surface. Related ideas have also been Recent interest in topological states2–4 has been stimu- used to topologically classify Fermi surfaces.43 lated by the realization that the combination of time-reversal Topological insulators and superconductors fit together symmetry and the spin-orbit interaction can lead to topologi- into an elegant mathematical framework that generalizes the cal insulating electronic phases5–10 and by the prediction11–13 above classifications.35,36 The topological classification of a and observation14–26 of these phases in real materials. A to- general Bloch or BdG theory is specified by the dimension d pological insulator is a two- or three-dimensional material and the ten Altland-Zirnbauer symmetry classes44 character- with a bulk energy gap that has gapless modes on the edge or izing the presence or absence of particle-hole, time-reversal, surface that are protected by time-reversal symmetry. The and/or chiral symmetry. The topological classifications, bulk boundary correspondence relates these modes to a Z2 given by Z, Z2, or 0 show a regular pattern as a function of topological invariant characterizing time-reversal invariant symmetry class and d, and can be arranged into a periodic Bloch Hamiltonians. Signatures of these protected boundary table of topological insulators and superconductors. Each modes have been observed in transport experiments on 2D nontrivial entry in the table is predicted, via the bulk- HgCdTe quantum wells14–16 and in photoemission and scan- boundary correspondence, to have gapless boundary states. ning tunnel microscope experiments on three-dimensional Topologically protected zero modes and gapless states can ͑ ͒ 17–19 20 22,23,25 3D crystals of Bi1−xSbx, Bi2Se3, Bi2Te3, and also occur at topological defects, and have deep implications 26 41,45–47 Sb2Te3. Topological insulator behavior has also been pre- in both field theory and condensed matter physics. A dicted in other classes of materials with strong spin-orbit simple example is the zero-energy Majorana mode that oc- 27–33 39 interactions. curs at a vortex in a px +ipy superconductor. Similar Majo- Superconductors, described within a Bogoliubov de rana bound states can be engineered using three dimensional Gennes ͑BdG͒ framework can similarly be classified heterostructures that combine ordinary superconductors and 34–37 H ͑ ͒ 48 topologically. The Bloch-BdG Hamiltonian BdG k has topological insulators, as well as semiconductor structures

␦ ͑ ͒ d=1 d=2 d=3 = d − D. 1.1

D=0 Thus, all line defects with ␦=2 have the same topological classification, irrespective of d, as do point defects with ␦ =1 and pumping cycles with ␦=0. Though the classifications D=1 t depend only on ␦, the formulas for the topological invariants depend on both d and D. This topological classification of H͑k,r͒ suggests a gen- D=2 eralization of the bulk-boundary correspondence that relates t the topological class of the Hamiltonian characterizing the defect to the structure of the protected modes associated with FIG. 1. ͑Color online͒ Topological defects characterized by a D the defect. This has a structure reminiscent of a mathematical 54 parameter family of d-dimensional Bloch-BdG Hamiltonians. Line index theorem that relates a topological index to an analyti- 41,45,46,55–59 defects correspond to d−D=2 while point defects correspond to d cal index that counts the number of zero modes. −D=1. Temporal cycles for point defects correspond to d−D=0. In this paper we will not attempt to prove the index theorem. Rather, we will observe that the topological classes for H͑k,r͒ coincide with the expected classes of gapless defect that combine superconductivity, magnetism, and strong spin- modes. In this regards the dependence of the classification on orbit interactions.49–52 Recently, we showed that the exis- ␦ in Eq. ͑1.1͒ is to be expected. For example, a point defect tence of a Majorana bound state at a point defect in a three at the end of a one-dimensional system ͑␦=1−0͒ has the ͑␦ dimensional Bogoliubov de Gennes theory is related to a Z2 same classification as a point defects in two dimensions topological invariant that characterizes a family of Bogoliu- =2−1͒ and three dimensions ͑␦=3−2͒. H ͑ ͒ bov de Gennes Hamiltonians BdG k,r defined for r on a We will begin in Sec. II by describing the generalized surface surrounding the defect.53 This suggests that a more periodic table. We will start with a review of the Altland general formulation of topological defects and their corre- Zirnbauer symmetry classes44 and a summary of the proper- sponding gapless modes should be possible. ties of the table. In Appendix A we will justify this generali- In this paper we develop a general theory of topological zation of the table by introducing a set of mathematical map- defects and their associated gapless modes in Bloch and pings that relate Hamiltonians in different dimensions and Bloch-BdG theories in all symmetry classes. As in Ref. 53, different symmetry classes. In addition to establishing that we assume that far away from the defect the Hamiltonian the classifications depend only on ␦=d−D, these mappings varies slowly in real space, allowing us to consider adiabatic allow other features of the table, already present for D=0 to changes in the Hamiltonian as a function of the real space be easily understood, such as the pattern in which the clas- position r. We thus seek to classify Hamiltonians H͑k,r͒, sifications vary as a function of symmetry class as well as the where k is defined in a d-dimensional Brillouin zone ͑a torus Bott periodicity of the classes as a function of d. Td͒, and r is defined on a D-dimensional surface SD sur- In Secs. III and IV we will outline the physical conse- rounding the defect. A similar approach can be used to clas- quences of this theory by discussing a number of examples sify cyclic temporal variations in the Hamiltonian, which de- of line and point defects in different symmetry classes and fine adiabatic pumping cycles. Hereafter we will drop the dimensions. The simplest example is that of a line defect in a BdG subscript on the Hamiltonian with the understanding 3D system with no symmetries. In Sec. III A we will show that the symmetry class dictates whether it is a Bloch or BdG that the presence of a 1D chiral Dirac fermion mode ͑analo- Hamiltonian. gous to an integer quantum-Hall edge state͒ on the defect is In Fig. 1 we illustrate the types of topological defects that associated with an integer topological invariant that may be can occur in d=1,2,or3.ForD=0 we regard S0 as two interpreted as the winding number of the “␪” term that char- points ͕͑−1,+1͖͒. Our topological classification then classi- acterizes the magnetoelectric polarizability.10 This descrip- fies the difference of H͑k,+1͒ and H͑k,−1͒. A nontrivial tion unifies a number of methods for “engineering” chiral difference corresponds to an interface between two topologi- Dirac fermions, which will be described in several illustra- cally distinct phases. For D=1 the one parameter families of tive examples. Hamiltonians describe line defects in d=3 and point defects Related topological invariants and illustrative examples in d=2. For d=1 it could correspond to an adiabatic tempo- will be presented in Secs. III B–III E for line defects in other ral cycle H͑k,t͒. Similarly for D=2, the two parameter fam- symmetry classes that are associated with gapless 1D helical ily describes a point defect for d=3 or an adiabatic cycle for Dirac fermions, 1D chiral Majorana fermions, and 1D helical a point defects in d=2. Majorana fermions. In Sec. IV we will consider point defects Classifying the D parameter families of d-dimensional in 1D models with chiral symmetry such as the Jackiw-Rebbi Bloch-BdG Hamiltonians subject to symmetries leads to a model45 or the Su, Schrieffer, Heeger model,47 and in super- generalization of the periodic table discussed above. The conductors without chiral symmetry that exhibit Majorana original table corresponds to D=0. For DϾ0 we find that for bound states or Majorana doublets. These will also be related ͑ 46 a given symmetry class the topological classification Z, Z2, to the early work of Jackiw and Rossi on Majorana modes or 0͒ depends only on at point defects in a model with chiral symmetry.

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TABLE I. Periodic table for the classiﬁcation of topological defects in insulators and superconductors. The rows correspond to the different Altland Zirnbauer ͑AZ͒ symmetry classes while the columns distinguish different dimensionalities, which depend only on ␦=d −D.

Symmetry ␦=d−D s AZ ⌰2 ⌶2 ⌸2 01234567 0A 000Z 0 Z 0 Z 0 Z 0 1 AIII 0 0 1 0 Z 0 Z 0 Z 0 Z

0AI100Z 0002Z 0 Z2 Z2 1BDI1 11 Z2 Z 0002Z 0 Z2

2D 010Z2 Z2 Z 0002Z 0 3 DIII −1 1 1 0 Z2 Z2 Z 0002Z 4 AII −1 0 0 2Z 0 Z2 Z2 Z 000

5 CII −1 −1 1 0 2Z 0 Z2 Z2 Z 00 6C 0−10002Z 0 Z2 Z2 Z 0 7CI1−110002Z 0 Z2 Z2 Z

Finally, in Sec. V we will regard r as including a temporal H͑k,r͒ = ⌰H͑− k,r͒⌰−1, ͑2.1͒ variable, and apply the considerations in this paper to classify cyclic pumping processes. The Thouless charge where the antiunitary time reversal operator may be written ␲ y/ប pump60,61 corresponds to a nontrivial cycle in a system with ⌰=ei S K. Sy is the spin and K is complex conjugation. For no symmetries and ␦=0 ͑d=D=1͒. A similar pumping sce- spin-1/2 fermions, ⌰2 =−1, which leads to Kramers theorem. nario can be applied to superconductors and defines a fer- In the absence of a spin-orbit interaction, the extra invariance mion parity pump. This, in turn, is related to the non-Abelian of the Hamiltonian under rotations in spin space allows an statistics of Ising anyons and provides a framework for un- additional time-reversal operator ⌰Ј=K to be defined, which derstanding braidless operations on systems of three- satisfies ⌰Ј2 =+1. dimensional superconductors hosting Majorana fermion Particle-hole symmetry is expressed by bound states. Details of several technical calculations can be H͑k r͒ ⌶H͑ k r͒⌶−1 ͑ ͒ found in the Appendices. An interesting recent preprint by , =− − , , 2.2 Freedman et al.,62 which appeared when this manuscript was where ⌶ is the antiunitary particle-hole operator. Fundamen- in its final stages discusses some aspects of the classification tally, ⌶2 =+1. However, as was the case for ⌰, the absence of topological defects in connection with a rigorous theory of of spin-orbit interactions introduces an additional particle- non-Abelian statistics in higher dimensions. hole symmetry, which can satisfy ⌶2 =−1. Finally, chiral symmetry is expressed by a unitary opera- II. PERIODIC TABLE FOR DEFECT CLASSIFICATION tor ⌸, satisfying Table I shows the generalized periodic table for the clas- H͑k,r͒ =−⌸H͑k,r͒⌸−1. ͑2.3͒ sification of topological defects in insulators and superconductors. It describes the equivalence classes of Hamiltonians A theory with both particle-hole and time-reversal symme- ⌸ i␹⌰⌶ H͑k,r͒, that can be continuously deformed into one another tries automatically has a chiral symmetry =e . The ␹ ⌸2 without closing the energy gap, subject to constraints of phase can be chosen so that =1. ⌰2 Ϯ ⌶2 Ϯ ⌸2 ͑ particle-hole and/or time-reversal symmetry. These are map- Specifying =0, 1, =0, 1, and =0,1 here 0 ͒ pings from a base space defined by ͑k,r͒ to a classifying denotes the absence of symmetry defines the ten Altland- space, which characterizes the set of gapped Hamiltonians. Zirnbauer symmetry classes. They can be divided into two In order to explain the table, we need to describe ͑i͒ the groups: eight real classes that have anti unitary symmetries ⌰ ⌶ symmetry classes, ͑ii͒ the base space, ͑iii͒ the classifying and or plus two complex classes that do not have anti space, and ͑iv͒ the notion of stable equivalence. The repeat- unitary symmetries. Altland and Zirnbauer’s notation for ing patterns in the table will be discussed in Sec. II C. Much these classes, which is based on Cartan’s classification of of this section is a review of material in Refs. 35 and 36. symmetric spaces, is shown in the left-hand part of Table I. What is new is the extension to DϾ0. To appreciate the mathematical structure of the eight real symmetry classes it is helpful to picture them on an 8 h “clock,” as shown in Fig. 2. The x and y axes of the clock A. Symmetry classes represent the values of ⌶2 and ⌰2. The “time” on the clock The presence or absence of time reversal symmetry, can be represented by an integer s defined modulo 8. particle-hole symmetry, and/or chiral symmetry define the Kitaev36 used a slightly different notation to label the sym- ten Altland-Zirnbauer symmetry classes.44 Time-reversal metry classes. In his formulation, class D is described by a symmetry implies that real Clifford algebra with no constraints, and in the other

115120-3 JEFFREY C. Y. TEO AND C. L. KANE PHYSICAL REVIEW B 82, 115120 ͑2010͒

Θ2 Two families of Hamiltonians are stably equivalent if they CI AI BDI can be deformed into one another after adding an arbitrary number of trivial bands. Thus, trivial insulators with different 7 0 1 numbers of core energy levels are stably equivalent. Stable C D Ξ2 equivalence can be implemented by considering an expanded classifying space that includes an infinite number of extra 6 2 C / ϫ ϵഫϱ conduction and valence bands, 0 =U U U k=0Gϱ,k. CII AII DIII With this notion of stable equivalence, the equivalence classes of Hamiltonians H͑k,r͒ can be formally added and 543 ͓H ͔ subtracted. The addition of two classes, denoted 1 ͓H ͔ + 2 is formed by simply combining two independent FIG. 2. The eight real symmetry classes that involve the anti- Hamiltonians into a single Hamiltonian given by the matrix ͓H H ͔ unitary symmetries ⌰ ͑time reversal͒ and/or ⌶ ͑particle hole͒ are direct sum, 1 2 . Additive inverses are constructed ͓H ͔ specified by the values of ⌰2 = Ϯ1 and ⌶2 = Ϯ1. They can be vi- through reversing conduction and valence bands, 1 ͓H ͔ ͓H H ͔ ͓H H͔ sualized on an eight-hour “clock.” − 2 = 1 − 2 . − is guaranteed to the be trivial class ͓0͔. Because of this property, the stable equivalence classes form an Abelian group, which is the key element of K classes Clifford algebra elements are constrained to anticom- 63–65 mute with q positive generators. The two formulations are theory. related by s=q+2 mod 8. The complex symmetry classes Symmetries impose constraints on the classifying space. For the symmetry classes with chiral symmetry, Eq. ͑2.3͒ can similarly be indexed by an integer s defined modulo 2. C restricts n=2k and the classifying space to a subset 1 For all classes, the presence of chiral symmetry is associated ͑ϱ͒ʚ / ϫ ͑ ͒ with odd s. =U U U U. The antiunitary symmetries in Eqs. 2.1 and ͑2.2͒ impose further constraints. At the special points where k and −k coincide, the allowed Hamiltonians are de- B. Base space, classifying space and stable equivalence R scribed by the 8 classifying spaces q of real K theory. The Hamiltonian is defined on a base space composed of d momentum k, defined in a d-dimensional Brillouin zone T C. Properties of the periodic table and real-space degrees of freedom r in a sphere SD ͑or SD−1ϫS1 for an adiabatic cycle͒. The total base space is For a given symmetry class s, the topological classifica- therefore Td ϫSD ͑or Td ϫSD−1ϫS1͒. As in Ref. 36, we will tion of defects is given by the set of stable equivalence D+d simplify the topological classification by treating the base classes of maps from the base space ͑k,r͒෈S to the clas- space as a sphere Sd+D. The “strong” topological invariants sifying space, subject to the symmetry constraints. These d D ͑ ͒ that characterize the sphere will also characterize T ϫS . form the K group, which we denote as KC s;D,d for the ͑ ͒ However, there may be additional topological structure in complex symmetry classes and KR s;D,d for the real sym- Td ϫSD that is absent in Sd+D. These correspond to “weak” metry classes. These are listed in Table I. topological invariants. For D=0 these arise in layered struc- Table I exhibits many remarkable patterns. Many can be tures. A weak topological insulator, for example, can be un- understood from the following basic periodicities: derstood as a layered two dimensional topological insulator. ͑ ͒ ͑ ͒ ͑ ͒ There are similar layered quantum-Hall states. For D0, KF s;D,d +1 = KF s −1;D,d , 2.5 then there will also be a weak invariant if the Hamiltonian H͑ ͒ k,r0 for fixed r=r0 is topologically nontrivial. As is the K ͑s;D +1,d͒ = K ͑s +1;D,d͒. ͑2.6͒ case for the classification of bulk phases D=0, we expect F F that the topologically protected gapless defect modes are as- Here s is understood to be defined modulo 2 for F=C and sociated with the strong topological invariants. modulo 8 for F=R. We will establish these identities math- The set of Hamiltonians that preserve the energy gap ematically in Appendix A. The basic idea is to start with separating positive and negative energy states can be simpli- some Hamiltonian in some symmetry class s and dimension- fied without losing any topological information. Consider the alities D and d. It is then possible to explicitly construct two retraction of the original Hamiltonian H͑k,r͒ to a simpler new Hamiltonians in one higher dimension which have ei- Hamiltonian whose eigenvalue spectrum is “flattened” so ther ͑i͒ d→d+1 or ͑ii͒ D→D+1. These new Hamiltonians that the positive- and negative-energy states all have the belongs to new symmetry classes that are shifted by 1 “hour” Ϯ same energy E0. The flattened Hamiltonian is then speci- on the symmetry clock and characterized by ͑i͒ s→s+1 or fied the set of all n eigenvectors ͓defining a U͑n͒ matrix͔ ͑ii͒ s→s−1. We then go on to show that this construction modulo unitary rotations within the k conduction bands or defines a 1–1 correspondence between the equivalence the n−k valence bands. The flattened Hamiltonian can thus classes of Hamiltonians with the new and old symmetry be identified with a point in the Grassmanian manifold classes and dimensions, thereby establishing Eqs. ͑2.5͒ and ͑2.6͒. G = U͑n͒/U͑k͒ ϫ U͑n − k͒. ͑2.4͒ n,k The periodicities in Eqs. ͑2.5͒ and ͑2.6͒ have a number of It is useful to broaden the notion of topological equiva- consequences. The most important for our present purposes lence to allow for the presence of extra trivial energy bands. is they can be combined to give

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͑ ͒ ͑ ͒ ͑ ͒ TABLE II. Symmetry classes that support topologically non- KF s;D +1,d +1 = KF s;D,d . 2.7 trivial line defects and their associated protected gapless modes. This ͑1,1͒ periodicity shows that the dependence on the dimensions d and D only occurs via ␦=d−D. Thus the depen- Symmetry Topological classes 1D gapless Fermion modes dence of the classifications on D can be deduced from the A Chiral Dirac table for D=0. This is one of our central results. Z In addition, the periodicities in Eqs. ͑2.5͒ and ͑2.6͒ ex- D Z Chiral Majorana plain other features of the table that are already present for DIII Z2 Helical Majorana D=0. In particular, the fact that s is defined modulo 2 ͑8͒ for AII Z2 Helical Dirac the complex ͑real͒ classes leads directly to the Bott period- C2Z Chiral Dirac icity of the dependence of the classifications on d

͑ ͒ ͑ ͒ ͑ ͒ KC s;D,d +2 = KC s;D,d , 2.8 A. Class A: Chiral Dirac fermion 1. Topological invariant ͑ ͒ ͑ ͒ ͑ ͒ KR s;D,d +8 = KR s;D,d . 2.9 A line defect in a generic 3D Bloch band theory with no ͑ ͒ ͑ ͒ ͑ ͒ symmetries is associated with an integer topological invari- Moreover, Eqs. 2.5 and 2.6 show that Ka s;D,d depends ant. This determines the number of chiral Dirac fermion only on d−D−s. This explains the diagonal pattern in Table modes associated with the defect. Since H͑k,r͒ is defined on I, in which the dependence of the classification on d is re- a compact four-dimensional space, this invariant is naturally peated in successive symmetry classes. Thus, the entire table expressed as a second Chern number could be deduced from a single row. Equations ͑2.5͒ and ͑2.6͒ do not explain the pattern of 1 ͵ ͓F ∧ F͔ ͑ ͒ classifications within a single row. Since this is a well- n = 2 Tr , 3.1 8␲ 3ϫ 1 studied math problem there are many routes to the T S answer.64,66,67 One approach is to notice that for d=0, where K ͑s,D,0͒ is simply the Dth homotopy group of the appro- F F A A ∧ A ͑ ͒ priate classifying space which incorporates the symmetry = d + 3.2 ͑ ⌶2 constraints. For example, for class BDI s=1, =+1, and is the curvature form associated with the non-Abelian Ber- ⌰2 =+1͒ the classifying space is the orthogonal group O͑ϱ͒. A ͗ ͉ ͘ ry’s connection ij= ui duj characterizing the valence-band Then, K ͑1,D,0͒=␲ ͓O͑ϱ͔͒, which are well known. This ͉ ͑ ͒͘ 1 R D eigenstates uj k,s defined on the loop S parameterized by implies s. It is instructive to rewrite this as an integral over s of a ͑ ͒ ␲ ͓ ͑ϱ͔͒ ͑ ͒ KR s;D,d = s+D−d−1 O . 2.10 quantity associated with the local band structure. To this end, ͓F∧F͔ Q Additional insight can be obtained by examining the in- it is useful to write Tr =d 3, where the Chern-Simons terconnections between different elements of the table. For 3 form is example, the structure within a column can be analyzed by 2 Q ͫA ∧ A A ∧ A ∧ Aͬ ͑ ͒ considering the effect of “forgetting” symmetries. Hamilto- 3 =Tr d + . 3.3 3 nians belonging to the real chiral ͑nonchiral͒ classes are automatically in complex class AIII ͑A͒. There are therefore K Now divide the integration volume into thin slices, T3 group homomorphisms that send any real entries in Table I to ϫ⌬S1, where ⌬S1 is the interval between s and s+⌬s.In complex ones directly above. In particular, as detailed in each slice, Stokes’ theorem may be used to write the integral Appendix B this distinguishes the Z and 2Z entries, which as a surface integral over the surfaces of the slice at s and indicate the possible values of Chern numbers ͓or U͑n͒ s+⌬s. In this manner, Eq. ͑3.1͒ may be written winding numbers͔ for even ͑or odd͒ ␦. In addition, the di- 1 d mensional reduction arguments given in Refs. 10 and 68 lead n = Ͷ ds ␪͑s͒, ͑3.4͒ ␲ to a dimensional hierarchy, which helps to explain the pattern 2 S1 ds within a single row as a function of d. where 1 III. LINE DEFECTS ␪͑ ͒ ͵ Q ͑ ͒ ͑ ͒ s = 3 k,s . 3.5 4␲ 3 Line defects can occur at the edge of a 2D system ͑␦=2 T −0͒ or in a 3D system ͑␦=3−1͒. From Table I, it can be seen Equation ͑3.5͒ is precisely the Qi, Hughes, and Zhang that there are five symmetry classes which can host non- formula10 for the “␪” term that characterizes the magneto- trivial line defects. These are expected to be associated with electric response of a band insulator. ␪=0 for an ordinary gapless fermion modes bound to the defect. Table II lists time reversal invariant insulator, and ␪=␲ in a strong topo- nontrivial classes, along with the character of the associated logical insulator. If parity and time-reversal symmetry are gapless modes. In the following sections we will discuss broken then ␪ can have any intermediate value. We thus each of these cases, along with physical examples. conclude that the topological invariant associated with a line

115120-5 JEFFREY C. Y. TEO AND C. L. KANE PHYSICAL REVIEW B 82, 115120 ͑2010͒

AF-I AF-I F-I F-I θ=−ε θ=+ε θ=0 θ=0

TI θ=π TI θ=π (a) (b) s B I θ=0 I θ=0

AF-TI AF-TI F-TI F-TI θ = π−ε θ = π+ε θ=π θ=π FIG. 3. A line dislocation in a three-dimensional quantum-Hall (c) (d) state characterized by Burgers vector B. FIG. 4. Heterostructure geometries for chiral Dirac fermions. ͑a͒ defect, which determines the number of chiral fermion and ͑b͒ show antiferromagnetic or ferromagnetic insulators on the branches is given by the winding number of ␪. We now surface of a topological insulator with chiral Dirac fermions at a consider several examples of 3D line defects that are associ- domain wall. ͑c͒ and ͑d͒ show a domain wall in an antiferromag- ated with chiral Dirac fermions. netic or ferromagnetic topological insulator. Chiral fermion modes are present when the domain wall intersects the surface. 2. Dislocation in a 3D integer quantum-Hall state A three-dimensional integer quantum-Hall state can be Tr͓F ∧ F͔ =Tr͓B · ͑2F0 ∧ dk − d͓F0,ap͔͒ ∧ ds͔. ͑3.9͒ thought of as a layered version of the two-dimensional inte- Upon integrating Tr͓F∧F͔ the total derivative term vanishes ger quantum-Hall state. This can be understood most simply due to the periodicity of ap. Evaluating the integral is then by considering the extreme limit where the layers are com- straightforward. The integral over s trivially gives 1. We are pletely decoupled 2D systems. A line dislocation, as shown then left with in Fig. 3 will then involve an edge of one of the planes and be associated with a chiral fermion edge state. Clearly, the 1 ͑ ͒ chiral fermion mode will remain when the layers are n = B · Gc, 3.10 2␲ coupled, provided the bulk gap remains finite. Here we wish to show how the topological invariant in Eq. ͑3.1͒ reflects where this fact. 1 On a loop surrounding the dislocation parameterized by G = ͵ dk ∧ Tr͓F0͔. ͑3.11͒ c ␲ s෈͓0,1͔ we may consider a family of Hamiltonians H͑k,s͒ 2 T3 given by the Hamiltonian of the original bulk crystal displaced by a distance sB, where B is a lattice vector equal to Gc is a reciprocal lattice vector that corresponds to the triad the Burgers vector of the defect. The corresponding Bloch of Chern numbers that characterize a 3D system. For in- ͑ ␲/ ͒͑ ͒ wave functions will thus be given by stance, in a cubic system Gc = 2 a nx ,ny ,nz , where, for ͑ ␲͒−1͐ ͓F0 ͔ ∧ example nz = 2 Tr xy dkx dky, for any value of kz. ͑ ͒ 0 ͑ ͒ ͑ ͒ umk,s r = umk r − sB , 3.6 An equivalent formulation is to characterize the displaced crystal in terms of ␪. Though Eq. ͑3.5͒ cannot be used, Eqs. 0 ͑ ͒ where umk r are Bloch functions for the original crystal. It ͑3.1͒ and ͑3.4͒ can be used to implicitly define ␪ up to an then follows that the Berry’s connection is arbitrary additive constant A A0 ͓ p͑ ͔͒ ͑ ͒ ␪͑ ͒ ͑ ͒ = + B · k − a k ds, 3.7 s = sB · Gc. 3.12 where 3. Topological insulator heterostructures A0 ͑ ͒ ͗ 0 ٌ͉ ͉ 0 ͘ mn k = umk k unk · dk Another method for engineering chiral Dirac fermions is use heterostructures that combine topological insulators and and magnetic materials. The simplest version is a topological in- -p ͑ ͒ ͗ 0 ٌ͉͑ ͉͒ 0 ͘ ͑ ͒ sulator coated with a magnetic film that opens a time amn k = umk r + k unk . 3.8 reversal symmetry breaking energy gap at the surface. A do- With this definition, ap͑k͒ is a periodic function: ap͑k+G͒ main wall is then associated with a chiral fermion mode. In =ap͑k͒ for any reciprocal lattice vector G.69 this section we will show how this structure, along with If the crystal is in a three dimensional quantum-Hall state, some variants on the theme, fits into our general framework. then the nonzero first Chern number is an obstruction to We first describe the structures qualitatively and then analyze finding the globally continuous gauge necessary to evaluate a model that describes them. Eq. ͑3.5͒. We therefore use Eq. ͑3.1͒, which can be evaluated Figure 4 shows four possible configurations. Figures 4͑a͒ by noting that and 4͑b͒ involve a topological insulator with magnetic mate-

115120-6 TOPOLOGICAL DEFECTS AND GAPLESS MODES IN… PHYSICAL REVIEW B 82, 115120 ͑2010͒

rials on the surface. The magnetic material could be either chiral Dirac modes occur at certain step edges in such crys- ferromagnetic or antiferromagnetic. We distinguish these two tals. These chiral modes can also be understood in terms of cases based on whether inversion symmetry is broken or not. the invariant in Eq. ͑3.1͒. Note that these chiral modes sur- Ferromagnetism does not violate inversion symmetry while vive in the presence of disorder even though the bulk state antiferromagnetism does ͑at least for inversion about the does not. Thus, the chiral mode, protected by the strong in- middle of a bond͒. This is relevant because ␪, discussed variant in Eq. ͑3.1͒, is more robust than the bulk state that above, is quantized unless both time reversal and inversion gave rise to it. symmetries are violated. Of course, for a noncentrosymmet- If one imagines weakening the coupling between the two ric crystal inversion is already broken so the distinction is antiferromagnetic topological insulators ͑using our terminol- unnecessary. 71͒ ͑ ͒ ogy, not that of Mong, et al. and taking them apart, then at Figure 4 a shows a topological insulator capped with an- some point the chiral mode has to disappear. At that point, tiferromagnetic insulators with ␪= Ϯ⑀ separated by a domain rather than taking the “shortest path” between ␲Ϯ⑀, ␪ takes wall. Around the junction where the three regions meet ␪ a path that passes through 0. At the transition between the cycles between ␲,+⑀, and −⑀. Of course this interface structure falls outside the adiabatic regime that Eq. ͑3.5͒ is based “short-path” and the “long-path” regimes, the gap on the on. However, it is natural to expect that the physics would domain wall must go to zero, allowing the chiral mode to not change if the interface was “smoothed out” with ␪ taking escape. This will have the character of a plateau transition in the shortest smooth path connecting its values on either side the 2D integer quantum-Hall effect. of the interface. Structures involving magnetic topological insulators Figure 4͑b͒ shows a similar device with ferromagnetic would be extremely interesting to study because with them it insulators, for which ␪=0 or ␲. In this case the adiabatic is possible to create chiral fermion states with a single ma- assumption again breaks down, however, as emphasized in terial. Indeed, one can imagine scenarios where a magnetic Ref. 10, the appropriate way to think about the surface is that memory, encoded in magnetic domains, could be read by ␪ connects 0 and ␲ along a path that is determined by the measuring the electrical transport in the domain wall chiral sign of the induced gap, which in turn is related to the mag- fermions. netization. In this sense, ␪ cycles by 2␲ around the junction. To model the chiral fermions in these structures we begin In Figs. 4͑c͒ and 4͑d͒ we consider topological insulators with the simple three-dimensional model for trivial and to- which have a weak magnetic instability. If in addition to pological insulators considered in Ref. 10 ␪ϳ␲Ϯ⑀ time-reversal, inversion symmetry is broken, then . H = v␮ ␴ជ · k + ͑m + ⑀͉k͉2͒␮ . ͑3.13͒ Recently Li, Wang, Qi, and Zhang70 have considered such 0 x z ␴ ␮ materials in connection with a theory of a dynamical axion Here ជ represents spin and z describes an orbital degree of and suggested that certain magnetically doped topological freedom. mϾ0 describes the trivial insulator and mϽ0 de- insulators may exhibit this behavior. They referred to such scribes the topological insulator. An interface where m materials as topological magnetic insulators. We prefer to changes sign is then associated with gapless surface states. call them magnetic topological insulators because as mag- Next consider time-reversal symmetry-breaking perturba- netic insulators they are topologically trivial. Rather, they are tions, which could arise from exchange ﬁelds due to the pres- topological insulators to which magnetism is added. Irre- ence of magnetic order. Two possibilities include spective of the name, such materials would be extremely H ␮ ͑ ͒ interesting to study, and as we discuss below, may have im- af = haf y, 3.14 portant technological utility. ͑ ͒ H ជ ␴ ͑ ͒ Figure 4 c shows two antiferromagnetic topological insu- f = hf · ជ . 3.15 lators with ␪=␲Ϯ⑀ separated by a domain wall, and Fig. 4͑d͒ shows a similar device with ferromagnetic topological Either haf or hf,z will introduce a gap in the surface states but H insulators. They form an interface with an insulator, which they have different physical content. 0 has an inversion H ͑ ͒ H ͑ ͒ ␮ could be vacuum. Under the same continuity assumptions as symmetry given by 0 k = P 0 −k P with P= z. Clearly, H H above the junction where the domain wall meets the surface f respects this inversion symmetry. af does not respect P ⌰ H will be associated with a chiral fermion mode. Like the struc- but does respect P . We therefore associate f with ferro- H ture in Fig. 4͑a͒, this may be interpreted as an edge state on magnetic order and af with antiferromagnetic order. a domain wall between the “half-quantized” quantum-Hall Within the adiabatic approximation, the topological in- ͑ ͒ states of the topological insulator surfaces. However, an variant in Eq. 3.1 can be evaluated in the presence of either ͑ ͒ ͑ ͒ equally valid interpretation is that the domain wall itself Eq. 3.14 or 3.15 . The antiferromagnetic perturbation in ͑ ͒ H forms a single two-dimensional integer quantum-Hall state Eq. 3.14 is most straightforward to analyze because 0 with an edge state. Our framework for topologically classi- +Haf is a combination of ﬁve anticommuting Dirac matrices. fying the line defects underlies the equivalence between On a circle surrounding the junction parameterized by s it these two points of view. can be written in the general form 71 Mong, Essen, and Moore have introduced a different H͑k,s͒ = h͑k,s͒ · ␥ជ , ͑3.16͒ kind of antiferromagnetic topological insulator that relies on ␥ ͑␮ ␴ ␮ ␴ ␮ ␴ ␮ ␮ ͒ ͑ ͒ ͓ ͑ ͒ the symmetry of time reversal combined with a lattice trans- where ជ = x x , x y , x z , z , y and h k,s = vk,m s ⑀͉ ͉2 ͑ ͔͒ lation. Due to the necessity of translation symmetry, how- + k ,haf s . For a model of this form, the second Chern ever, such a phase is not robust to disorder. They found that number in Eq. ͑3.1͒ is given simply by the winding number

115120-7 JEFFREY C. Y. TEO AND C. L. KANE PHYSICAL REVIEW B 82, 115120 ͑2010͒

of the unit vector dˆ ͑k,s͒=h/͉h͉෈S4 as a function of k and s. MM SM This is most straightforward to evaluate in the limit ⑀→0, where dˆ is confined to the “equator” ͑d ,d ,d ,0,0͒ every- 1 2 3 TS TI where except near kϳ0 and ͉k͉տ1/⑀. The winding number (a) (b) is determined by the behavior at kϳ0, and may be expressed by Eq. ͑3.4͒ with ␪ given by FIG. 5. Heterostructure geometries for Chiral Majorana fermions. ͑a͒ shows a magnetic domain wall on the surface of a topo- m + ih logical superconductor while ͑b͒ shows an interface between a su- ei␪ = af . ͑3.17͒ ͱ 2 2 perconductor and a magnet on the surface of a topological insulator. m + haf We therefore expect a topological line defect to occur at 2. Dislocation in a layered topological superconductor an intersection between planes where m and haf change sign. The simplest example to consider is a dislocation in a The chiral fermion mode associated with this defect can seen three-dimensional superconductor. The discussion closely explicitly if we solve a simple linear model, m= fzz, haf parallels Sec. IIIA2and we find = fyy. This model, which has the form of a harmonic oscillator, is solved in Appendix C, and explicitly gives the chiral 1 ñ = B · G˜ , ͑3.21͒ Dirac fermion mode with dispersion 2␲ c ͑ ͒ ͑ ͒ ͑ ͒ ˜ E kx = v sgn fzfy kx. 3.18 where B is the Burgers vector of the dislocation and Gc characterizes the triad of first Chern numbers characterizing the 3D BdG Hamiltonian. A 3D system consisting of layers B. Class D: Chiral Majorana fermions of a 2D topological superconductor will be characterized by ˜ 1. Topological invariant a nonzero Gc. Since, as a 3D superconductor, the layered structure is in the topologically trivial class, such a state A line defect in a superconductor without time-reversal could be referred to as a weak topological superconductor. symmetry is characterized by an integer topological invariant The simplest model system in this class is a stack of 2D that determines the number of associated chiral Majorana px +ipy superconductors. A dislocation would then have ñ fermion modes. Since the BdG Hamiltonian characterizing a =1 and a single chiral Majorana fermion branch. A possible superconductor has the same structure as the Bloch Hamil- physical realization of the weak topological superconductor tonian, we can analyze the problem by “forgetting” about the state is Sr2RuO4, which may exhibit triplet px +ipy pairing. particle-hole symmetry and treating the BdG Hamiltonian as Since the spin-up and spin-down electrons make two copies if it was a Bloch Hamiltonian. The second Chern number, of the spinless state, a dislocation will be associated with ñ given by Eq. ͑3.1͒ can be defined. It can be verified that any =2. Thus, we predict that there will be two chiral Majorana value of the Chern number is even under particle-hole sym- modes bound to the dislocation, which is the same as a single metry so that particle-hole symmetry does not rule out a chiral Dirac fermion mode. nonzero Chern number. We may follow the same steps as Eqs. ͑3.1͒–͑3.5͒ to express the integer topological invariant 3. Superconductor heterostructures as We now consider heterostructures with associated chiral Majorana modes. The simplest to consider is a BdG analog 1 ñ = ͵ Tr͓F˜ ∧ F˜ ͔, ͑3.19͒ of the structures considered in Fig. 5. These would involve, ␲2 8 T3ϫS1 for example, an interface between a 3D time-reversal invariant topological superconductor with a magnetic material with F˜ a magnetic domain wall. The analysis of such a structure is where is the curvature form characterizing the BdG theory. ͑ ͒ As in Eq. ͑3.4͒, ñ may be expressed as a winding number of similar to that in Eq. 3.13 if we replace the Pauli matrices ␮ជ ˜␪ describing the orbital degree of freedom with Pauli matri- , which is expressed as an integral over the Brillouin zone ces describing Nambu space ␶ជ. Protected chiral Majorana ˜ of the Chern-Simons 3 form. The difference between n and n fermion modes of this sort on the surface of 3He-B with a is that ñ characterizes a BdG Hamiltonian. If we considered magnetic domain wall have been recently discussed by the BdG Hamiltonian for a nonsuperconducting insulator, Volovik.72 then due to the doubling in the BdG equation, we would find In Ref. 48 a different method for engineering chiral Ma- jorana fermions was introduced by combining an interface ͑ ͒ ñ =2n. 3.20 between superconducting and magnetic regions on the surface of a topological insulator. To describe this requires the ␲ In this case, the chiral Dirac fermion that occurs for a 2 eight-band model introduced in Ref. 53 ͑n=1͒ winding of ␪ corresponds to a 4␲ ͑n=2͒ winding of ˜␪. ␶ ␮ ␴ជ ͑ ⑀͉ ͉2͒␶ ␮ ⌬␶ ␮ ͑ ͒ Superconductivity allows for the possibility of a 2␲ winding H = z x · k + m + k z z + x + h y. 3.22 in ˜␪: a chiral Dirac fermion can be split into a pair of chiral ͑Here, for simplicity we consider only the antiferromagnetic Majorana fermions. term͒. The surface of the topological insulator occurs at a

115120-8 TOPOLOGICAL DEFECTS AND GAPLESS MODES IN… PHYSICAL REVIEW B 82, 115120 ͑2010͒ domain wall ͑say, in the x-y plane͒, where m͑z͒ changes sign. 1 ␯ = ͩ͵ Tr͓F ∧ F͔ − ͵ Q ͪmod 2, The superconducting order parameter ⌬ and magnetic pertur- 2 3 8␲ 3 1 3 1 1/2͒T ϫS͑ץ bation h both lead to an energy gap in the surface states. This ͑1/2͒T ϫS Hamiltonian is straightforward to analyze because ͑3.25͒ ͓H,␶ ␮ ͔=0, which allows the 8ϫ8 problem to be divided x y where F and Q are expressed in terms of the Berry’s con- into two 4ϫ4 problems, which have superconducting/ 3 nection A using Eqs. ͑3.2͒ and ͑3.3͒. The integral is over half magnetic mass terms ⌬Ϯh. Near a defect where ⌬=h the of the base space ͑1/2͒͑T3 ϫS1͒, defined such that ͑k,r͒ and ⌬+h gap never closes while the ⌬−h gap can be critical. ͑−k,r͒ are never both included. The second term is over the ⌬Ͼh leads to a superconducting state while ⌬Ͻh leads to a boundary of ͑1/2͒͑T3 ϫS1͒, which is closed under ͑k,r͒ quantum-Hall-type state. There is a transition between the →͑−k,r͒. Equation ͑3.25͒ must be used with care because two at ⌬=h. the Chern Simons form in the second term depends on the An explicit model for the line defect can be formulated gauge. A different continuous gauge can give a different ␯, with m͑z͒= f z, ⌬−h= f y and ⌬+h=M. The topological in- z y but due to Eq. ͑3.24͒, they must be related by an even inte- variant in Eq. ͑3.19͒ can be evaluated using a method similar ger. Thus, an odd number is distinct. to Eq. ͑3.16͒ and the chiral Majorana states can be explicitly In addition to satisfying Eq. ͑3.24͒, it is essential to use a solved along the lines of Eq. ͑3.18͒. ϫ 1 3 1 ץ Q gauge in which at least 3 is continuous on 2 T S ͑ 1 3 ϫ 1͒ though not necessarily on all of 2 T S . This continuous C. Class AII: Helical Dirac fermions gauge can always be found if the base space is a sphere S4. 3 ϫ 1 1. Topological invariant However for T S , the weak topological invariants can pose an obstruction to finding a continuous gauge. We will Line defects for class AII are characterized by a Z2 topo- show how to work around this difficulty at the end of the logical invariant. To develop a formula for this invariant we following section. follow the approach used in Ref. 73 to describe the invariant characterizing the quantum spin-Hall insulator. 2. Dislocation in a weak topological insulator As in the previous section, a line defect in three dimen- ͑ ͒෈ 3 Ran, Zhang, and Vishwanath recently studied the problem sions is associated with a four parameter space k,r T 75 ϫS1. Due to time-reversal symmetry, the second Chern num- of a line dislocation in a topological insulator. They found ber that characterized the line defects in Eq. ͑3.1͒ must be that an insulator with nontrivial weak topological invariants zero. Thus there is no obstruction to defining Bloch basis can exhibit topologically protected helical modes at an ap- functions ͉u͑k,r͒͘ continuously over the entire base space. propriate line dislocation. In this section we will show that ͑ ͒ these protected modes are associated with a nontrivial Z2 However, the time reversal relation between −k,r and ͑ ͒ ͑k,r͒ allows for an additional constraint so that the state is invariant in Eq. 3.25 . In addition to providing an explicit specified by the degrees of freedom in half the Brillouin example for this invariant, this formulation provides addi- zone. tional insight into why protected modes can exist in a weak As in Ref. 73 it is useful to define a matrix topological insulator. As argued in Ref. 9 and 12, the weak topological invariants lose their meaning in the presence of ͑ ͒ ͗ ͑ ͉͒⌰͉ ͑ ͒͘ ͑ ͒ disorder. The present considerations show that the helical wmn k,r = um k,r un − k,r . 3.23 modes associated with the dislocation are protected by the strong topological invariant associated with the line defect. ͉ ͑ ͒͘ ͉ ͑ ͒͘ Because um k,r and un −k,r are related by time- Thus if we start with a perfect crystal and add disorder, then ͑ ͒ reversal symmetry w k,r is a unitary matrix that depends the helical modes remain, even though the crystal is no on the gauge choice for the basis functions. Locally it is longer a weak topological insulator. The helical modes re- possible to choose a basis in which main even if the disorder destroys the crystalline order, so that dislocations become ill defined, provided the mobility ͑ ͒ ͑ ͒ w k,r = w0, 3.24 gap remains finite in the bulk crystal. In this case, the Hamil- tonian has a nontrivial winding around the line defect, even ͑Ϯ ͒ where w0 is independent of k and r so that states at k,r though the defect has no obvious structural origin. Thus, the T have a fixed relation. Since for k=0 w=−w , w0 must be weak topological insulator provides a route to realizing the ␴ antisymmetric. A natural choice is thus w0 =i 2 1. topologically protected line defect. But once present, the line The Z2 topological invariant is an obstruction to finding defect is more robust than the weak topological insulator. ͑ ͒ such a constrained basis globally. The constrained basis can To evaluate the Z2 invariant in Eq. 3.25 for a line dislo- be defined on two patches but the basis functions on the two cation we repeat the analysis in Sec. IIIA2. Because of the patches are necessarily related by a topologically nontrivial subtlety with the application of Eq. ͑3.25͒ we will first con- transition function. In this sense, the Z2 invariant resembles sider the simplest case of a dislocation in a weak topological the second Chern number in Eq. ͑3.1͒. insulator. Afterward we will discuss the case of a crystal with In Appendix E we will generalize the argument developed both weak and strong invariants. in Ref. 73 to show that the transition function relating the The Bloch functions on a circle surrounding a dislocation ͑ ͒ ͓F∧F͔ two patches defines the Z2 topological invariant, which may are described by Eq. 3.6 and the evaluation of Tr be written74 proceeds exactly as in Eqs. ͑3.7͒–͑3.9͒. To evaluate the sec-

115120-9 JEFFREY C. Y. TEO AND C. L. KANE PHYSICAL REVIEW B 82, 115120 ͑2010͒

ond term in Eq. ͑3.25͒ we need the Chern-Simons 3 form. is the same as the criterion for the existence of protected One approach is to use Eqs. ͑3.3͒ and ͑3.7͒. However, this is helical modes on a dislocation Ran, Zhang, and .1/2͒͑T3 ϫS1͒ because A has a Vishwanath75 derived using a different method͑ץ not continuously defined on term B·kds that is discontinuous at the Brillouin zone Evaluating Eq. ͑3.28͒ in a crystal that is both a strong boundary. An alternative is to write topological insulator and a weak topological insulator ͑such ͒ Q ͓ ͑ A0 ∧ ͓F0 p͔͒ ∧ ͔ ͑ ͒ as Bi1−xSbx is problematic because the 2D invariants evalu- 3 =Tr B · 2 dk − ,a ds . 3.26 / ated on the planes x1 =0 and x1 =1 2 are necessarily different From Eq. ͑3.9͒ this clearly satisfies Tr͓F∧F͔=dQ and it is in a strong topological insulator. This arises because a non- 3 ␯ 1/2͒͑T3 ϫS1͒ as long as A0 is trivial strong topological invariant 0 is an obstruction to͑ץ defined continuously on 0 3 1/2͒T so Eq. ͑3.29͒ cannot͑ץ 1/2͒T3. For a weak topological continuously defining A on͑ץ continuously defined on / insulator this is always possible, provided ͑1/2͒T3 is defined be evaluated continuously between x1 =0 and x1 =1 2. From appropriately. Equation ͑3.26͒ differs from Eq. ͑3.3͒ by a the point of view of the topological classification of the de- 3 ϫ 1 ␯ total derivative. fect on T S , 0 is like a weak topological invariant be- 3 Combining Eqs. ͑3.9͒, ͑3.25͒, and ͑3.26͒, the terms in- cause it a property of T and is independent of the real-space 1 volving ap cancel because ap is globally defined. ͑Note that parameter s in S . Thus this complication is a manifestation ap is unchanged by a k-dependent—but of the fact that topological classification of Hamiltonians on 3 1 4 r-independent—gauge transformation͒. This cannot be said T ϫS has more structure than those on S . The problem is 3 1 of the term involving A0, however, because in a weak topo- not with the existence of the invariant ␯ on T ϫS but rather logical insulator A0 is not globally defined on ͑1/2͒T3. Per- with applying the formulas ͑3.25͒ and ͑3.28͒. The problem forming the trivial integral over s we then find can be circumvented with the following trick. Consider an auxiliary Hamiltonian H˜ ͑k,r͒=H͑k,r͒ 1 H ͑ ͒ H ␯ = B · G␯ mod 2, ͑3.27͒ STI k , where STI is a simple model Hamiltonian for a 2␲ strong topological insulator such as Eq. ͑3.13͒, which can be where chosen such that it is a constant independent of k everywhere except in a small neighborhood close to k=0 where a band inversion occurs. Adding such a Hamiltonian that is indepen- ͵ ͓F0͔ ∧ ͵ ͓A0͔ ∧ G␯ = Tr dk − Tr dk. dent of r will have no effect on the topologically protected 1/2T3ץ 1/2T3 modes associated with a line defect so we expect the invari- ͑3.28͒ ant ␯ to be the same for both H͑k,r͒ and H˜ ͑k,r͒.IfH͑k,r͒ ␯ The simplest case to consider is a weak topological insu- has a nontrivial strong topological invariant 0 =1 then H˜ ͑ ͒ ␯ ͑ ͒ lator consisting of decoupled layers of 2D quantum spin-Hall k,r will have 0 =0, so that Eq. 3.28 can be applied. G␯ insulator stacked with a lattice constant a in the z direction. will then be given by the 2D invariant in Eq. ͑3.29͒ evaluated F0 F0͑ ͒ In this case = kx ,ky is independent of kz so the kz H˜ H ͑ ͒ for , which will be independent of x1. Since STI k is integral can be performed trivially. This leads to G␯ k-independent everywhere except a neighborhood of k=0, =͑2␲/a͒␯zˆ, where H this will agree with the 2D invariant evaluated for at x1 / i =1 2, but not x1 =0. It then follows that even in a strong ␯ = ͫ͵ Tr͓F0͔ − ͵ Tr͓A0͔ͬ ͑3.29͒ ␲ topological insulator the invariant characterizing a line dis- 1/2T2ץ 1/2T2 2 location is given by Eq. ͑3.27͒, where G␯ is given by Eq. ͑3.30͒ in terms of the weak topological invariants. is the 2D Z2 topological invariant characterizing the individual layers. Equation ͑3.28͒ also applies to a more general 3D weak 3. Heterostructure geometries topological insulator. A weak topological insulator is charac- In principle, it may be possible to realize 1D helical fer- ͑␯ ␯ ␯ ͒ terized by a triad of Z2 invariants 1 2 3 that define a mod mions in a 3D system that does not rely on a weak topologi- 9,12 2 reciprocal lattice vector cal insulating state. It is possible to write down a 3D model, ␯ ␯ ␯ ͑ ͒ analogous to Eq. ͑3.13͒ that has bound helical modes. How- G␯ = 1b1 + 2b2 + 3b3, 3.30 ever, it is not clear how to physically implement this model. where bi are primitive reciprocal lattice vectors correspond- This model will appear in a more physical context as a BdG ͑ ␲␦ ͒ ing to primitive lattice vectors ai such that ai ·bj =2 ij . theory in the following section. ␯ The indices i can be determined by evaluating the 2D invariant in Eq. ͑3.29͒ on the time-reversal invariant plane ␲ D. Class DIII: Helical Majorana fermions k·ai = . To show that G␯ in Eqs. ͑3.28͒ and ͑3.30͒ are equivalent, Line defects for class DIII are characterized by a Z2 topo- ͑ ͒ consider G␯ ·a1 in Eq. 3.28 . If we write k=x1b1 +x2b2 logical invariant that signals the presence or absence of 1D +x3b3, then the integrals over x2 and x3 have the form of Eq. helical Majorana fermion modes. As in Sec. III B, the BdG ͑ ͒ 3.29 . Since this is quantized, it must be independent of x1 Hamiltonian has the same structure as a Bloch Hamiltonian, / and will be given by its value at x1 =1 2. This then gives and the Z2 invariant can be deduced by “forgetting” the ␲␯ G␯ ·a1 =2 1. A similar analysis of the other components particle-hole symmetry, and treating the problem as if it was establishes the equivalence. A nontrivial value of Eq. ͑3.27͒ a Bloch Hamiltonian in class AII.

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TABLE III. Symmetry classes supporting nontrivial point topological defects and their associated E=0 modes. SSϕ=π ϕ=0 Symmetry Topological classes E=0 bound states TI AIII Z Chiral Dirac BDI Z Chiral Majorana

D Z2 Majorana FIG. 6. Helical Majorana fermions at a linear Josephson junc- Majorana Kramers doublet ͑ ͒ tion with phase difference ␲ on the surface of a topological DIII Z2 =Dirac insulator. CII 2Z Chiral Majorana Kramers Doublet ͑=Chiral Dirac͒ There are several ways to realize helical Majorana fermions. The simplest is to consider the edge of a 2D time- reversal invariant superconductor or superfluid, or equiva- superconductor without spin orbit interactions that has even lently a dislocation in a layered version of that 2D state. A parity singlet pairing. Line defects are characterized by an second is to consider a topological line defect in a 3D class integer topological invariant that determines the number of DIII topological superconductor or superfluid. Such line de- chiral Majorana fermion modes associated with the line. As fects are well known in of 3He B ͑Refs. 41 and 76͒ and have in class D, this may be evaluated by forgetting the particle- recently been revisited in Refs. 37 and 77. hole symmetry and evaluating the corresponding Chern num- Here we will consider a different realization that uses to- ber that would characterize class A. The 2Z in Table II for pological insulators and superconductors. Consider a linear this case, however, means that the Chern integer computed in junction between two superconductors on the surface of a this manner is necessarily even. This means that there will topological insulator as shown in Fig. 6. In Ref. 48 it was necessarily be an even number 2n of chiral Majorana fer- shown that when the phase difference between the supercon- mion modes, which may equivalently viewed as n chiral ductors is ␲ there are gapless helical Majorana modes that Dirac fermion modes. An example of such a system would 78 propagate along the junction. This can be described by an bea2Ddx2−y2 +idxy superconductor, which exhibits chiral eight-band minimal model that describes a topological insu- Dirac fermion edge states, or equivalently a dislocation in a lator surface with a superconducting proximity effect 3D layered version of that state. H ␶ ␮ ␴ ͑ ⑀͉ ͉2͒␶ ␮ ⌬ ␶ ͑ ͒ = v z x ជ · k + m + k z z + 1 x. 3.31 Here m is the mass describing the band inversion of a topo- IV. POINT DEFECTS ͑ ͒ ⌬ logical insulator, as in Eq. 3.22 , and 1 is the real part of Point defects can occur at the end of a 1D system ͑␦=1 the superconducting gap parameter. This model has time re- ͒ ͑␦ ͒ ͑␦ ⌰ ␴ −0 or at topological defects in 2D =2−1 or 3D =3 versal symmetry with =i yK and particle-hole symmetry ͒ ␦ ⌶ ␴ ␶ −2 systems. From the =1 column Table I, it can be seen with = y yK. The imaginary part of the superconducting ⌬ ␶ that there are five symmetry classes that can have topologi- gap, 2 y violates time-reversal symmetry. A line junction ␲ cally nontrivial point defects. These are expected to be asso- along the x direction with phase difference at the surface ciated with protected zero-energy bound states. Table III lists of a topological insulator corresponds to the intersection of ͑ ͒ ⌬ ͑ ͒ the nontrivial classes, along with the character of the associ- planes where m z and 1 y change sign. ated zero modes. In this section we will discuss each of these The Z2 invariant characterizing such a line defect is ͓H ␮ ␶ ͔ cases. straightforward to evaluate because , y x =0. This extra symmetry allows a “spin Chern number” to be defined, n␴ ͑ ␲2͒−1͐ ͓␮ ␶ F∧F͔ = 16 Tr y x . Since the system decouples into A. Classes AIII, BDI, and CII: Chiral zero modes two time reversed versions of Eq. ͑3.13͒, n␴ =1. By repeating ␯ 1. Topological invariant and zero modes the formulation in Appendix E of the Z2 invariant ,itis ␯ straightforward to show that this means =1. Point defects in classes AIII, BDI, and CII are character- The helical modes can be explicitly seen by solving the ized by integer topological invariants. The formula for this ⌬ linear theory, m= fzz, 1 = fyy, which leads to the harmonic integer invariant can be formulated by exploiting the chiral oscillator model studied in Appendix C. In the space of the symmetry in each class. In a basis where the chiral symmetry two zero modes the Hamiltonian has the form ⌸ ␶ operator is = z, the Hamiltonian may be written H ␴ ͑ ͒ = vkx x 3.32 0 q͑k,r͒ H͑k,r͒ = ͩ ͪ. ͑4.1͒ and describes 1D helical Majorana fermions. q͑k,r͒† 0 When the Hamiltonian has a flattened eigenvalue spectrum E. Class C: Chiral Dirac fermions H2 =1, q͑k,r͒ is a unitary matrix. For a point defect in d We finally briefly consider line defects in class C. Class C dimensions, the Hamiltonian as a function of d momentum can be realized when time-reversal symmetry is broken in a variables and D=d−1 position variables is characterized by

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the winding number associated with the homotopy In either case, the invariant in Eq. ͑4.2͒ changes by 1 when m ␲ ͓ ͑ →ϱ͔͒ 2d−1 U n =Z, which is given by changes sign. ͑d −1͒! 3. Jackiw-Rossi Model in d=2 n = ͵ Tr͓͑qdq†͒2d−1͔. ͑4.2͒ ͑ ͒ ͑ ␲ ͒d 2d −1 ! 2 i TdϫSd−1 Jackiw and Rossi introduced a two-dimensional model that has protected zero modes.46 The Hamiltonian can be For a Hamiltonian that is built from anticommuting Dirac written matrices, H͑k,r͒=dˆ ͑k,r͒·␥ជ, this invariant is given simply H͑ ͒ ␥ជ ⌫ជ ␾ជ ͑ ͒ ͑ ͒ by the winding degree of the mapping dˆ ͑k,r͒ from Td k,r = v · k + · r , 4.6 ϫSd−1 to S2d−1, which is expressed as an integral of the Jaco- ͑ ͒ ͑␥ ␥ ͒ ͑⌫ ⌫ ͒ where k= kx ,ky , and 1 , 2 and 1 , 2 are anticommut- bian ing Dirac matrices. They showed that the core of a vortex where ␾=␾ +i␾ winds by 2␲n is associated with n zero dˆ ͑k,r͒ 1 2ץ !d −1͒͑ n = ͵ ddkdd−1r . ͑4.3͒ modes that are protected by the chiral symmetry. Viewed as a d−1ץ dץ ␲d 2 TdϫSd−1 k r BdG Hamiltonian, these zero modes are Majorana bound states. ͑ ͒ In class AIII there are no constraints on q k,r other than This can be interpreted as a Hamiltonian describing su- unitarity so all possible values of n are possible. There are perconductivity in Dirac fermions. In this interpretation the additional constraints for the chiral classes with antiunitary ͑␥ ␥ ͒ ␶ ͑␴ ␴ ͒ Dirac matrices are expressed as 1 , 2 = z x , y and symmetries. As shown in Appendix B, this is simplest to see ͑⌫ ⌫ ͒ ͑␶ ␶ ͒ ␴ 1 , 2 = x , y , where ជ is a Pauli matrix describing spin by analyzing the constraints on the winding degree discussed and ␶ជ describes particle-hole space. The superconducting above. n must be zero in classes CI and DIII. There is no ⌬ ␾ ␾ pairing term is = 1 +i 2. In this interpretation a vortex constraint on n in class BDI while n must be even in class ⌰ ␴ violates the physical time-reversal symmetry =i yK. How- CII. ever, even in the presence of a vortex this model has a ficti- The topological invariant is related to an index that char- ⌰˜ ␴ ␶ acterizes the chirality of the zero modes tious “time-reversal symmetry” = x xK which satisfies ⌰˜ 2 =+1. This symmetry would be violated by a finite chemi- ͑ ͒ ␮␶ n = N+ − N−, 4.4 cal potential term z. Combined with particle-hole symme- ⌶ ␴ ␶ ͑⌶2 ͒ ⌰˜ where NϮ are the number of zero modes that are eigenstates try = y yK =+1 , defines the BDI class with chiral ⌸ ␴ ␶ of ⌸ with eigenvalue Ϯ1. To see that these zero modes are symmetry = z z. indeed protected consider N =nϾ0 and N =0. Any term in Evaluating the topological invariant in Eq. ͑4.2͒ again re- + − ͉ ͉→ϱ the Hamiltonian that could shift any of the N+ degenerate quires a k regularization. One possibility is to add ⑀͉ ͉2␶ ͉ ͉→ϱ states would have to have a nonzero matrix element connect- k x so that k can be replaced by a single point. In ing states with the same chirality. Such terms are forbidden, this case the invariant can be determined by computing the though, by the chiral symmetry ͕H,⌸͖=0. In the supercon- winding degree of dˆ ͑k,r͒ on S3. In the limit ⑀→0 the k ducting classes BDI and CII the zero energy states are Ma- integral can be performed so that Eq. ͑4.2͒ can be expressed ␾ ␾ ͉⌬͉ i␸ jorana bound states. In class CII, however, since time rever- as the winding number of the phase of 1 +i 2 = e sal symmetry requires that n must be even, the paired 1 Majorana states can be regarded as zero-energy Dirac fer- n = ͵ d␸. ͑4.7͒ ␲ mion states. 2 S1 In the special case where H͑k,r͒ has the form of a mas- sive Dirac Hamiltonian, by introducing a suitable regulariza- 4. Hedgehogs in d=3 tion for ͉k͉→ϱ the topological invariant in Eqs. ͑4.2͒ and ͑4.3͒ can be expressed in a simpler manner as a topological In Ref. 53 we introduced a three dimensional model for invariant characterizing the mass term. In the following sec- Majorana bound states that can be interpreted as a theory of tions we consider this in the three specific cases d=1,2,3. a vortex at the interface between a superconductor and a topological insulator. In the special case that the chemical potential is equal to zero, model has the same form as Eq. 2. Solitons in d=1 ͑4.6͒, except that now all of the vectors are three dimen- The simplest topological zero mode occurs in the Jackiw- sional. In the topological insulator model we have ␥ជ 45 ͑␥ ␥ ␥ ͒ ␮ ␶ ␴ ⌫ជ ͑⌫ ⌫ ⌫ ͒ ͑␮ ␶ ␶ ␶ ͒ ␶ Rebbi model, which is closely related to the Su, Schrieffer, = 1 , 2 , 3 = x z ជ and = 1 , 2 , 3 = z z , x , y . ជ and and Heeger model.47 Consider ␴ជ are defined as before while ␮ជ describes a orbital degree of ⌸ ␮ ␶ H͑ ͒ ␴ ␴ ͑ ͒ freedom. The chiral symmetry, = y z is violated if a k,x = vk x + m y. 4.5 ␮␶ chemical potential term z is included. Domain walls where m͑x͒ changes sign as a function of x are Following the same steps that led to Eq. ͑4.7͒ the invari- associated with the well known zero-energy soliton states. ant Eq. ͑4.2͒ is given by the winding number of ␾ˆ =␾ជ /͉␾ជ ͉ on To analyze the topological class requires a regularization S2 for ͉k͉→ϱ. This can either be done with a lattice, as in the ⑀ 2␴ 1 Su, Schrieffer, Heeger model or by adding a term k y,asin n = ͵ ␾ˆ · ͑d␾ˆ ϫ d␾ˆ ͒. ͑4.8͒ ␲ Eq. ͑3.13͒ so that ͉k͉→ϱ can be replaced by a single point. 4 S2

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B. Class D: Majorana bound states seen by considering the edge states of the topological superconductor in the presence of a hole.39 Particle-hole symmetry 1. Topological invariant requires that the quantized edge states come in pairs. When Point defects in class D are characterized by a Z2 topo- the flux is an odd multiple of h/2e, the edge states are quan- logical invariant that determines the presence or absence of a tized such that a zero mode is present. In this section we will Majorana bound state associated with the defect. These in- evaluate the topological invariant in Eq. ͑4.9͒ associated with clude the well-known end states in a 1D p-wave supercon- a loop surrounding the vortex. ductor and vortex states in a 2D p +ip superconductor. In H0͑ ͒ x y We begin with the class D BdG Hamiltonian p kx ,ky Ref. 53 we considered such zero modes in a three- characterizing the topological superconductor when the su- dimensional BdG theory describing Majorana zero modes in perconducting phase is zero. We include the subscript p to topological insulator structures. Here we develop a unified denote the first Chern number that classifies the topological description of all of these cases. superconductor. We can then introduce a nonzero supercon- For a point defect in d dimensions, the Hamiltonian de- ducting phase by a gauge transformation pends on d momentum variables and D=d−1 position vari- ␸␶ / ␸␶ / H ͑k ␸͒ e−i z 2H0͑k͒ei z 2 ͑ ͒ ables. In Appendix D we show that the Z2 invariant is given p , = p , 4.14 by ␶ where z operates in the Nambu particle-hole space. We now 2 i d wish to evaluate Eq. ͑4.9͒ for this Hamiltonian when phase ␯ ͩ ͪ ͵ Q ͑ ͒ = 2d−1 mod 2, 4.9 ␸͑s͒ winds around a vortex. There is, however, a difficulty d! 2␲ d d−1 T ϫS because the Chern Simons formula requires a gauge that is 2 1 where Q is the Chern Simons form. The specific cases of continuous throughout the entire base space T ϫS . The 2d−1 H0͑ ͒ interest are nonzero Chern number p characterizing p k is an obstruction to constructing such a gauge. A similar problem arose in Q ͓A͔ ͑ ͒ 1 =Tr , 4.10 Sec. IIIC2, when we discussed a line dislocation in a weak topological superconductor. We can adapt the trick we used 2 there to get around the present problem. We thus double the Q =TrͫAdA + A3ͬ, ͑4.11͒ 3 3 Hilbert space to include two copies of our Hamiltonian, one with Chern number p and one with Chern number −p 3 3 H0͑k͒ 0 Q ͫA͑ A͒2 A3 A A5ͬ ͑ ͒ 0 p 5 =Tr d + d + . 4.12 H˜ ͑k͒ = ͩ ͪ. ͑4.15͒ 2 5 H0 ͑ ͒ 0 −p k ͑ ͒ ͑ ͒ It is instructive to see that Eq. 4.9 reduces to Eq. 4.2 in We then put the vortex in only the +p component the case in which a system also has particle-hole symmetry. In this case, as detailed in Appendix D it is possible to H˜ ͑k,␸͒ = e−i␸qH˜ 0͑k͒ei␸q, ͑4.16͒ A † / Q choose a gauge in which =q dq 2, so that 2d−1 ϰ͑qdq†͒2d−1. where 1+␶ 10 2. End states in a 1D superconductor q = z ͩ ͪ. ͑4.17͒ 2 00 The simplest example of a point defect in a supercon- ␶ ␶ ductor occurs in Kitaev’s model38 of a one-dimensional We added an extra phase factor by replacing z by 1+ z in i␸q p-wave superconductor. This is described by a simple 1D order to make e periodic under ␸→␸+2␲. tight binding model for spinless electrons, which includes a Since the Chern number characterizing H˜ 0͑k͒ is zero, † nearest-neighbor hopping term tci ci+1+H.c. and a nearest- there exists a continuous gauge neighbor p-wave pairing term ⌬c c +H.c.. The Bogoliubov i i+1 ͉˜ ͑ ␸͒͘ i␸q͉˜0͑ ͒͘ ͑ ͒ de Gennes Hamiltonian can then be written as ui k, = e ui k , 4.18 H͑k͒ = ͑t cos k − ␮͒␶ + ⌬ sin k␶ . ͑4.13͒ which allows us to evaluate the Chern-Simons integral. The z x A˜ ͗ ͉ ͘ Berry’s connection ij= ˜ui du˜ j is given by This model exhibits a weak pairing phase for ͉␮͉Ͻt and a strong pairing phase for ͉␮͉Ͼt. The weak pairing phase will A˜ = A˜ 0 + iQd␸, ͑4.19͒ have zero-energy Majorana states at its ends. A˜ 0͑ ͒ H˜ 0͑ ͒ ͑ ͒ The topological invariant in Eq. ͑4.9͒ can be easily evalu- where k is the connection describing k and Qij k ͗ 0͑ ͉͒ ͉ 0͑ ͒͘ ͑ ͒ ated. We find A=d␪/2, where ␪ is the polar angle of d͑k͒ = ˜ui k q ˜uj k . Inserting this into Eq. 4.11 and rearrang- =͑t cos k−␮,⌬ sin k͒. It follows that for ͉␮͉Ͻt, the topo- ing terms we find logical invariant is ␯=1 mod 2. Q ͓ F˜ 0 ͑ A˜ 0͔͒ ∧ ␸ ͑ ͒ 3 =Tr 2Q − d Q d , 4.20 3. Vortex in a 2D topological superconductor where F˜ 0 =dA˜ 0 +A˜ 0 ∧A˜ 0. Since the second term is a total In two dimensions, a Majorana bound state occurs at a derivative it can be discarded. For the first term there are two ␶ ͑ ͒ vortex in a topological superconductor. This can be easily contributions from the 1 and the z in Eq. 4.17 . Upon inte-

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␶ φ=2π φ=2π φ=2π grating over k, the z term can be shown to vanish as a consequence of particle-hole symmetry. The 1 term simply φ=0 φ=0 φ=0 projects out the Berry curvature of the original Hamiltonian H0͑ ͒ p k so that Q ͓F0͔ ∧ ␸ ͑ ͒ 3 =Tr d . 4.21 ν=1 ν=0 ν=0 (a) (b) (c) ͑ ͒ It follows from Eq. 4.9 that the Z2 invariant characterizing ͑ ͒ the vortex is FIG. 7. Color online Visualization of the Z2 Pontrjagin invariant characterizing maps from ͑k,␾͒෈S2 ϫS1 to S2 when the wind- ␯ = pm mod 2, ͑4.22͒ ing degree for each ␾ is 1. The inner sphere corresponds to k at ␾=0 while the outer sphere is ␾=2␲. The lines depict inverse where p is the Chern number characterizing the topological 2 superconductor and m is the phase winding number associ- images of two specific points on S , which are lines connecting the inner and outer spheres. In ͑a͒ they have one twist, which cannot be ated with the vortex. eliminated. The double twist in ͑b͒ can be unwound by smoothly It is also instructive to consider this invariant in the con- dragging the paths around the sphere to arrive at ͑c͒, which has no text of the simple two-band model introduced by Read and twist. Green.39 This can be written as a simple tight-binding model H0͑ ͒ ͓ ͑ ͒ ␮͔␶ ⌬͑ ␶ ␶ ͒ kx,ky = t cos kx + cos ky − z + sin kx x + sin ky y , vortex in the superconducting state at the interface between a ͑4.23͒ superconductor and a topological insulator. As shown in Ref. 53, this can be described by the simple Hamiltonian where the superconducting order parameter ⌬ is real. As in ͑ ͒ Eq. 4.13 , this model exhibits weak and strong pairing H = v␶ ␮ ␴ជ · k − ␮␶ + ͑m + ⑀͉k͉2͒␶ ␮ + ⌬ ␶ + ⌬ ␶ . phases for ͉␮͉Ͻt and ͉␮͉Ͼt. These are distinguished by the z x z z z 1 x 2 y Chern invariant, which in turn is related to the winding num- ͑4.24͒ ber on S2 of the unit vector dˆ ͑k͒, where dជ͑k͒ are the coefficients of ␶ជ in Eq. ͑4.23͒. A nonzero superconducting phase is Here m is a mass which distinguishes a topological insulator ␶ ͑ ͒ from a trivial insulator, and ⌬=⌬ +i⌬ is a superconducting again introduced by rotating about z as in Eq. 4.14 . Here 1 2 order parameter. For ␮=0, this Hamiltonian has the form of we wish to show that in this two band model the Z2 invariant ␯ can be understood from a geometrical point of view. the three dimensional version of Eq. ͑4.6͒ discussed in Sec. IVA4, where the mass term is characterized by the vector The Z2 invariant characterizing a vortex can be under- ␾ជ ͑ ⌬ ⌬ ͒ ⌬ ˆ ͑ ␾͒ = m, 1 , 2 . A vortex in at the interface where m stood in terms of the topology of the maps d kx ,ky , . from T2 ϫS1 to S2. These maps were first classified by Pontrjagin79 changes sign then corresponds to a hedgehog singularity in ជ and have also appeared in other physical contexts.53,80,81 ␾. From Eq. ͑4.3͒, it can be seen that the class BDI Z invari- Without losing generality, we can reduce the torus T2 to a ant is n=1. This then establishes that the class D Z invariant ␯ sphere S2 so the mappings are S2 ϫS1 →S2. When for fixed ␾ is =1. The Z2 survives when a nonzero chemical potential ˆ ͑ ␾͒ 2 Ϯ reduces the symmetry from class BDI to class D. d kx ,ky , has an S winding number of p, the topological classification is Z2p. In the case of interest, p=1, so there are two classes. C. Class DIII: Majorana doublets This Z2 Pontrjagin invariant can be understood pictorially by considering inverse image paths in ͑k,␾͒ space, which Point defects in class DIII are characterized by a Z2 topo- map to two specific points on S2. These correspond to 1D logical invariant. These are associated with zero modes, but curves in S2 ϫS1. Figure 7 shows three examples of such unlike class D, the zero modes are required by Kramers theo- curves. The inner sphere corresponds to ␾=0 while the outer rem to be doubly degenerate. The zero modes thus form a sphere corresponds to ␾=2␲. Since p=1, for every point on Majorana doublet, which is equivalent to a single Dirac fer- S2 the inverse image path is a single curve connecting the mion. ␦ inner and outer spheres. The key point is to examine the In Table I, Class DIII, =1 is an entry that is similar to Class AII, ␦=2. The Z for DIII invariant bears a resem- linking properties of these curves. The Z2 invariant describes 2 the number of twists in a pair of inverse image paths, which blance to the invariant for AII, which is a generalization of is1in͑a͒,2in͑b͒,and0in͑c͒. The configuration in ͑b͒ can the Z2 invariant characterizing the 2D quantum spin-Hall in- be continuously deformed into that in ͑c͒ by dragging the sulator. In Appendix B we will establish a formula that em- paths around the inner sphere. This can be verified by a ploys the same gauge constraint simple demonstration using your belt. The twist in ͑a͒, how- ͑ ͒ ͑ ͒ ever, cannot be undone. The number of twists thus defines w k,r = w0, 4.25 the Z2 Pontrjagin invariant. ͑ ͒ where w0 is a constant independent of k and r. w k,r re- 4. Superconductor heterostructures lates the time-reversed states at k and −k and is given by Eq. Finally, in three dimensions, a nontrivial point defect can ͑3.23͒. Provided we choose a gauge that satisfies this con- occur at a superconductor heterostructure. An example is a straint, the Z2 invariant is given by

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H ␶ ␴ ⌬ ␶ ͑ ͒ = vk z z + x. 4.28 SSϕ=0 ϕ=π 1 ␴ Here z describes the spin of the quantum spin-Hall edge ⌬ state and 1 is the real superconducting order parameter. ⌶ ␴ ␶ This model has particle-hole symmetry = y yK and time- QSHI ⌰ ␴ ␲ reversal symmetry =i yK and is in class DIII. A junction ⌬ corresponds to a domain wall where 1 changes sign. Fol- FIG. 8. A Josephson junction in proximity with the helical edge lowing appendix C, it is straightforward to see that this will states of a quantum spin-Hall insulator. When the phase difference involve a degenerate pair of zero modes indexed by the spin ␴ ␶ ␶ ␴ is ␲, there is a zero energy Majorana doublet at the junction. z and chirality y constrained by y z =−1. The Hamiltonian ͑4.28͒ should be viewed as a low-energy d theory describing the edge of a 2D quantum spin-Hall insu- 1 i ⌬ ˜␯ = ͩ ͪ ͵ Q mod 2. ͑4.26͒ lator. Nonetheless, we may describe a domain wall where 1 ␲ 2d−1 d! 2 TdϫSd−1 changes sign using an effective one dimensional theory by ⌬ ⌬ ⑀ 2 introducing a regularization replacing 1 by 1 + k . This regularization will not affect the topological structure of a This formula is almost identical to the formula for a point ⌬ domain wall, where 1 changes sign. A topologically equiva- defect in class D but they differ by an important factor of lent lattice version of the theory then has the form two. Due to the combination of time-reversal and particle- ͑ ͒ H ␶ ␴ ͓⌬ ͑ ͔͒␶ ͑ ͒ hole symmetries the Chern-Simons integral in Eq. 4.26 is = t sin k z z + 1 + u 1 − cos k x. 4.29 guaranteed to be an integer but the integer is not gauge in- ͉⌬ ͉Ͻ variant. When the time-reversal constraint is satisfied, the where we assume 1 2u. parity ˜␯ is gauge invariant. It then follows that the class D The topological invariant can be evaluated using either ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ invariant in Eq. ͑4.9͒, ␯=0 mod 2. Eq. 4.26 or 4.27 . To use Eq. 4.26 , note that Eq. 4.29 ͑ In the special case d=1 there is a formula that does not has exactly the same form as two copies distinguished by ␴ Ϯ ͒ ͑ ͒ ͑ ͒ rely on the gauge constraint, though it still requires a glo- z = 1 of Eq. 4.13 . The evaluation of Eq. 4.26 then bally defined gauge. It is related to the similar “fixed-point” proceeds along the same lines. It is straightforward to check formula for the invariant for the 2D quantum spin-Hall that in a basis where the time-reversal constraint in Eq. ͑ ͒ ͑ ␴ insulator,73 and has recently been employed by Qi, Hughes, 4.25 is satisfied this fixes the relative phases of the z Ϯ ͒ A ␪ ␪ ͑ ͒ and Zhang82 to classify one-dimensional time-reversal in- = 1 states , =d , where is the polar angle of d k ͓ ⌬ ͑ ͔͒ ⌬ variant superconductors. In class DIII, it is possible to = t sin k, 1 +u 1−cos k . It follows that a defect where 1 ␯ choose a basis in which the time-reversal and particle-hole changes sign has ˜ =1. ͑ ͒ ⌰ operators are given by ⌰=␶ K and ⌶=␶ K so that the chiral To use Eq. 4.27 , we transform to a basis in which y x ␶ ⌶ ␶ ⌸ ␶ operator is ⌸=␶ . In this basis, the Hamiltonian has the form = yK, = xK, and = z. This is accomplished by the uni- z ͓ ͑␲/ ͒␴ ␶ ͔ ͓ ͑␲/ ͒␶ ͔ ͑4.1͒, where q͑k,r͒→q͑k͒ satisfies q͑−k͒=−q͑k͒T. Thus, tary transformation U=exp i 4 y z exp i 4 x . Then, H ͑ ͒ ͑ ͒ ͕ ␴ ͓⌬ Pf͓q͑k͔͒ is defined for the time-reversal invariant points k has the form of Eq. 3.13 with q k =−i t sin k z + 1 ͑ ͔͒␴͖ ͓ ͑ ͔͒ =0 and k=␲. q͑k͒ is related to w͑k͒ because in a particular +u 1−cos k y. It follows that det q k is real and positive ͱ ͓ ͑ ͔͒/ͱ ͓ ͑ ͔͒ ͓⌬ ͔ gauge it is possible to choose w͑k͒=q͑k͒/ ͉Det͓q͑k͔͉͒. The for all k. Moreover, Pf q 0 det q 0 =sgn 1 while ͓ ͑␲͔͒/ͱ ͓ ͑␲͔͒ ⌬ Z2 invariant is then given by Pf q det q =1. Again, a defect where 1 changes sign has ˜␯=1.

Pf͓q͑␲͔͒ ͱDet͓q͑0͔͒ ͑−1͒˜␯ = , ͑4.27͒ Pf͓q͑0͔͒ ͱDet͓q͑␲͔͒ V. ADIABATIC PUMPS In this section we will consider time-dependent Hamilto- where the branch ͱDet͓q͑k͔͒ is chosen continuously between nians H͑k,r,t͒, where in additional to having adiabatic spa- k=0 and k=␲. The equivalence of Eqs. ͑4.26͒ and ͑4.27͒ for tial variation r there is a cyclic adiabatic temporal variation d=1 is demonstrated in Appendix A. Unlike Eq. ͑4.26͒, how- parameterized by t. We will focus on pointlike spatial de- ever, the fixed-point formula ͑4.27͒ does not have a natural fects, in which the dimensions of k and r are related by generalization for dϾ1. d−D=1. Majorana doublets can occur at topological defects in Adiabatic cycles in which H͑k,r,t=T͒=H͑k,r,t=0͒ can time-reversal invariant topological superconductors, or in be classified topologically by considering t to be an addi- Helium 3B. Here we consider a different configuration at a tional “spacelike” variable, defining D˜ =D+1. Such cycles Josephson junction at the edge of a quantum spin-Hall insu- will be classified by the ␦=0 column of Table I. Topologi- lator ͑Fig. 8͒. When the phase difference across the Joseph- cally nontrivial cycles correspond to adiabatic pumps. Table son junction is ␲, it was shown in Refs. 38 and 83 that there IV shows the symmetry classes which host nontrivial pump- is a level crossing in the Andreev bound states at the junc- ing cycles, along with the character of the adiabatic pump. tion. This corresponds precisely to a Majorana doublet. There are two general cases. Classes A, AI, and AII define a This can be described by the simple continuum 1D theory charge pump, where after one cycle an integer number of introduced in Ref. 83 charges is transported toward or away from the point defect.

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TABLE IV. Symmetry classes that support nontrivial charge or B. Class D, BDI: Fermion parity pump fermion parity pumping cycles. Adiabatic cycles of point defects in class D and BDI are Symmetry Topological classes Adiabatic Pump characterized by a Z2 topological invariant. In this section we will argue that a nontrivial pumping cycle transfers a unit of A Z Charge fermion parity to the point defect. This is intimately related AI Z Charge to the Ising non-Abelian statistics associated with defects supporting Majorana bound states. BDI Z2 Fermion parity Like the point defect in class DIII ͑␦=1͒, the temporal D Z Fermion parity 2 pump ͑␦=0͒ in class D occupies an entry in Table I similar to AII 2 Charge Kramers doublet Z the line defect ͑␦=2͒ in class AII so we expect a formula that is similar to the formula for the 2D quantum spin-Hall insulator. This is indeed the case, though the situation is slightly Classes BDI and D define a fermion parity pump. We will more complicated. The Hamiltonian H͑k,r,t͒ is defined on a discuss these two cases separately. base space Td ϫSd−1ϫS1. In Appendix F we will show that We note in passing that the ␦=0 column of Table I also the invariant can be written in a form that resembles Eq. applies to topological textures, for which d=D. For example, ͑3.25͒ a spatially dependent three-dimensional band structure H͑k,r͒ can have topological textures analogous to Skyrmi- id ons in a 2D magnet. Such textures have recently been ana- ␯ = ͫ͵ Tr͑Fd͒ − Ͷ Q ͬmod 2, 84 d 2d−1 Tץ lyzed by Ran, Hosur, and Vishwanath for the case of class d!͑2␲͒ T 1/2 1/2 D, where they showed that the Z2 invariant characterizing the ͑ ͒ texture corresponds to the fermion parity associated with the 5.2 texture. Thus, nontrivial textures are fermions. T ෈͓ ␲͔ where 1/2 is half of the base manifold, say, k1 0, , and Q the Chern-Simons form 2d−1 is generated by a continuous A. Classes A, AI, and AII: Thouless charge pumps valence frame u ͑k,r,t͉͒ ␲ that obeys certain particle- v k1=0, hole gauge constraint. This is more subtle than the time- The integer topological invariant characterizing a pump- reversal gauge condition in Eq. ͑3.24͒ for line defects in AII ing cycle in class A is simply the Chern number characteriz- and point defects in DIII. Unlike Eq. ͑3.24͒, we do not have H͑k r t͒ 60,61 ing the Hamiltonian , , . Imposing time-reversal a computational way of checking whether or not a given symmetry has only a minor effect on this. For ⌰2 =−1 ͑Class ͒ frame satisfies the constraint. Nevertheless, it can be defined, AII , an odd Chern number violates time-reversal symmetry and in certain simple examples, the particle-hole constraint is so that only even Chern numbers are allowed. This means automatically satisfied. that the pumping cycle can only pump Kramers pairs of elec- ⌰2 ͑ ͒ The origin of the difficulty is that unlike time-reversal trons. For =+1 Class AI all Chern numbers are consis- symmetry, particle-hole symmetry connects the conduction tent with time-reversal symmetry. and valence bands. The gauge constraint therefore involves The simplest charge pump is the 1D model introduced by 60 both. Valence and conduction frames can be combined to Thouless. A continuum version of this model can be written form a unitary matrix in the form ͉͉ H͑ ͒ ␴ ͓ ͑ ͒ ⑀ 2͔␴ ͑ ͒␴ ͑ ͒ k,t = vk z + m1 t + k x + m2 t y. 5.1 ͑ ͒ ͑ ͒ ෈ ͑ ͒ ͑ ͒ Gk,r,t = ΂uv k,r,t uc k,r,t ΃ U 2n . 5.3 ͉͉ When the masses undergo a cycle such that the phase of m1 +im2 a single electron is transmitted down the wire. In this case, H͑k,t͒ has a nonzero first Chern number. The The orthogonality of conduction and valence band states im- change in the charge associated with a point in a 1D system plies that is given by the difference in the Chern numbers associated † ⌶ ͑ ͒ with either side of the point. Thus, after a cycle a charge e Gk,r,t G−k,r,t =0. 5.4 accumulates at the end of a Thouless pump. ͒ ͑ → Tץ A two-dimensional version of the charge pump can be In general, we call a frame G: 1/2 U 2n particle-hole developed based on Laughlin’s argument85 for the integer trivial if it can continuously be deformed to a constant while quantum-Hall effect. Consider a 2D ␯=1 integer quantum- satisfying Eq. (5.4) throughout the deformation. The Chern Hall state and change the magnetic flux threading a hole Simons term in Eq. ͑5.2͒ requires a gauge that is built from from 0 to h/e. In the process, a charge e is pumped to the the valence-band part of a particle-hole trivial frame. edge states surrounding the hole. This pumping process can Though the subtlety of the gauge condition makes a gen- be characterized by the second Chern number characterizing eral computation of the invariant difficult, it is possible to H͑ ␪ ͒ ␪ the 2D Hamiltonian kx ,ky , ,t , where parameterizes a understand the invariant in the context of specific models. circle surrounding the hole. A similar pump in 3D can be Consider, a theory based on a point defect in the considered and is characterized by the third Chern number. d-dimensional version of Eq. ͑4.6͒

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H͑k,r,t͒ = v␥ជ · k + ⌫ជ · ␾ជ ͑r,t͒. ͑5.5͒ ϕ=0 ϕ(t) ϕ=0 ⌫ជ ␥ជ d ϫ d Here and are 2 2 Dirac matrices, and we suppose that (a) fermion parity ␾ជ ͑ ͒ for fixed t, the d-dimensional mass vector r,t has a point E ͑ ͒ topological defect at r0 t . If Ref. 53 we argued that adiabatic cycles for such point defects are classified by a Pontrjagin invariant similar to that discussed in Sec. IVB3. This may also be understood in terms of the rotation of the “orienta- ␾ជ ͑ ͒ ͑ ͓͒ tion” of the defect. Near the defect, suppose r,t =O t r ϕ(t) ͑ ͔͒ ͑ ͒ ͑ ͒ (b) −r0 t , where O t is a time-dependent O d rotation. In the course of the cycle, the orientation of the topological defect, 0 2π characterized by O͑t͒ goes through a cycle. Since for dՆ3, ␲ ͓ ͑ ͔͒ FIG. 9. A one-dimensional fermion parity pump based on a 1D 1 O d =Z2, there are two classes of cycles. As shown in ␲ topological superconductor, which has Majorana states at its ends. Ref. 53, the nontrivial cycle, which corresponds to a 2 ␲ rotation changes the sign of the Majorana fermion wave When the phase of the central superconductor is advanced by 2 the fermion parity associated with the pairs of Majorana states in- function associated with the topological defect. We will ar- side each circle changes. Thus fermion parity has been pumped gue below that this corresponds to a change in the local from one circle to the other. ͑b͒ shows the evolution of the energy fermion parity in the vicinity of the defect. For d=2, ␲ ͓ ͑ ͔͒ levels associated with a weakly coupled pair of Majorana states as a 1 O 2 =Z. However, the change in the sign of the Majo- function of phase. The level crossing at ␾=␲ is protected by the ͑ ͒ rana bound state is given by the parity of the O 2 winding local conservation of fermion parity. number. In theories with more bands, it is only this parity that is topologically robust. by considering Bloch/BdG Hamiltonians that vary adiabati- In d=1, the single ⌫ matrix in the two-band model does cally with spatial ͑and/or temporal͒ parameters. This led to a not allow for continuous rotations. Consider instead Kitaev’s generalization of the bulk-boundary correspondence, which model38 for a 1D topological superconductor with at time- identifies protected gapless fermion excitations with topo- dependent phase logical invariants characterizing the defect. This leads to a number of additional questions to be addressed in future H͑ ͒ ͑ ␮͒␶ ⌬ ͑ ͒ ␶ ⌬ ͑ ͒ ␶ k,t = t cos k − z + 1 t sin k x + 2 t sin k y. work. ͑5.6͒ The generalized bulk-boundary correspondence has the flavor of a mathematical index theorem, which relates an ͑ ͒ In this case it is possible to apply the formula 5.2 because analytic index that characterizes the zero modes of a system T ␲ץ on the boundary , which is k=0 or k= the Hamiltonian is to a topological index. It would be interesting to see a more ͑ ͒ independent of t, so that the gauge condition in Eq. 5.4 is general formulation of this relation86,87 that applies to the automatically satisfied. Moreover, the second term in Eq. classes without chiral symmetry that have invariants and ͑ ͒ Z2 5.2 involving the Chern Simons integral is equal to zero so goes beyond the adiabatic approximation we used in this F͑ ͒ T that the invariant is simply the integral of x,t over 1/2.It paper. Though the structure of the gapless modes associated ␯ is straightforward to check that this gives =1. with defects make it clear that such states are robust in the In order to see why this corresponds to a pump for fer- presence of disorder and interactions, it would be desirable mion parity, suppose a topological superconductor is broken to have a more general formulation of the topological invari- in two places, as shown in Fig. 9. At the ends where the ants characterizing a defect that can be applied to interacting superconductor is cut there will be Majorana bound states. and/or disordered problems. The pair of bound states associated with each cut defines two An important lesson we have learned is that topologically quantum states which differ by the parity of the number of protected modes can occur in a context somewhat more gen- electrons. If the two ends are weakly coupled by electron eral than simply boundary modes. This expands the possibili- tunneling then the pair of states will split. Now consider ties for engineering these states in physical systems. It is thus ␲ advancing the phase of the central superconductor by 2 .As an important future direction to explore the possibilities for shown in Refs. 38 and 83, the states interchange as depicted heterostructures that realize topologically protected modes. ␲ in Fig. 9. The level crossing that occurs at phase difference The simplest version of this would be to engineer protected is protected by the conservation of fermion parity. Thus, at chiral fermion modes using a magnetic topological insulator. the end of the cycle, one unit of fermion parity has been The perfect electrical transport in such states could have far transmitted from one circled region to the other. The pump- reaching implications at both the fundamental and practical ing of fermion parity also applies to adiabatic cycles of point level. In addition, it is worth considering the expanded pos- defects in higher dimensions and is deeply connected with sibilities for realizing Majorana bound states in supercon- the Ising non-Abelian statistics associated with those 53 ductor heterostructures, which could have implications for defects. quantum computing. VI. CONCLUSION Finally, it will be interesting to generalize these topological considerations to describe inherently correlated states, In this paper we developed a unified framework for clas- such as the Laughlin state. Could a fractional quantum-Hall sifying topological defects in insulators and superconductors edge state arise as a topological line defect in a 3D system?

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θ /2 compactify 7 0 1 7 0 1

6 2 6 2 0 T d × S D 5 4 3 5 4 3

/2 compactify Σ(T d × S D ) FIG. 11. Hamiltonian mappings in Eqs. ͑A1͒ and ͑A3͒ are drawn on the left and right clocks, respectively. Solid ͑dotted͒ ar- ͑ ͒ FIG. 10. Suspension ⌺͑Td ϫSD͒. The top and bottom of the rows represent addition of one momentum spatial dimension. cylinder ⌺͑Td ϫSD͒ϫ͓−␲/2,␲/2͔ are identified to two points. base space as a d+D-dimensional sphere, then the suspen- Understanding the topological invariants that would charac- sion is a d+D+1-dimensional sphere. H terize such a defect would lead to a deeper understanding of Without loss of generality we assume c is flattened so H2 ͕H ⌸͖ H2 topological states of matter. that c =1. Since c , =0 it follows that nc=1 as well. The second term in Eq. ͑A1͒ violates the chiral symmetry. H ͑ Thus, if c belongs to the complex class AIII with no anti- ACKNOWLEDGMENTS ͒ H unitary symmetries , then nc belongs to class A. Equation ͑ ͒ We thank Claudio Chamon, Liang Fu, Takahiro Fukui, A1 thus provides a mapping from class AIII to class A. and Roman Jackiw for helpful discussions. This work was For the real classes, which have antiunitary symmetries, supported by NSF under Grant No. DMR-0906175. We the second term will violate either particle-hole symmetry or ␪ thank R. Jackiw for asking the question that inspired the time-reversal symmetry, depending on whether is a mo- ͑ ⌰ calculation that led to Eq. ͑4.22͒. mentum or position-type variable odd or even under and ⌶͒. This will lead to a new nonchiral symmetry class related to the original class by either a clockwise or counterclock- APPENDIX A: PERIODICITY IN SYMMETRY wise turn on the symmetry clock ͑Fig. 11͒. To determine AND DIMENSION which it is, note that if we require ͓⌰,⌶͔=0 then ͑⌰⌶͒2 ⌰2⌶2 ͑ ͒͑s−1͒/2 In this appendix we will establish the relations in Eqs. = = −1 . The unitary chiral symmetry operator ͑ ⌸2 ͒ ͑2.5͒ and ͑2.6͒ between the K groups in different position- satisfying =1 can then be written ͑ ͒ momentum dimensions D,d and different symmetry ⌸ = i͑s−1͒/2⌰⌶. ͑A2͒ classes s. We will do so by starting with an arbitrary Hamil- ͑ ͒ ␪ tonian in KF s;D,d and then explicitly constructing new It follows that if is momentumlike, then time-reversal sym- Hamiltonians in one higher position or momentum dimen- metry is violated when s=1 mod 4, while particle hole is sion, which have a symmetry either added or removed. The violated when s=3 mod 4. This corresponds to corresponds ͑ ͒ → ␪ new Hamiltonians will then belong to KF s+1;D,d+1 or to a clockwise rotation on the symmetry clock, s s+1. If ͑ ͒ → KF s−1;D+1,d . The first step is to identify the mappings is position like then s s−1. and show they preserve the group structure. This defines We next build a chiral Hamiltonian from a nonchiral one group homomorphisms relating the K groups. The next step by adding a symmetry. This is accomplished by doubling the is to show they are isomporphisms by showing that the maps number of bands in a manner similar to the doubling em- have an inverse, up to homotopic equivalence. ployed in the Bogoliubov de Gennes description of a superconductor. We thus write

1. Hamiltonian mappings H ͑ ␪͒ ␪H ͑ ͒ ␶ ␪ ␶ ͑ ͒ c k,r, = cos nc k,r z + sin 1 a, A3 There are two classes of mappings: those that add sym- where a=x or y. Here ␶ជ are Pauli matrices that act on the metries and those that remove symmetries. These need to be doubled degree of freedom. As in Eqs. ͑A1͒, Eq. ͑A3͒ gives considered separately. a new Hamiltonian defined on a base space that is the sus- We consider first the symmetry removing mappings that pension of the original base space. If H2 =1 it follows that H nc send a Hamiltonian c with chiral symmetry to a Hamil- H2 =1 so the energy gap is preserved. It is also clear that the H ͕H ͑ ͒ ⌸͖ c tonian nc without chiral symmetry. Suppose c k,r , new Hamiltonian has a chiral symmetry because it anticom- =0, where ⌸ is the chiral operator. Then define ⌸ ␶ ␶ H H mutes with =i z a. Thus, if nc is in class A, then c is in H ͑k,r,␪͒ = cos ␪H ͑k,r͒ + sin ␪⌸ ͑A1͒ class AIII. nc c For the real-symmetry classes a=x or y must be chosen so for −␲/2Յ␪Յ␲/2. This has the property that at ␪= Ϯ␲/2 that the second term in Eq. ͑A3͒ preserves the original anti- Ϯ⌸ H the new Hamiltonian is , independent of k and r. Thus, unitary symmetry of nc. This depends on the original anti- at each of these points we may consider the base space Td unitary symmetry and whether ␪ is chosen to be a momen- ϫ D H S defined by k and r to be contracted to a point. The new tum or a position variable. For example, if nc has time- Hamiltonian is then defined on the suspension ⌺͑Td ϫSD͒ of reversal symmetry, ⌰, and ␪ is a momentum ͑position͒ ͑ ͒ ͑ ͒ H the original base space see Fig. 10 . If we treat the original variable, then we require a=y a=x . In this case, c has the

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⌶ ␶ ⌰ ͑⌶ ␶ ⌰͒ ␪ Ϯ␲/ additional particle-hole symmetry = x =i y that base space is compactified to points at = 2. A minimal ⌶2 ⌰2 ͑⌶2 ⌰2͒ H satisfies = =− . A similar analysis when nc has Hamiltonian thus has the form particle-hole symmetry allows us to conclude that the sym- H͑ ␪͒ ␪H ͑ ͒ ␪H ͑ ͒ H k,r, = cos 1 k,r + sin 0. A8 metry class of c is given by a clockwise rotation on the → ␪ symmetry clock, s s+1, when is a momentum variable. The constraint H͑k,r,␪͒2 =1 requires When ␪ is a position variable, s→s−1 gives a counterclock- H2 H ͑ ͒2 ͕H H ͑ ͖͒ ͑ ͒ wise rotation. 0 = 1 k,r =1, 0, 1 k,r =0. A9 Equations ͑A1͒ and ͑A3͒ map a Hamiltonian into a new If H͑k,r,␪͒ is nonchiral, then Eq. ͑A8͒ is already in the Hamiltonian in a different dimension and different symmetry ͑ ͒ ⌸ H H ͑ ͒ H ͑ ͒ H form of Eq. A1 with = 0 and c k,r = 1 k,r . 1 class. It is clear that two Hamiltonians that are topologically ͑ ͒ equivalent will be mapped to topologically equivalent automatically has chiral symmetry due to Eq. A9 . This shows that Eqs. ͑A4͒ and ͑A5͒ are invertible when s is odd. Hamiltonians since the mapping can be done continuously H͑ ␪͒ H H ͑ ͒ If k,r, is chiral, then both 0 and 1 k,r anticom- on a smooth interpolation between the original Hamiltonians. ⌸ H ␶ ͑ ͒ ͑ ͒ mute with the chiral symmetry operator . Rename 0 = a Thus, Eqs. A1 and A3 define a mapping between equiva- ⌸ ␶ ␶ ͑ ͒ ␪ ͑ and =i z a, where a=x a=y when is a position mo- lence classes of Hamiltonians. Moreover, since the direct ͒ ͕H ␶ ͖ ͕H ␶ ͖ sum of two Hamiltonians is mapped to the direct sum of the mentum variable. It follows that 1 , x = 1 , y =0 so we new Hamiltonians, the group property of the equivalence can write classes is preserved. Equations ͑A1͒ and ͑A3͒ thus define a H ͑ ͒ ͑ ͒ ␶ ͑ ͒ 1 k,r = h k,r z. A10 K-group homomorphism ͑ ͒ ͑ ͒ H Equation A8 thus takes the form of Eq. A3 with nc=h. K ͑s;D,d͒ → K ͑s +1;D,d +1͒, ͑A4͒ ␶ ⌰ ⌶ ͑ ͒ F F Since z anticommutes with either or , h k,r carries exactly one antiunitary symmetry and is therefore nonchiral. ͑ ͒ → ͑ ͒͑͒ KF s;D,d KF s −1;D +1,d A5 This shows that Eqs. ͑A4͒ and ͑A5͒ are invertible when s is even. for F=R,C. 2. Invertibility APPENDIX B: REPRESENTATIVE HAMILTONIANS In order to establish that Eqs. ͑A4͒ and ͑A5͒ are isomor- AND CLASSIFICATION BY WINDING NUMBERS phisms we need to show that there exists an inverse. This is not true of the Hamiltonian mappings. A general Hamiltonian In this appendix we construct representative Hamiltonians cannot be built from a lower dimensional Hamiltonian using for each of the symmetry classes that are built as linear com- Eqs. ͑A1͒ and ͑A3͒. However, we will argue that it is pos- binations of Clifford algebra generators that can be repre- sible to continuously deform any Hamiltonian into the form sented as anticommuting Dirac matrices. This allows us to given by Eq. ͑A1͒ or ͑A3͒. Thus, the mappings between relate the integer topological invariants, corresponding to the equivalence classes have an inverse. To show this we will Z and 2Z entries in Table I, to the winding degree in maps use a mathematical method borrowed from Morse theory.67 between spheres. Similar construction for defectless bulk Without loss of generality we again consider flattened Hamiltonians can be found in Ref. 68 by Ryu, et al..In Hamiltonians having equal number of conduction and va- general, Hamiltonians do not have this specific form. How- lence bands with energies Ϯ1. Consider H͑k,r,␪͒, where ever, since each topological class of Hamiltonians includes ␪෈͓−␲/2,␲/2͔ is either a position or momentum variable representatives of this form, it is always possible to smoothly and H is independent of k and r at ␪= Ϯ␲/2. We wish to deform H͑k,r͒ into this form. show that H͑k,r,␪͒ can be continuously deformed into the The simplest example of this approach is the familiar case form ͑A1͒ or ͑A3͒. To do so we define an artificial “action” of a two-dimensional Hamiltonian with no symmetries ͑class A͒. A topologically nontrivial Hamiltonian can be repre- H͒ ͑ ͒ sented as a 2ϫ2 matrix that can be expressed in terms of ץH ץ͑ H͑ ␪͔͒ ͵ ␪ d D͓ S k,r, = d d kd r Tr ␪ ␪ . A6 Pauli matrices as H͑k͒=h͑k͒·␴ជ . The Hamiltonian can then ˆ 2 S can be interpreted as a “height” function in the space of be associated with a unit vector d͑k͒=h͑k͒/͉h͑k͉͒෈S .Itis gapped symmetry preserving Hamiltonians. Given any then well known that the Chern number characterizing H͑k͒ Hamiltonian there is always a downhill direction. These in two dimensions is related to the degree, or winding num- 2 downhill vectors can then be integrated into a deformation ber, of the mapping from k to S . This approach also applies trajectory. Since the action is positive definite, it is bounded to higher Chern numbers characterizing Hamiltonians in below. The deformation trajectory must end at a Hamiltonian even dimensions d=2n. In this case, a Hamiltonian that is a n n that locally minimizes the action. combination of 2n+1 2 ϫ2 Dirac matrices, and can be as- Under the flatness constraint H2 =1, minimal Hamilto- sociated with a unit vector dˆ ෈S2n. nians satisfy the Euler-Lagrange equation For the complex chiral class AIII, the U͑n͒ winding num- 2H H ͑ ͒ ber characterizing a family of Hamiltonians can similarly beץ ␪ + =0. A7 expressed as a winding number on spheres. For example, in H͑ ͒ ͑ ͒␴ The solutions must be a linear combination of sin ␪ and d=1, a chiral Hamiltonian can be written k =hx k x ͑ ͒␴ ͑ ͕H ␴ ͖ ͒ ˆ ͑ ͒෈ 1 cos ␪. The coefficient of sin ␪ must be constant because the +hy k y so , z =0 , and is characterized by d k S .

115120-19 JEFFREY C. Y. TEO AND C. L. KANE PHYSICAL REVIEW B 82, 115120 ͑2010͒

TABLE V. Examples of Dirac matrices for ͑p,q͒=͑s,0͒.

Classes Dirac matrices Symmetry operators ⌫ ␥ ⌰⌶⌸ s AZ 0 ជ

0AI 1 K

1BDI␴z ␴y K ␴xK ␴x

2D␴z ␴y ␴x ␴xK 3 DIII ␶z␴z ␶z␴y ␶z␴x ␶x i␶y␴xK ␴xK ␶y 4AII␶z␴z ␶z␴y ␶z␴x ␶x ␶y i␶y␴xK

The integer topological invariant can then be expressed by pings that correspond to counterclockwise rotations ͑s→s the winding number of dˆ ͑k͒. Similar considerations apply to −1͒ introduce an additional momentumlike generator ͑q the integer invariants for chiral Hamiltonians in higher odd →q+1͒. Equation ͑B7͒ follows because this procedure can dimensions. be repeated to generate Hamiltonians with any indices ͑p,q͒. For the real symmetry classes we introduce “position- Some examples are listed in Table V. ͑ type” Dirac matrices ⌫␮ and “momentum-type” Dirac matri- The integer topological invariants in Table I which occur ␥ ͕⌫ ⌫ ͖ ␦ ͕␥ ␥ ͖ ␦ when s−␦ is even͒ can be related to the winding degree of ces i. These satisfy ␮ , ␯ =2 ␮␯, i , j =2 ij, and ͕⌫ ␥ ͖ ˆ D+d → p+q ␮ , j =0, and are distinguished by their symmetry under the maps d:S S . This can be nonzero when the antiunitary symmetries. If there is time-reversal symmetry spheres have the same total dimensions. In light of Eq. ͑B7͒, we require ͑p,q͒ can always be chosen so that d+D=p+q. The antiunitary symmetries impose constraints on the possible values of ͓⌫␮,⌰͔ = ͕␥ ,⌰͖ =0 ͑B1͒ i these winding numbers, which depend on the relation be- while with particle-hole symmetry tween ␦=d−D and s=p−q. The involutions on Sd+D and Sp+q have opposite orienta- ͕⌫ ⌶͖ ͓␥ ⌶͔ ͑ ͒ ␮, = i, =0. B2 tions when ␦−sϵ2 or 6 mod 8, and therefore an involution d+D p+q For a Hamiltonian that is a combination of p momentumlike preserving map S →S can have nonzero winding de- ␥ ⌫ gree only when ␦−sϵ0 or 4 mod 8. Symmetry gives a fur- matrices 1,...,p and q+1 positionlike matrices 0,...,q ther constraint on the latter case. Consider a sphere map H͑ ͒ ͑ ͒ ⌫ជ ͑ ͒ ␥ជ ͑ ͒ 2 → 2 k,r = R k,r · + K k,r · B3 S␪,␾ S␽,␸, where the involutions on the spheres send ␽,␸͒. In order for ␸͑␪,␾͒͑ۋ␪,␾+␲͒ and ͑␽,␸͒͑ۋ␪,␾͒͑ the coefficients must satisfy the involution =␸͑␪,␾+␲͒, the winding number must be even. Together, R͑− k,r͒ = R͑k,r͒, ͑B4͒ these show for ␦ − s ϵ 0͑mod 8͒ K͑− k,r͒ =−K͑k,r͒. ͑B5͒ Z deg ෈ Ά2Z for ␦ − s ϵ 4͑mod 8͒· ͑B8͒ This can be characterized by a unit vector 0 otherwise. ͑K,R͒ dˆ ͑k,r͒ = ෈ Sp+q, ͑B6͒ This gives a topological understanding of the ’s and 2 ’s on ͱ͉ ͉2 ͉ ͉2 Z Z K + R the periodic table in terms of winding number, which can be where Sp+q is a ͑p+q͒ sphere in which p of the dimensions identified with the more general analytic invariants, namely, are odd under the involution in Eq. ͑B5͒. Chern numbers for nonchiral classes The symmetry class s of H͑k,r͒ is related to the indices ͑ ͒/ 1 i d+D 2 ͑p,q͒ characterizing the numbers of Dirac matrices by n = ͩ ͪ ͵ Tr͑F͑d+D͒/2͒͑B9͒ d + D 2␲ dϫ D p − q = s mod 8. ͑B7͒ ͩ ͪ! T S 2 H ͑ ͒⌫ To see this, start with a Hamiltonian 0 =R0 k,r 0 that ϫ ⌫ and winding numbers of the chiral flipping operator q͑k,r͒ involves a single 1 1 position like “Dirac matrix” 0 =1 so ͓ ͑ ͒ ͑ ͔͒ ͑p,q͒=͑0,0͒. This clearly has time-reversal symmetry, with for chiral ones see Eqs. 4.1 and 4.2 . ⌰ =K, and corresponds to class AI with s=0. Next, generate d + D −1 Hamiltonians H with different symmetries s by using the ͩ ͪ! s 2 Hamiltonian mappings introduced in Appendix A. Both the n = ͵ Tr͓͑qdq†͒d+D͔. ͑ ͒ ͑ ␲ ͒͑d+D+1͒/2 mappings in Eqs. ͑A1͒ and ͑A3͒ define a new Clifford alge- d + D ! 2 i TdϫSD bra with one extra generator that is either position or mo- ͑B10͒ mentum type. The mappings that correspond to clockwise rotations on the symmetry clock ͑s→s+1͒ introduce an ad- The Z2’s on the periodic table are not directly characterized ditional positionlike generator ͑p→p+1͒ while the map- by winding degree but rather through dimensional reduction.

115120-20 TOPOLOGICAL DEFECTS AND GAPLESS MODES IN… PHYSICAL REVIEW B 82, 115120 ͑2010͒

H͑ ͒ ϵ␦ E Given a Hamiltonian k,k1 ,k2 ,r with s mod 8, its winding degree mod 2 determines the Z2 classification of its equatorial offspring H ͑k,k ,r͒ and H ͑k,r͒. For k2=0 1 k1,2=0 example, topological insulators in two and three dimensions k are equatorial restrictions of a four-dimensional model = dˆ :S4→S4 with unit winding number. Around the north pole, the Hamiltonian has the form ͑ ͒ ͑ ͒ H͑ ͒ ͑ ␧ 2͒␮ ␮ ␴ជ ␮ ͑ ͒ FIG. 12. Color online Energy spectrum of Hamiltonian C4 . k,k4 = m + k 1 + k · 3 + k4 2 B11 ͉⌿ ͘ The zero mode 0 of positive chirality at kʈ =0 corresponds the chiral mode that generates the midgap kʈ-linear energy spectrum. and on the equator k4 =0, this gives a three-dimensional Dirac theory of mass m around k=0 that locally describes ⌫ 3D topological insulators Bi2Se3 and Bi2Te3 around . ries far away from the defect. The single zero mode of Eq. ͑C1͒ and spectral flow of Eq. ͑C4͒ are equated to unit winding degree of an adiabatic limit. In general, bulk-boundary APPENDIX C: ZERO MODES IN THE HARMONIC correspondence is mathematically summarized by index OSCILLATOR MODEL theorems that associate certain analytic and topological indi- We present exact solvable soliton states of Dirac-type de- ces of Hamiltonians.41,45,46,56,86,87 fect Hamiltonians. These include zero modes at a point defect of a Hamiltonian in the chiral class AIII and chiral modes along a line defect of a Hamiltonian in the nonchiral APPENDIX D: INVARIANT FOR POINT DEFECTS class A. We establish the connection between the two kinds IN CLASS D AND BDI of boundary modes through the Hamiltonian mapping in Eq. ͑A1͒. We follow the derivation given in Ref. 53, which was A nontrivial chiral Hamiltonian isotropic around a point based on Qi, Hughes, and Zhang’s formulation of the topo- defect at r=0 is a Dirac operator logical invariant characterizing a three-dimensional topological insulator.10 For a point defect in d dimensions, the Hamil- H =−i␥ជ · ٌ+ r · ⌫ជ , ͑C1͒ tonian H͑k,r͒ depends on d momentum variables and d−1 position variables. We introduce a one parameter deforma- where the chiral operator is ⌸=id͟d ␥ ⌫ and its adiabatic j=1 j j tion H˜ ͑␭,k,r͒ that connects H˜ ͑k,r͒ at ␭=0 to a constant −ik·rH ik·r ␥ជ ⌫ជ limit e e =k· +r· has unit winding degree on Hamiltonian at ␭=1 while breaking particle-hole symmetry. 2d−1 ͕͑ ͒ 2 2 ͖ S = k,r :k +r =1 . The particle-hole symmetry can be restored by including a ˜ −1 .H2 =−ٌ2 + r2 − i␥ជ · ⌫ជ ͑C2͒ mirror image H͑␭,k,r͒=−⌶H͑−␭,k,r͒⌶ for −1Ͻ␭Ͻ0 For ␭=Ϯ, ͑k,r͒ can be replaced by a single point, so the 2d d d−1 and the spectrum is determined by the quantum numbers nj parameter space ͑␭,k,r͒ is the suspension ⌺͑T ϫS ͒ of Ն ␰ 0 of the harmonic oscillator and the parities j of the mu- the original space. The Hamiltonian defined on this space is ␥ ⌫ tually commuting matrices i j j, for j=1,...,d. characterized by its dth Chern character d E2 ␰ ͑ ͒ = ͚ 2nj +1− j. C3 1 i d j=1 ␯ = ͩ ͪ ͵ Tr͓Fd͔. ͑D1͒ d! 2␲ d d−1 ͉⌿ ͘ ␰ ⌺͑T ϫS ͒ The unique zero-energy state 0 , indexed by nj =0 and j =1, has positive chirality ⌸=+1, and is exponentially local- ⌿ ͑ ͒ϰ −1/2r2 ized at the point defect as 0 r e . Due to particle-hole symmetry, the contributions from the Next we consider a nonchiral Hamiltonian isotropic along two hemispheres ␭Ͼ0, ␭Ͻ0 are equal. Using the fact that a line defect. the integrand is the derivative of the Chern Simons form, Tr͓Fd͔=dQ , we can therefore write ជ 2d−1 H͑kʈ͒ = kʈ⌸ − i␥ជ · ٌ+ r · ⌫, ͑C4͒

d where kʈ is parallel to the defect line, r and ٌ are normal 2 i −ik·rH͑ ͒ ik·r ␯ = ͩ ͪ ͵ Q . ͑D2͒ position and derivative. Its adiabatic limit e kʈ e ␲ 2d−1 ជ d! 2 TdϫSd−1 =kʈ⌸+k·␥ជ +r·⌫ is related to that of Eq. ͑C1͒ by ͑1,1͒ peri- 2d 2 odicity and has unit winding degree on S =͕͑kʈ ,k,r͒:kʈ ͉ ͉2 ͉ ͉2 ͖ ͉⌿ ͘ ͑ ͒ + k + r =1 . The zero mode 0 of Eq. C1 gives rise As was the case in Refs. 10 and 53, ␯ can be different for H͑ ͉͒⌿ ͘ ⌸͉⌿ ͘ ͉⌿ ͘ to a positive chiral mode, kʈ 0 =kʈ 0 =+kʈ 0 different deformations H͑␭,k,r͒. However, particle-hole ͑Fig. 12͒. symmetry requires the difference is an even integer. Thus, ͑ ͒ The two examples veriﬁed bulk-boundary correspondence the parity of Eq. D2 deﬁnes the Z2 invariant. Q through identifying analytic information of the defect-bound The Chern Simons form 2d−1 can be expressed in terms solitons and the topology of slowly spatial modulated theo- of the connection A via the general formula

115120-21 JEFFREY C. Y. TEO AND C. L. KANE PHYSICAL REVIEW B 82, 115120 ͑2010͒

2d–2 d−1 (a) ∂T1/2 (b) i ␯ = ͫ͵ Tr͑Fd−1͒ − Ͷ Q ͬ 2d−3 ͑ ͒ ͑ ␲͒d−1 2d−2 2d−2 Tץ B d −1 ! 2 T 2

− 1/2 1/2 d S 2d–2 ͑E1͒ × T1/2 A 1 − d

T B topologically classifies line defects in AII, where the Chern- Simons form, defined by Eq. ͑D3͒, is generated by the Berry k =- k =0 k = A ͗ ͑ ͉͒ ͑ ͒͘ d d d connection mn= um k,r dun k,r , and the valence frame ͑ ͒ um k,r satisfies the gauge condition FIG. 13. ͑a͒ Schematic of the base space T2d−2=Td ϫSd−2, split into two halves. ͑b͒ Division of T2d−2 into patches A and B, each w ͑k,r͒ = ͗u ͑k,r͉͒⌰u ͑− k,r͒͘ = constant ͑E2͒ ͉ A/B͘ mn m n closed under TR and has an individual valence frame um that T2d−2ץsatisfies the gauge condition in Eq. ͑E2͒. ͑ ͒෈ on the boundary k,r 1/2 . The nontriviality of the Z2 invariant is a topological ob- 1 struction to choosing a global continuous valence frame Q ͵ ͓A͑ A 2A2͒d−1͔ ͑ ͒ ͉ ͑ ͒͘ ͑ ͒ 2d−1 = d dt Tr td + t . D3 um k,r that satisfies the gauge condition in Eq. E2 on the 0 whole base space T2d−2. If there is no topological obstruction 88 In the addition of time-reversal symmetry ⌰2 =1 or equiva- from the bulk, the gauge condition forces the valence frame ⌸ ⌰⌸ ␶ to be singular at two points, depicted in Fig. 13͑b͒, related to lently a chiral symmetry = = z, a valence frame of the BDI Hamiltonian ͑4.1͒ can be chosen to be each other by time reversal. One removes the singularity by picking another valence frame locally defined on two small 1 q͑k,r͒ balls enclosing the two singular points, denoted by B in Fig. u͑k,r͒ = ͩ ͪ, ͑D4͒ ͑ ͒ ͉ A/B͑ ͒͘ ͱ2 − 1 13 b . We therefore have two valence frames um k,r defined on two patches of the base space, A=T2d−2\B and B, where q is unitary and 1 is the identity matrix. This corre- each obeying the gauge condition in Eq. ͑E2͒. A † 1 † sponds the Berry connection =u du= 2 q dq and Chern- The wave functions on the two patches translate into each Simons form other through transition function

1 d−1 d t t tAB͑k,r͒ = ͗uA ͑k,r͉͒uB͑k,r͒͘ ෈ U͑k͒͑E3͒ Q = ͵ dtͫ ͩ −1ͪͬ Tr͓͑q†dq͒2d−1͔ mn m n 2d−1 2 2 2 BϷS2d−3ഫS2d−3. The function behavior onץ on the boundary 0 ͑ ͒d ͑ ͒ ͑ ͒ −1 d! d −1 ! † 2d−1 the two disjoint 2d−3 spheres is related by time reversal. = Tr͓͑q dq͒ ͔. ͑D5͒ AB 2d−3 2 ͑2d −1͒! The topology is characterized by the winding of t :S →U͑k͒ on one of the spheres This equates the winding number of q in Eq. ͑4.2͒ to the Chern Simons invariant in Eq. ͑4.9͒. ͑d −2͒! ␯ = Ͷ Tr͓͑tABd͑tAB͒†͒2d−3͔, ͑E4͒ ͑ ͒ ͑ ␲ ͒d−1 2d −3 ! 2 i S2d−3 APPENDIX E: INVARIANT FOR LINE DEFECTS IN CLASS AII AND POINT DEFECTS IN DIII ͑−1͒d = Ͷ ͑QA − QB ͒. ͑E5͒ ͑ ͒ ͑ ␲ ͒d−1 2d−3 2d−3 We formulate a topological invariant that characterizes d −1 ! 2 i S2d−3 line defects in class AII in all dimensions that is analogous to the integral formula invariant characterizing the quantum The two integrals can be evaluated separately. Since Q ͑Fd−1͒ spin-Hall insulator introduced in Ref. 73. This can be applied d 2d−3=Tr , Stokes’ theorem tells us to weak topological insulators in three dimensions with dislocation around a line defect. The invariant can be indirectly ͵ ͑Fd−1͒ ͑Ͷ Ͷ ͒QA applied to strong topological insulators through decomposi- Tr = − 2d−3, T2d−2 S2d−3ץ AപT2d−2 tion into strong and weak components. As a consequence of 1/2 1/2 the Hamiltonian mapping in Eq. ͑A1͒ that identifies ͑s =4, ␦=2͒ and ͑s=3, ␦=1͒, this gives a new topological ͵ Tr͑Fd−1͒ = Ͷ QB . invariant that classified point defects in class DIII in all di- 2d−3 BപT2d−2 S2d−3 mensions. 1/2 ͑ ͒ Combining these into Eq. E5 identifies the Z2 invariant in ͑ ͒ 1. Line defects in class AII Eq. E1 with the winding number of the transition function. The curvature term in Eq. ͑E1͒ is gauge invariant. Any T2d−2ץ T2d−2 d ϫ d−2 d The base space manifold is =T S , where T is gauge transformation on the boundary 1/2 respecting the the Brillouin zone and Sd−2ϫR is a cylindrical neighborhood gauge condition in Eq. ͑E2͒ has even winding number and that wraps around the line defect in real space. Divide the would alter the Chern-Simons integral by an even integer. T2d−2 base space into two pieces, 1/2 and its time reversal coun- The gauge condition is therefore essential to make the for- ͓ ͑ ͔͒ terpart see Fig. 13 a . We will show the Z2 invariant mula nonvacuous.

115120-22 TOPOLOGICAL DEFECTS AND GAPLESS MODES IN… PHYSICAL REVIEW B 82, 115120 ͑2010͒

Spin Chern number θ /2 (a) (b) A quantum spin-Hall insulator is characterized by its spin B Chern number n␴ =͑n↑ −n↓͒/2. We generalize this to time- reversal invariant line defects of all dimensions by equating 0 2d–1 A it with Eq. ͑E1͒. This applies in particular to a model we T considered for a linear Josephson junction in Sec. III D. B A spin operator S is a unitary operator, square to unity, /2 commutes with the Hamiltonian, and anticommutes with the time reversal operator. The valence spin frame FIG. 14. ͑a͒ Schematic of the suspension ⌺T2d−1. ͑b͒ Decompo- ͉ ↓ ͑ ͒͘ ⌰͉ ↑ ͑ ͒͘ ͑ ͒ sition into patches A and B. um k,r = um − k,r E6 automatically satisfies the time reversal gauge constraint in Set ⌰=␶ K and ⌶=␶ K under an appropriate choice of ͑ ͒ y x Eq. E2 . It is straightforward to check that the curvature and basis. A canonical valence frame of H͑k,r,␪͒ can be chosen Chern-Simons form can be split as direct sums according to to be spins. E7͒ ␲ ␪͑ ,ءF͑k,r͒ = F↑͑k,r͒ F↑͑− k,r͒ sinͩ − ͪq͑k,r͒ B 4 2 u ͑k,r,␪͒ = , ͑E12͒ ء ↑ ↑ Q ͑ ͒ Q ͑ ͒ Q ͑ ͒ ͑ ͒ + ␲ ␪ 2d−3 k,r = 2d−3 k,r 2d−3 − k,r . E8 ΂ ΃ − cosͩ − ͪ1 Again assuming that there is no lower dimensional “weak” 4 2 topology, the ↑ frame can be defined everywhere on T2d−2 T2d−2 ↓ where q͑k,r͒෈U͑k͒ is from the canonical form of the chiral with a singularity at one point, say, in 1/2 , and the frame is singular only at the time reversal of that point. Hamiltonian H͑k,r͒ in Eq. ͑4.1͒, 1 is the kϫk identity ma- The curvature term of Eq. ͑E1͒ splits into two terms trix, and the valence frame is nonsingular everywhere except ␪ ␲/ B → A B BA at =− 2. There is a gauge transformation u+ u =u+t ␪ Ϯ␲/ A ͵ ͑Fd−1͒ ͵ ͵ ͑Fd−1͒͑ ͒ everywhere except = 2 such that the new frame u Tr = ͫ − ͬTr ↑ E9 ͑ ͒ 89 T2d−2 T2d−2 T2d−2 T2d−2 satisfies the gauge condition in Eq. E2 . A valence frame 1/2 1/2 \ 1/2 B͑␪͒ ⌰ B͑ ␪͒ on patch B can be constructed by requiring u− = u+ − and the two spin components of the Chern-Simons term around ␪=−␲/2. ͛ Q -T2d−2 d add up into The topology is characterized by the evenness or oddץ 1/2 2 −3 Z2 ness of the winding number of tAB as in Eq. ͑E4͒. This can be Ͷ Q↑ ͵ ͑Fd−1͒͑͒ evaluated by the integral along the equator ␪=0 2 2d−3 =−2 Tr ↑ E10 T2d−2 T2d−2 T2d−2ץ 1/2 \ 1/2 ͑d −1͒! by Stokes theorem. ˜␯ = ͵ Tr͕͓tABd͑tAB͒†͔2d−1͖, ͑ ͒ ͑ ␲ ͒d Combining these two, we equate Eq. ͑E1͒ to the spin 2d −1 ! 2 i TdϫSd−1 Chern number ͑E13͒ id−1 ͵ ͑Fd−1͒ ͑ ͒ where uA =uBtBA is a solution to the gauge condition in Eq. n↑ = Tr . E11 ͑ ͒ ␲d−1 ↑ + d −1 !2 T2d−2 ͑E2͒ or equivalently tBA satisfies ͑ Time reversal requires ntot=n↑ +n↓ =0 and therefore n␴ = n↑ q͑k,r͒ = tBA͑− k,r͒␴ tBA͑k,r͒T, ͑E14͒ −n↓͒/2=n↑. y ͑ ͒ ␴ where the constant in Eq. E2 is chosen to be i y. ͑ ͒ 2. Point defects in class DIII The winding number in Eq. E13 can also be expressed as a Chern-Simons integral. The base space manifold is T2d−1=Td ϫSd−1. The Hamil- ͑ ͒ tonian mapping in Eq. A1 relates a point-defect Hamil- id tonian H͑k,r͒ in class DIII to a line-defect Hamiltonian ˜␯ = ͵ ͑QB − QA ͒, ͑E15͒ ͑ ␲͒d 2d−1 2d−1 H͑k,r,␪͒=cos ␪H͑k,r͒+sin ␪⌸ in class AII, where ⌸ d! 2 TdϫSd−1 =i⌰⌶ is the chiral operator, ͑k,r,␪͒෈⌺T2d−1 ͓see Fig. / 14͑a͔͒ and ␪ is odd under time reversal. The line defect where QA B are the Chern-Simons form generated by valence / Hamiltonian H͑k,r,␪͒ is topologically characterized by the frames uA B. Restricted to ␪=0, Eq. ͑E12͒ gives uB͑k,r͒ ͑ ͒ 1 ͑ ͑ ͒ ͒ ͑ ͒ generalization of Eq. E1 , which was proven to be identical = ͱ2 q k,r ,−1 . Following Eq. D5 , the first term of Eq. to the winding number in Eq. ͑E4͒ of the transition function ͑E15͒ equals half of the winding number of q, which is guar- tAB ͓see Fig. 14͑b͒ for the definition of patches A and B͔.We anteed to be zero by time-reversal and particle-hole symme- will utilize this to construct a topological invariant that char- tries. And therefore point defects in DIII are classified by the acterizes point defects in class DIII. Chern-Simons invariant

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k d d−1 1 1 i d 1 T × S × S ˜␯ = ͩ ͪ ͵ Q mod 2, ͑E16͒ ␲ 2d−1 d! 2 TdϫSd−1 + T1/2 where the Chern-Simons form is generated by a valence 0 ∂T frame that satisfies the time-reversal gauge constraint in Eq. 1/2 T – ͑E2͒. 1/2 Note that the integrality of the Chern-Simons integral in s Eq. ͑E16͒ is a result of particle-hole symmetry. Forgetting 0 deformation 1 time reversal symmetry, point defects in class D are classified by the Chern-Simons invariant ␯=2˜␯ in Eq. ͑D2͒ with a FIG. 15. Deformation of Hamiltonian and its base manifold so Tץ factor of 2. Time-reversal symmetry requires the zero modes that the boundary 1/2 is shrink to a point. to form Kramers doublets and therefore ␯=2˜␯ must be even. A gauge transformation in general can alter ˜␯ by any integer. APPENDIX F: INVARIANT FOR FERMION Thus, similar to the formula in class AII, the time-reversal PARITY PUMPS gauge constraint in Eq. ͑E2͒ is essential so that Eq. ͑E16͒ is In the appendix, we will show that a invariant in Eq. nonvacuous. Z2 ͑5.2͒ under a particle-hole gauge constraint topologically ␦ Fixed points formula in 1D classified fermion parity pumps in dimension =0 and class D or BDI. ͑See Sec. VB for the full statement.͒ We will We here identify Eq. ͑E16͒, or equivalently Eq. ͑E13͒,to show Eq. ͑5.2͒ using a construction similar to a reasoning in a fixed point invariant in 1 dimension. In Ref. 82, Qi, Moore and Balents.7 We consider a deformation of the Hughes, and Zhang showed that 1D time reversal invariant Hamiltonian along with the base manifold so that the bound- ͒ ͑ Tץ superconductors are classified by the topological invariant Z2 ary 1/2 is deformed into a single point see Fig. 15 . Let ෈͓ ͔ T+ ͑ ͒ ͑ ͒ ␲ s 0,1 be the deformation variable, and denote 1/2 s and ͒ ͑ Tץ † ␯ Pf qk=␲ 1˜ ͑−1͒ = expͫ ͵ Tr͑q dq ͒ͬ ͑E17͒ 1/2 s be the corresponding deformation slices. Chern in- ͑ ͒ k k Pf qk=0 2 0 variant under the basis ⌰=␶ K and ⌶=␶ K, where q is the chiral y x k 1 i d flipping operator in Eq. ͑4.1͒. Time-reversal and particle-hole n = ͩ ͪ ͵ Tr͑Fd͒͑F1͒ T d! 2␲ T+ ͑ ͒ symmetries requires qk =−q−k. Hence the Pfaffians are well 1/2 s=1 ␲ defined as qk is antisymmetric at the fixed points k=0, . Using the gauge condition in Eq. ͑E14͒, we can expressed integrally classifies Hamiltonians on the half-manifold ͑⌰ ͒ ͑ ͒ ͑␴ ͒ BA T+ ͑s=1͒. Particle-hole symmetry requires opposite Chern the Pfaffians as Pf qk=0,␲ =det tk=0,␲ Pf y , where tk is 1/2 abbreviated to tk. invariant on the other half. A different choice of deformation could only change the Chern invariant by an even integer.90 ͑ ͒ ␲ Pf qk=␲ Hence the Chern invariant modulo 2 defines a invariant. = expͫ− ͵ Tr͑t dt†͒ͬ. ͑E18͒ Z2 ͑ ͒ k k Pf qk=0 0 The Chern integral can be further deformed and decom- posed into ͑ ͒ ͑ †͒ Substitute Eq. E14 into the Cartan form Tr qkdqk gives 1 Tr͑q dq†͒ =Tr͑t dt† ͒ +Tr͑t dt†͒. ͑E19͒ ͵ ͑Fd͒ ͵ ͵ ͑Fd͒ k k −k −k k k Tr + ds Tr T+ ͑s͒ץ T+ ͑s=0͒ 0 Combining these into Eq. ͑E17͒ 1/2 1/2 ␲ ͵ ͑Fd͒ Ͷ Q ͑ ͒ ˜␯ 1 † † = Tr − 2d−1, F2 ͒ ͑ +Tץ ͒ ͑ +expͫ ͵ Tr͑t dt − t dt ͒ͬ, ͑E20͒ T = −1͒͑ −k −k k k 1/2 s=0 1/2 s=0 2 0 where Stokes’ theorem is used and the negative sign is from ␲ 1 a change in orientation of the boundary. This proves Eq. =expͫ− ͵ Tr͑t dt†͒ͬ, ͑E21͒ k k ͑ ͒ 2 5.2 . The particle-hole gauge constraint is built-in since the −␲ ͑ ͒ Gk,r,t s deforms into a constant at s=1 while respecting which agrees Eq. ͑E13͒. particle-hole symmetry in Eq. ͑5.4͒ at all s.

1 D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, 4 M. Z. Hasan and C. L. Kane, arXiv:1002.3895v1 ͑unpublished͒. Phys. Rev. Lett. 49, 405 ͑1982͒. 5 C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 ͑2005͒. 2 X. L. Qi and S. C. Zhang, Phys. Today 63͑1͒,33͑2010͒. 6 C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 ͑2005͒. 3 J. E. Moore, Nature ͑London͒ 464, 194 ͑2010͒. 7 J. E. Moore and L. Balents, Phys. Rev. B 75, 121306͑R͒͑2007͒.

115120-24 TOPOLOGICAL DEFECTS AND GAPLESS MODES IN… PHYSICAL REVIEW B 82, 115120 ͑2010͒

8 R. Roy, Phys. Rev. B 79, 195322 ͑2009͒. 33 B. Yan, C. X. Liu, H. J. Zhang, C. Y. Yam, X. L. Qi, T. Frauen- 9 L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 heim, and S. C. Zhang, arXiv:1003.0074 ͑unpublished͒. ͑2007͒. 34 R. Roy, arXiv:0803.2868 ͑unpublished͒. 10 X. L. Qi, T. L. Hughes, and S. C. Zhang, Phys. Rev. B 78, 35 A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, 195424 ͑2008͒. Phys. Rev. B 78, 195125 ͑2008͒; AIP Conf. Proc. 1134,10 11 A. Bernevig, T. Hughes, and S. C. Zhang, Science 314, 1757 ͑2009͒. ͑2006͒. 36 A. Kitaev, AIP Conf. Proc. 1134,22͑2009͒. 12 L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 ͑2007͒. 37 X. L. Qi, T. L. Hughes, S. Raghu, and S. C. Zhang, Phys. Rev. 13 H. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, Lett. 102, 187001 ͑2009͒. Nat. Phys. 5, 438 ͑2009͒. 38 A. Kitaev, arXiv:cond-mat/0010440 ͑unpublished͒. 14 M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. 39 N. Read and D. Green, Phys. Rev. B 61, 10267 ͑2000͒. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 40 A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 ͑2003͒. ͑2007͒. 41 G. E. Volovik, The Universe in a Helium Droplet ͑Clarendon, 15 M. König, H. Buhmann, L. W. Molenkamp, T. Hughes, C.-X. Oxford, 2003͒. Liu, X.-L. Qi, and S.-C. Zhang, J. Phys. Soc. Jpn. 77, 031007 42 G. E. Volovik, JETP Lett. 90, 587 ͑2009͒. ͑2008͒. 43 P. Hořava, Phys. Rev. Lett. 95, 016405 ͑2005͒. 16 A. Roth, C. Brüne, H. Buhmann, L. W. Molenkamp, J. Maciejko, 44 A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 ͑1997͒. X. L. Qi, and S. C. Zhang, Science 325, 294 ͑2009͒. 45 R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 ͑1976͒. 17 D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and 46 R. Jackiw and P. Rossi, Nucl. Phys. B 190, 681 ͑1981͒. M. Z. Hasan, Nature ͑London͒ 452, 970 ͑2008͒. 47 W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 18 D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Oster- 1698 ͑1979͒. walder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. 48 L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 ͑2008͒. Cava, and M. Z. Hasan, Science 323, 919 ͑2009͒. 49 J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. 19 P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, D. Qian, A. Rev. Lett. 104, 040502 ͑2010͒. Richardella, M. Z. Hasan, R. J. Cava, and A. Yazdani, Nature 50 J. Alicea, Phys. Rev. B 81, 125318 ͑2010͒. ͑London͒ 460, 1106 ͑2009͒. 51 R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 20 Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. 105, 077001 ͑2010͒. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nat. Phys. 5, 52 Y. Oreg, G. Refael, and F. von Oppen, arXiv:1003.1145 ͑unpub- 398 ͑2009͒. lished͒. 21 Y. S. Hor, A. Richardella, P. Roushan, Y. Xia, J. G. Checkelsky, 53 J. C. Y. Teo and C. L. Kane, Phys. Rev. Lett. 104, 046401 A. Yazdani, M. Z. Hasan, N. P. Ong, and R. J. Cava, Phys. Rev. ͑2010͒. B 79, 195208 ͑2009͒. 54 M. Nakahara, Geometry, Topology and Physics ͑Adam Hilger, 22 Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S. K. Mo, X. L. Bristol, 1990͒. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. 55 J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, 986 ͑1981͒. Fisher, Z. Hussain, and Z. X. Shen, Science 325, 178 ͑2009͒. 56 E. J. Weinberg, Phys. Rev. D 24, 2669 ͑1981͒. 23 D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. 57 E. Witten, Phys. Lett. 117B, 324 ͑1982͒. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. 58 E. Witten, J. Diff. Geom. 17,611͑1982͒. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and 59 S. C. Davis, A. C. Davis, and W. B. Perkins, Phys. Lett. 408B, M. Z. Hasan, Nature ͑London͒ 460, 1101 ͑2009͒. 81 ͑1997͒. 24 S. R. Park, W. S. Jung, C. Kim, D. J. Song, C. Kim, S. Kimura, 60 D. J. Thouless, Phys. Rev. B 27, 6083 ͑1983͒. K. D. Lee, and N. Hur, Phys. Rev. B 81, 041405͑R͒͑2010͒. 61 Q. Niu and D. J. Thouless, J. Phys. A 17, 2453 ͑1984͒. 25 Z. Alpichshev, J. G. Analytis, J. H. Chu, I. R. Fisher, Y. L. Chen, 62 M. Freedman, M. B. Hastings, C. Nayak, X. L. Qi, K. Walker, Z. X. Shen, A. Fang, and A. Kapitulnik, Phys. Rev. Lett. 104, and Z. Wang, arXiv:1005.0583 ͑unpublished͒. 016401 ͑2010͒. 63 M. Karoubi, K-Theory: An Introduction ͑Springer-Verlag, Berlin, 26 D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. 1978͒. Osterwalder, L. Patthey, A. V. Fedorov, H. Lin, A. Bansil, D. 64 H. B. Lawson and M. L. Michelsohn, Spin Geometry ͑Princeton Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Phys. Rev. Lett. University Press, Princeton, 1989͒. 103, 146401 ͑2009͒. 65 M. Atiyah, K Theory ͑Westview Press, Boulder, CO, 1994͒. 27 A. Shitade, H. Katsura, J. Kunes, X. L. Qi, S. C. Zhang, and N. 66 R. Bott, Ann. Math. 70, 313 ͑1959͒. Nagaosa, Phys. Rev. Lett. 102, 256403 ͑2009͒. 67 J. Milnor, Morse Theory ͑Princeton University Press, Princeton, 28 R. Li, J. Wang, X. L. Qi, and S. C. Zhang, Nat. Phys. 6, 376 1963͒. ͑2010͒. 68 S. Ryu, A. Schnyder, A. Furusaki, and A. W. W. Ludwig, New J. 29 S. Chadov, X. L. Qi, J. Kübler, G. H. Fecher, C. Felser, and S. C. Phys. 12, 065010 ͑2010͒. Zhang, Nature Mater. 9, 541 ͑2010͒. 69 E. I. Blount, Solid State Phys. 13, 305 ͑1962͒. 30 H. Lin, L. A. Wray, Y. Xia, S. Xu, S. Jia, R. J. Cava, A. Bansil, 70 R. Li, J. Wang, X. L. Qi, and S. C. Zhang, Nat. Phys. 6, 284 and M. Z. Hasan, Nature Mater. 9, 546 ͑2010͒. ͑2010͒. 31 H. Lin, L. A. Wray, Y. Xia, S. Y. Xu, S. Jia, R. J. Cava, A. 71 R. S. K. Mong, A. M. Essin, and J. E. Moore, Phys. Rev. B 81, Bansil, and M. Z. Hasan, arXiv:1003.2615 ͑unpublished͒. 245209 ͑2010͒. 32 H. Lin, L. A. Wray, Y. Xia, S. Y. Xu, S. Jia, R. J. Cava, A. 72 G. E. Volovik, Pis’ma Zh. Eksp. Teor. Fiz. 91,61͑2010͒; Bansil, and M. Z. Hasan, arXiv:1004.0999 ͑unpublished͒. arXiv:1001.1514 ͑unpublished͒.

115120-25 JEFFREY C. Y. TEO AND C. L. KANE PHYSICAL REVIEW B 82, 115120 ͑2010͒

73 L. Fu and C. L. Kane, Phys. Rev. B 74, 195312 ͑2006͒. 84 Y. Ran, P. Hosur, and A. Vishwanath, arXiv:1003.1964 ͑unpub- 74 T. Fukui and T. Fujiwara, J. Phys. A 42, 362003 ͑2009͒. lished͒. 75 Y. Ran, Y. Zhang, and A. Vishwanath, Nat. Phys. 5, 298 ͑2009͒. 85 R. B. Laughlin, Phys. Rev. B 23, 5632͑R͒͑1981͒. 76 P. G. Grinevich and G. E. Volovik, J. Low Temp. Phys. 72, 371 86 T. Fukui and T. Fujiwara, J. Phys. Soc. Jpn. 79, 033701 ͑2010͒. ͑1988͒. 87 T. Fukui, Phys. Rev. B 81, 214516 ͑2010͒. 77 M. A. Silaev and G. E. Volovik, arXiv:1005.4672 ͑unpublished͒. 88 Bulk topology is a lower dimenional topological obstruction 78 R. B. Laughlin, Phys. Rev. Lett. 80, 5188 ͑1998͒. coming purely from the momentum part Td of the base space 79 L. S. Pontrjagin, Rec. Math. ͓Mat. Sbornik͔ N.S. 9, 331 ͑1941͒ Td ϫSd−2. One could mathematically remove the bulk topologi- ͓http://mathnet.ru/mst6073͔. cal obstruction by adding a defectless Hamiltonian H˜ ͑k,r͒ 80 ͑ ͒ H͑ ͒ H ͑ ͒ J. Jäykkä and J. Hietarinta, Phys. Rev. D 79, 125027 2009 . = k,r 0 k . 81 L. Kapitanski, London Mathematical Society Durham Sympo- 89 Again, assume the bulk has trivial topology. Or otherwise re- sium ͑unpublished͒. move the topological obstruction mathematically by adding a 82 X. L. Qi, T. L. Hughes, and S. C. Zhang, Phys. Rev. B 81, defectless Hamiltonian. ͔ T ͑ ͒ ෈͓ץ 90 ͒ ͑ 134508 2010 . Deformation of Hamiltonians on 1/2 s , for s 0,1 has di- 83 L. Fu, and C. L. Kane, Phys. Rev. B 79, 161408͑R͒͑2009͒. mension ␦=͑d−1͒−͑d−1+1+1͒=−2, and is classiﬁed by 2Z.

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