Topological insulators and superconductors: ten-fold way and dimensional hierarchy
Shinsei Ryu, Andreas Schnyder, Akira Furusaki, Andreas Ludwig
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Topological insulators and superconductors:ten-fold way and dimensional hierarchy
Shinsei Ryu , Andreas P Schnyder , , Akira Furusaki andAndreas W W Ludwig Department of Physics, University of California, Berkeley, California 94720, USA Kavli Institute for Theoretical Physics, University of California, Santa Barbara,California 93106, USA Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569Stuttgart, Germany Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan Department of Physics, University of California, Santa Barbara, California 93106,USAE-mail: [email protected], [email protected]
Abstract.
It has recently been shown that in every spatial dimension there existprecisely five distinct classes of topological insulators or superconductors. Within agiven class, the different topological sectors can be distinguished, depending on thecase, by a Z or a Z topological invariant. This is an exhaustive classification. Herewe construct representatives of topological insulators and superconductors for all fiveclasses and in arbitrary spatial dimension d , in terms of Dirac Hamiltonians. Usingthese representatives we demonstrate how topological insulators (superconductors) indifferent dimensions and different classes can be related via “dimensional reduction” bycompactifying one or more spatial dimensions (in “Kaluza-Klein”-like fashion). For Z -topological insulators (superconductors) this proceeds by descending by one dimensionat a time into a different class. The Z -topological insulators (superconductors),on the other hand, are shown to be lower-dimensional descendants of parent Z -topological insulators in the same class, from which they inherit their topologicalproperties. The 8-fold periodicity in dimension d that exists for topological insulators(superconductors) with Hamiltonians satisfying at least one reality condition (arisingfrom time-reversal or charge-conjugation/particle-hole symmetries) is a reflection ofthe 8-fold periodicity of the spinor representations of the orthogonal groups SO( N )(a form of Bott periodicity). Furthermore, we derive for general spatial dimensionsa relation between the topological invariant that characterizes topological insulatorsand superconductors with chiral symmetry (i.e., the winding number) and the Chern-Simons invariant. For lower dimensional cases, this formula relates the winding numberto the electric polarization ( d = 1 spatial dimensions), or to the magnetoelectricpolarizability ( d = 3 spatial dimensions). Finally, we also discuss topological fieldtheories describing the space time theory of linear responses in topological insulators(superconductors), and study how the presence of inversion symmetry modifies theclassification of topological insulators (superconductors).PACS numbers: 73.43.-f, 74.20.Rp, 74.45.+c, 72.25.Dc ONTENTS Contents1 Introduction 4 Z type. . . . . . . . . . . . . . . . 161.2.3 Classifying space. . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.4 Weak topological insulators and superconductors. . . . . . . . . . 171.2.5 Zero modes localized on topological defects. . . . . . . . . . . . . 181.3 Outline of the present article . . . . . . . . . . . . . . . . . . . . . . . . . 19 A → AIII ) 20 d = 2 n + 2 dimensions . . . . . . . . . . . . . . . . . . . . . . 202.2 Class AIII in d = 2 n + 1 dimensions . . . . . . . . . . . . . . . . . . . . . 222.2.1 Winding number. . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.2 Chern-Simons invariant. . . . . . . . . . . . . . . . . . . . . . . . 232.3 Dirac insulators and dimensional reduction . . . . . . . . . . . . . . . . . 262.4 Example: d = 3 → → d = 5 → → D = d + 1space-time dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Z topological insulators: real case 36 d = 2 n + 3 to d = 2 n + 2. . . . . . . . . . . . . . . . . . . . 373.1.2 From d = 2 n + 2 to d = 2 n + 1. . . . . . . . . . . . . . . . . . . . 373.1.3 From d = 2 n + 3 to d = 2 n . . . . . . . . . . . . . . . . . . . . . . 383.1.4 From d = 2 n to d = 2 n −
1. . . . . . . . . . . . . . . . . . . . . . 383.2 Example: d = 3 → → d = 5 → Z topological insulators and dimensional reduction 40 Z classification of first descendants. . . . . . . . . . . . . . . . . 434.1.3 Z classification of second descendants. . . . . . . . . . . . . . . . 454.1.4 Example: d = 3 → →
1. . . . . . . . . . . . . . . . . . . . . . . 464.2 Topological insulators lacking chiral symmetry . . . . . . . . . . . . . . . 48
ONTENTS
A Cartan symmetric spaces: generic Hamiltonians, NL σ M field theories,and classifying spaces of K -theory 52B Spinor representations of SO( N ) B.1 Spinors of SO(2 n + 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53B.2 Discrete symmetries of Dirac Hamiltonians . . . . . . . . . . . . . . . . . 55 C Topological insulators protected by a combination of spatial inversionand either time-reversal or charge-conjugation symmetry 57
C.1 Inversion symmetry combined with another discrete symmetry . . . . . . 57C.2 Classification of gapped Hamiltonians in the presence of a combination ofspatial inversion and either time-reversal or charge-conjugation symmetry 60C.2.1 Example: ( ǫ W , η W ) = (1 ,
1) in d = 2. . . . . . . . . . . . . . . . . 61 ONTENTS
1. Introduction
Topological insulators (superconductors) are gapped phases of non-interacting fermionswhich exhibit topologically protected boundary modes. These boundary states aregapless, with extended wavefunctions, and protected against arbitrary deformations(or perturbations) of the Hamiltonian, as long as the generic symmetries (such as e.g.time-reversal symmetry) of the Hamiltonian are preserved and the bulk gap is notclosed ‡ . The different phases within a given topological insulator (superconductor)are characterized by a topological invariant, which is either an integer Chern/windingnumber or a Z quantity. The integer quantum Hall effect (IQHE) is the best knownexample of such a phase. In the IQHE, the transverse (Hall) electrical conductance σ xy carried by edge states is quantized and proportional to the topologically invariantChern number [1, 2] § . As a consequence of the quantization of the Hall conductance σ xy ,two different quantum Hall states with different values of σ xy cannot be adiabaticallyconnected without closing the energy gap in the bulk. Therefore they representdistinct “topologically ordered” phases that are separated by a quantum phase transition(figure 1a).Other examples of gapped topological phases are the (spinless) chiral p x + i p y superconductor [8], which breaks time-reversal symmetry ( T ), and the quantum spinHall (QSH) states [9, 10, 11, 12, 13, 14, 15], which are time-reversal invariant. In theformer case, the topological features in question are those of the fermionic quasiparticleexcitations deep in the superconducting phase, whose dynamics is described bythe Bogoliubov-de Gennes (BdG) Hamiltonian k . The BdG Hamiltonian of anysuperconductor has, by construction, a “built-in” charge-conjugation (or: particle-hole) symmetry ( C ). In analogy to the IQHE, the different topological phases of the p x + i p y superconductor can be labeled by an integer ( Z ) topological invariant [16].The corresponding quantized conductance is the transverse (Hall) thermal, or “Leduc-Righi” conductance (divided by temperature). The quantum spin Hall effect (QSHE),on the other hand, which occurs in two-dimensional and three-dimensional time-reversalinvariant insulators, is characterized by a Z topological number ν [9, 12, 13, 14]. Thisbinary quantity distinguishes the non-trivial state ( ν = 1), whose boundary modesconsist of an odd number of Dirac fermion modes (i.e., odd number of Kramers’ pairs),from the trivial state ( ν = 0), which is characterized by an even number of Kramers’pairs of boundary (surface or edge) states (figure 1b). A physical realization of the QSHEwas theoretically predicted [17] and experimentally observed [18, 19] in HgTe/(Hg,Ce)Te ‡ In the presence of disorder (= lack of translational symmetry) a sufficient condition is that all bulkstates near the Fermi energy are localized [3]. § A profound implication of a non-vanishing Chern number is that exponentially localized Wannierwave functions cannot be constructed [4, 5, 6]. Yet another manifestation of the non-trivial natureof the band structure when σ xy = 0 appears in the scaling of the entanglement entropy (and theentanglement entropy spectrum) [7]. k Physically, the quasiparticles are responsible for heat (energy) transport, and not for the electricaltransport properties of the superconductor.
ONTENTS quantum phase transition quantum phase transition Figure 1.
Topological distinction among quantum ground states. (a) Z classificationand (b) Z classification. semiconductor quantum wells. Subsequently, based both on theoretical considerations[15, 20] and experimental measurements [21, 22, 23, 24, 25], the three-dimensional Z topological insulator phase was shown to occur in Bismuth-related materials, such asBiSb alloys, Bi Se and Bi Te .The key difference among the IQHE, the QSHE and three-dimensional Z topological insulator, and the chiral p x + i p y superconductor lies in their genericsymmetry properties (to be discussed in more detail below). All these states aretopological in the sense that they cannot be continuously deformed into a trivialinsulating state (i.e., a state without any gapless boundary modes) while keepingthe generic symmetries of the system intact and without closing the bulk energygap. The key property of the QSH states, for example, is the spin-orbit interactionin combination with time-reversal symmetry ( T ). The latter protects the boundarystates from acquiring a gap or becoming localized in the presence of disorder (= termsadded to the Hamiltonian which break translational symmetry). The chiral p x + i p y superconductor, on the other hand, is characterized by C but the lack of T .The aforementioned examples of “symmetry protected” topological states arepart of a larger scheme ¶ that fully classifies [26, 27] all topological insulators(superconductors) in terms of symmetry and spatial dimension. This classificationscheme is summarized in table 3. The first column in this table provides the completelist of all possible “symmetry classes” of single-particle Hamiltonians. There areprecisely ten such “symmetry classes”, a fundamental result due to Zirnbauer, andAltland and Zirnbauer [29, 30, 31]. These authors recognized that there is a one-to-one correspondence between single-particle Hamiltonians and the set of (“large”)symmetric spaces, of which there are precisely ten, a classic result obtained in 1926 by ¶ A complete classification of all topological insulators (superconductors) in dimensionalities upto d = 3 was given in Ref. [26]. A systematic regularity (periodicity) of the classification as thedimensionality is varied was discovered by Kitaev in Ref. [27] for all dimensions through the use ofK-Theory. A certain systematic pattern as the dimensionality is varied was discovered for some of thetopological insulators by Qi, Hughes and Zhang in Ref. [28] using a Chern-Simons action in extendedspace. Below we will use a version of the ideas of “dimensional reduction” employed in Ref. [28]. ONTENTS
Consider the gapped first-quantized Hamiltonian describing the topological insulator(superconductor) in the bulk ( d spatial dimensions). The topological features ofthis Hamiltonian, which we are interested in characterizing, are (by definition) notchanged if we modify (deform) the Hamiltonian by adding terms (or perturbations) toit which break any translational symmetry that may be present. Thus the topologicalfeatures must be properties of a translationally not invariant Hamiltonian. Seekinga classification scheme, one must have some framework within which to classify suchtranslationally not invariant Hamiltonians. Such a classification cannot involve thenotion of ordinary symmetries, i.e., of unitary operators which commute with theHamiltonian. Consider, e.g., (Pauli-) spin rotation symmetry, ubiquitous in condensedmatter systems. The notion of spin-rotation invariance can be eliminated by writing theHamiltonian in block form, and by focussing attention on a block of the Hamiltonian.Such a block decomposition can be performed for any symmetry which commutes withthe Hamiltonian. The notion of “symmetry classes” mentioned above refers to theproperties of these blocks of symmetry-less “irreducible” Hamiltonians: as it turns out,there are only ten of them. The basic idea why this is the case is simple to understand.The only properties that the blocks can satisfy are certain reality conditions that followfrom the “extremely generic symmetries” of time-reversal and charge-conjugation (or:particle-hole symmetry). These are not symmetries in the above sense, since they areboth represented by anti- unitary operators, when acting on the single-particle Hilbertspace. Specifically, consider a general system of non-interacting fermions described by a“second-quantized” Hamiltonian H . For a non-superconducting system, e.g., this reads H = X A,B ψ † A H A,B ψ B , (1)with fermion creation and annihilation operators satisfying canonical anti-commutationrelations { ψ A , ψ † B } = δ A,B . (2)Here we imagine for convenience of notation that we have “regularized” the systemon a lattice, and A, B are combined labels for the lattice sites i, j , and if relevant, ofadditional quantum numbers such as e.g., a Pauli-spin quantum number (e.g., A = ( i, a )with a, b = ± / H A,B is an N × N matrix, the “first quantized” Hamiltonian. ONTENTS ψ A , ψ † A and whose firstquantized form is again a matrix H when discretized on a lattice.)Now, time reversal symmetry can be expressed in terms of H : the system is invariantunder time reversal symmetry if and only if the complex conjugate of the first quantizedHamiltonian H ∗ is equal to H up to a unitary rotation U T , i.e. T : U † T H ∗ U T = + H . (3)Moreover, the system is invariant under charge-conjugation (or: particle-hole) symmetryif and only if the complex conjugate of the Hamiltonian H ∗ = H T is equal to minus H up to a unitary rotation U C , i.e. C : U † C H ∗ U C = −H . (4)(This property may be less familiar, but it is easy to check [26] that it is acharacterization of charge-conjugation (particle-hole) symmetry for non-interactingsystems of fermions). A look at equations (3,4) reveals that T and C , when actingon the single particle Hilbert space, are not unitary symmetries, but rather realityconditions on the Hamiltonian H modulo unitary rotations + ∗ . + In second-quantized language, time-reversal and particle-hole operations can be written in terms oftheir action on the canonical fermion creation and annihilation operators, T ψ A T − = X B ( U T ) A,B ψ B , C ψ A C − = X B ( U ∗ C ) A,B ψ † B . (5)While the particle-hole transformation is unitary, the time-reversal operation is anti-unitary, T i T − = − i. The system is time-reversal invariant (particle-hole symmetric), if and only if T H T − = H ( C H C − = H ). This leads directly to the conditions (3) and (4) for the first quantized Hamiltonian.Note that T H T − = H implies T ψ A ( t ) T − = T e +i Ht ψ A e − i Ht T − = P B ( U T ) A,B ψ B ( − t ).Iterating T and C twice, one obtains T ψ A T − = P B ( U ∗ T U T ) A,B ψ B , and C ψ A C − = P B ( U ∗ C U C ) A,B ψ B . When acting on the first quantized Hamiltonian this reads ( U ∗ T U T ) † H ( U ∗ T U T )= H and ( U ∗ C U C ) † H ( U ∗ C U C )= H , respectively. The first quantized Hamiltonians H are seen (below) to runover an irreducible representation space, and thus ( U ∗ T U T ) and ( U ∗ C U C ) are both multiples of theidentity matrix I N (by Schur’s lemma). Since U T and U C are unitary matrices, there are only twopossibilities for each, i.e., U ∗ T U T = ± I N and U ∗ C U C = ± I N . The time-reversal operation T and theparticle-hole transformation C can then each square to plus or to minus the identity, T = ±
1, and C = ± ∗ It may also be worth noting that we may assume without loss of generality that there is only a single time-reversal operator T and a single charge-conjugation operator C . If the (first quantized)Hamiltonian H was invariant under, say, two charge-conjugation operations C and C , then thecomposition C · C of these two symmetry operations would be a unitary symmetry when acting onthe first quantized Hamiltonian H , i.e., the product U C · U ∗ C would commute with H . By bringing theHamiltonian H in block form, U C · U ∗ C would then be constant on each block. Thus, on each block U C and U C would then be trivially related to each other, and it would suffice to consider one of the twocharge conjugation operations. – On the other hand, note that the product T ·C corresponds to a unitary symmetry operation when acting on the first quantized Hamiltonian H . But in this case the unitarymatrix U T · U ∗ C does not commute but anti- commutes with H . Therefore, T · C does not correspond toan an “ordinary” symmetry of H . It is for this reason that we need to consider the product T · C (called“chiral”, or “sublattice” symmetry S below) as an additional essential ingredient for the classificationof the blocks, besides time-reversal T and charge-conjugation (particle-hole) symmetries C . ONTENTS T ), the Hamiltonian can either be (i): not time-reversal invariant,in which case we write T = 0, or (ii): it may be time-reversal invariant and the anti-unitary time-reversal symmetry operator T squares to plus the identity operator, inwhich case we write T = +1, or (iii): it may be time-reversal invariant and the anti-unitary time-reversal symmetry operator T squares to minus the identity, in which casewe write T = −
1. Similarly, there are three possible ways for the Hamiltonian H torespond to charge-conjugation (particle-hole symmetry) C (again, C may square to plusor minus the identity operator). For these three possibilities we write C = 0 , +1 , − × H to respond to time-reversal and charge-conjugation (particle-hole transformation). These are not yet allten cases because it is also necessary to consider the behavior of the Hamiltonian underthe product S = T · C which is a unitary operation. A moment’s thought (see table 1)shows that for 8 of the 9 possibilities the behavior of the Hamiltonian under the product S = T · C is uniquely fixed ♯ . (We write S = 0 if the operation S is not a symmetryof the Hamiltonian, and S = 1 if it is.) The only case when the behavior under thecombined transformation S = T · C is not determined by the behavior under T and C is the case where T = 0 and C = 0. In this case either S = 0 or S = 1 is possible. Thisthen yields (3 × −
1) + 2 = 10 possible behaviors of the Hamiltonian.The list of ten possible behaviors of the first quantized Hamiltonian under T , C and S is listed in table 1. These are the ten generic symmetry classes (the “ten-fold way”)which are the framework within which the classification scheme of topological insulators(superconductors) is formulated.Let us first point out a very general structure seen in table 1. This is listedin the column entitled “Hamiltonian”. When the first quantized Hamiltonian H is“regularized” (or: “put on”) a finite lattice, it becomes an N × N matrix (as discussedabove). The entries in the column “Hamiltonian” specify the type of N × N matrix thatthe quantum mechanical time-evolution operator exp(i t H ) is. For example, for systemswhich have no time-reversal or charge-conjugation symmetry properties at all, i.e., forwhich T = 0, C = 0, S = 0, which are listed in the first row of the table, there are noconstraints on the Hamiltonian except for Hermiticity. Thus H is a generic Hermitianmatrix and the time-evolution operator is a generic unitary matrix, so that exp(i t H ) isan element of the unitary group U( N ) of unitary N × N matrices. By imposing time-reversal symmetry (for a system that has, e.g., no other degree of freedom such as, e.g.,spin), there exists a basis in which H is represented by a real symmetric N × N matrix.This, in turn, can be expressed as saying that the time-evolution operator is an elementof the coset of groups, U( N ) / O( N ). All other entries of the column “Hamiltonian” can ♯ The symmetry operation S is sometimes called “sublattice symmetry”, hence the notation S .However, in many instances S is not realized as a sublattice symmetry, but is just simply the productof T and C . Therefore the term “chiral symmetry” to describe the symmetry operation S is sometimesmore appropriate. ONTENTS Cartan label T C S Hamiltonian
G/H (ferm. NL σ M)A (unitary) 0 0 0 U( N ) U(2 n ) / U( n ) × U( n )AI (orthogonal) +1 0 0 U( N ) / O( N ) Sp(2 n ) / Sp( n ) × Sp( n )AII (symplectic) − N ) / Sp(2 N ) O(2 n ) / O( n ) × O( n )AIII (ch. unit.) 0 0 1 U( N + M ) / U( N ) × U( M ) U( n )BDI (ch. orth.) +1 +1 1 O( N + M ) / O( N ) × O( M ) U(2 n ) / Sp(2 n )CII (ch. sympl.) − − N + M ) / Sp( N ) × Sp( M ) U(2 n ) / O(2 n )D (BdG) 0 +1 0 SO(2 N ) O(2 n ) / U( n )C (BdG) 0 − N ) Sp(2 n ) / U( n )DIII (BdG) − N ) / U( N ) O(2 n )CI (BdG) +1 − N ) / U( N ) Sp(2 n ) Table 1.
Listed are the ten generic symmetry classes of single-particle Hamiltonians H , classified according to their behavior under time-reversal symmetry ( T ), charge-conjugation (or: particle-hole) symmetry ( C ), as well as “sublattice” (or: “chiral”)symmetry ( S ). The labels T, C and S, represent the presence/absence of time-reversal, particle-hole, and chiral symmetries, respectively, as well as the types of thesesymmetries. The column entitled “Hamiltonian” lists, for each of the ten symmetryclasses, the symmetric space of which the quantum mechanical time-evolution operatorexp(i t H ) is an element. The column “Cartan label” is the name given to thecorresponding symmetric space listed in the column “Hamiltonian” in ´Elie Cartan’sclassification scheme (dating back to the year 1926). The last column entitled “ G/H (ferm. NL σ M)” lists the (compact sectors of the) target space of the NL σ M describingAnderson localization physics at long wavelength in this given symmetry class. be obtained from analogous considerations †† . What is interesting about this columnis that its entries run precisely over what is known as the complete set of ten (“large”)symmetric spaces † , classified in 1926 in fundamental work by the mathematician ´ElieCartan. Thus, as the first quantized Hamiltonian runs over all ten possible symmetryclasses, the corresponding quantum mechanical time-evolution operator runs over all tensymmetric spaces. Thus, the appearance of the Cartan symmetric spaces is a reflectionof fundamental aspects of (single-particle) quantum mechanics. We will discuss the lastcolumn entitled “ G/H (ferm. NL σ M)” in the following subsection. †† Possible realizations of the chiral symmetry classes AIII, BDI, CII possessing time-evolutionoperators in table 1 with N = M are tight-binding models on bipartite graphs whose two (disjoint)subgraphs contain N and M lattice sites. † A symmetric space is a finite-dimensional Riemannian manifold of constant curvature (its Riemanncurvature tensor is covariantly constant) which has only one parameter, its radius of curvature. Thereare also so-called exceptional symmetric spaces which, however, are not relevant for the problem athand, because for them the number N would be a fixed finite number, which would prevent us frombeing able to take the thermodynamic (infinite-volume) limit of interest for all the physical systemsunder consideration. ONTENTS The approach used in Ref. [26] to classify all possible topological insulators(superconductors) rested on the existence of protected extended degrees of freedomat the system’s boundary. These are, in particular, also protected in the presence ofarbitrarily strong perturbations at the boundary which break translational symmetry(commonly referred to as “random”, or “disordered”). The existence of extended,gapless degrees of freedom even in strongly random fermionic systems is highly unusual,because of the phenomenon of Anderson localization ‡ . Thus, the degrees of freedom atthe boundary of topological insulators (superconductors) must be of a very special kind,in that they entirely evade the phenomenon of Anderson localization. Our approachconsists in classifying precisely all those problems of (non-interacting) fermionic systemswhich completely evade the phenomenon of Anderson localization. We have completedthis task in Ref. [26]. Thus, we have reduced the problem of classifying all topologicalinsulators (superconductors) in d spatial bulk dimensions to a classification problem ofAnderson localization at the ( d −
1) dimensional boundary. By solving the mentionedproblem of Anderson localization, we have thereby solved the problem of classifying alltopological insulators (superconductors).We will now focus attention on the above-mentioned problem of Andersonlocalization at the ( d −
1) dimensional boundary of a d dimensional topological insulator(superconductor). Specifically, we will now review a solution of this problem [26, 36]which allows us to see directly the dependence on dimensionality d of the classification oftopological insulators (superconductors). In the later, main part of this article, we willpresent another point of view of this dependence on dimensionality d (using “dimensionalreduction”).The theoretical description of problems of Anderson localization is well known tobe very systematic and geometrical [37, 38]. A problem of Anderson localization is ingeneral described by a random Hamiltonian (i.e., one that lacks translational symmetry).That Hamiltonian will be in one of the ten symmetry classes listed in table 1 andwe are currently focusing on Hamiltonians describing the boundary of the topologicalinsulator (superconductor). Now, as it turns out, at long length scales (much larger thanthe “mean free path”) a description in terms of a “non-linear-sigma-model” (NL σ M)emerges. A NL σ M is a system like that describing the classical statistical mechanicsof a Heisenberg magnet. The only difference is that while the magnet is formulated interms of unit vector spins, pointing to the surface of a two-dimensional sphere, for ageneral NL σ M that spin is replaced § by an element of one of the ten symmetric spaceslisted in the last column of table 1, called the “target space” of the NL σ M, denoted by ‡ This is the phenomenon that, at least for sufficiently strong disorder potentials, spatially extended(= delocalized) eigenstates of the Hamiltonian tend to become localized (i.e., exponentially decayingin space) [34, 35, 38, 33]. § The two-dimensional sphere S is a particularly simple example of a symmetric space, namely it canbe written as the space S = U(2) / U(1) × U(1), i.e., the first row in the last column of table 1, when n = 1. ONTENTS G/H k . For a given symmetry class of the original Hamiltonian, whose time-evolutionoperator is characterized by the penultimate column, the specific “target space” thatneeds to be used for the NL σ M describing Anderson localization in this symmetry classis listed in the last column of table 1 (see also appendix A).Now, the NL σ M on the ( d −
1) dimensional boundary of the d dimensionaltopological insulator (superconductor) completely evades Anderson localization if acertain extra term of topological origin can be added to the action of the NL σ M whichhas no adjustable parameter [26, 36]. Whether such an extra term is allowed dependson (i): the “target space” of the NL σ M in the symmetry class in question ¶ , and (ii):the dimensionality ¯ d := ( d −
1) of the boundary on which the NL σ M is defined. Thereare only three terms of topological origin which can possibly be added to the action ofthe NL σ M: these are a θ -term (Pruisken term) [39], a Z topological term [40], or aWess-Zumino-Witten (WZW) term [41, 42]. It is the homotopy groups of the NL σ Mtarget spaces
G/H which determine whether it is possible to add such a topological termto a given NL σ M action (see table 2). Specifically, a θ -term (Pruisken term) can appearwhen π ¯ d ( G/H ) = Z . Similarly, a Z topological term is allowed when π ¯ d ( G/H ) = Z ,and a WZW term can be included + when π ¯ d +1 ( G/H ) = Z . Note that, on the onehand, one obtains, upon addition of a θ -term (Pruisken term), a one-parameter familyof theories (on the boundary) depending on the value of θ , all of which reside in thesame symmetry class. On the other hand, however, it is known that only for a specialvalue of the parameter θ Anderson localization is avoided; for generic values of θ , thisis not the case. It is for this reason that the ability to add a θ -term (Pruisken term)is not of interest to the question we are asking. One is therefore left only with a Z topological term or a WZW term as the only terms of topological origin which have nofreely adjustable parameter, and which are thus of relevance here.The homotopy groups for all ten NL σ M target spaces
G/H listed in the last columnof table 1 are well known from the literature (see, e.g., Ref. [43] for a summary) andthis information is summarized in table 2 for the convenience of the reader. In orderto make a certain regular structure of this table apparent, the rows of table 1 havebeen re-ordered in a specific way [44]. Part of this re-ordering is a subdivision of allsymmetry classes into what is called “ complex case ” and “ real case ” in table 2. Thephysical origin of this subdivision is simple to understand. In the category “ complexcase ” appear precisely those symmetry classes in which there is no reality condition( T or C ) whatsoever imposed on the Hamiltonian. We see from table 1 that these arethe classes which carry Cartan label A and AIII. The Hamiltonians in these symmetryclasses are therefore “complex” (in this sense). The other category called “ real case ” in k Both G and H are classical Lie groups, and H is a maximal subgroup of G . ¶ The relevant “target space” is, as already mentioned above, listed in the last column of table 1. + Observe that, while the homotopy group determines if a term of topological origin for a given NL σ Maction is allowed in principle, it depends on the specific disorder model considered, whether such a termis actually present. In any case, if a term of topological origin is possible in a specific dimension andsymmetry class, then a corresponding topological insulator (superconductor) can exist in this symmetryclass (in one dimension higher).
ONTENTS T or C – cf. equations (3,4)] imposed on theHamiltonian (see table 1). In this sense the Hamiltonians in these symmetry classes aretherefore “real”. complex case: AZ G/H ¯ d = 0 ¯ d = 1 ¯ d = 2 ¯ d = 3 ¯ d = 4 ¯ d = 5 ¯ d = 6 ¯ d = 7A U( N + M ) / U( N ) × U( M ) ← Z ←− Z ←− Z ←− Z ← AIII U( N ) ←− Z ←− Z ←− Z ←− Z real case: AZ G/H ¯ d = 0 ¯ d = 1 ¯ d = 2 ¯ d = 3 ¯ d = 4 ¯ d = 5 ¯ d = 6 ¯ d = 7AI Sp( N + M ) / Sp( N ) × Sp( M ) ← Z ←− Z Z Z ← BDI
U(2 N ) / Sp(2 N ) ←− Z ←− Z Z Z D O(2 N ) / U( N ) Z ←− Z ←− Z Z DIII O( N ) Z Z ←− Z ←− Z AII O( N + M ) / O( N ) × O( M ) ← Z Z Z ←− Z ← CII U( N ) / O( N ) ←− Z Z Z ←− Z Sp(2 N ) / U( N ) ←− Z Z Z ←− Z CI Sp(2 N ) ←− Z Z Z ←− Z Table 2.
Table of homotopy groups π ¯ d ( G/H ) for symmetric spaces
G/H , taken fromthe standard mathematical literature (see e.g. Ref. [43] for a summary). [Here, N mustbe sufficiently large for a given ¯ d . Cartan labels of those symmetry classes invariantunder the chiral symmetry operation S = T · C from table 1 are indicated by boldface letters.]. The pattern continues for higher ¯ d , with periodicity 2 for the complexcase, and 8 for the real case. The entries corresponding to Z topological insulators(superconductors) in ( ¯ d + 1) dimensions (see e.g. table 3) are indicated by blue colorboldface symbols. There is a ( ¯ d + 1)-dimensional Z topological insulator, whenever π ¯ d ( G/H ) = Z (also indicated in blue). A ( ¯ d + 1)-dimensional Z topological insulator,on the other hand, can be realized, whenever π ¯ d +1 ( G/H ) = Z . – This is a way [26, 36]to directly relate the classification of topological insulators (superconductors) to thetable of homotopy groups. Table 2 now tells us directly in which symmetry class there exist topologicalinsulators or superconductors (and of which type, Z or Z ): according to the abovediscussion we know that a topological insulator (superconductor) exists in a givensymmetry class in d = ¯ d + 1 spatial dimensions if and only if the target space of theNL σ M on the ¯ d -dimensional boundary allows either (a): for Z topological term, whichis the case when π ¯ d ( G/H ) = π d − ( G/H ) = Z , or (b): allows for a WZW term, which isthe case when π d ( G/H ) = π ¯ d +1 ( G/H ) = Z . By using this rule in conjunction with table2 of homotopy groups, we arrive with the help of table 1 at the table 3 of topological ONTENTS complex case: Cartan \ d · · · A Z Z Z Z Z Z · · · AIII 0 Z Z Z Z Z Z · · · real case: Cartan \ d · · · AI Z Z Z Z Z · · · BDI Z Z Z Z Z Z · · · D Z Z Z Z Z Z Z · · · DIII 0 Z Z Z Z Z Z Z · · · AII 2 Z Z Z Z Z Z Z · · · CII 0 2 Z Z Z Z Z Z · · · C 0 0 2 Z Z Z Z Z · · · CI 0 0 0 2 Z Z Z Z Z · · · Table 3.
Classification of topological insulators and superconductors as a function ofspatial dimension d and symmetry class, indicated by the “Cartan label” (first column).The definition of the ten generic symmetry classes of single particle Hamiltonians(due to Altland and Zirnbauer[29, 30]) is given in table 1. The symmetry classesare grouped in two separate lists, complex and real cases, depending on whether theHamiltonian is complex, or whether one (or more) reality conditions (arising from time-reversal or charge-conjugation symmetries) are imposed on it; the symmetry classesare ordered in such a way that a periodic pattern in dimensionality becomes visible[27]. (See also the discussion in subsection 1.1 and table 2.) The symbols Z and Z indicate that the topologically distinct phases within a given symmetry class oftopological insulators (superconductors) are characterized by an integer invariant ( Z ),or a Z quantity, respectively. The symbol “0” denotes the case when there existsno topological insulator (superconductor), i.e., when all quantum ground states aretopologically equivalent to the trivial state. insulators and superconductors ∗ .A look at table 3 reveals that in each spatial dimension there exist five distinctclasses of topological insulators (superconductors), three of which are characterized byan integral ( Z ) topological number, while the remaining two possess a binary ( Z )topological quantity ♯ .The topological insulators (superconductors) appear in table 3 along diagonal lines.That is, as the spatial dimension is increased by one, locations where the topological ∗ To be explicit, one has to move all the entries Z in table 2 into the locations indicated by the arrows,and has to replace the column label ¯ d by d = ¯ d + 1. The result is table 3. ♯ Note that while d = 0 , , , ONTENTS †† .Second, we want to draw the reader’s attention to the fact that for a given symmetryclass in table 3 any Z topological insulator (superconductor) always appears as part ofa “triplet” that consists of a d -dimensional Z topological insulator (superconductor) andtwo Z topological insulators (superconductors) in ( d −
1) and ( d −
2) dimensions. Thissuggests that the topological characteristics of the members of such a “triplet” are closelyrelated and indeed, it was shown in Ref. [28] that the Z classification in symmetry classAII in d = 2 and 3 can be derived from the four-dimensional Z topological insulatorby a process of “dimensional reduction”. The same procedure has been applied [28] toderive Z classifications in those symmetry classes that do not possess a form of chiralsymmetry (i.e., for which S = 0) – there are five of them. In section 4 we will extendthis approach to the Z topological insulators (superconductors) in the remaining fivesymmetry classes which possess a form of chiral symmetry (i.e., for which S = 1).Let us now discuss explicit forms of the “terms of topological origin” which canbe added to the action of the NL σ M with “target space”
G/H , at the boundary of thetopological insulator (superconductor).
Here there is a pattern, alternating in the dimension¯ d of the boundary, for the way these topological terms can be constructed. (i): Fora ¯ d = odd-dimensional boundary, and for the NL σ Ms which describe the Anderson-delocalized boundary of the topological bulk, the field (denoted by Q ∈ G/H below † †† The IQHE is not quite an anomaly in the sense that the parity and T is explicitly broken.However, the quantization of σ xy in the IQHE can be viewed as essentially the same phenomenonas the appearance of the Chern-Simons term in QED in D = 2 + 1-dimensional space-time, wheremassless Dirac fermions are coupled with the electromagnetic U(1) gauge field. † While looking similar, the field Q in the NL σ M has nothing to do with the spectral projector inmomentum space defined in equation (7).
ONTENTS
15– cf. last column of table 1) of the NL σ M field theory is a Hermitean matrix fieldwhich satisfies certain constraints. The WZW term responsible for the lack of Andersonlocalization takes on the form of the ¯ d -dimensional integral over the boundary of the¯ d -dimensional Chern-Simons form (compare equation (22) of the main text below).Alternatively, it can be written also as an integral over a ¯ d + 1-dimensional region whoseboundary coincides with the physical ¯ d -dimensional space, by smoothly interpolatingthe NL σ M field configuration Q into one dimension higher. The fact that π ¯ d +1 ( G/H ) = Z guarantees that different ways of interpolations do not matter, and the action dependsonly on the physical field configurations in ¯ d -dimensional space. (ii): For a ¯ d = even-dimensional boundary, and for the NL σ Ms which describe the Anderson-delocalizedboundary of the topological bulk, the field in the NL σ M field theory is either a groupelement or a unitary matrix field (denoted by g ∈ G/H below – cf. last column of table1) which may, depending on the case, be subject to certain constraints. The WZWterm responsible for the lack of Anderson localization takes on the form of a ¯ d + 1-dimensional integral of the winding number density, as defined in equation (20) of themain text below.To illustrate these terms, we will now give explicit examples for them in lowdimensionalities: • d = 2: For the d = 2-dimensional topological insulator of the IQHE, the ¯ d = 1-dimensional NL σ M describing the edge states is the one on U(2 n ) / U( n ) × U( n ).The field Q of the NL σ M field theory can be parameterized as Q = U † Λ U ∈ U(2 n ) / U( n ) × U( n ) where U ∈ U(2 n ) and Λ = diag ( I n , − I n ). The term of relevancefor the absence of Anderson localization at the edges of the integer quantum Hallinsulator is ∝ σ xy R dx tr[ A x ] where A x := U † ∂ x U ; this is a ¯ d = 1 dimensionalanalogue of the three-dimensional Chern-Simons term. This term can also berewritten as a WZW type two-dimensional integral ∝ σ xy R D dxdu ǫ µν tr [ Q∂ µ Q∂ ν Q ]( µ, ν = x, u ). Here, we have extended the original ¯ d = 1 dimensional space toa two-dimensional region D by adding a fictitious space direction, parameterizedby u ∈ [0 , ∂D of the two-dimensional region coincides with theoriginal ¯ d = 1 dimensional space. Accordingly, the original NL σ M field Q ( x ) issmoothly extended to Q ( x, u ) such that it coincides with Q ( x ) when u = 0. Since π [U(2 n ) / U( n ) × U( n )] = Z the two-dimensional integral turns out to depend onlyon the field configuration on ∂D . Note that when we specialize to the case of n = 1, all these expressions are well-known from the coherent state path integralof an SU(2) quantum spin whose path integral is described by a d = 1 dimensionalNL σ M on the target space O(3) = U(2) / U(1) × U(1). • d = 3: For d = 3 topological insulators (superconductors) in classes AIII, DIII andCI, the relevant NL σ Ms at their ¯ d = 2-dimensional boundary (surface) are the oneswith a group manifold as target manifold [U( n ), O( n ), Sp( n ) for classes AIII, DIII,and CI, respectively]. The relevant WZW type term can be written as a three-dimensional integral, R D d x du ǫ µνλ tr [( g − ∂ µ g )( g − ∂ ν g )( g − ∂ λ g )], where g ∈ U( n ), ONTENTS n ), Sp( n ) [41]. Here again, the original ¯ d = 2 dimensional space is smoothlyextended to the three-dimensional region D , in such a way that ∂D coincideswith the physical ¯ d = 2 dimensional space. Accordingly, the field configuration g ( x, y ) defined on the ¯ d = 2 dimensional space is smoothly extended to the three-dimensional one g ( x, y, u ) such that g ( x, y, u = 0) = g ( x, y ). • d = 4: For the d = 4 topological insulators in classes A, AII, and AI, the relevant¯ d = 3 NL σ Ms have target spaces
G/H = G(2 n ) / G( n ) × G( n ) with G = U, O, andSp respectively. The field of the NL σ M can be parameterized by Q = U † Λ U where U ∈ G(2 n ). The relevant “term of topological origin” is the Chern-Simons term ∝ R d x ǫ µνλ tr [ A µ ∂ ν A λ + (2 / A µ A ν A λ ] where A µ = U † ∂ µ U ( µ = x, y, z ). Theappearance of this term at the boundary of the d = 4 topological insulators in classAII was discussed in Ref. [48]. • d = 5: For the d = 5 topological insulators in symmetry classes AIII, BDI, CII,the ¯ d = 4 dimensional NL σ Ms have target spaces with a matrix field g which isan element of G/H = U( n ), U(2 n ) / Sp(2 n ), U(2 n ) / O(2 n ), respectively. For thesespaces the WZW term takes the form of the ¯ d + 1 = 5 dimensional integral (withboundary) of the winding number density defined in equation (21) of the main textbelow.Note that those entries in table 2 of homotopy groups which are integers, π ¯ d ( G/H ) = Z , have a periodicity in ¯ d equal to four. Therefore, the forms of the “termsof topological origin” of WZW type listed in the above low-dimensional cases will repeat,with the appropriate replacement of ¯ d . Z type. The Z topological term for the NL σ Mat the ¯ d = 2 dimensional surface of the d = 3-dimensional Z topological insulatorsin symmetry class AII was discussed in Refs. [48, 49, 50]. Similarly, at the ¯ d = 2dimensional surface of the d = 3-dimensional Z topological insulators in symmetryclass CII, the Z topological term is added to the NL σ M [51].
There is a third way in which the set of Cartan symmetricspaces listed in table 1 appear in the context of the classification of topological insulators(superconductors). This arises from Fermi statistics which allows for a distinctionbetween those states of the (first quantized) Hamiltonian H with energies which lieabove, and those that lie below the Fermi energy E F . In a topological insulator(superconductor) there is always an energy gap between the Fermi energy E F andthe eigenstates of H lying above as well as those lying below E F . Therefore onemay, without closing the bulk gap , continuously deform the Hamiltonian H into onein which all eigenstates below the Fermi energy E F have the same energy (say) E = − E = +1. By ONTENTS ‡ , which we often denote by Q , hasthe same topological properties as the original Hamiltonian. The structure of Q for allsymmetry classes is listed in table III (second column) of the first article of Ref. [26] forsystems with translational symmetry. ( Q will also be used again in the present article.)If we were to consider a “zero-dimensional” topological insulator (superconductor) where“space consisted of a single point”, then for each of the ten symmetry classes we obtain § a matrix Q of a certain kind. It turns out that the so-obtained matrix Q for eachsymmetry class is again an element of one of the ten symmetric spaces listed in table 1,except that the order in which these spaces appear is different from the order in the twocolumns of table 1. The resulting assignment is listed k in the last column in table A1.The symmetric spaces to which the matrices Q belong are what is called the “classifyingspace” in the classification scheme of Ref. [27] employing K-Theory ¶ . In appendix Awe collect a number of interesting relationships between these three lists of the tensymmetric spaces: (i) the unitary time evolution operator, (ii) the NL σ M target space,and (iii) the classifying space.We end this subsection by commenting on additional information that can be readoff from our main results, listed in Table 3. This pertains to the existence of so-called weak topological insulators (superconductors), as well as of zero-energy or extendedmodes localized on topological defects in topological insulators (superconductors).
Table 3 classifies topologicalfeatures of gapped free fermion Hamiltonians that do not depend on the presence oftranslational symmetries of a crystal lattice + . In particular, these properties are notdestroyed when translation symmetry is broken, e.g., by the introduction of positionaldisorder. Systems which exhibit such robust topological properties are often also referredto as strong topological insulators (superconductors). This is to contrast them withso-called weak topological insulators (superconductors) which only possess topologicalfeatures when translational symmetry is present. As soon as translational symmetryis broken, such weak topological features are no longer guaranteed to exist, and thesystem is allowed to become topologically trivial. For example, systems defined ona d -dimensional lattice whose momentum space is the d -dimensional torus T d , allowfor weak topological insulators which are not strong topological insulators. Suchsystems are topologically equivalent to parallel stacks of lower dimensional strong ‡ This “simplified Hamiltonian” Q is often referred to in Ref. [26] and in the present article as a“projector”, since it is trivially related to the projection operator P onto all filled (with E < E F )eigenstates of H by Q = 1 − P . (The matrix in subsection 1.2.1 that is denoted by the same letter Q is an entirely different object, not to be confused with Q = 1 − P appearing here.) § From table III of the first article of Ref. [26], by setting the wavevector k → k The last row entitled “classifying space” in table A1, is nothing but a list of the matrices Q , and thislist follows from the second column of table III of the first article of Ref. [26] by letting k = 0. ¶ The list of “classifying spaces” was obtained in Ref. [27] through the use of Clifford algebras. + As we will explain below, in order to classify such so-called “strong topological insulators” it issufficient to only consider continuous
Hamiltonians with momentum space S d , the d -dimensional sphere. ONTENTS T , which is a subspace of the three-dimensional momentumspace T , onto the complex Grassmannian, G m,m + n ( C ). Similarly, in symmetry classAII, there exists a three-dimensional weak topological insulator [12, 13, 14, 15], whichconsists of layered two-dimensional Z topological quantum states, and whose weaktopological features are described by a triplet of Z invariants.The three-dimensional “weak” integer quantum Hall insulators, as well as theweak topological insulators of Refs. [12, 13, 14, 15] are both examples of d = 3-dimensional weak topological insulators of “codimension” one, i.e., they can be viewedas one-dimensional arrays of d = 2-dimensional strong topological insulators. Theexistence of these states can be read off from the d = 2-column and the rows of Table3 labeled A and AII, respectively. In general, d -dimensional weak topological statescan be of any “codimension” k , with 0 < k ≤ d . The existence of weak topologicalinsulators and superconductors in d spatial dimensions and for a given symmetryclass can be readily inferred from table 3. That is, d -dimensional weak topologicalinsulators (superconductors) of “codimension” k can occur whenever there exists astrong topological state in the same symmetry class, but in d − k dimensions. Specifically,the topological order (characterized by elements in Z or in Z ) can be specified foreach of the (cid:0) dk (cid:1) “orientations” of the submanifold. Moreover, it is also possible that inaddition “strong” topological order, i.e., in the full space dimension d , exists. In the K-Theory description of Ref. [27] the presence of weak topological features is described byadditional summands in the Abelian group of topological invariants (i.e., Z or Z ) whichappears when homotopy classes of maps from the sphere (strong topological insulators)are replaced by maps from the torus T d (weak and strong topological insulators). Another interesting applicationof table 3 is to determine whether r -dimensional topological defects in topologicalinsulators (superconductors) can support localized (isolated) zero modes. For example,let us first consider a point-like defect (i.e., r = 0), which is embedded in a d -dimensionalsystem ( d ≥
1) of any symmetry class. Furthermore, let us assume that at the defecttime-reversal symmetry is broken (T = 0) but particle-hole symmetry is preserved withC = +1, i.e., we are dealing with symmetry class D. From table 3 we infer that insymmetry class D in d = r + 1 = 1 dimensions there is a Z classification, which impliesthat the ( r = 0)-dimensional boundary can support gapless states, and, therefore, apoint-like defect with the symmetries of class D can bind zero modes. This situation isrealized, e.g., for a vortex core in a ( p ± i p )-wave superconductor, which can bind isolatedMajorana modes. In general, whether it is possible for a r -dimensional topological defectof a given symmetry class to support gapless states or not is determined by the entryof table 3 at the intersection of column d = r + 1 and the row of the given symmetry ONTENTS r = 1) in a d = 3-dimensional lattice hostinga weak topological insulator in symmetry class AII was found to bind an extendedgapless zero mode. (This corresponds in table 3 to the intersection of the row denotingsymmetry class AII with the column d = r + 1 = 1 + 1 = 2, the latter representing the d = 3-dimensional weak topological insulator, which corresponds to a strong topologicalinsulator in “codimension” k = 1.) To illustrate the use of table 3, we can for examplepredict that a similar binding of extended zero modes to lattice dislocation lines ( r = 1)can also occur in weak d = 3-dimensional topological insulators of “codimension” k = 1in symmetry classes A, D, DIII, and C. In this paper we discuss in detail the mechanism behind the dimensional periodicityand shift property appearing in table 3. We first demonstrate that there exists,for all five symmetry classes of topological insulators or superconductors, and in alldimensions d , a representative of the Hamiltonian in this class which has the formof a Dirac Hamiltonian, by constructing explicitly such a representative. We usethese Dirac Hamiltonian representatives in the five symmetry classes with topologicalinsulators (superconductors) to obtain relationships between these five symmetry classesin different dimensions. First we construct dimensional hierarchies (“dimensionalladders”) relating Z topological insulators (superconductors) in different dimensionsand symmetry classes (see sections 2 and 3). This is done by using a process of“dimensional reduction” in which spatial dimensions are compactified (in “Kaluza-Kleinlike” fashion), thus relating higher to lower dimensional theories. By employing thesame dimensional reduction process, we derive in section 4 the topological classificationof the Z topological insulators (superconductors) from their higher-dimensional parent Z topological insulators (superconductors) in the same symmetry class. Sections 2,3, 4 taken together provide a complete, independent derivation of table 3 of topologicalinsulators (superconductors).In section 2.2 we establish a connection between the topological invariant(winding number) defined for topological insulators and superconductors with chiralsymmetry and yet another topological invariant, the Chern-Simons invariant, for generaldimensions [equation (43)]. For lower dimensional cases, the formula (43) relates thewinding number to the electric polarization ( d = 1 spatial dimensions), and to the themagnetoelectric polarizability ( d = 3 spatial dimensions).In section 2.6 we list, for topological insulators in symmetry classes A and AIII thetopological field theory describing the space time theory of linear responses in all spacetime dimensions D = d + 1. For symmetry class A this is the Chern-Simons action, andin class AIII it is the theta term with theta angle θ = π . (Generalizations to topologicalsinglet superconductors with SU(2) spin-rotation symmetry are mentioned in section 5).Finally, as already mentioned in subsection 1.2, we have discussed for all ONTENTS σ M field theories on symmetric spaces. In particular, in odddimensions, these are the Chern-Simons terms.In appendix A we collect a number of interesting relationships between theCartan classification of generic Hamiltonians due to Altland and Zirnbauer, the NL σ Mtarget space of Anderson localization, and the classifying space appearing in the K-theory approach to topological insulators. The representation theory of the spinorrepresentations of the orthogonal groups SO( N ) is briefly reviewed in appendix B.Furthermore, we study in appendix C the influence of inversion symmetry on theclassification of topological insulators (superconductors), combined with either time-reversal or charge-conjugation (particle-hole) symmetry.
2. Dimensional hierarchy: complex case ( A → AIII ) In this section, we consider the relationship between 2 n -dimensional class A topologicalinsulator and (2 n − T and C ), but Hamiltonians in classAIII possess “chiral symmetry” called S in the table, i.e., they anti- commute witha unitary operator.) First we present a general discussion about class A and AIIItopological insulators (sections 2.1 and 2.2) and then we will focus on Dirac Hamiltonianrepresentatives in these symmetry classes (sections 2.3, 2.4, and 2.5). The insights gainedfrom the study of these Dirac Hamiltonian examples are applicable to a wider class ofinsulators. Finally, in section 2.6 we discuss effective topological field theories describingtopological response functions of class A and AIII topological insulators. d = 2 n + 2 dimensions The class of problems that we will consider below is non-interacting fermionic systemswith translation invariance. For such systems, the eigenvalue problem at eachmomentum k in the Brillouin zone (BZ) is described by H ( k ) | u a ( k ) i = E a ( k ) | u a ( k ) i , a = 1 , . . . , N tot . (6)Here, H ( k ) is a N tot × N tot single-particle Hamiltonian in momentum space, and | u a ( k ) i is the a -th Bloch wavefunction with energy E a ( k ). We assume that there is a finite gapat the Fermi level, and therefore, we obtain a unique ground state by filling all statesbelow the Fermi level. (In this paper, we always adjust E a ( k ) in such a way that theFermi level is at zero energy.)We assume there are N − ( N + ) occupied (unoccupied) Bloch wavefunctions for each k with N + + N − = N tot . We call the set of filled Bloch wavefunctions {| u − ˆ a ( k ) i} , wherehatted indices ˆ a = 1 , . . . , N − labels the occupied bands only. We introduce the spectral ONTENTS Q -matrix” by P ( k ) = X ˆ a | u − ˆ a ( k ) ih u − ˆ a ( k ) | , Q ( k ) = 1 − P ( k ) . (7)The ground state (filled Fermi sea) is characterized at each k by the set ofnormalized vectors {| u − ˆ a ( k ) i} , which is a member of U( N tot ), or, equivalently, by Q ( k ), up to basis transformations within occupied and unoccupied bands. Thus Q ( k ) can be viewed as an element of the complex Grassmannian G N + ,N + + N − ( C ) =U( N + + N − ) / [U( N + ) × U( N − )]. For a given system, Q ( k ) defines a map from BZ intothe complex Grassmannian, and hence classifying topological classes of band insulatorsis equivalent to counting how many distinct classes there are for the space of all suchmappings. The answer to this question is given by the homotopy group π d ( G m,m + n ( C )),which is nontrivial in even dimensions d = 2 n + 2.Define for the occupied bands, the non-Abelian Berry connection [54] A ˆ a ˆ b ( k ) = A ˆ a ˆ bµ ( k )d k µ = h u − ˆ a ( k ) | d u − ˆ b ( k ) i , µ = 1 , · · · , d, ˆ a, ˆ b = 1 , · · · , N − , (8)where A ˆ a ˆ bµ = − ( A ˆ b ˆ aµ ) ∗ . The Berry curvature is defined by F ˆ a ˆ b ( k ) = d A ˆ a ˆ b + (cid:0) A (cid:1) ˆ a ˆ b = 12 F ˆ a ˆ bµν ( k )d k µ ∧ d k ν . (9)Class A insulators in even spatial dimensions d = 2 n + 2 ( n = 0 , , , . . . ) can becharacterized by the Chern form of the Berry connection in momentum space. The( n + 1)-th Chern character isch n +1 ( F ) = 1( n + 1)! tr (cid:18) i F π (cid:19) n +1 . (10)The integral of the Chern character in d = 2 n + 2 dimensions is an integer, the ( n + 1)stChern number,Ch n +1 [ F ] = Z BZ d =2 n +2 ch n +1 ( F ) = Z BZ d =2 n +2 n + 1)! tr (cid:18) i F π (cid:19) n +1 ∈ Z . (11)Here, R BZ d denotes the integration over d -dimensional k -space. For lattice models, itcan be taken as a Wigner-Seitz cell in the reciprocal lattice space. On the other hand,for continuum models, it can be taken as R d . Assuming that the asymptotic behaviorof the Bloch wavefunctions approach a k -independent value as | k | → ∞ , the domain ofintegration can be regarded as S d ∗ .When, d = 2 ( n = 0), Ch [ F ] is the TKNN integer [1],Ch [ F ] = i2 π Z BZ d =2 tr ( F ) = i2 π Z d k tr ( F ) , (12) ∗ For Dirac insulators with linear dispersion, discussed below, this assumption is not entirely correct.If the k -linear behavior of the Dirac spectrum persists to infinitely large momentum, the behavior ofthe wavefunctions at | k | → ∞ is not trivial: the one-point compactification thus does not work. In thiscase, the domain of integration becomes effectively a half of S d , and accordingly, integer topologicalnumbers become half integers. However, the Dirac spectrum can be properly regularized, in such a waythat the behavior of the wavefunctions becomes trivial at larger k , see for example equation (50). Thisis also the case when the Dirac Hamiltonians are formulated on a lattice [see equation (82)]. ONTENTS σ xy in units of ( e /h ). The secondChern-number ( n = 1), Ch [ F ],Ch [ F ] = − π Z BZ d =4 tr (cid:0) F (cid:1) = − π Z d kǫ κλµν tr ( F κλ F µν ) , (13)can be used to describe a topological insulator in d = 4, or, alternatively, a certainadiabatic “pumping process” in lower spatial dimension [28].The Chern character ch n +1 can be written in terms of its Chern-Simons form,ch n +1 ( F ) = d Q n +1) − ( A , F ) . (14)Here the Chern-Simons form is defined as Q n +1 ( A , F ) := 1 n ! (cid:18) i2 π (cid:19) n +1 Z d t tr ( AF nt ) , (15)where F t = t d A + t A = t F + ( t − t ) A . (16)For example, Q ( A , F ) = i2 π tr A , Q ( A , F ) = − π tr (cid:18) A d A + 23 A (cid:19) . (17) d = 2 n + 1 dimensions2.2.1. Winding number. We now discuss band insulators in symmetry class AIII. Bydefinition, for all class AIII band insulators, we can find a unitary matrix Γ whichanticommutes with the Hamiltonians, {H ( k ) , Γ } = 0 , Γ = 1 . (18)It follows that the spectrum is symmetric with respect to zero energy and N + = N − =: N . As a consequence of the chiral symmetry (18), all class AIII Hamiltonians, as wellas their Q -matrix (7), can be brought into block off-diagonal form, Q ( k ) = q ( k ) q † ( k ) 0 ! , q ∈ U( N ) , (19)in the basis in which Γ is diagonal. The off-diagonal component q ( k ) defines a map fromBZ onto U( N ), and classifying class AIII topological insulators reduces to considering thehomotopy group π d (U( N )). The homotopy group is nontrivial in odd spatial dimensions d = 2 n + 1, while it is trivial in even spatial dimensions, i.e., there is no non-trivialtopological insulator in class AIII in even spatial dimensions.In odd spatial dimensions d = 2 n + 1, class AIII topological insulators arecharacterized by the winding number [26] ν n +1 [ q ] := Z BZ d =2 n +1 ω n +1 [ q ] , (20) ONTENTS ω n +1 [ q ] := ( − n n !(2 n + 1)! (cid:18) i2 π (cid:19) n +1 tr (cid:2) ( q − d q ) n +1 (cid:3) (21)= ( − n n !(2 n + 1)! (cid:18) i2 π (cid:19) n +1 ǫ α α ··· tr (cid:2) q − ∂ α q · q − ∂ α q · · · (cid:3) d n +1 k. While derived from the Chern form, the Chern-Simonsforms, introduced in equation (15), themselves define a characteristic class for an odd-dimensional manifold. This suggests that the Chern-Simons forms can be used tocharacterize also AIII topological insulators in d = 2 n + 1 dimensions. Integratingthe Chern-Simons form over the BZ, we introduceCS n +1 [ A , F ] := Z BZ n +1 Q n +1 ( A , F ) . (22)Unlike the winding number, CS n +1 [ A , F ] is defined without assuming chiral symmetryand can be used for non-chiral topological insulators (superconductors).Before discussing how useful Chern-Simons forms are, however, observe that theyare not gauge invariant. Neither are the integral of the Chern-Simons forms over theBZ. However, for two different choices of gauge A and A ′ , which are connected by agauge transformation A ′ = g − A g + g − d g, F ′ = g − F g, (23)we have Q n +1 ( A ′ , F ′ ) − Q n +1 ( A , F ) = Q n +1 ( g − d g,
0) + d α n , (24)where α n is some 2 n form [46]. We note that Q n +1 (cid:0) g − d g, (cid:1) = 1 n ! (cid:18) i2 π (cid:19) n +1 Z d t tr (cid:0) A ( t − t ) n A n (cid:1) = 1 n ! (cid:18) i2 π (cid:19) n +1 tr (cid:2) ( g − d g ) n +1 (cid:3) Z d t ( t − t ) n = ω n +1 [ g ] . (25)This is nothing but the winding number density, and its integral ν n +1 [ g ] := Z BZ n +1 Q n +1 (cid:0) g − d g, (cid:1) = Z BZ n +1 ω n +1 [ g ] (26)is an integer, which counts the non-trivial winding of the map g ( k ) : BZ n +1 → U( N ).Note that π n +1 [U( N )] = Z (for large enough N ). We thus concludeCS n +1 [ A ′ , F ′ ] − CS n +1 [ A , F ] = integer , (27)and hence the exponential W n +1 := exp { π iCS n +1 [ A , F ] } (28)is a well-defined, gauge invariant quantity, although it is not necessarily quantized. ONTENTS not restricted to chiral topological insulators. We now compute the Chern-Simons invariant for class AIIItopological insulators in d = 2 n + 1. We first explicitly write down the Berry connectionfor chiral symmetric Hamiltonians. To this end, we observe that for unoccupied andoccupied bands, the Bloch wavefucntions satisfy Q ( k ) | u +ˆ a ( k ) i = + | u +ˆ a ( k ) i , Q ( k ) | u − ˆ a ( k ) i = −| u − ˆ a ( k ) i , (29)respectively. Introducing | u ǫ ˆ a ( k ) i = 1 √ χ ǫ ˆ a ( k ) η ǫ ˆ a ( k ) ! , ǫ = ± , (30)we rewrite (29) as q ( k ) q † ( k ) 0 ! χ ± ˆ a ( k ) η ± ˆ a ( k ) ! = ± χ ± ˆ a ( k ) η ± ˆ a ( k ) ! . (31)We can then construct a set of eigen Bloch functions as | u ǫ ˆ a ( k ) i = 1 √ χ ǫ ˆ a ǫq † ( k ) χ ǫ ˆ a ! . (32)The N -dimesional space spanned by the occupied states {| u − ˆ a ( k ) i} can be obtained byfirst choosing N independent orthonormal vectors n ǫ ˆ a which are k -independent, and thenfrom n ǫ ˆ a , | u ǫ ˆ a ( k ) i N = 1 √ n ǫ ˆ a ǫq † ( k ) n ǫ ˆ a ! . (33)From these Bloch functions, the Berry connection is computed as N h u ǫ ˆ a ( k ) | ∂ µ u ǫ ˆ b ( k ) i N d k µ = 12 (cid:2) h n ǫ ˆ a | ∂ µ | n ǫ ˆ b i + ǫ h n ǫ ˆ a q ( k ) | ∂ µ | q † ( k ) n ǫ ˆ b i (cid:3) d k µ = 12 (cid:2) ǫ h n ǫ ˆ a q ( k ) | ∂ µ | q † ( k ) n ǫ ˆ b i (cid:3) d k µ = 12 [ q ( k ) ∂ µ q † ( k )] ˆ a ˆ b d k µ =: A Nˆ a ˆ b , (34)where we have made a convenient choice ( n ǫ ˆ a ) ˆ b = δ ˆ a ˆ b . While this looks almost likea pure gauge, it is not exactly so because of the factor 1 /
2. Note also that thiscalculation shows that the Berry connection for the occupied and unoccupied bandsare identical. With equations (29) through (33), we have succeeded in constructingeigen Bloch wavefuctions of the “ Q -matrix”, equation (19), in block-off diagonal basis,that are free from any singularity. I.e., we have explicitly demonstrated that there is noobstruction to constructing eigen wavefuctions globally. We emphasize that this appliesto all symmetry classes with chiral symmetry (AIII, BDI, DIII, CII, CI) for any spatialdimension ( d = 2 n as well as d = 2 n + 1).Alternatively, one can construct another set of eigen Bloch functions | u ǫ ˆ a ( k ) i S = 1 √ ǫq ( k ) n ǫ ˆ a n ǫ ˆ a ! . (35) ONTENTS S h u ǫ ˆ a ( k ) | ∂ µ u ǫ ˆ b ( k ) i S d k µ = 12 (cid:2) ǫ h n ǫ ˆ a q † ( k ) | ∂ µ | q ( k ) n ǫ ˆ b i + h n ǫ ˆ a | ∂ µ | n ǫ ˆ b i (cid:3) d k µ = 12 (cid:2) ǫ h n ǫ ˆ a q † ( k ) | ∂ µ | q ( k ) n ǫ ˆ b i + 0 (cid:3) d k µ = 12 [ q † ( k ) ∂ µ q ( k )] ˆ a ˆ b d k µ =: A Sˆ a ˆ b , (36)where we have made a convenient choice ( n ǫ ˆ a ) ˆ b = δ ˆ a ˆ b . The two gauges are related toeach other by A S = g − A N g + g − d g, (37)where the transition function g is the block off-diagonal projector, g = q : q † ( k ) (cid:20) q ( k ) ∂ µ q † ( k ) (cid:21) q ( k ) + q † ( k ) ∂ µ q ( k ) = 12 (cid:2) ∂ µ q † ( k ) (cid:3) q ( k ) + q † ( k ) ∂ µ q ( k )= − q † ( k ) ∂ µ q ( k ) + q † ( k ) ∂ µ q ( k )= 12 q † ( k ) ∂ µ q ( k ) . (38)We now compute the Chern-Simons invariant for class AIII topological insulatorsin d = 2 n + 1. In the gauge where A = A S = (1 / q − d q ), we haved A = 12 d( q − d q ) = −
12 ( q − d q )( q − d q ) , F t = t d A + t A = (cid:18) − t t (cid:19) ( q − d q ) . (39)Then, Q n +1 ( A , F ) = 1 n ! (cid:18) i2 π (cid:19) n +1 Z d t tr ( AF nt )= 1 n ! (cid:18) i2 π (cid:19) n +1 Z (cid:18) − t t (cid:19) n d t (cid:2) ( q − d q ) n +1 (cid:3) = 1 n ! (cid:18) i2 π (cid:19) n +1 Z u n ( u − n d u tr (cid:2) ( q − d q ) n +1 (cid:3) , (40)which is a half of the winding number density, Q n +1 ( A S , F S ) = 12 ω n +1 [ q ] . (41)Hence we concludeCS n +1 (cid:2) A S , F S (cid:3) = Z BZ d =2 n +1 Q n +1 ( A S , F S ) = 12 ν n +1 [ q ] . (42)As a corollary, W n +1 = exp { π i CS n +1 [ A , F ] } = exp { π i ν n +1 [ q ] } . (43)Thus, while in general the quantity W n +1 is not quantized, for class AIII Hamiltoniansin d = 2 n + 1, W n +1 can take only two values, W n +1 = ± ONTENTS n +1 [ A , F ] in d = 1 ( n = 0) spatial dimension takes the form ofthe U(1) Wilson loop defined for BZ d =1 ≃ S . It is quantized for chiral symmetricsystems [56]. Also, the logarithm of W represents the electric polarization [57, 58].For a lattice system, the non-invariance of CS [ A , F ] [i.e., it can change under a gaugetransformation, CS [ A , F ] → CS [ A , F ] + (integer)] has a clear meaning: since thesystem is periodic, the displacement of electron coordinates has a meaning only withina unit cell, and two electron coordinates that differ by an integer multiple of the latticeconstant should be identified.In d = 3 ( n = 1) spatial dimensions CS represents the magnetoelectricpolarizability [28, 59, 60]. While this was discussed originally for three-dimensional Z topological insulators in symplectic symmetry (class AII), the magnetoelectricpolarizability is also quantized for class AIII. We start from a d = 2 n +3-dimensional gapless Dirac Hamiltonian defined in momentumspace, H d =2 n +3(2 n +3) ( k ) = d =2 n +3 X a =1 k a Γ a (2 n +3) . (44)Here, k a =1 ,...,d are d -dimensional momenta, and Γ a =1 ,..., n +3(2 n +3) are (2 n +1 × n +1 )-dimensionalHermitian matrices that satisfy { Γ a (2 n +3) , Γ b (2 n +3) } = 2 δ a,b . Some results for the gammamatrices, on which the following discussion is based, are summarized in appendix B.The massless Dirac Hamiltonian H d =2 n +3(2 n +3) ( k ) cannot be realized on a lattice in a naiveway, because of the fermion doubling problem.By replacing k n +3 by a mass term, we obtain a d = 2 n + 2 dimensional class Atopological Dirac insulator, H d =2 n +2(2 n +3) ( k, m ) = d =2 n +2 X a =1 k a Γ a (2 n +3) + m Γ n +3(2 n +3) . (45)The topological character of this Dirac insulator will be further discussed below.By setting in addition k n +2 = 0, we obtain an insulator in one dimension lower, H d =2 n +1(2 n +3) ( k, m ) = d =2 n +1 X a =1 k a Γ a (2 n +3) + m Γ n +3(2 n +3) . (46)By construction, H d =2 n +1(2 n +3) ( k, m ) anticommutes with Γ n +2(2 n +3) , n H d =2 n +1(2 n +3) ( k, m ) , Γ n +2(2 n +3) o = 0 , (47)and hence it is a member of class AIII.This construction of the lower-dimensional models from their higher-dimensional“parent” is an example of the Kaluza-Klein dimensional reduction. To obtain H d =2 n +1(2 n +3) ( k, m ) from H d =2 n +2(2 n +3) ( k, m ), one can first compactify the (2 n + 2)-th spatialdirection into a circle S . The (2 n +2)-th component of the momentum is then quantized, ONTENTS k n +2 = 2 πN n +2 /ℓ , where N n +2 ∈ Z , and ℓ is the radius of S . The energy eigenvaluesnow carry the integral label N n +2 , in addition to the continuous label, k i =1 ,..., n +1 . Bymaking the radius of the circle very small, all levels with N n +2 = 0 have very largeenergies while the levels with N n +2 = 0 are not affected by the small radius limit. Inthis way, in the limit ℓ →
0, there is a separation of energy scales, and we only keepthe states with N n +2 = 0, while neglecting all states with N n +2 = 0 (the Kaluza-Kleinmodes).Similarly, H d =2 n +2(2 n +3) ( k, m ) is naturally obtained from H d =2 n +3(2 n +3) ( k ) by dimensionalreduction, if we also include a U(1) gauge field. We can introduce a (external,electromagnetic) U(1) gauge field a i =1 , ··· , n +3 by minimal coupling, H d =2 n +3(2 n +3) ( k ) →H d =2 n +3(2 n +3) ( k + a ). In the S compactification of the (2 n + 3)-th spatial coordinate, weagain keep modes with k n +3 = 0 only, while neglecting all the Kaluza-Klein modes with k n +3 = 0. The vector field (gauge field) a i =1 ,... n +3 in d = 2 n + 3 dimensions can bedecomposed into a vector field ( a i =1 ,... n +2 ) and a scalar field ( a n +3 ≡ ϕ ) in d = 2 n + 2,leading to H d =2 n +2(2 n +3) ( k + a, ϕ ) = d =2 n +2 X i =1 ( k i + a i )Γ i (2 n +3) + ϕ Γ n +3(2 n +3) . (48)Switching off the d = 2 n + 2-dimensional gauge field, and assuming the scalar field isconstant ϕ = m , we obtain H d =2 n +2(2 n +3) ( k, m ).For the massive Dirac insulator (45) in class A in 2 n + 2 dimensions, the n -th Chernnumber Ch n +1 is nonvanishing [61]. Accordingly, for the massive Dirac insulator (46)in class AIII in 2 n + 1 dimensions, the winding number and the Chern-Simons form arenon-zero.As the Chern-Simons form can be derived from its higher-dimensional parent, theChern form, one may wonder if there is a connection between CS n +1 = ν n +1 of a classAIII Dirac topological insulator in d = 2 n + 1 and Ch n +1 of a class A Dirac topologicalinsulator in d = 2 n + 2. We now try to answer this question.Let us start from H d =2 n +2(2 n +3) ( k, m ) = d =2 n +2 X a =1 k a Γ a (2 n +3) + m ( k )Γ n +3(2 n +3) . (49)Here we have regularized the Dirac spectrum by making the mass term k dependent, m ( k ) = m − Ck , with sgn( C ) = sgn( m ) . (50)With this regularization, as | k | → ∞ , H d =2 n +2(2 n +3) ( k, m ) ∝ Γ n +3(2 n +3) , and hence thewavefunctions become k -independent (i.e., the BZ is topologically S d ). The sign choiceof the constant C , sgn( C ) = sgn( m ), makes the Chern invariant for the Dirac insulatornon-zero.By dropping k n +2 , we get the (regularized) Dirac insulator in one dimension lower, H d =2 n +1(2 n +3) (˜ k, m ) = d =2 n +1 X a =1 k a Γ a (2 n +3) + m (˜ k, n +3(2 n +3) , (51) ONTENTS k = ( k , . . . , k n +1 ). As mentioned before, H d =2 n +1(2 n +3) (˜ k, m ) is chiral symmetric, {H d =2 n +1(2 n +3) (˜ k, m ) , Γ n +2(2 n +3) } = 0. We work in the basis where Γ n +2(2 n +3) is diagonal. In thisbasis, H d =2 n +1(2 n +3) is block-off diagonal, H d =2 n +1(2 n +3) (˜ k, m ) = D (˜ k )˜ D † (˜ k ) 0 ! . (52)Correspondingly, H d =2 n +2(2 n +3) ( k ) = ∆( k n +2 ) D ( k ) D † ( k ) − ∆( k n +2 ) ! , (53)where ∆( k n +2 ) = k n +2 , and D ( k ) = D (˜ k, k n +2 ) satisfies D (˜ k, k n +2 ) = ˜ D (˜ k ) , when k n +2 = 0 . (54)We now look for eigenfunctions of H d =2 n +2(2 n +3) with negative eigenvalues, ∆( k n +2 ) D ( k ) D † ( k ) − ∆( k n +2 ) ! χη ! = − λ ( k ) χη ! . (55)To this end, we first solve the following auxiliary eigenvalue problem D ( k ) D † ( k ) 0 ! χη ! = − ˜ λ ( k ) χη ! , (56)where ˜ λ = q ˜ k + [ m ( k )] . We see from (33) and (35) that the solutions are given by1 √ − n ˆ a q † ( k ) n ˆ a ! or 1 √ q ( k ) n ˆ a − n ˆ a ! , (57)where q ( k ) and q † ( k ) are the off-diagonal blocks of the Q -matrix derived from theauxiliary Hamiltonian. If we specialize to the case where k = (˜ k, H d =2 n +1(2 n +3) (˜ k ) is given in terms of q (˜ k ) and q † (˜ k ). We then construct two different setsof normalized eigenfunctions of H d =2 n +2(2 n +3) | u − ˆ a ( k ) i N = 1 p λ ( λ + ∆) − ˜ λn ˆ a ( λ + ∆) q † n ˆ a ! , (58)and | u − ˆ a ( k ) i S = 1 p λ ( λ − ∆) (∆ − λ ) qn ˆ a ˜ λn ˆ a ! , (59)with the eigenvalue − λ = − p ˜ λ + ∆ . When specialized to k = (˜ k, H d =2 n +1(2 n +3) .Since the Hamiltonian H d =2 n +2 ( k ) is characterized by the non-zero Chern number,Ch n +1 [ F ] = 0, it is not possible to define wavefunctions globally. We thus need to splitthe BZ (BZ d =2 n +2 ) into two parts, BZ d =2 n +2N and BZ d =2 n +2S . We choose the interfaceof these two patches as ∂ BZ d =2 n +2N = − ∂ BZ d =2 n +2S = BZ d =2 n +1 , on which a lower ONTENTS Figure 2. (a) The d = 2 n + 2 dimensional Brillouin zone, BZ d =2 n +2 ∼ S n +2 = S n +1 ∧ S , and its d = 2 n + 1 dimensional descendant BZ d =2 n +1 ∼ S n +1 located atthe equator of S n +2 . (b) Splitting the d = 2 n + 2 dimensional Brillouin zone in half,or embedding d = 2 n + 1 dimensional Brillouin zone into a higher-dimensional one. dimensional Hamiltonian “lives”. For each patch, BZ d =2 n +2N , S , we can choose a globalgauge. Observe that the wavefunction | u − ˆ a ( k ) i N is well-defined for k n +2 >
0, and has asingularity at (˜ k, k n +2 ) = (0 , − p m/C ). Similarly, | u − ˆ a ( k ) i S is well-defined for k n +2 < k, k n +2 ) = (0 , p m/C ).The two gauges are thus complementary and at the boundary ∂ BZ d =2 n +2N = − ∂ BZ d =2 n +2S they are glued by the transition function A S = g − A N g + g − d g, (60)where g (˜ k ) ∈ U( N − ) with ˜ k ∈ ∂ BZ d =2 n +2N . Then,Ch n +1 [ F ] = Z BZ d =2 n +2 ch n +1 ( F ) = Z BZ d =2 n +2 d Q n +1 ( A , F )= Z BZ d =2 n +2N d Q n +1 ( A N , F ) + Z BZ d =2 n +2S d Q n +1 ( A S , F )= Z ∂ BZ d =2 n +2N Q n +1 ( A N , F ) − Z ∂ BZ d =2 n +2N Q n +1 ( A S , F ) , (61)where we have used Stokes theorem. From formula (24), it follows thatCh n +1 [ F ] = Z ∂ BZ d =2 n +1N Q n +1 (cid:0) g − d g, (cid:1) . (62)By use of equation (25) we conclude that the RHS of the above equation is nothing butthe winding number. Thus, the Chern number can be expressed as a winding numberof the transition function g . For the case we are interested in, i.e., when a d = 2 n + 1topological insulator “lives” on ∂ BZ d =2 n +1N , the transition function is given by the off-diagonal block of the projector, g (˜ k ) = q (˜ k ). This is how two topological insulators in d = 2 n + 2 and d = 2 n + 1 are related. d = 3 → → a =1 , , = { σ x , σ y , σ z } , (63) ONTENTS σ x,y,z are the 2 × d = 3 gapless (Weyl) chiral fermion H d =3(3) ( k ) = X a =1 k a Γ a (3) . (64)By replacing the third component of the momemtum by a mass term, k a =3 → m , weget a d = 2 dimensional Dirac Hamiltonian, H d =2(3) ( k, m ) = X a =1 k a Γ a (3) + m Γ = k x σ x + k y σ y + mσ z . (65)This is a class A Hamiltonian. The Bloch wavefunctions are given by [ λ ( k ) := √ k + m ] | u + ( k ) i = 1 p λ ( λ − m ) k x − i k y λ − m ! , | u − ( k ) i = 1 p λ ( λ + m ) − k x + i k y λ + m ! . (66)We note that | u − ( k ) i is well-defined for all k when m >
0. (When m <
0, by choosinga gauge properly, we can obtain a similar well-defined wavefunction.) Assuming m > | u − ( k ) i , for which we obtain the Berry connection A x ( k, m ) = + i k y λ ( λ + m ) , A y ( k, m ) = − i k x λ ( λ + m ) , (67)and the Berry curvature F xy ( k, m ) = ∂ k x A y − ∂ k y A x = − i m λ . (68)The Chern number is non-zero,Ch [ F ] = i2 π Z d kF xy = i2 π Z d k − i m λ = 12 m | m | , (69)which is nothing but the Hall conductance σ xy . This model can be realized, at lowenergies, in the honeycomb lattice model introduced by Haldane [62].Finally, setting k = 0, we get a d = 1 dimensional Dirac Hamiltonian, H d =1(3) ( k, m ) = k x σ x + mσ z . (70)This is nothing but the chiral topological Dirac insulator in class AIII, as it anticommuteswith σ y . A topological invariant can be defined for the d = 1 dimensional DiracHamiltonian as follows. We first go to the canonical form by ( σ x , σ y , σ z ) → ( σ x , σ z , − σ y ), H d =1(3) ( k, m ) → k x σ x − mσ y . This d = 1 dimensional Hamiltonian describes the physics ofpolyacetylene (see, for example, Refs. [63, 64].) The Bloch wavefunction with negativeeigenvalue is | u − ( k x ) i = 1 p k x + m ) − k x − i mλ ! . (71) ONTENTS Q ( k ) in this basis is q ( k ) = − k x + i m p k x + m , (72)and the winding number is given by ν [ q ] = i2 π Z BZ q − d q = i2 π Z ∞−∞ d k x − i mk x + m = 12 m | m | . (73)From equation (71) the Berry connection is obtained, A ( k x ) = h u − ( k x ) | d u − ( k x ) i = 12 − i mk x + m d k x . (74)Then, we find CS [ A , F ] = i2 π Z BZ tr A = i2 π Z ∞−∞ d k x − i mk x + m = 14 m | m | . (75)This is a half of the winding number ν [ q ] as expected. Hence, the Wilson “loop” isgiven by W = exp { π iCS [ A , F ] } = exp Z BZ tr A = e ± π i / . (76)Here, we mention, however, that there is a subtlety in computing the Wilson loop.In the basis where the Hamiltonian in momentum space is real, H d =1(3) ( k, m ) = k x σ x + mσ z (70), the Bloch wavefunction is real for a given k , and as a consequence the Berryconnection A ( k x ) vanishes identically. One would then conclude W = 1, not W = − ν = sgn( m ). This puzzle can be solved by properly regularizing the Diracinsulator. For simplicity, let us replace k x by sin k x , and m by m − k x ). Withsuch a regularization, the BZ is topologically S , and one finds a singularity in thewavefunction at k x = π , where the phase of the wavefunction jumps by π (i.e., thereis a Dirac string at k x = π .). If, on the other hand, we use a different basis (71), wecan avoid to have a Dirac string in the BZ, and the Wilson loop can be obtained byintegrating the Berry connection over the BZ. d = 5 → → a =1 ,..., = { α x , α y , α z , β, − i βγ } , (77)where we are using the Dirac representation, α i = σ i σ i ! , β = − ! , γ = ! . (78)Observe that α x α y α z β = − i βγ . These matrices can be used to construct a ( d = 5)-dimensional gapless chiral fermion H d =5(5) ( k ) = X a =1 k a Γ a (5) . (79) ONTENTS +2 +40-2-4-2-1+10 w i nd i ng nu m be r m (a) k y / p k x / p (b) E ne r g y Figure 3. (a) Winding number ν for Hamiltonian (82) as a function of m . (b) Two-dimensional energy spectrum of the surface states of model (82) with mass m = +0 . ν ( m = +0 .
5) = − By replacing the fifth component of the momemtum by a mass term, k a =5 → m , weobtain a d = 4 dimensional Dirac Hamiltonian, H d =4(5) ( k, m ) = X a =1 k a Γ a (5) + m Γ . (80)It is known that for this gapped Hamiltonian, the second Chern number is non-zero[61, 28].Finally, setting k = 0, we get a d = 3 dimensional Dirac Hamiltonian, H d =3(5) ( k, m ) = X a =1 k a Γ a (5) + m Γ = X a =1 k a α a − m i βγ . (81)This is nothing but the chiral topological Dirac insulator in class AIII discussed in Ref.[26]. The tight-binding version of this model for a simple cubic lattice is given by H d =3(5) ( k, m ) = X a =1 sin k a Γ a (5) + m + X a =1 cos k a ! Γ . (82)By taking open boundary conditions in the z direction and periodic boundary conditionsin the x and y directions, we can study the surface states of this model. When thewinding number ν is non-vanishing, there are | ν | surface Dirac states, which cross thebulk band gap (see figure 3).We now study the Berry connection of the above class A ( d = 4) and class AIII( d = 3) Dirac insulators. To treat these two cases in a unified fashion, let us considerthe following d = 4 Dirac Hamiltonian H ( k, m ) = X a =1 k a α a + m β + k w ( − i βγ ) . (83) ONTENTS k w → m it can be also viewed as a d = 3 dimensional DiracHamiltonian with two masses. The four eigenvalues are E ( k ) = ± λ ( k ) , λ ( k ) = q k + m . (84)The two normalized negative energy eigenstates with E ( k ) = − λ ( k ) are | u − ( k ) i = 1 p λ ( λ + m ) − k x + i k y i k w + k z λ + m , | u − ( k ) i = 1 p λ ( λ + m ) − k z + i k w − k x − i k y λ + m . (85)The two normalized positive energy eigenstates with E ( k ) = + λ ( k ) are | u +1 ( k ) i = 1 p λ ( λ − m ) k x − i k y − i k w − k z λ − m , | u +2 ( k ) i = 1 p λ ( λ − m ) − i k w + k z k x + i k y λ − m . (86)Note that if m > | u +1 , ( k ) i are not well-defined at λ ( k ) = m (i.e, k = 0). Observethat when m > | u − , ( k ) i have the form of wavefunctions expected from the generalconsideration (corresponding to the A N gauge choice).When m = 0, the Hamiltonian anticommutes with β and is block off-diagonal.The off-diagonal component of the Q -matrix is given by q ( k ) = − λ ( k · σ − i m ) (87)By noting ( µ = 1 , , ∂ µ q = k µ λ ( k · σ − i m ) − λ σ µ , q † ∂ µ q = − λ ( k µ − k · σσ µ − i mσ µ ) , (88)and also, Z d k ǫ µνρ tr (cid:2) q † ∂ µ qq † ∂ ν qq † ∂ ρ q (cid:3) = Z d k − ǫ µνρ λ tr (cid:2) − m ( k · σ ) σ µ ( k · σ ) σ ν σ ρ + i m σ µ σ ν σ ρ (cid:3) = ( − m ) ǫ µνρ Z d k ǫ µνρ k ( k + m ) + i m ǫ µνρ Z d k − ǫ µνρ ( k + m ) = 12 m Z d k k + m ) , (89) ONTENTS ν [ q ] = 1224 π m Z ∞ d k πk ( k + m ) = 12 π π × π m | m | = 12 m | m | . (90)We now compute the Chern-Simons invariant (magnetoelectric polarizability) forthe massive Dirac Hamiltonian. For the lower two occupied bands, we can introduce aU(2) gauge field by A ˆ a ˆ bµ ( k )d k µ = h u − ˆ a ( k ) | d u − ˆ b ( k ) i (ˆ a, ˆ b = 1 , A x = i2 λ ( λ + m ) + k y k w + i k z k w − i k z − k y ! ,A y = i2 λ ( λ + m ) − k x i k w − k z − i k w − k z + k x ! ,A z = i2 λ ( λ + m ) − k w − i k x + k y i k x + k y + k w ! . (91)The gauge field can be decomposed into U(1) ( a ) and SU(2) ( a j = x,y,z ) parts as A µ ( k ) = a µ ( k ) σ a jµ ( k ) σ j . (92)The U(1) part of the Berry connection is trivial, whereas the SU(2) part is given by a xx = i k w λ ( λ + m ) , a yx = i − k z λ ( λ + m ) , a zx = i + k y λ ( λ + m ) ,a xy = i − k z λ ( λ + m ) , a yy = i − k w λ ( λ + m ) , a zy = i − k x λ ( λ + m ) ,a xz = i + k y λ ( λ + m ) , a yz = i + k x λ ( λ + m ) , a zz = i − k w λ ( λ + m ) . (93)The Chern-Simons form can be computed asCS = − π Z d kǫ µνρ tr (cid:18) A µ ∂ ν A ρ + 23 A µ A ν A ρ (cid:19) = 18 π Z d k m (2 λ + m ) λ ( λ + m ) = 1 π m | m | arctan (cid:20) ( m + m ) / − m ( m + m ) / + m (cid:21) / . (94)When m = 0,CS = 14 m | m | . (95)As the CS term is determined only modulo 1, the fractional part sgn( m ) is an intrinsicproperty. On a lattice, as there are two-copies of the 4 × = sgn( m ), and e π iCS = − and W , which is similar to the case of theWilson loop W . When m = 0 and m = 0, the Chern-Simons form is zero identically,and one would conclude W = 1, while chiral symmetry for the case of m = 0 and ONTENTS m = 0 allows us to define ν , and ν = sgn( m ). As before, this puzzle can be solvedby properly regularizing the Dirac insulator. With such a regularization, the BZ istopologically S , and one finds a singularity in the wavefunctions. If, on the other hand,we use a different basis, we can avoid to have a Dirac string in the BZ. D = d + 1 space-time dimensions For class A and AIII topological insulators or superconductors, there is always aconserved U(1) quantity; either electric charge or one component (the z -component,say) of spin is conserved [65]. Transport properties of these conserved quantities inthese topological insulators or superconductors in d spatial dimensions can be describedin terms of an effective field theory for the linear responses in D = d + 1 space-time dimensions, which can be obtained by coupling an external space-time dependentgauge field a µ to the topological insulator or superconductor (we will choose the Diracrepresentative; see the discussion near (48)), and by integrating out the gapped fermions.One can then read off the theory of the linear responses from the so-obtained (classical)effective action for a µ .The theory for the linear responses of a topological insulator in symmetry class Aand in d spatial dimensions is then given by the Chern-Simons action in D = d + 1space-time dimensions, S D =2 n +1CS [ a ] = 2 π i k Z R D =2 n +1 Q n +1 [ a ] , k = Ch n . (96)Here, Q [ a ] is the Chern-Simons form of the U(1) gauge field a µ defined in D = 2 n + 1dimensional (Euclidean) spacetime, and the level k of the Chern-Simons term is givenin terms of Ch n , which is the Chern invariant computed from the Bloch wavefunctionsin d = 2 n space dimensions.On the other hand, the theory of linear responses of a topological insulator insymmetry class AIII in d = 2 n − D = 2 n space-time dimensions, S D =2 nθ [ a ] = i θ Z R D =2 n ch n [ f ] , θ = πν n − , (97)where f is the field strength of the external gauge field a µ , and the theta angle θ is relatedto the winding number ν n − , which can be computed from the Bloch wavefunctions. Theintegration of fermions in D = 2 n dimensions is trickier than in D = 2 n + 1 dimensions;while a simple derivative expansion of the fermion determinant would appear to yieldthe Chern-Simons action (96), a more careful evaluation yields in fact the theta term,which is obtained from the Fujikawa Jacobian [66].The theta angle in the theta term (97) can take on a priori any value, if there is nodiscrete symmetry that pins its value. In class AII Z topological insulators in D = 4 n dimensions, time-reversal symmetry demands θ to be an integer multiple of π . Similarly,in class AIII, chiral symmetry pins θ to be an integer multiple of π . This can be seen ONTENTS T : S D =2 n +1CS → ( − n S D =2 n +1CS , S D =2 nθ → ( − n +1 S D =2 nθ , C : S D =2 n +1CS → ( − n +1 S D =2 n +1CS , S D =2 nθ → ( − n S D =2 nθ . (98)Here, the (anti-unitary) time-reversal symmetry operation T may either square to plusor minus the identity. Similarly, the charge-conjugation symmetry operation C mayeither square to plus or minus the identity. Since the chiral symmetry S can be expressedas the product of these two symmetry operations, i.e., S = T · C , one obtains S : S D =2 nθ → − S D =2 nθ . (99)Together with the 2 π periodicity of θ , it follows that θ is either 0 or π .
3. Dimensional hierarchy for Z topological insulators: real case We now consider eight “real” symmetry classes. In particular, we consider topologicalinsulators (superconductors) with integer classification, Z or 2 Z . ( Z cases will bediscussed in section 4.) A connection similar to the one between classes A and AIII canbe established.The massive Dirac Hamiltonians relevant to this section are all obtained from thefollowing massless Dirac Hamiltonian(0) : H d =2 n +3(2 n +3) ( k ) = d =2 n +3 X a =1 k a Γ a (2 n +3) , (100)by replacing some momenta by a mass, or by simply dropping them. Specifically, wewill consider ( i ) : H d =2 n +2(2 n +3) ( k, m ) = d =2 n +2 X a =1 k a Γ a (2 n +3) + m Γ n +3(2 n +3) , (101)( ii ) : H d =2 n +1(2 n +3) ( k, m ) = d =2 n +1 X a =1 k a Γ a (2 n +3) + m Γ n +3(2 n +3) , (102)( iii ) : H d =2 n (2 n +3) ( k, m ) = d =2 n X a =1 k a Γ a (2 n +3) + m M , (103)( iv ) : H d =2 n − n +3) ( k, m ) = d =2 n − X a =1 k a Γ a (2 n +3) + m M , (104)where M := iΓ n +3(2 n +3) Γ n +2(2 n +3) Γ n +1(2 n +3) . (105)The massive Dirac Hamiltonian H d =2 n +2(2 n +3) ( k, m ) is obtained from its gapless parent H d =2 n +3(2 n +3) ( k ) by replacing k n +3 with a mass m . By further removing k n +2 , we obtain H d =2 n +1(2 n +3) ( k, m ). This is essentially the same procedure as in the previous section, ONTENTS \ d · · · AI i iii i · · · BDI ii iv ii · · ·
D 0 i iii i · · · DIII ii iv ii · · ·
AII iii i iii · · · CII iv ii iv · · · C iii i iii · · · CI iv ii iv · · · Table 4.
Symmetry class of Dirac Hamiltonians H d =2 n +3(2 n +3) ( k ) (100), H d =2 n +2(2 n +3) ( k, m )(101), H d =2 n +1(2 n +3) ( k, m ) (102), H d =2 n (2 n +3) ( k, m ) (103) and, H d =2 n − n +3) ( k, m ) (104), markedby “0”, “ i ”, “ ii ”, “ iii ”, and “ iv ”, respectively. Chiral and non-chiral symmetric classesare colored in blue and red, respectively. where we obtained the class AIII topological Dirac Hamiltonian from its parent classA Hamiltonian. On the other hand, H d =2 n (2 n +3) ( k, m ) can be obtained from H d =2 n +3(2 n +3) ( k ) byfirst removing three components of the momentum k n +3 , n +2 , n +1 , and then adding M as a mass term. Finally, H d =2 n − n +3) ( k, m ) is obtained from H d =2 n (2 n +3) ( k, m ) by dimensionalreduction.Here, an important difference from the dimensional reduction in the complex caseis the shifting of symmetry classes. The parent Hamiltonian H d =2 n +3(2 n +3) ( k ) has one (andonly one) of four discrete symmetries, T (squaring to plus or minus the identity), or C (squaring to plus or minus the identity), depending on the dimensionality d (see table4 and appendix B). That is, it is a member of any one of AI, D, AII, and C. d = 2 n + 3 to d = 2 n + 2 . When dimensionally reducing H d =2 n +3(2 n +3) ( k ) to H d =2 n +2(2 n +3) ( k, m ), the symmetry of the parent Hamiltonian H d =2 n +3(2 n +3) ( k ) is broken by themass term: While the sign of the kinetic term is reversed under the discrete symmetry,the sign of the mass term remains unchanged. Thus, the new massive Hamiltonian is amember of a different symmetry class. By this procedure, we get an even dimensionalHamiltonian H d =2 n +2(2 n +3) ( k, m ) with the “shifted” symmetry,AI → C , C → AII , AII → D , D → AI . (106)This can be easily proved by taking an explict form of gamma matrices (see appendix B).Observe that, while we have chosen to remove k n +3 and hence Γ n +3(2 n +3) , any othercomponent could have been removed instead of k n +3 and Γ n +3(2 n +3) . However, all thesedifferent choices are unitarily equivalent, as they are simply related by a permutationof Clifford generators, Γ a (2 n +3) . d = 2 n +2 to d = 2 n +1 . Further dimensionally reducing H d =2 n +2(2 n +3) ( k, m ) to H d =2 n +1(2 n +3) ( k, m ), we obtain a chiral symmetric Dirac Hamiltonian. Together with existing ONTENTS → BDI , AII → DIII , C → CII , AI → CI . (107)This again can be easily proved by taking an explict form of gamma matrices (seeappendix B). d = 2 n + 3 to d = 2 n . Let us now consider H d =2 n (2 n +3) ( k, m ). One can checkthat this Hamiltonian belongs to the same symmetry class as its parent H d =2 n +3(2 n +3) ( k ).To study the topological properties of H d =2 n (2 n +3) ( k, m ), observe that, by construction, H d =2 n (2 n +3) ( k, m ) commutes with a product made of any two of Γ n +1 , n +2 , n +3(2 n +3) , (cid:2) H d =2 n (2 n +3) ( k, m ) , Γ a (2 n +3) Γ b (2 n +3) (cid:3) = 0 (108)where ( a, b ) = (2 n + 1 , n + 2), (2 n + 2 , n + 3), or (2 n + 3 , n + 1). This suggests thatthe Hamiltonian can be made block-diagonal. We note that in the representation of thegamma matrices we are using, the Hamiltonian is block diagonal, H d =2 n (2 n +3) ( k, m ) = d =2 n X a =1 k a Γ a (2 n +1) ⊗ σ − m Γ n +1(2 n +1) ⊗ σ = H d =2 n (2 n +1) ( k, − m ) 00 −H d =2 n (2 n +1) ( k, m ) ! (109)By use of a unitary transformation for the lower right block,Γ n +1(2 n +1) (cid:2) −H d =2 n (2 n +1) ( k, m ) (cid:3) Γ n +1(2 n +1) = H d =2 n (2 n +1) ( k, − m ) , (110) H d =2 n (2 n +3) ( k, m ) reduces to two copies of H d =2 n (2 n +1) ( k, − m ). On the other hand, we havealready seen that each H d =2 n (2 n +1) ( k, − m ) = P d =2 na =1 k a Γ a (2 n +1) − m Γ n +1(2 n +1) has a non-trivialChern number ±
1. We thus conclude that the Chern number characterizing H d =2 n (2 n +3) ( k )is ± d = 2 n to d = 2 n − . Finally, we discuss the topological insulators(superconductors) H d =2 n − n +3) ( k, m ) marked by “ iv ” in table 4. One can check that thesymmetry class of this Hamiltonian is shifted from that of H d =2 n (2 n +3) ( k, m ) asAI → CI , C → CII , AII → DIII , D → BDI . (111)Just as H d =2 n (2 n +3) ( k, m ) is characterized by a non-zero Chern invariant which is even, thedescendant H d =2 n − n +3) ( k, m ) is also characterized by an even winding number.The symmetry properties of all the Dirac Hamiltonians discussed here aresummarized in table 4. d = 3 → → d = 3-dimensional gapless Dirac fermion H ( k ) = k x σ x + k y σ y + k z σ z . (112) ONTENTS σ y H ( k ) σ y = H ∗ ( − k ) . (113)Starting from this Hamiltonian, we dimensionally reduce this Hamiltonian successivelyand obtain the lower dimensional Hamiltonians with shifted symmetry classes, AII → D → BDI. We replace k z by a mass term, and consider a d = 2-dimensional massiveDirac fermion H ( k, m ) = k x σ x + k y σ y + mσ z . (114)This Hamiltonian is now a member of class D since σ x H ∗ ( k, m ) σ x = −H ( − k, m ) . (115)As mentioned before, the Chern number is non-zero for this model, Ch = sgn( m ).This Hamiltonian is in the same universality class as the chiral p + i p -wave topologicalsuperconductor [8, 67].The single-particle Hamiltonian (114), of course, looks identical to (65). Thisdegeneracy is lifted once we consider more generic types of perturbations than themass term, which can in principle be spatially inhomogeneous. Also, because of the C symmetry [see equation (115)], the Hamiltonian (114) can be viewed as a single-particleHamiltonian for real (Majorana) fermions, while this is not the case for a class AIIIsingle-particle Hamiltonian. In other words, a class D system can be written in thesecond quantized form as H = 12 X k ∈ BZ (cid:16) a † k , a − k (cid:17) H ( k, m ) a k a †− k ! . (116)where a † k /a k is a fermion creation/annihilation operator with momentum k .By further switching off k y , or by the dimensional reduction in the y -direction, H ( k, m ) = k x σ x + mσ z . (117)This Hamiltonian is a member of class BDI since σ x H ∗ ( k, m ) σ x = −H ( − k, m ) , σ z H ∗ ( k, m ) σ z = H ( − k, m ) . (118)This model can be realized as a continuum limit of the lattice Majorana fermion modeldiscussed in Ref. [68]. For this d = 1 topological massive Dirac Hamiltonian, thewinding number and the Wilson loop can be computed in the same way as we did forthe d = 1 class AIII topological Dirac insulators (70). From (73), we immediately see ν = sgn( m ). d = 5 → d = 2 · a =1 ,..., = { τ y ⊗ σ x,y,z , τ x ⊗ σ , τ z ⊗ σ } (119) ONTENTS σ y (cid:0) Γ a (5) (cid:1) T σ y = +Γ a (5) . (120)Thus, the gapless Hamiltonian H ( k ) = X a =1 k a Γ a (5) (121)is a member of class C, as seen from σ y k a (cid:0) Γ a (5) (cid:1) T σ y = − ( − k a )Γ a (5) . (122)We now remove three momenta, and add one mass term. There are four possible massterms that are compatible with the class C symmetry τ x,z, ⊗ σ y , τ y ⊗ σ . (123)To construct a kinetic term we pick two gamma matrices from the list (119). Forany choice of two gamma matrices there is only one mass term in equation (123) thatanticommutes with the chosen kinetic term. In this way, we obtain, for example, thefollowing massive Hamiltonian H ( k, m ) = τ y ⊗ σ x k x + τ y ⊗ σ z k z + mτ ⊗ σ y . (124)Observe that i( τ y ⊗ σ y )( τ x ⊗ σ )( τ z ⊗ σ ) = τ ⊗ σ y . It is easy to see that this Hamiltonianhas the doubled Chern number, Ch = ±
2. To see this, first rotate τ y → τ z , H ( k, m ) → τ z ⊗ σ x k x + τ z ⊗ σ z k z + mτ ⊗ σ y = σ x k x + σ z k z + σ y m − σ x k x − σ z k z + σ y m ! , (125)and then apply a unitary transformation, σ σ y ! σ x k x + σ z k z + σ y m − σ x k x − σ z k z + σ y m ! σ σ y ! = σ x k x + σ z k z + σ y m σ x k x + σ z k z + σ y m ! . (126)The d = 2 dimensional Hamiltonian constructed here has the same topologicalproperties as the d = 2 dimensional d + i d -wave superconductor discussed in Ref. [69]. Z topological insulators and dimensional reduction In the previous sections we have studied Z topological insulators (superconductors),which are characterized by an integer – the Chern or winding number. In this sectionwe discuss Z topological insulators (superconductors) and show how they can be derivedas lower dimensional descendants of parent Z topological insulators (superconductors)(see table 5). Recently, Qi et al. [28] have employed this procedure to derive descendant Z topological insulators (superconductors) for those symmetry classes that break chiral ONTENTS \ d · · · AI Z Z Z Z Z · · · BDI Z Z Z Z Z Z · · · D Z Z Z Z Z Z Z · · · DIII Z Z Z Z Z Z Z · · · AII 2 Z Z Z Z Z Z Z · · · CII Z Z Z Z Z Z · · · C 0 0 2 Z Z Z Z Z · · · CI Z Z Z Z Z · · · Table 5.
All Z topological insulators (superconductors) can be described throughdimensional reduction as lower dimensional descendants of parent Z topologicalinsulators (superconductor) in the same symmetry class, which are characterized bynon-trivial winding/Chern numbers. Symmetry classes in which chiral (“sublattice”)symmetry is present (S=1) are colored blue (the Cartan label marked in bold-face),whereas those where it is absent (S=0) are colored red. Four among the eight symmetry classes of table 5 (see also table 1) are invariant underchiral (“sublattice”) symmetry, which is a combination of particle-hole and time-reversalsymmetry. These four symmetry classes are called BDI, CI, CII, and DIII. It is possibleto characterize the topological properties of chiral symmetric Z topological insulatorsby a winding number (topological invariant), which is defined in terms of the block-offdiagonal projector (see section 2.2.1). From this topological invariant we can derivea Z classification by imposing the constraint of additional discrete symmetries (suchas C and T ) for the lower-dimensional descendants. For a d -dimensional Z topologicalinsulator (superconductor) in a given symmetry class, we distinguish between first andsecond descendants, whose (reduced) dimensionality is d − d −
2, respectively.
In order to better understand the roleplayed by T and C , we first study the transformation properties of the winding number(20) and the winding number density (21) under these discrete symmetries. First of all,we note that the winding number density is purely real, w ∗ n +1 [ q ] = w n +1 [ q ], which canbe checked by direct calculation. Without any additional discrete symmetries, i.e., forsymmetry class AIII, the off-diagonal projector q , and hence the winding number density(21) are not subject to additional constraints. However, for the symmetry classes BDI, ONTENTS q at wavevector k to the one at wavevector − k . As a consequence, theconfigurations of the winding number density w in momentum space are restrictedby time-reversal and particle-hole symmetries. Let us now derive these symmetryconstraints on the winding number density w n +1 [ q ] for any odd spatial dimension d = 2 n + 1. Classes DIII and CI:
First we consider symmetry classes DIII and CI, where the blockoff-diagonal projector satisfies [26] q T ( − k ) = ǫq ( k ) , ǫ = ( − , DIII+1 , CI . (127)We introduce the auxiliary functions v µ ( k ) := (cid:18) q † ∂∂k µ q (cid:19) ( k ) and ˜ v µ ( k ) := (cid:18) q ∂∂k µ q † (cid:19) ( k ) . (128)From the symmetry property (127) it follows that v µ ( k ) = − ˜ v ∗ µ ( − k ), irrespective of thevalue taken by ǫ . Noting that (d q † ) q = − q † d q we find ǫ α α ··· α n +1 tr (cid:2) v α ( k ) v α ( k ) · · · v α n +1 ( k ) (cid:3) = ǫ α α ··· α n +1 tr (cid:2) v α ( − k ) v α ( − k ) · · · v α n +1 ( − k ) (cid:3) ∗ . (129)Thus, the winding number density for symmetry class DIII and CI is subject to theconstraint w n +1 [ q ( k )] = ( − n +1 w ∗ n +1 [ q ( − k )] . (130)Consequently, the winding number ν n +1 [ q ] is vanishing in d = 4 n + 1 dimensions, i.e.,there exists no non-trivial topological state characterized by an integer winding numberin (4 n + 1)-dimensional systems belonging to symmetry class DIII or CI (cf. table 5).Conversely, in d = (4 n + 3) dimensions both in class DIII and CI there are topologicallynon-trivial states ( ii or iv in table 4), and one would naively expect that for each ofthese states there are lower dimensional descendants characterized by Z topologicalinvariants. However, this is not the case, as we will explain below. Classes BDI and CII:
Next we consider symmetry classes BDI and CII, where theprojector satisfies [26] Gq ∗ ( − k ) G − = q ( k ) , G = ( σ , BDIi σ y , CII . (131)Introducing the auxiliary function v µ ( k ), equation (128), as before, we find that v µ ( k ) = − Gv ∗ µ ( − k ) G − , for both class BDI and CII. From the cyclic property of thetrace it follows that ǫ α α ··· α n +1 tr (cid:2) v α ( k ) v α ( k ) · · · v α n +1 ( k ) (cid:3) = ǫ α α ··· α n +1 ( − n +1 tr (cid:2) v α ( − k ) v α ( − k ) · · · v α n +1 ( − k ) (cid:3) ∗ . (132) ONTENTS q(k, t)q(k, t)q ' (k, t) r (k, t)r (k, t)q (k) q (k)q (k) q (k)q (k)q (k) q (k)q (k) Figure 4. (a) One-parameter interpolation of two topological insulators representedby the projectors q and q . In (b), the one-way interpolation (a) is extended bymaking use of the discrete symmetry. (c) Two different interpolations, each coloredred and blue, respectively. In (d) the original interpolations are rearranged into twodifferent interpolations. Hence, the winding number density in class BDI and CII is constrained by w n +1 [ q ( k )] = ( − n w ∗ n +1 [ q ( − k )] . (133)As a result, the winding number ν n − [ q ] is vanishing in d = 4 n − n − Z topological insulators belonging to symmetryclass BDI or CII (see table 5). On the other hand, in (4 n + 1) dimensions there exist Z topological states in both class BDI and CII ( ii or iv in table 4), and one wouldexpect, as before, that for each of these states there are lower-dimensional topologicallynon-trivial descendants. Again, this expectation turns out to be incorrect, as we willexplain below. Z classification of first descendants. In this section, we show how a Z topological classification is obtained for 2 n -dimensional, chiral symmetric insulators(supercondcutors) under the constraint of additional discrete symmetries (i.e., C and T ). To uncover the Z topological characteristics of chiral symmetric systems, we studythe topological distinctions among the block off-diagonal projectors q ( k ). Let us considertwo projectors q ( k ) and q ( k ), whose momentum space configuration is restricted bythe symmetry constraint of one of the classes DIII/CI, equation (127), or BDI/CII,equation (131). We introduce a continuous interpolation q ( k, t ), t ∈ [0 , π ] between thesetwo projectors (see figure 4a) with q ( k, t = 0) = q ( k ) and q ( k, t = π ) = q ( k ) . (134)Since the topological space of 2 n -dimensional class AIII insulators is simply connected,the continuous deformation q ( k, t ) is well defined. In general, q ( k, t ) does not satisfythe symmetry constraints encoded by equation (127) or equation (131), respectively. ONTENTS t ∈ [ π, π ] q ( k, t ) = ǫq T ( − k, π − t ) , ǫ = ( − , DIII+1 , CI , (135)while, in the case of symmetry class BDI/CII we set for t ∈ [ π, π ] q ( k, t ) = Gq ∗ ( − k, π − t ) G − , G = ( σ , BDIi σ y , CII . (136)Equations (134)–(136) with the parameter t replaced by the wavevector component k n +1 represent a (2 n +1)-dimensional projection operator respecting the symmetry constraintsof the corresponding symmetry class. Consequently, a winding number for q ( k, t ), ν n +1 [ q ( k, t )], can be defined in the ( k, t ) space. Two different interpolations q ( k, t )and q ′ ( k, t ) generally give different winding numbers, ν n +1 [ q ( k, t )] = ν n +1 [ q ′ ( k, t )].However, we can show that symmetry constraint (135) or (136), respectively, leadsto ν n +1 [ q ( k, t )] − ν n +1 [ q ′ ( k, t )] = 0 mod 2 . (137)To prove equation (137) we introduce two new interpolations r ( k, t ) and r ( k, t ) thattransform into each other under the discrete symmetry operations (see figure 4) r ( k, t ) = ( q ( k, t ) , t ∈ [0 , π ] q ′ ( k, π − t ) , t ∈ [ π, π ] ,r ( k, t ) = ( q ′ ( k, π − t ) , t ∈ [0 , π ] q ( k, t ) , t ∈ [ π, π ] , (138)These are recombinations of the deformations q ( k, t ) and q ′ ( k, t ) with ν n +1 [ q ( k, t )] − ν n +1 [ q ′ ( k, t )] = ν n +1 [ r ( k, t )] + ν n +1 [ r ( k, t )] . (139)Now, we make use of the result from section 4.1.1. Namely, for symmetry class BDI/CIIwe found w n +1 [ q ( k, t )] = w ∗ n +1 [ q ( − k, − t )] (140)in d = 4 n + 1 spatial dimensions, whereas for symmetry class DIII/CI w n − [ q ( k, t )] = w ∗ n − [ q ( − k, − t )] (141)in d = 4 n − d = 4 n + 1dimensions. With equation (20) and (140) we obtain ν n +1 [ r ] = Z d n k d t w n +1 [ r ( k, t )]= Z d n k d t w ∗ n +1 [ r ( − k, − t )]= Z d n k d t w ∗ n +1 [ r ( k, t )] = ν n +1 [ r ] , (142) ONTENTS r transforms into r under the discrete symmetryoperations of class BDI/CII. In conclusion, we have shown ν n +1 [ q ( k, t )] − ν n +1 [ q ′ ( k, t )] =2 ν n +1 [ r ( k, t )] ∈ Z for any two interpolations q ( k, t ) and q ′ ( k, t ) belonging to classBDI/CII. Hence, we can define a relative invariant for the 4 n -dimensional projectionoperators q ( k ) and q ( k ) ν n [ q ( k ) , q ( k )] = ( − ν n +1 [ q ( k,t )] , (143)which is independent of the particular choice of the interpolation q ( k, t ) between q ( k )and q ( k ). Once we have identified a “vacuum” projection operator, e.g., q ( k ) ≡ q ,we can construct with equation (143) a Z invariant: non-trivial Hamiltonians arecharacterized by ν n [ q, q ] = −
1, whereas trivial ones satisfy ν n [ q, q ] = +1. Finally, wenote that the calculation leading to the relative invariant ν n [ q , q ], equation (143), canbe repeated for symmetry class DIII/CI in d = 4 n − ν n − [ q , q ].For d = 4 n − d = 4 n + 1) the dimensional reduction seems to be possible bothfor class DIII and CI (BDI and CII). However, for a given n the dimensional reductionseems to be meaningful only for one of them; while for one of them (corresponding to ii in table 4), there is a decendent Z insulator in one dimension lower, for the other(corresponding to iv in table 4), there are no decendent Z insulators. In other words,for a given n = odd (even), there are lower dimensional Z topological insulators foreither one of class DIII and CI (BDI and CII), but not both. This is because the aboveprocedure does not apply when the classification of topological insulator is not Z , but2 Z . Z classification of second descendants. The dimensional reduction procedurepresented in the previous subsection can be repeated once more to obtain a Z classification of the second descendants. As before, we first focus on symmetry classBDI/CII and study the topological distinctions among the block off-diagonal projectors.We consider two (4 n − q ( k ) and q ( k ), which satisfy thesymmetry constraints imposed by class BDI/CII. We define an adiabatic interpolation q ( k, t ), t ∈ [0 , π ] q ( k, t = 0) = q ( k ) , q ( k, t = π ) = q ( k ) ,q ( k, t ) = Gq ∗ ( − k, − t ) G − , G = ( σ , BDIi σ y , CII . . (144)We can interpret q ( k, t ) as a 4 n -dimensional projector belonging to symmetry classBDI/CII. Therefore, for two deformations q ( k, t ) and q ′ ( k, t ) of the form (144) arelative invariant ν n [ q ( k, t ) , q ′ ( k, t )] can be defined, as discussed in section 4.1.2. Itturns out that due to condition (144) the invariant ν n [ q ( k, t ) , q ′ ( k, t )] is independentof the particular choice of interpolations, i.e., ν n [ q ( k, t ) , q ′ ( k, t )] = +1 for any twodeformations q ( k, t ) and q ′ ( k, t ) satisfying condition (144). In order to prove this, weconsider a continuous interpolation r ( k, t, s ) between the two deformations q ( k, t ) and ONTENTS q ′ ( k, t ) with r ( k, t, s = 0) = q ( k, t ) , r ( k, t, s = π ) = q ′ ( k, t ) ,r ( k, t = 0 , s ) = q ( k ) , r ( k, t = π, s ) = q ( k ) ,r ( k, t, s ) = Gr ∗ ( − k, − t, − s ) G − , G = ( σ , BDIi σ y , CII . (145)This represents a (4 n +1)-dimensional off-diagonal projector in symmetry class BDI/CIIwith the winding number ν n +1 [ r ( k, t, s )]. We note that r ( k, t, s ) is not only adeformation between q ( k, t ) and q ′ ( k, t ), but can also be viewed as a continuousinterpolation between r ( k, , s ) ≡ q ( k ) and r ( k, π, s ) ≡ q ( k ) (for any s ∈ [0 , π ]).Therefore, we find ν n [ q ( k, t ) , q ′ ( k, t )] = ν n [ r ( k, t, , r ( k, t, π )] = ν n [ r ( k, , s ) , r ( k, π, s )]Since r ( k, , s ) ≡ q ( k ) and r ( k, π, s ) ≡ q ( k ) are independent of s , we findthat ν n [ r ( k, , s ) , r ( k, π, s )] = ( − ν n +1 [ r ] = +1. Hence, we have shown that ν n [ q ( k, t ) , q ′ ( k, t )] only depends on q ( k ) and q ( k ). Therefore, ν n [ q , q ( k, t )] togetherwith a reference (“vacuum”) projector q constitutes a well-defined Z invariant in 4 n − d = 3 → → . As an example we consider a 3-dimensionaltopological Dirac superconductor belonging to symmetry class DIII. Consider a d = 3dimensional Dirac Hamiltonian, H d =3(5) ( k, m ) = k x α x + k y α y + k z α z − m i βγ . (146)This is nothing but the chiral topological Dirac superconductor in class DIII. ThisHamiltonian is essentially identical to the BdG Hamiltonian describing the Bogoliubovquasiparticles in the B phase of superfluid He [26, 70, 71] ♯ . It also describes an auxiliaryMajorana hopping problem for an interacting bosonic model on the diamond lattice [72].In this basis, discrete symmetries are given by T : ( σ y ⊗ τ x ) (cid:2) H d =3(5) ( − k, m ) (cid:3) ∗ ( σ y ⊗ τ x ) = H d =3(5) ( k, m ) , C : ( σ y ⊗ τ y ) (cid:2) H d =3(5) ( − k, m ) (cid:3) ∗ ( σ y ⊗ τ y ) = −H d =3(5) ( k, m ) . (147)Combining these two, we have chiral symmetry, which allows us to define the windingnumber ν . From the calculations in Sec. 2.5, the winding number is non-zero, ν = ± /
2, depending on the sign of the mass.By dimensional reduction, we obtain the Hamiltonian in one-dimension lower, H d =2(5) ( k, m ) = k x α x + k y α y − m i βγ . (148)This Hamiltonian is a 2D analogue of He-B, and is unitarily equivalent to the directproduct of spinless p + i p and p − i p wave superconductors. As this is obtained from ♯ In Ref. [55], a topological invariant counting the number of gap-closing Dirac points in an extended,higher-dimensional parameter space was discussed for a four-band example (146), as opposed to ν [ q ]defined for a given topological phase. Moreover, the crucial role for the protection of topologicalproperties arising from the combined time-reversal and charge-conjugation symmetries of symmetryclass DIII was not discussed. ONTENTS Z state.Finally, by further reducing dimensions, H d =1(5) ( k, m ) = k x α x − m i βγ . (149)This is a 1D p x -wave superconductor. Again, this is a Z state.We now discuss the topological character of these states in more detail. The Z nature of the state in this d = 2 example can be studied by the Kane-Mele invariant[9, 15]. It can be expressed as an SU(2) Wilson loop W SU(2) [ L ] := 12 tr P exp (cid:20)I L A ( k ) (cid:21) . (150)Here A ( k ) is the SU(2) Berry connection, P represents the path-ordering. By definition, W SU(2) [ L ] is a well-defined and gauge invariant quantity for any loop L in the BZ. Fortime-reversal invariant systems, it is useful to consider a loop L that satisfies time-reversal symmetry; i.e., a loop which is mapped onto itself (up to reparameterization)under k → − k . For these time-reversal invariant loops, the SU(2) Wilson loop isquantized, W SU(2) [ L ] = ± , (151)and provides a way to distinguish different Z states [9, 15, 73]. The quantization of theWilson loop can be proved by first noting that because of T , Bloch wavefunctions at k and − k are related to each other by a unitary transformation w ( k ), | Θ u − ˆ a ( k ) i = w ˆ a ˆ b ( k ) | u − ˆ a ( − k ) i (152)where Θ represents T operation and w ˆ a ˆ b ( k ) := h u − ˆ a ( − k ) | Θ u − ˆ b ( k ) i . (153)Accordingly, the Berry connection at k is gauge equivalent to A Tµ at − k , A µ ( − k ) = + w ( k ) A Tµ ( k ) w † ( k ) − w ( k ) ∂ µ w † ( k )= − w ( k ) A ∗ µ ( k ) w † ( k ) − w ( k ) ∂ µ w † ( k ) . (154)Because of this sewing condition (154) of the gauge field, when plugged in (150),contributions at k and − k in the path integral cancel pairwise, except at those momentawhich are invariant under T by themselves. Thus, W SU(2) [ L ] = Y K Pf [ w ( K )] , (155)where K is a momentum which is invariant under k → − k .We now compute the Z number for the d = 2 class DIII topological superconductor(148). From (85), | u − ( k ) i = 1 √ λ − k x + i k y i m λ , | u − ( k ) i = 1 √ λ i m − k x − i k y λ . (156) ONTENTS | Θ u − ( k ) i = − i √ λ λ − i mk x + i k y , | Θ u − ( k ) i = − i √ λ − λ − k x + i k y i m . (157)where Θ = ( σ y ⊗ τ x ) K with K being the complex conjugation. The “sewing matrix” w ( k ) can then be computed as w ( k ) := h u ˆ a ( − k ) | Θ u ˆ b ( k ) i = 1 λ ( k ) − i k x + k y m ( k ) − m ( k ) +i k x + k y ! . (158)Here, we have regularized Dirac Hamiltonian properly, by making the mass k -dependent,e.g.,. m → m ( k ) = m − Ck . With this regularization, k = 0 and k = ∞ are the twotime-reversal invariant momenta. One findsPf w (0) = m (0) λ (0) Pf (i σ y ) = m | m | Pf (i σ y ) , Pf w ( ∞ ) = m ( ∞ ) λ ( ∞ ) Pf (i σ y ) = − C | C | Pf (i σ y ) , (159)and hence W SU(2) [ L ] = − sign( m )sign( C ) . (160)That is, when sign( m ) = sign( C ) the Hamiltonian (146) is a Z topologicalsuperconductor. Four among the eight (“real”) symmetry classes of table 5 break chiral (“sublattice”)symmetry. The Z topological insulators (superconductors) which break chiral symmetryare characterized by a Chern number [see equation (11)]. Using this topologicalinvariant one can derive the Z classification for the lower dimensional descendants. Thisderivation is analogous to the discussion in section 4.1 and has been performed previouslyin Ref. [28] for the Z topological insulators which break chiral symmetry. For thesereasons we do not repeat the argument here and refer the reader to Ref. [28] for details.For the Z topological insulators (superconductors) which break chiral symmetry, onecan construct a Z index similar to the one constructed by Moore and Balents [12], inall (even) dimensions [74].
5. Discussion
In this paper we have performed an exhaustive study of all topological insulators andsuperconductors in arbitrary dimensions. The main part of this paper deals withdimensional reduction procedures which relate topological insulators (superconductors)in different dimensions and symmetry classes. We also discussed topological field
ONTENTS D = d + 1 space time dimensions describing linear responses of topologicalinsulators (superconductors). Furthermore, we studied how the presence of inversionsymmetry modifies the classification of topological insulators (superconductors) (seeappendix C). In the following we give a brief summary of the main results of the paper. We have constructed for all five symmetry classes of topological insulators orsuperconductors a Dirac Hamiltonian representative, in all spatial dimensions. Usingthese Dirac Hamiltonians as canonical examples, we have demonstrated that topologicalinsulators (superconductors) in different spatial dimensions and symmetry classes can berelated to each other by dimensional reduction procedures. These Dirac representativeshave been useful for constructing string theory realization of topological insulatorsand superconductors: In Ref. [75], a one-to-one correspondence between the ten-foldclassification of topological insulators and superconductors and a K-theory classificationof D-branes was established, where open string excitations between two D-branes ofvarious dimensions are shown to reproduce all Dirac representatives constructed in thispaper.
We first studied topological insulators (superconductors)with Hamiltonians which are complex (i.e., belonging to the “complex case” of table 3,that is, classes A and AIII from table 1). Here, starting from a topological insulatorwhich lacks chiral symmetry (i.e., class A) in even spatial dimensions d = 2 n , weobtain a topological insulator (superconductor) in d = 2 n − d = 2 n − ν n − ) are inherited from its higher dimensionalparent, the d = 2 n class A topological insulator (characterized by the Chern numberCh n ), it is important to first point out a few properties of the momentum space topologyof Bloch wavefunctions in symmetry classes A and AIII. First we note that the non-zeroChern number in class A leads to an obstruction to the existence of globally definedBloch eigenfunctions. I.e., it is not possible to construct Bloch eigenfunctions for theparent class A topological insulator globally on the d = 2 n dimensional Brillouin zone.Hence, the Bloch eigenfunctions can only be defined locally on some suitably chosencoordinate patches. For the descendant topological insulator (superconductor) withchiral symmetry, on the other hand, since there is no Chern invariant in d = 2 n −
1, therealways exists a basis (i.e., a gauge) in which the Bloch eigenfunctions are well-definedglobally on the entire d = 2 n − ONTENTS d = 2 n − d = 2 n dimensional Brillouin zone of the parent class A topological insulator.The transition function between these two coordinate patches is given by the block off-diagonal projector † q of the descendant Hamiltonian, and the Chern number Ch n ofthe parent Hamiltonian can be written in terms of a winding number of this transitionfunction (see section 2.3). Also demonstrated is, using the Dirac Hamiltonianrepresentatives, the periodicity 8 shift of symmetry classes for the topological insulators(superconductors) in the 8 symmetry classes in which the Hamiltonian possesses at leastone reality condition (arising from T or C ). Once one has constructed a representativein terms of a Dirac Hamiltonian, the reality properties of the spinor representationsof the orthogonal groups SO( N ) (which are linked to the reality properties of therepresentations of the Clifford algebra formed by the gamma matrices) leads directlyto this 8-fold periodicty in the spatial dimension d (see section 3). From the DiracHamiltonian representatives of topological insulators (superconductors) in all d spatialdimensions (all of which are of course massive), one can realize a d − In the low-energy limit, all topological insulators (superconductors) can be describedby topological field theories for the linear responses in space-time, which characterizeuniversal (in principle) experimentally accessible observables of the topological features,such as, e.g., in transport. For example, for symmetry class A in even spatial dimensions d = 2 n , the topological field theory of the linear responses in D = d + 1 space-time † of the type described in equation (34) of the first article in Ref. [26] ONTENTS n thChern number Ch n . For symmetry class AIII in odd spatial dimensions d = 2 n −
1, onthe other hand, the topological field theory in D = 2 n space-time dimensions is givenby the θ term, whose coefficient (the θ angle) is given by the winding number ν n +1 (see section 2.6). For topological singlet superconductors (symmetry classes C andCI), one can couple external SU(2) gauge field to the conserved spin current operatorof the BdG quasiparticles. The spin response in topological singlet superconductors,can be described by SU(2) gauge theory with Chern-Simons type topological term in d = 2 n [8] and with the θ -type-term in d = 2 n + 1 [77]. In passing we note, that it isalso possible to establish a connection among these topological field theories in variousspace-time dimensions and symmetry classes via dimensional reduction procedures. Thetopological field theory formulation may be a good starting point to explore, moregenerally, topological phases in interacting systems beyond those that are presentlyknown. We have also studied the restrictions imposed on groundstate properties of topologicalinsulators (superconductors) by the presence of inversion symmetry. That is, we studiedtopological states that are protected by a combination of spatial inversion (denoted by I ) and an additional discrete symmetry (i.e., T or C ). These systems are invariant underthe combined symmetry operations T · I or C · I , but all three symmetries T , C and I ,are assumed to be absent. We have determined the space of projectors describing thesetopological states. From the homotopy groups of these space of projectors follows theclassification of topological states that are protected by T · I or C · I (see table C2).
An important direction for future study is the search for experimental realizationsof three-dimensional topological singlet or triplet superconductors. Given how fastexperimental realizations of the QSHE in d = 2 and the Z topological insulatorsin d = 3 have been found, we anticipate a similar development for the three-dimensional topological singlet or triplet superconductors. For example, unconventionalsuperconductors in heavy fermion systems, typically possessing strong spin-orbit effects,have been studied extensively over the years. They might be good candidates fortopological superconductors. Moreover, the search for non-trivial topological quantumground states is not restricted to free fermions, or BCS quasiparticles, but includesalso, more generally, strongly interacting systems other than BCS with emergent freefermion behavior at low energies. Furthermore, it should be noted that the classificationscheme given by table 3 is also applicable outside the realm of condensed matter physics.For example, topological properties of color superconducting phases [78, 79], which arepredicted to occur in quark matter, can be discussed in terms of the present classification ONTENTS
Acknowledgments
This work is supported in part by NSF Grants No. PHY05-51164 (APS) and No. DMR-0706140 (AWWL), and by a Grant-in-Aid for Scientific Research from the Japan Societyfor the Promotion of Science (Grant No. 21540332) (AF). SR thanks the Center forCondensed Matter Theory at University of California, Berkeley for its support. SRthanks A. M. Essin, J. E. Moore, and A. Vishwanath for useful discussion.
A. Cartan symmetric spaces: generic Hamiltonians, NL σ M field theories,and classifying spaces of K -theory In this appendix we review the appearance of Cartan’s ten-fold list of symmetric spacesin the context of (i) basic quantum mechanics: where they describe the time evolutionoperators exp(i t H ) of generic Hamiltonians H , (ii) NL σ M field theories: where theydescribe the “target space” manifold of the NL σ M, and (iii) K-Theory: where theydescribe the “classifying space” (briefly discussed at the end of subsection 1.2 of theintroduction). As reviewed in the introduction, the homotopy groups of these symmetricspaces play a key role in the classification of topological insulators and superconductorsboth in the approach of Ref. [26], which makes use of results from Anderson localizationphysics, as well as in the approach of Ref. [27], which is based on K-theory.
Cartan time evolution op. fermionic replica classifyinglabel exp { i t H} NL σ M target space spaceA U( N ) × U( N ) / U( N ) U(2 n ) / U( n ) × U( n ) U( N + M ) / U( N ) × U( M ) = C AIII U( N + M ) / U( N ) × U( M ) U( n ) × U( n ) / U( n ) U( N ) × U( N ) / U( N ) = C AI U( N ) / O( N ) Sp(2 n ) / Sp( n ) × Sp( n ) O( N + M ) / O( N ) × O( M ) = R BDI O( N + M ) / O( N ) × O( M ) U(2 n ) / Sp(2 n ) O( N ) × O( N ) / O( N ) = R D O( N ) × O( N ) / O( N ) O(2 n ) / U( n ) O(2 N ) / U( N ) = R DIII
SO(2 N ) / U( N ) O( n ) × O( n ) / O( n ) U(2 N ) / Sp(2 N ) = R AII U(2 N ) / Sp(2 N ) O(2 n ) / O( n ) × O( n ) Sp( N + M ) / Sp( N ) × Sp( M ) = R CII
Sp( N + M ) / Sp( N ) × Sp( M ) U( n ) / O( n ) Sp( N ) × Sp( N ) / Sp( N ) = R C Sp(2 N ) × Sp(2 N ) / Sp(2 N ) Sp(2 n ) / U( n ) Sp(2 N ) / U( N ) = R CI Sp(2 N ) / U( N ) Sp(2 n ) × Sp(2 n ) / Sp(2 n ) U( N ) / O( N ) = R Table A1.
Three appearances of the list of Cartan’s ten symmetric spaces. Firstcolumn: unitary time evolution operator, second column: (compact) target space ofNL σ Ms, third column: classifying space. (We use the convention in which m = evenin Sp( m ). – Moreover, the Cartan labels of those symmetry classes invariant under thechiral symmetry operation S = T · C from table 1 are indicated by bold face letters.)
In table A1 we list for each symmetry class denoted by its Cartan label in thefirst column, the time evolution operator in this symmetry class (penultimate column
ONTENTS ‡ of the NL σ M field theories in thethird column, and in the last column the classifying space appearing in K-Theory [27].Here we would like to re-emphasize the remarkable fact that the same ten Cartansymmetric spaces describe all three objects listed in the three last column of table A1.Furthermore, there are remarkable relations between these columns. First, the secondcolumn (“time evolution op.”) can be obtained from the last column (“classifyingspace”) by shifting the entries in the last column down by one entry (modulo eight, andmodulo two in the “real” and “complex” cases, respectively). Second, the third column(“NL σ M target space”) is obtained from the last column by performing a reflection(modulo 8) in the last column about the entry in the row labeled D (and an exchangefor the complex case). Consequently, there is then also a resulting relationship betweenthe second and the third columns [80].As reviewed in subsection 1.2 of the introduction, the classifying spaces listed inthe last column of table A1 describe topological insulators (superconductors) in zerodimensions, i.e., at one point in space § . The disconnected components of this space ofHamiltonians Q (which cannot be continuously deformed into each other) are labeledby Z or Z appearing in the d = 0 column of table 3, which denotes the list of zerothhomotopy groups of the spaces in the last column of table A1. All higher homotopygroups of the same spaces can then be inferred from table 2 due to the dimensionalperiodicity and shift properties visible in that table. B. Spinor representations of
SO( N )In this appendix we review spinor representations of SO( N ) and their properties underreality conditions [81, 82]. It is most convenient to discuss spinor representations interms of Clifford algebras, i.e., in terms of gamma matrices { Γ a ( N ) } a =1 ,...,N satisfying { Γ j ( N ) , Γ k ( N ) } = 2 δ jk , with j, k = 1 , . . . , N . Given such a set of gamma matrices a spinorrepresentation of SO( N ) can be readily obtained M jk = − i4 h Γ j ( N ) , Γ k ( N ) i , (B.1)with the SO( N ) generators M jk . B.1. Spinors of
SO(2 n + 1)In what follows we will focus on SO(2 n + 1), which has a 2 n -dimensional irreduciblespinor representation. The gamma matrices in the Dirac representation for N = 2 n + 1 ‡ There are three different varieties of NL σ M field theories: supersymmetric, fermionic replica, andbosonic replica models. In the latter two formulations, the replica limit must be taken, where one lets thenumber of replicas, N , tend to zero at the end of the calculations. While the target spaces of fermionicreplica models are compact, those of bosonic replica models are non-compact. In the supersymmetricformulation, on the other hand, the target spaces are supermanifolds with both bosonic and fermioniccoordinates. The fermionic and bosonic replica NL σ Ms, with N finite, can be viewed as fermion-fermionand boson-boson subsectors, respectively, of the corresponding supersymmetric NL σ Ms [38]. § Ref. [27]; compare also table III (second column, with k = 0) of the first article of [26]. ONTENTS a (2 n +1) = Γ a (2 n − ⊗ σ , a = 1 , · · · , n − , Γ n − n +1) = I n − ⊗ σ , Γ n (2 n +1) = I n − ⊗ σ , Γ n +1(2 n +1) = ( − i) n Γ n +1) Γ n +1) · · · Γ n (2 n +1) , (B.2)where I n − is the 2 n − × n − identity matrix. [The gamma matrices in the Diracrepresentation for N = 2 n can be constructed by just leaving out Γ n +1(2 n +1) , i.e.,Γ a (2 n ) = Γ a (2 n +1) , with a = 1 , · · · , n .] To be more explicit,Γ n +1) = σ ⊗ σ ⊗ · · · ⊗ σ | {z } n − , Γ n +1) = σ ⊗ σ ⊗ · · · ⊗ σ | {z } n − , Γ n +1) = σ ⊗ σ ⊗ σ ⊗ · · · ⊗ σ | {z } n − , Γ n +1) = σ ⊗ σ ⊗ σ ⊗ · · · ⊗ σ | {z } n − , ...Γ n − n +1) = σ ⊗ · · · ⊗ σ | {z } n − ⊗ σ , Γ n (2 n +1) = σ ⊗ · · · ⊗ σ | {z } n − ⊗ σ , (B.3)and Γ n +1(2 n +1) = σ ⊗ · · · ⊗ σ | {z } n . (B.4)From the explicit construction of the gamma matrices, we infer that Γ , , ··· , n +1(2 n +1) areall real, and Γ , , ··· , n (2 n +1) are purely imaginary. In order to implement discrete symmetrieson the space of Dirac Hamiltonians we define the matrices B n +1) := Γ n +1) Γ n +1) · · · Γ n − n +1) ,B n +1) := Γ n +1) Γ n +1) · · · Γ n (2 n +1) . (B.5)By use of the reality and anti-commutation properties of the gamma matrices, one findsthat (cid:2) B n +1) (cid:3) ∗ B n +1) = ( − n ( n − / , (cid:2) B n +1) (cid:3) ∗ B n +1) = ( − n ( n +1) / . (B.6)The operator B n +1) is used to construct Majorana (real) representations of SO(2 n + 1).For odd N = 2 n + 1, they are possible when [81, 82] (cid:2) B n +1) (cid:3) ∗ B n +1) = ( − n ( n +1) / = 1= ⇒ n = 0 , N = 2 n + 1 = 1 , . (B.7) ONTENTS n = even all gamma matrices can be made real and symmetric,whereas when n = odd, they can be made purely imaginary and skew-symmetric.For later use, we also introduce B n +1) := (cid:2) B n +1) (cid:3) − B n +1) = ( − i) n Γ n +1(2 n +1) , ˜ B n +1) := B n +1) Γ n (2 n +1) , ˜ B n +1) := B n +1) Γ n (2 n +1) , (B.8)Note that these matrices satisfy B n +1) Γ a (2 n +1) (cid:2) B n +1) (cid:3) − = ( ( − n +1 Γ a ∗ (2 n +1) , a = 1 , · · · , n, ( − n Γ a ∗ (2 n +1) , a = 2 n + 1 ,B n +1) Γ a (2 n +1) (cid:2) B n +1) (cid:3) − = ( − n Γ a ∗ (2 n +1) , a = 1 , · · · , n + 1 ,B n +1) Γ a (2 n +1) (cid:2) B n +1) (cid:3) − = ( − Γ a (2 n +1) , a = 1 , · · · , n, +Γ a (2 n +1) , a = 2 n + 1 . (B.9)In the following subsection we will use the “ B -matrices”, equations (B.5) and (B.8),as symmetry operators in order to implement discrete symmetries on the space of DiracHamiltonians. To identify the character of these discrete symmetries we first computethe sign η B (2 n +1) picked up by B (2 n +1) under transposition, [ B (2 n +1) ] T = η B (2 n +1) B (2 n +1) .Note that the sign η B (2 n +1) is independent on the choice of basis for the gammamatrices (see Ref. [81]). From equations (B.6) and (B.7) and by use of the property[ B n +1) ] † B n +1) = 1, it follows that η B n +1) = ( − n ( n +1) / . (B.10)Similarly, η B n +1) = ( − n ( n − / ,η ˜ B n +1) = − ( − n ( n +1) / ,η ˜ B n +1) = ( − n ( n +3) / . (B.11) B.2. Discrete symmetries of Dirac Hamiltonians
Let us now determine the symmetry properties of the Dirac Hamiltonians H d =2 n +3(2 n +3) ( k ), H d =2 n +2(2 n +3) ( k ), . . . , H d =2 n − n +3) ( k, m ), defined in equations (100) through (104). We find thatthese Dirac Hamiltonians satisfy the following symmetry conditions(0) : B n +1) H d =2 n +1(2 n +1) ( k ) (cid:2) B n +1) (cid:3) − = ( − n +1 h H d =2 n +1(2 n +1) ( − k ) i ∗ , (B.12)( i ) : B n +1) H d =2 n (2 n +1) ( k, m ) (cid:2) B n +1) (cid:3) − = ( − n (cid:2) H d =2 n (2 n +1) ( − k, m ) (cid:3) ∗ , (B.13)( ii ) : B n +1) H d =2 n − n +1) ( k, m ) (cid:2) B n +1) (cid:3) − = ( − n h H d =2 n − n +1) ( − k, m ) i ∗ , ˜ B n +1) H d =2 n − n +1) ( k, m ) h ˜ B n +1) i − = ( − n +1 h H d =2 n − n +1) ( − k, m ) i ∗ . (B.14) ONTENTS iii ) : B n +1) H d =2 n − n +1) ( k, m ) (cid:2) B n +1) (cid:3) − = ( − n +1 h H d =2 n − n +1) ( − k, m ) i ∗ , (B.15)( iv ) : B n +1) H d =2 n − n +1) ( k, m ) (cid:2) B n +1) (cid:3) − = ( − n +1 h H d =2 n − n +1) ( − k, m ) i ∗ , ˜ B n +1) H d =2 n − n +1) ( k, m ) h ˜ B n +1) i − = ( − n h H d =2 n − n +1) ( − k, m ) i ∗ . (B.16)A few remarks are in order. • The Hamiltonians H d =2 n − n +1) ( k, m ) and H d =2 n − n +1) ( k, m ) satisfy two different discretesymmetry conditions, and are therefore also left invariant under the combinationof these two symmetries, which defines a chiral symmetry. In other words, both H d =2 n − n +1) ( k, m ) and H d =2 n − n +1) ( k, m ) anticommute with a unitary matrix n H d =2 n − n +1) ( k, m ) , Γ n (2 n +1) o = 0 , n H d =2 n − n +1) ( k, m ) , Γ n − n +1) o = 0 . (B.17) • The Hamiltonians H d =2 n (2 n +3) ( k, m ) and H d =2 n − n +3) ( k, m ) can be made block diagonalbecause (cid:2) H d =2 n (2 n +3) ( k, m ) , Γ a (2 n +3) Γ b (2 n +3) (cid:3) = 0 , h H d =2 n − n +3) ( k, m ) , Γ a (2 n +3) Γ b (2 n +3) i = 0 , (B.18)where ( a, b ) = (2 n + 1 , n + 2), (2 n + 2 , n + 3), or (2 n + 3 , n + 1). • In the case of the massive Hamiltonian H d =2 n (2 n +1) ( k, m ), the parity transformation k can be implemented by B n +1) B n +1) H d =2 n (2 n +1) ( k, m ) (cid:2) B n +1) (cid:3) − = H d =2 n (2 n +1) ( − k, m ) . (B.19)By combining the discrete symmetry B n +1) and the parity symmetry B n +1) , theHamiltonian also satisfies B n +1) H d =2 n (2 n +1) ( k, m ) (cid:2) B n +1) (cid:3) − = ( − n (cid:2) H d =2 n (2 n +1) ( k, m ) (cid:3) ∗ . (B.20)This discrete symmetry is unique in that, unlike T or C which relates BlochHamiltonians at k and − k , it constrains the form of the Hamiltonian at a given k .This is nothing but the real/pseudo-real condition for SO(2 n +1). As a consequence,the Hamiltonian in k -space can be written as a real/pseudo-real matrix. Hence,classifying topological insulators (superconductors) satisfying the combination of B n +1) and B n +1) amounts to classifying real bundles. (It is not necessaryto satisfy both, though.) An example of a topological insulator/superconductorsatisfying the combination of B n +1) and B n +1) is constructed in appendix C. k Parity here simply means k → − k symmetry. In odd space time dimensions, it is actually not calledparity. ONTENTS C. Topological insulators protected by a combination of spatial inversionand either time-reversal or charge-conjugation symmetry
In the main text, we have focused on the role of generic symmetries such as time reversaland charge conjugation, which are not related to any spatial symmetries. However,real systems often have discrete spatial symmetries, such as parity, reflection, discreterotations, etc. Hence, it is meaningful to study the restrictions imposed by thesesymmetries on the Bloch wavefunctions, and thus on the ground state properties. Inthis appendix, we briefly discuss topological states that are protected by a combinationof spatial inversion and an additional discrete symmetry, such as T or C . C.1. Inversion symmetry combined with another discrete symmetry
Consider a tight-binding Hamiltonian, H = X r,r ′ ψ † ( r ) H ( r, r ′ ) ψ ( r ′ ) , H † ( r ′ , r ) = H ( r, r ′ ) , (C.1)where ψ ( r ) is an n f -component fermion operator, and index r labels the lattice sites.(The internal indices are suppressed.) Each block in the single-particle Hamiltonian H ( r, r ′ ) is a n f × n f matrix, and we assume the total size of the single particleHamiltonian is n f V × n f V , where V is the total number of lattice sites. The componentsin ψ ( r ) can describe, e.g., orbitals or spin degrees of freedom, as well as different siteswithin a crystal unit cell centered at r .Provided the system has translational symmetry, H ( r, r ′ ) = H ( r − r ′ ) , (C.2)with periodic boundary conditions in each spatial direction (i.e., the system is definedon a torus T d ), we can perform the Fourier transformation and obtain in momentumspace H = X k ∈ BZ ψ † ( k ) H ( k ) ψ ( k ) , (C.3)where ψ ( r ) = √ V − X k ∈ BZ e i k · r ψ ( k ) , H ( k )= X r e − i k · r H ( r ) . (C.4)[Here, if different components in ψ ( r ) were to represent different sites within a unitcell centered at r , we could include the phase e i k · ( r − r a ) in the definition of ψ ( k ), ψ ( k ) → diag (cid:0) e i k · ( r − r a ) (cid:1) a =1 , ··· ,n f ψ ( k ) where r a represents a location within a unit cell.]In the main text of this paper, we considered topological insulators (superconduc-tors) protected by discrete symmetries, such as T and C . We now focus on the effect ofinversion symmetry. By definition, an inversion Ω relates fermion operators at r and at − r , Ω ψ ( r ) Ω − = U ψ ( − r ) , (C.5) ONTENTS − − ǫ V , η V ) (1 , −
1) (1 ,
1) ( − , −
1) ( − , × × × × × AI × × (cid:13) × × AII × (cid:13) × × × chiral AIII (cid:13) × × × × (sublattice) BDI (cid:13) × (cid:13) × (cid:13)
CII (cid:13) (cid:13) × (cid:13) ×
BdG D × × × × (cid:13) C × × × (cid:13) × DIII (cid:13) (cid:13) × × (cid:13) CI (cid:13) × (cid:13) (cid:13) × Table C1.
The ten generic symmetry classes of single-particle Hamiltonians classifiedin terms of the presence ( (cid:13) ) or absence ( × ) of the time-reversal symmetry ( ǫ V = +1)with T= − η V = −
1) and T= +1 ( η V = +1), and the particle-hole symmetry( ǫ V = −
1) with C= − η V = −
1) and C= +1 ( η V = +1), as well as chiral (orsublattice) symmetry denoted by S; see table 1. For the notation of these symmetryclasses in terms of ( ǫ V , η V ), see the main text and Ref. [26]. where U is a constant n f × n f unitary matrix. (It is important to distinguish betweeninversion and parity symmetry. The former means r → − r for any space-time dimension.Parity transformation in even space-time dimension agrees with inversion, but in oddspace-time dimension, it is different from inversion.) One can imagine, for example, U represents a parity eigenvalue of each orbital at site r ( s, p, . . . , orbitals), which can beeither +1 or −
1. Similarly if different components in ψ ( r ) represent different sites inthe unit cell centered at r , inversion transformation sends r to − r and at the same timecan reshuffle locations of electron operators within a unit cell, which can be describedby the unitary matrix U . The invariance of H under Ω implies U − H ( r, r ′ ) U = H ( − r, − r ′ ) , U − H ( r ) U = H ( − r ) . (C.6)Thus, the Bloch Hamiltonians at k and − k are unitarily equivalent, U − H ( k ) U = H ( − k ) . (C.7)To summarize, definition (C.5) implies that (i) the spectrum at k is identical to the oneat − k , and (ii) the unitary matrix U relating the Hamiltonian at k to the one at − k is k -independent. This form of symmetry does not necessarily correspond to inversionsymmetry verbatim, but might describe an inversion symmetry which is realized as aprojective symmetry (e.g., invariance of the Hamiltonian under inversion followed bya gauge transformation). The only big assumption here is that we have assumed U is k -independent.We now combine the inversion symmetry with another discrete symmetry, V − H ( k ) V = ǫ V H ( − k ) T , ǫ V = ± , ONTENTS V T = η V V, η V = ± . (C.8)As before we distinguish four different cases [see section 1.1, and in particular equations(3) and (4)]: ( ǫ V , η V ) = (1 ,
1) corresponds to a time-reversal symmetry with T= +1,( ǫ V , η V ) = (1 , −
1) denotes a time-reversal symmetry with T= −
1, ( ǫ V , η V ) = ( − , ǫ V , η V ) = ( − , −
1) is a particle-holesymmetry with C= −
1. These four cases together with the corresponding “Cartanlabel” are summarized in table C1. Then, for the combined transformation, W − H ( k ) W = ǫ W H ( k ) T , W := U V, with ǫ W = ǫ V . (C.9)Iterating this transformation twice, H ( k ) → W H ( k ) T W − → W ( W − ) T H ( k ) W T W − , (C.10)which suggests, from Schur’s lemma, W T W − ∝ I n f . Hence, as before, for W , wedistinguish the following two cases, W T = η W W, η W = ± . (C.11)The signature η W is invariant under unitary transformations. It depends on thesignature η V , the signature η U for the inversion U T = η U U , and the commutationrelation of U and V . Here, note that the signature η U for inversion alone is not invariantunder unitary transformation, while the signature η W is invariant. As an illustration, letus consider spinless fermions. Time-reversal symmetry T for spinless fermions implies H ( i, j ) ∗ = H ( i, j ) . (C.12)Thus, when inversion and T are combined, H ∗ ( r ) = W H ( − r ) W − , H ∗ ( k ) = W − H ( k ) W, where W = U. (C.13)In this example and in this basis, the signature of W solely depends on the signature of U , η W = η U .Interesting examples are provided by Dirac Hamiltonians. For a massive DiracHamiltonian, the mass matrix itself can be taken as the unitary matrix U which wasintroduced in equation (C.5). Put differently, any massive Dirac Hamiltonian possessesa U -type symmetry. These U -type symmetries can be, however, different from theinversion symmetry which is imposed at the microscopic level. However, if one startsfrom a microscopic lattice model with inversion symmetry, and arrives at a DiracHamiltonian with a mass in the continuum, inversion symmetry in the continuum mustbe implemented by a mass matrix.First, consider Hamiltonian (101)( i ) : H d =2 n (2 n +1) ( k, m ) = d =2 n X a =1 k a Γ a (2 n +1) + m Γ n +1(2 n +1) . (C.14)This Hamiltonian has a V -type symmetry with V = B n +1) , ǫ V = ( − n , η V = η B n +1) = ( − n ( n − / . (C.15) ONTENTS U = Γ n +1(2 n +1) , η U = +1 . (C.16)The combined transformation W is given by W = B n +1) , ǫ W = ( − n , η W = η B n +1) = ( − n ( n +1) / . (C.17)Second, we consider the massive Dirac Hamiltonian (103)( iii ) : H d =2 n − n +1) ( k, m ) = d =2 n − X a =1 k a Γ a (2 n +1) + m M , with M := iΓ n +1(2 n +1) Γ n (2 n +1) Γ n − n +1) . (C.18)The V -type discrete symmetry for this is V = B n +1) , ǫ V = ( − n +1 , η V = η B n +1) = ( − n ( n +1) / . (C.19)The inversion symmetry is represented by U = M , η U = +1 . (C.20)Thus, the combined transformation is W = M B n +1) = iΓ n +1(2 n +1) Γ n (2 n +1) Γ n − n +1) × Γ n +1) Γ n +1) · · · Γ n (2 n +1) . with ǫ W = ( − n +1 , η W = ( − n ( n +1) / . (C.21)where we have noted U V = V U . C.2. Classification of gapped Hamiltonians in the presence of a combination of spatialinversion and either time-reversal or charge-conjugation symmetry
We now classify the ground states of gapped Hamiltonians protected by a W -typesymmetry. We distinguish four different cases, ( ǫ W , η W ) = ( ± , ± ǫ W , η W ) = (1 , W is a symmetricunitary matrix, W = W T ( η W = +1). Such a matrix is an element of U( n f ) / O( n f ). Anyelement in U( n f ) / O( n f ) can be written as W = XX T where X is a unitary matrix.One can then take a basis in which the Bloch Hamiltonian for any given k is a realsymmetric matrix. In other words, if we define ˜ H := X † H X , then ˜ H ∗ = ˜ H . Thus, when( ǫ W , η W ) = (+1 , +1), the Hamiltonian is real symmetric and hence Bloch wavefunctionscan be taken real at each k . We assume there are N − ( N + ) occupied (unoccupied) Blochwavefunctions for each k with N + + N − = N tot (= n f ). The spectral projector onto thefilled Bloch states or the “ Q -matrix” in this case can be viewed as an element of the realGrassmannian G N − ,N + + N − ( R ) = O( N + + N − ) / [O( N + ) × O( N − )]. For a given system, the“ Q -matrix” defines a map from the BZ onto the real Grassmannian. Hence, classifyingtopological classes of band insulators amounts to counting the number of topologicallyinequivalent mappings from the BZ to G N − ,N + + N − ( R ). Mathematically, this is given bythe homotopy group of the space of projectors (i.e., of the real Grassmannian in thepresent case), which can be read off from table 2. ONTENTS d ( ǫ W , η W ) Projectors 0 1 2 3 4 5 6 7 · · · (+1 , −
1) Sp( N + M ) / Sp( N ) × Sp( M ) Z Z Z Z · · · ( − , +1) O(2 N ) / U( N ) Z Z Z Z · · · (+1 , +1) O( N + M ) / O( N ) × O( M ) Z Z Z Z · · · ( − , −
1) Sp( N ) / U( N ) 0 0 Z Z Z Z · · · Table C2.
Classification of topological insulators (superconductors) protected by acombination of spatial inversion and an additional discrete symmetry.
The same reasoning can be repeated for the case ( ǫ W , η W ) = (+1 , − G N − ,N + + N − ( H ) =Sp( N + + N − ) / [Sp( N + ) × Sp( N − )], whose homotopy group is again given in table 2.When ( ǫ W , η W ) = ( − , +1), the Hamiltonian can be taken, in a suitable basis,real and skew symmetric, i.e., an element of so( n f ). Any element of so( n f ) can betransformed into a canonical form by an orthogonal transformation. For the Q -matrix,its canonical form ˜ Q is a matrix whose matrix elements are ± Q = S ˜ QS T , S ∈ SO( n f ) , where ˜ Q = · · ·− − , (C.22)where we have assumed n f is an even integer. We thus identified the space of projectorsas SO( n f ) / U( n f / ǫ W , η W ) = ( − , − n f ) / U( n f ).To summarize, we have listed in table C2, for each of the four cases of W -type symmetries, the space of projectors “ G/H ” together with their homotopy groups π d ( G/H ), which yields the classification of topological insulators (superconductors)protected by a combination of inversion and an additional discrete symmetry. Notice, wefocus here on the implication of the combination of inversion and a discrete symmetries,but we do not require each symmetry separately.
C.2.1. Example: ( ǫ W , η W ) = (1 , in d = 2 . We now specialize to the caseof ( ǫ W , ǫ W ) = (1 , G N + ,N + + N − ( R ) and the projection operator defines a map from the BZonto the real Grassmannian. In particular, when d = 2, the projector Q ( k x , k y ) defines amap from S or T onto G N + ,N − + N + ( R ). The homotopy group π [ G N + ,N − + N + ( R )] = Z tells us that the space of quantum ground states is partitioned into two topologicallydistinct sectors. ONTENTS
62A representative Q -field configurations which is a non-trivial element of π [ G N + ,N − + N + ( R )] = Z can be constructed following Ref. [83]. For simplicity, takethe case where we have four bands in total and two filled bands; take N + = N − = 2.Then such a representative Q -matrix is given by Q ( l ) ( k ) = n ( l ) z − n ( l ) x − n ( l ) y n ( l ) z n ( l ) y − n ( l ) x − n ( l ) x n ( l ) y − n ( l ) z − n ( l ) y − n ( l ) x − n ( l ) z with n ( l ) ( k ) := (cid:16) n ( l ) x , n ( l ) y , n ( l ) z (cid:17) = (cid:16) cos( lφ ) sin( θ ) , sin( lφ ) sin( θ ) , cos θ (cid:17) . (C.23)where l ∈ Z and θ and φ is spherical coordinates parameterizing the BZ ≃ S . Theprojector is an non-trivial element of the second homotopy group when l is odd, whileit is trivial when l is even. Observe that the normalized vector n ( l ) ( k ) itself defines amap from S onto S , and wraps S integer (= l ) times.We can construct a Hamiltonian which has Q ( l ) ( k ) as a projector. We consider thefollowing four band tight-binding Hamiltonian on the square lattice: H = X r (cid:2) ψ † ( r ) h ψ ( r )+ ψ † ( r ) h x ψ ( r + ˆ x ) + h . c . + ψ † ( r ) h y ψ ( r + ˆ y ) + h . c . (cid:3) , (C.24)where ˆ x = (1 ,
0) and ˆ y = (0 , h = − µ σ − σ ! , h x = tσ ∆ σ y − ∆ σ y − tσ ! ,h y = tσ − i∆ σ − i∆ σ − tσ ! , (C.25)and t, ∆ , µ ∈ R are a parameter of the model. In k -space, H ( k ) = R z − R x − R y R z R y − R x − R x R y − R z − R y − R x − R z (C.26)where R ( k ) = (cid:16) −
2∆ sin k y , −
2∆ sin k x , t (cos k x + cos k y ) − µ (cid:17) . (C.27)We will set t = ∆ = 1 henceforth. As we change µ , there are four phases separatedby quantum phase transitions at µ = ± µ = 0. When | µ | >
4, the normalizedvector R ( k ) / | R ( k ) | does not wrap S as we sweep the momentum and the system is in atrivial phase. On the other hand, for | µ | < R ( k ) / | R ( k ) | wrapsthe sphere S n = +1 times (or n = − ONTENTS (a) E ne r g y E ne r g y (b) k y +4-4 0 2 pp k y pp Figure C1.
The energy spectrum with t = ∆ = 1 and µ = − µ = +1[panel (b)]. phase. The projector constructed from (C.26) is topologically equivalent to Q ( l ) ( k ) with l odd.For topological insulators and superconductors protected by T , C , or combinationof both, a diagnostics of bulk topological character is the existence of gapless edgemodes. For insulators and superconductors with an inversion symmetry, it is less clearif looking for edge modes is useful to characterize the bulk topological character sinceboundary breaks inversion symmetry. [Nevertheless, as discussed recently in Ref. [84],for the entanglement entropy spectrum, an inversion symmetry can protect the gaplessentanglement entropy spectrum.] For the present case, however, the Hamiltonian hasseveral accidental symmetries (see below), in particular T (odd), which may protect thegapless nature of edge states, if they exist, and if the number of Kramers pairs is odd.In Fig. C1, the energy spectrum for µ = ± y -direction (located at x = 0 and x = N x ) are created, and the energyeigenvalues are plotted as a function of k y . Each eigenvalue is doubly degenerate foreach k y . For µ = ±
1, there are four edge modes, two of which localized at x = 0 andthe other two at x = N x , while for | µ | > W -type symmetry with ( ǫ W , η W ) = (1 , T A − H ( − k ) ∗ A = H ( k ) , A = τ z σ or τ z σ y , inversion symmetries I B − H ( − k ) B = H ( k ) , B = τ z σ or τ z σ y , and charge-conjugation symmetries C C − H ( − k ) ∗ C = −H ( k ) , C = τ x σ x or τ x σ z . ONTENTS T , A = τ z σ y with A T = − A , it is a member of symmetry class AII. Furthermore,it supports two branches of modes per edge which are counter propagating. Therepresentative Hamiltonian so constructed is thus a Z topological insulator in classAII (i.e., it is in the QSH phase). With further imposing a C , C = τ x σ z with C T = + C ,the single particle Hamiltonian can be interpreted as a member of symmetry class DIII.Again, it is a non-trivial Z topological superconductor in class DIII. Thus, non-trivial Z topological insulators (AII) and superconductors (DIII) in d = 2 dimensions can beinterpreted, somewhat accidentally, a non-trivial element of π [ G N + ,N − + N + ( R )] = Z .To implement a W -type symmetry, T for symmetry classes AII and DIII ( A = τ z σ y with A T = − A ) can be combined with an inversion symmetry represented by B = τ z σ y .This inversion symmetry is, however, an artificial symmetry in the sense that it flipsspin. References [1] Thouless D J, Kohmoto M, Nightingale P, and den Nijs M 1982
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