Approximate symmetries and conservation laws in topological insulators and associated Z -invariants
aa r X i v : . [ m a t h - ph ] A p r Approximate symmetries and conservation lawsin topological insulators and associated Z -invariants Nora Doll and Hermann Schulz-Baldes
Department Mathematik, Friedrich-Alexander-Universit¨at Erlangen-N¨urnbergCauerstr. 11, D-91056 Erlangen, GermanyEmail: [email protected], [email protected]
Abstract
Solid state systems with time reversal symmetry and/or particle-hole symmetry oftenonly have Z -valued strong invariants for which no general local formula is known. Forphysically relevant values of the parameters, there may exist approximate symmetriesor almost conserved observables, such as the spin in a quantum spin Hall system withsmall Rashba coupling. It is shown in a general setting how this allows to define robustinteger-valued strong invariants stemming from the complex theory, such as the spinChern numbers, which modulo 2 are equal to the Z -invariants. Moreover, these integerinvariants can be computed using twisted versions of the spectral localizer.Keywords: approximate laws, strong invariants, spectral localizer Physical systems can be distinguished by symmetry properties. If one considers merely chiralsymmetry (CHS), time-reversal symmetry (TRS) and particle-hole symmetry (PHS), then oneobtains the so-called Cartan-Altland-Zirnbauer (CAZ) classes widely used in solid state physics.All insulators within one such CAZ class can further be distinguished by topological invariantswhich take values either in the integers or just in Z = { , } . The most stable of suchinvariants are called strong invariants and their possible values make up Kitaev’s periodic tableof topological insulators, see Table 1. The periodicity of this table can be explained as amanifestation of Bott periodicity of K -theory [12]. As can be seen, there are two diagonalswith eight Z -invariants each. While there are index theorems for these Z -invariants [24, 9, 4]even in the mobility gap regime, the standard ways to compute them numerically via edgestate crossings [3, 8] or the spectral localizer [14, 15, 17] have only been justified in particularsituations, even though a rigorous general bulk-boundary correspondence is available [4, 1]. Oneof the difficulties is that there is no general cohomological formula for Z -invariants [7, 11, 19](other than for the integer-valued entries of the periodic table which are given by Chern numbersand winding numbers, see [22]). 1 TRS PHS CHS CAZ d = 1 d = 2 d = 3 d = 4 d = 5 d = 6 d = 7 d = 80 +1 0 0 AI 2 Z Z S Z I Z Z Z Z A Z A2 0 +1 0 D Z C Z Z Z Q3 − Z S Z A Z Z − Z S Z I Z Z − − Z Z A Z Q Z − Z Z Q Z I Z − Z Z C Z C Z Table 1:
List of the real symmetry classes ordered by
TRS , PHS and
CHS as well as the
CAZ label. Then follow the strong invariants in dimension d = 1 , . . . , . They constitute Kitaev’speriodic table of topological insulators [12] . The roman letters indicate the approximate law thatleads to integer valued strong invariants. Here I is an approximate spin inversion symmetry, S and Q are approximate spin and charge conservation laws, and finally C and A designatean approximate chiral symmetry and an approximate conservation law, possibly involving spin,a sublattice structure or the particle-hole degree of freedom. All these approximate laws aredescribed in detail in Section 5.Furthermore the physical importance of Z -invariants is disputable. Solid state systems aredirty so that conservation laws and symmetry relations may hold only up to error terms (anexception is the particle-hole symmetry resulting merely from the one-particle approximation).Moreover, there are potentially more interesting and stable invariants taking integer values.This last point is best explained on the particular case of a two-dimensional quantum spin Hallsystem (case j = 4 and d = 2 in Table 1, hence with odd TRS). Historically, non-vanishing Z -invariants were theoretically found in such systems [10]. However, experiments have clearlyshown that conductive properties within the new phase are even stable under perturbationsbreaking TRS [13]. Most likely, the reason for this is the existence of non-vanishing spin Chernnumbers introduced by Sheng et al. [26] and Prodan [20, 21]. These invariants are connectedto an approximately conserved quantity in the physical system, here given by the z -componentof the spin. Almost conservation means that the commutator of the spin component with theHamiltonian is small, which reflects that the Rashba spin orbit coupling is small [20]. On theother hand, there is a rigorous argument showing that spin Chern numbers lead to conductingedge states [3] and that the associated edge currents have stability properties [23, 19].The first question addressed in this paper is how to construct integer valued invariants forsymmetry classes having a Z -invariant according to the periodic table. This will be achievedunder a supplementary assumption that either there is an approximately conserved quantity(like the spin component in the quantum spin Hall effect) or an approximate symmetry. Anexample for the latter is an approximate chiral symmetry in odd dimensional systems of CAZclass A that was already discussed in [22]. More precisely, such systems do not have chiralsymmetries, but the terms in the Hamiltonian breaking these symmetries are small. This still2 CHS CAZ d = 1 mod 2 d = 0 mod 20 0 A C Z A1 1 AIII Z C,A ATable 2:
List of the complex CAZ classes in dimension d mod and the strong invariants. The C and A designate an approximate chiral symmetry and an approximate conservation law. allows to define integer valued invariants as (higher) winding numbers. Similarly, if an evendimensional system of CAZ class AIII has an approximate conservation law, it can have anassociated non-vanishing Chern number. In the two cases ( j, d ) = (0 ,
0) and ( j, d ) = (1 , Z -invariants. One reason is that it isstill a challenge to compute the Z -invariants numerically and that the Z -invariants associatedto the approximate laws give a relatively easy handle which may, moreover, shed light on thephysical properties of such topologically non-trivial systems. The analysis of these Z -invariantscorresponding to the Z -invariants in the Kitaev table will essentially be particular applicationsof the complex cases described in Section 3.The second contribution of this paper is to show how these invariants can be computedby a suitable (twisted) modification of the spectral localizer (defined in [15, 17] and Section 2below). Also in the case of quantum spin Hall systems, this is a considerable improvement onnumerical procedures to calculate spin Chern numbers [21]. The numerical implementation ofthe twisted spectral localizer is the object of another study.The paper is organized as follows: Section 2 reviews earlier results that are needed to stateand prove the new contributions of this paper. This includes a description of the index theoreticcharacterization of the strong invariants for the complex CAZ classes [22] and of the spectrallocalizer for integer valued pairings [15, 17]. Section 3 then develops the two main ideas ofthis paper in the complex CAZ classes. More precisely, it is shown how an approximate chiralsymmetry or an approximate conservation law allow to define an integer valued strong invariantand it is shown how to compute this invariant with a twisted spectral localizer. Section 4 thenreviews how the strong invariants of the Kitaev table are given in terms of index pairings withreal symmetries. This is essentially based on [9]. The next Section 5 then shows how for eachof the Z -entries in Table 1 a specific approximate law allows to construct a strong integervalued invariant which mod 2 is equal to the given Z -entry. In each case we attempt to arguethat the approximate symmetry or approximate conservation is physically reasonable and showhow to construct a twisted spectral localizer allowing to compute these integer valued stronginvariants. Acknowledgement:
This work was partially supported by the DFG.3
Review of prior results on complex CAZ classes
Throughout H = H ∗ will be a Hamiltonian acting on the d -dimensional tight-binding Hilbertspace ℓ ( Z d , C L ). The L -dimensional fiber may contain spin, particle-hole, sublattice and otherinternal degrees of freedom. Moreover, H is supposed to be of finite range or, more generally,have decaying matrix entries in the following sense: Definition 1
A linear operator A on ℓ ( Z d , C L ) is said to be local if for all k ∈ N there existssome constant C k < ∞ such that kh x | A | y ik ≤ C k | x − y | k . (1) The norm on the l.h.s. is the operator norm on matrices from C L × L and | x | denotes theeuclidean length of x ∈ Z d . Locality in one or another form is, of course, a crucial assumption from a physical point ofview. Based on the above, it was shown in [25] that the set of local operators is a ∗ -algbra thatis invariant under smooth functional calculus. For covariant operators, locality is equivalent tosmoothness and these two facts are also well-known (see [22]).The second assumption on the Hamiltonian is the existence of a spectral gap at the Fermilevel. After a shift in energy, the Fermi level can be chosen to be at 0 so that the H is invertible.This implies that the Hamiltonian describes an insulator. For some of the claims below it issufficient to suppose that H has a mobility gap, but this case will not be dealt with here as itleads to supplementary technical difficulties. Associated to H is now a Fermi projection P = χ ( H ≤ . As H has a gap, P can be written as a smooth function of H and is hence also local.In the complex cases of the CAZ classification, the Hamiltonian may also have a so-calledchiral (or sublattice) symmetry. For the implementation of this symmetry, let us suppose that H acts on ℓ ( Z d , C L ), that is, the dimension of the fiber is doubled. Further suppose that on C L act the Pauli matrices: σ = (cid:18) (cid:19) , σ = (cid:18) − ı ı (cid:19) , σ = (cid:18) − (cid:19) , where each entry involves the identity on ℓ ( Z d , C L ). Then H is said to have a chiral symmetryif ( σ i ) ∗ H σ i = − H . (2)If i = 3, then this is equivalent to H being block off-diagonal:( σ ) ∗ H σ = − H ⇐⇒ H = (cid:18) AA ∗ (cid:19) . (3)4ere A is an operator on ℓ ( Z d , C L ) which is invertible as H is invertible. If H has a chiralsymmetry, then ( σ ) ∗ H σ = − H = ⇒ P = 12 (cid:18) − U − U ∗ (cid:19) , (4)and U = A | A | − is called the Fermi unitary [22]. The topological invariant of the Hamiltonian will be given in terms of (Noether) indices ofFredholm operators. There is a by now standard way [22] to construct them. One considersthe d -dimensional (unbounded self-adjoint) Dirac operator D = d X i =1 X i ⊗ L ⊗ γ i . (5)Here X , . . . , X d are the self-adjoint position operators on ℓ ( Z d ) defined by X j | x i = x j | x i , and γ , . . . , γ d ∈ C d ′ × d ′ are anti-commuting self-adjoint matrices of size d ′ = 2 ⌊ d ⌋ which square to , namely they form an irreducible representation of the complex Clifford algebra C d with d generators. If d is even, there exists a symmetry γ ∈ C d ′ × d ′ anticommuting with γ , . . . , γ d , sothat γDγ = − D . Actually, γ = γ d +1 from the representation of C d +1 . The representation canand will be assumed such that γ = diag( , − ) is block diagonal and thus the Dirac operatoris off-diagonal in the grading of γ : D = (cid:18) D D ∗ (cid:19) , d even . (6)Moreover, the identity L in (5) acts on the matrix degrees of freedom. Then set L ′ = (cid:26) Ld ′ , d odd ,L d ′ , d even . Then D acts on ℓ ( Z d , C L ′ ) for d odd, while for even d , D acts ℓ ( Z d , C L ′ ) and D on ℓ ( Z d , C L ′ ).As the commutator of D with any local operator is bounded, the Dirac operator specifies aneven or odd Fredholm module for the algebra of local operators. Hence one has standard indexpairings that are described next. From the data of the Dirac operator, one can define the Hardyprojection E = χ ( D ≥ , and for d even the (unitary) Dirac phase F by2 E − = (cid:18) FF ∗ (cid:19) , d even . (7)5rom E and F and the data of the Hamiltonian given by the Fermi projection or Fermi unitary(if H has a chiral symmetry) one now defines the index pairings as the operator T = (cid:26) P F P + − P , d even ,E U E + − E , d odd and H chiral . (8)Then T is a Fredholm operator, namely it has finite dimension kernel and cokernel. Theassociated index Ind( T ) = dim (cid:0) Ker( T ) (cid:1) − dim (cid:0) Ker( T ∗ ) (cid:1) , is called the (strong) topological invariant. It takes values in Z and provides the entries inTable 2. Admittedly, this may seem like an awkward way to introduce topological invariants.However, if the Hamiltonian is part of a covariant family of Hamiltonians [22], then the indexis almost surely constant and equal to either Chern numbers (for even d ) or higher windingnumbers (for odd d ) which are both used as the standard definition of the strong invariants.This latter fact is the statement of an index theorem [22]. One of the advantages of the abovedefinition is that it does not require covariant Hamiltonians and can thus be associated to asingle Hamiltonian. The topological content of the index can then be read off the spectral flowupon the insertion of a monopole [5]. For the present paper, the index approach is technicallyadvantageous. The spectral localizer is another Dirac-like operator combining the Hamiltonian from Section 2.1with the Dirac operator from Section 2.2, just as the index pairing (8). The main interest in thisobject is that its finate dimensional approximations have a spectral asymmetry that is equal tothe invariant given by the index pairing, see Theorem 1 below. Let us describe the constructionand result in some detail, as one of the aims of this paper is to modify the spectral localizer.As for the index pairings, the odd and even cases have to be treated separately. For d odd, theDirac operator D acts on ℓ ( Z d , C L ′ ) where L ′ = Ld ′ . On the other hand, a chiral Hamiltonian H acting on ℓ ( Z d , C L ) is in the form (3) with matrix entries A acting on ℓ ( Z d , C L ). Theyare identified with H ⊗ and A ⊗ acting on ℓ ( Z d , C L ′ ) and ℓ ( Z d , C L ′ ) respectively. For atuning parameter κ >
0, the spectral localizer is then defined to be the operator L od κ = (cid:18) κ D AA ∗ − κ D (cid:19) = κ D ⊗ σ + H , (9)acting on ℓ ( Z d , C L ′ ). For d even, with D acting on ℓ ( Z d , C L ′ ) and given by (6) and H ∼ = H ⊗ also acting on ℓ ( Z d , C L ′ ), the spectral localizer is defined again as an operator on ℓ ( Z d , C L ′ )by L ev κ = (cid:18) H κ D κ D ∗ − H (cid:19) = κ D + H ⊗ γ . (10)Let us note that both L od κ and L ev κ are self-adjoint (domain issues are not discussed here). Nowthe finite volume restriction of the spectral localizer are constructed using the partial isometry6 ρ from ℓ ( Z d , C L ′ ) onto Ran( χ ( | D | ≤ ρ )) and Ran( χ ( | D | ≤ ρ )), respectively for odd andeven d . Here ρ > D ). As D has compact resolvent, therange of π ρ is finite dimensional. Note that, for even d , the operator D is normal so that | D | = | D ∗ | . It is now natural to consider also π ρ ⊕ π ρ which will also simply be denoted by π ρ .Now for any operator B on ℓ ( Z d , C L ′ ) or ℓ ( Z d , C L ′ ), the finite volume restriction is definedby B ρ = π ρ Bπ ∗ ρ . The finite volume spectral localizer is then defined by L od κ,ρ = ( L od κ ) ρ = (cid:18) κ D ρ A ρ A ∗ ρ − κ D ρ (cid:19) , L ev κ,ρ = ( L ev κ ) ρ = (cid:18) H ρ κ D ,ρ κ D ∗ ,ρ − H ρ (cid:19) . These are both finite dimensional self-adjoint matrices. In the following, the upper index on L od κ,ρ and L ev κ,ρ will be dropped. Theorem 1 ([16, 17])
Let g be the invertibility gap of H , namely g = k H − k − . For odd d suppose the H is of the form (3) . For odd and even d respectively, suppose that the tuningparameter κ and radius ρ are admissible in the sense that the bounds κ ≤ g k H k k [ D, A ] k , κ ≤ g k H k k [ D , H ] k , (11) and ρ > gκ (12) hold. Then ( L κ,ρ ) ≥ g ρ , (13) namely L κ,ρ is invertible so that its signature is well-defined. Furthermore, for the index pairings T given by (8) , one has Ind( T ) = 12 Sig ( L κ,ρ ) . The main interest of the localizer is that it allows to access the topological invariants withoutheavy numerical computations. Indeed, the finite-dimensional matrix L κ,ρ is merely built fromthe matrix entries of the Hamiltonian and the position operator. No spectral calculus of theHamiltonian is needed (namely, one does not need to compute the Fermi projection), not evenof the spectral localizer itself because one can compute the signature directly with the blockChualesky decomposition. As stressed in [16, 17] as well as [15], the conditions (11) and (12)are not optimal, but are likely not far from optimal. On the other hand, even if H has merelya mobility gap, numerics have shown that the (fluctuation) signature of the spectral localizeris still linked to the strong invariant (see [18]). In this section, approximate symmetries and the associated invariants are discussed for thecomplex CAZ classes (hence no real structure is involved). Let us first give a short overview by7iscussing the entries of Table 2. For d even, there is a Z -valued strong invariant for systemswithout symmetry, namely CAZ class A. It is known to be equal to the Chern numbers [22]. Forodd d , there is a Z -valued strong invariant for systems with a chiral symmetry. This invariantis known to be equal to the (higher) winding numbers [22]. Now for odd d and CAZ classA, the table has no entry, indicating that there is no non-trivial strong invariant. If there is,however, an approximate chiral symmetry then it was already noted in [22] and will be furtherdiscussed in Section 3.1 that the strong invariant from odd-dimensional chiral systems is stillwell-defined. Similarly, for even dimensional chiral systems an approximate conservation lawstill allows to define a strong invariant, see Section 3.3. Furthermore, in the two cases withpossible strong invariant, namely ( j, d ) = (1 ,
1) and ( j, d ) = (0 , j, d ) = (0 ,
1) the approximate chiral symmetry only leads to avanishing invariant, one may look for a further approximate symmetry. This is not investigatedhere though.After having defined the strong invariants associated to approximate laws, the second aim ofthis section is to show how a twisted modification of the spectral localizer allows to determinethe invariant in an efficient manner, which is also susceptible to numerical evaluation.
Let us begin with the case of an odd dimensional system described by a Hamiltonian H on ℓ ( Z d , C L ) without chiral symmetry, but for which η = k σ Hσ + H k is small. This will be called an approximate chiral symmetry. Strictly speaking, H is not inCAZ class AIII and hence according to the Kitaev periodic table there would not be any stronginvariant associated with H , but as in [22] one can readily argue that the Z -valued invariantcan still exist. Indeed, if H with approximate chiral symmetry is written in the grading of thePauli matrices as H = (cid:18) H + AA ∗ H − (cid:19) , (14)then η = 2 max {k H + k , k H − k} . If g = k H − k − , then η < g assures that A is invertible.Hence from the operator A one can construct the odd index pairing from the CAZ class AIIIand in odd dimension it can take arbitrary integer values. Why is this of any use? First ofall, the index allows to distinguish different topological ground states. Second of all, the bulk-boundary correspondence is still valid in an approximate manner. This is best illustrated indimension d = 1 for which the topological systems in class AIII are then given by stacking Su-Schrieffer-Heeger models. The integer index pairing is then equal to the number of zero-energychiral edge modes for a half-line system ( e.g. [22]). If now H + and H − are added, these zeromodes will be shifted away from 0, but for small enough η they can still be found in the gap8f the bulk spectrum. Hence in a weak perturbative sense, the bulk-boundary correspondencepersists. In the following, the bulk-boundary correspondence will not be discussed for the otherapproximate symmetries, but we do expect similar weak forms for all of them.The second focus of this paper is rather on showing how a modification of the odd spectrallocalizer allows to determine the index in an efficient manner, also in numerical computations.This modification of (9) is given by L κ = (cid:18) κ σ D HH κ σ D (cid:19) . (15)Here σ is a ”twist” of the Dirac operator and such twist will appear in other spectral localizersbelow. The proof of the following result is then deferred to the Appendix A. Proposition 1
Let d be odd and g = k H − k − . Suppose that H is of the form (3) . Let thetuning parameter satisfy κ ≤ g k H k k [ D, A ] k , (16) and the radius ρ and the perturbation size η satisfy ρ > g κ , η < g . (17) Then
Ind( EA | A | − E + − E ) = 14 Sig ( L κ,ρ ) . (18)Let us stress that, as in the cases described in Section 2.3, the conditions (16) and (17) aresufficient for (18) to hold, but the r.h.s. may well be of interest in more general situations, inparticular, even in a mobility gap regime. A possible modification of (15) is to work with aspectral localizer e L κ,ρ = (cid:18) H + + κ D AA ∗ H − − κ D (cid:19) . Still one has for admissible ( κ, ρ ) thatInd( EA | A | − E + − E ) = 12 Sig (cid:16) e L κ,ρ (cid:17) . The matrix dimension of e L κ,ρ is by a factor 2 smaller than that of L κ,ρ , which is advantagousfor numerics. On the other hand, the twisted version (15) is more intuitive. ( j, d ) = (0 , A conservation law of a Hamiltonian H is a commutation relation of the form[ H, σ i ] = 0 , (19)9here σ i is one of the Pauli matrices σ , σ , σ . This requires that H acts on ℓ ( Z d , C L ) withan even dimensional fiber on which again the Pauli matrices act. An approximate conservationlaw then requires η = k ( σ i ) ∗ H σ i − H k = k [ H, σ i ] k to be small. Note the difference of sign in (19) w.r.t. the chiral symmetry (2). In a spinfulsystem, σ i could be a component of a spin and as such it appears later on ( e.g. Section 5.5dealing with the quantum spin Hall effect). Furthermore, for a BdG operator it can represent thecharge operator, see Section 4.1. Finally let us add that one could also require the approximateconservation of a self-adjoint operator other than a Pauli matrix σ i ; if the observable is gapped,one can then consider the symmetry built from the projection below the gap. This slightgeneralization is not spelled out in detail here. If now (19) holds with i = 3, the conservationlaw is equivalent to H being block diagonal, namely if the block entries of H are as in (14),then ( σ ) ∗ H σ = H ⇐⇒ H = (cid:18) H + H − (cid:19) , (20)with H ± being self-adjoint operators on ℓ ( Z d , C L ). In view of (20), an approximate conserva-tion law with σ allows to split the system into a sum of two subsystems, up to errors. Moreprecisely, with matrix entries as in (14), one has η = 2 k A k and then (cid:18) H + H − (cid:19) = H (cid:20) − H − (cid:18) AA ∗ (cid:19)(cid:21) (21)and a bound η < g with g = k H − k − implies by a Neumann series that diag( H + , H − ) isinvertible. Then both H ± have a spectral projection P ± = χ ( H ± <
0) which is local. Thus ineven dimensions there are two invariants Ind( P ± F P ± + − P ± ). Considering the homotopy t ∈ [0 , H ( t ) = (cid:18) H + t At A ∗ H − (cid:19) inside the gapped local operators, one has a path t ∈ [0 , P ( t ) = χ ( H ( t ) <
0) of localprojections, thus of constant index so thatInd( P + F P + + − P + ) + Ind( P − F P − + − P − ) = Ind( P F P + − P ) . (22)Hence even if the standard strong invariant Ind( P F P + − P ) vanishes, one can have non-trivial strong invariants Ind( P ± F P ± + − P ± ) adding up to 0. The spectral localizer suitablefor the computation of these invariants is L κ = (cid:18) H κ D σ κ σ D − H (cid:19) . (23)Again the proof of the following result is given in Appendix A.10 roposition 2 Let d be even and g = k H − k − . Suppose that the tuning parameter satisfies κ ≤ g k H k max {k [ D, H + ⊕ H + ] k , k [ D, H − ⊕ H − ] k} , (24) and that ρ and η satisfy (17) . Then Ind( P + F P + + − P + ) − Ind( P − F P − + − P − ) = 12 Sig( L κ,ρ ) . Following Prodan [20], another way to split P is to consider P σ P as a self-adjoint operatoron the range of P . If [ H, σ ] = 0, the spectrum of P σ P is equal to {− , } and has, inparticular, a spectral gap at 0. Now due to the existence of the gap of H , P can be written as acontour integral along a path Γ which has minimal distance g from the spectrum of H . Hence[ σ , P ] = I Γ dz πı [ σ , ( z − H ) − ] = I Γ dz πı ( z − H ) − [ H, σ ]( z − H ) − , so that k [ σ , P ] k ≤ Cg − η for a constant C that is essentially the length of the path (roughly thesize of the spectrum). If now Cg − η <
1, one concludes that 0 is not in the spectrum of
P σ i P because ( P σ P ) = P (cid:0) − ( ı [ σ , P ]) (cid:1) P . Then one can set Q ± = χ (cid:0) ± P σ P > (cid:1) which is thena local projection and has, in even dimensions, a well-defined index Ind( Q ± F Q ± + − Q ± ).Now the above P ± are not equal to Q ± , however, they are homotopic to each other within theset of local projections. One has P = Q + + Q − , but in general P = P + + P − . Let us stressthough that the condition η < g C − needed to define Q ± is considerably more stringent than η < g needed for the definition of P ± . ( j, d ) = (1 , This section deals with even dimensional systems with a chiral symmetry which also have anapproximate conservation law. Moreover, the two associated symmetry operators are supposedto be anticommuting. On first sight, this may appear to be an awkward situation to consider,but it turns out to be relevant for several BdG systems, e.g. the CAZ class ( j, d ) = (3 , σ Hσ = − H , η = k [ H, σ ] k small . However, it is more convenient to conjugate these equations by the eigenbasis of σ , namely tosuppose σ Hσ = − H , η = k [ H, σ ] k small . (25)Writing H then in the block form (14), this becomes H = (cid:18) H + A − A − H + (cid:19) , η = 2 k A k small . (26)11rguing as in (21) one concludes that H + is invertible if η < k H − k − . Hence there is a localprojection P + = χ ( H + ≤
0) which in even dimensions has an invariant Ind( P + F P + + − P + ).The twisted spectral localizer suitable for the computation of this index is L κ = (cid:18) H κ D σ κ σ D − H (cid:19) (27)Then by Proposition 2, Ind( P + F P + + − P + ) = 14 Sig( L κ,ρ ) , provided that κ ≤ g (cid:0) k H kk [ D, H + ⊕ H + ] k (cid:1) − and (17) holds. Section 3.2 considered an even-dimensional system without symmetry, namely ( j, d ) = (0 , Z -invariant is well-defined, but vanishes. Given an approximate conser-vation law, it is nevertheless possible to write this as a sum of two potentially non-vanishinginvariants, see (22). A similar situation may also appear for odd-dimensional chiral systems,that is, ( j, d ) = (1 , σ Hσ = − H .Hence it will be implemented by a second set of Pauli matrices ν , ν , ν commuting with the σ i so that the Hamiltonian is supposed to act on ℓ ( Z d , C L ) to accommodate both the ν i and σ i . Now let us suppose smallness of either(i) η = k ν H ν + H k or (ii) η = k ν H ν − H k . As H is chiral, it is of the form (3). In the grading of the ν i , the operator A on ℓ ( Z d , C L ) cannow be decomposed as A = (cid:18) A + BC A − (cid:19) . (28)Then the two above cases become(i) η = 2 max {k A + k , k A − k} or (ii) η = 2 max {k B k , k C k} , and the odd index pairings become, provided that η < g ,(i) Ind( EB | B | − E + − E ) = − Ind( EC | C | − E + − E ) , (ii) Ind( EA + | A + | − E + − E ) = − Ind( EA − | A − | − E + − E ) . Provided that conditions similar to those in Propositions 1 and 2 hold, these invariants can becomputed with the following spectral localizers:(i) Ind( EB | B | − E + − E ) = 14 Sig( L κ,ρ ) , L κ = (cid:18) κ ν D AA ∗ κ ν D (cid:19) , (29)(ii) Ind( EA + | A + | − E + − E ) = 14 Sig( L κ,ρ ) , L κ = (cid:18) κ ν D AA ∗ − κ ν D (cid:19) . (30)12 Review of prior results on real CAZ classes
In the CAZ classification the Hamiltonian can have further symmetries, namely a particle-holesymmetry leading to Bogoliubov-de Gennes operators or a time-reversal symmetry. Both arereal symmetries invoking a complex conjugation. All this is recalled in Section 4.1. Similarly,the Dirac operator has real symmetries depending on the dimension. This is inherited from thereal Clifford representation and reviewed in Section 4.2. The combination of these symmetrieshave implications on the index pairings. Particular focus will be on Z -valued index pairings,see Section 4.3. Finally Section 4.4 discusses real symmetries of the spectral localizer. The Hamiltonian H = H ∗ is supposed to be as in Section 2.1, namely it acts on a d -dimensionaltight-binding Hilbert space with finite dimensional fiber and satisfies the locality bound (1).The Hilbert space is ℓ ( Z d , C lL ) with l = 2 , , s i , τ i and ν i , commuting with each other, can be implemented. We will not keep trackof the size index l explicitly in the following, hoping that the reader has little difficulties todetermine it. Furthermore, the Hilbert space is equipped with an anti-unitary involution C called complex conjugation. It is simply defined fiberwise. For any operator A , we will alsodenote its complex conjugate by A = C A C . It is a linear operator. Also let us denote thetranspose of A by A T = C A ∗ C . The time-reversal symmetry (TRS) will be implemented byPauli matrices s i acting on spin degrees of freedom. A Hamiltonian is said to have TRS if( s i ) ∗ H s i = H . (31)The TRS is called odd if i = 2. Otherwise, the TRS is called even. The particle-hole symmetry(PHS) is implemented by Pauli matrices τ i :( τ i ) ∗ H τ i = − H . (32)Again it is called odd if i = 2 and even otherwise. For the Fermi projection P this implies( τ i ) ∗ P τ i = − P .
The natural form of PHS stemming from fermionic quadratic many-body Hamiltonians is evenand implemented by τ . This leads to the (even) Bogoliubov-de Gennes (BdG) operators:( τ ) ∗ H τ = − H ⇐⇒ H = (cid:18) h ∆∆ ∗ − h (cid:19) , (33)with the so-called pair creation operator ∆ satisfying the BdG equation ∆ T = − ∆ (see [2, 6]for details). Another representation of the Hamiltonian is obtained by Cayley transform on theparticle-hole fiber: H Maj = ( τ C ) ∗ H τ C , τ C = √ (cid:18) ı − ı (cid:19) . (34)13hen H Maj = − H Maj = − ( H Maj ) T is a purely imaginary and anti-symmetric operator. Of course,it is also possible to combine PHS with TRS which leads to the 8 real CAZ classes, see Table 1.If a Hamiltonian has both a PHS and a TRS, they can be combined to a chiral symmetry( s i τ j ) ∗ H ( s i τ j ) = − H . Hence the Fermi projection is described by a Fermi unitary in thesecases, but this requires diagonalizing the chiral symmetry operator s i τ j . Furthermore, theFermi unitary then has a further symmetry property (symmetric, real, etc. ). Let us regroup allthese informations in a small table. The index j corresponds to that of Table 1, and s i and σ i square to , namely i = 0 , , j = 0 j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7 τ ∗ Hτ = − H τ ∗ Hτ = − H τ ∗ Hτ = − H τ ∗ Hτ = − H τ ∗ Hτ = − H τ ∗ Hτ = − Hs ∗ i Hs i = H s ∗ i Hs i = H s ∗ Hs = H s ∗ Hs = H s ∗ Hs = H s ∗ i Hs i = Hs ∗ i P s i = P σ ∗ i Uσ i = U τ ∗ P τ = − P σ ∗ U T σ = U s ∗ P s = P σ ∗ Uσ = U τ ∗ P τ = − P σ ∗ i U T σ i = U (35)Here the Pauli matrices σ , σ , σ appear after reducing out the two symmetries in the cor-responding column. These symmetry properties do, of course, have an important impact onthe index pairing, in particular, when combined with similar symmetry properties of E and F stemming from D . Before starting with this analysis in Section 4.2, let us first discuss furthersymmetries of even BdG operators. The charge conservation symmetry( τ ) ∗ H τ = H (36)is equivalent to having a vanishing pair creation ∆ = 0. Then H is classified by h which can bein CAZ class A, or AI or AII pending on whether H also has a TRS. As explained in [2] (seealso [6]), a global spin rotation invariance for a BdG operator leads to operators with odd PHS.Finally let us note that operators with PHS and/or TRS can also have another chiral symmetryor conservation law which are then implemented by further Pauli matrices (commuting with τ i and s i as stated above). The Dirac operator is still given by (5) with a representation γ , . . . , γ d of the complex Cliffordalgebra. In particular, there is an associated Hardy projection E which in even dimension d is chiral and described the Dirac phase F , see (7). On top of that, the Clifford representationcan be chosen such that γ i +1 = γ i +1 is real and γ i = − γ i is purely imaginary for all i . Thenit is possible to construct for every d two commuting representations Γ , Γ , Γ and Σ , Σ , Σ of the Pauli matrices acting on the Clifford representation space such that the following holds: d = 1 d = 2 d = 3 d = 4 d = 5 d = 6 d = 7 d = 8 Γ ∗ D Γ = − D Γ ∗ D Γ = − D Γ ∗ D Γ = − D Γ ∗ D Γ = − D Γ ∗ D Γ = − D Γ ∗ D Γ = − D Σ ∗ i D Σ i = D Σ ∗ i D Σ i = D Σ ∗ D Σ = D Σ ∗ D Σ = D Σ ∗ D Σ = D Σ ∗ i D Σ i = D Σ ∗ E Σ= E Σ ∗ F T Σ= F Σ ∗ E Σ= − E Σ ∗ F Σ= F Σ ∗ E Σ= E Σ ∗ F T Σ= F Σ ∗ E Σ= − E Σ ∗ F Σ= F Σ ∗ D T Σ= D Σ ∗ D Σ= D Σ ∗ D T Σ= D Σ ∗ D Σ= D Σ = Σ = Σ = − Σ = − Σ = − Σ = − Σ = Σ = (37)14ere Σ i can be Σ = , Σ or Σ , and in the case Σ = no matrix degree of freedom maybe needed so that Σ = 1 can be scalar. The first row is not explicitly spelled out in [9], butcan readily be obtained from the results of Section 3.4 in [9], by applying in two of the 8 casesa Cayley transform to obtain commuting symmetry operators. Let us be explicit about thispoint for d = 2. The standard representation of the Dirac opertor is, cf. Section 2.2, D = (cid:18) X − ı X X + ı X (cid:19) , D = − γ D γ = γ D γ , with γ , γ , γ being the Pauli matrices. From this one can read off the second line in d = 2,namely F = X − ı X and Σ = 1 scalar. Then with the Cayley transform γ C defined as in(34), the transformed Dirac operator D ′ = ( γ C ) ∗ Dγ C is given by D ′ = (cid:18) X − X − X − X (cid:19) , D ′ = − ( γ ) ∗ D ′ γ = D ′ . Hence the equations in the d = 2 entry in the first line hold for D ′ with Γ = γ and Σ i = .The other case d = 6 can be dealt with similarly. The last line of (37) contains the symmetry ofthe Hardy projection and Dirac phase, obtained after reducing out. Here Σ is a real symmetryon the Clifford representation space (or half of it), namely Σ = Σ and Σ = ± . Let us stressthat the tables (37) and (35) are essentially the same, if one identifies d ∼ = 1 − j mod 8. Z -index As in Section 2.2, one can now build from the Fermi projection or Fermi unitary (stemmingfrom the Hamiltonian) and the Hardy projection or the Dirac phase (stemming from the Diracoperator) an index pairing. Taking into account all the possible real symmetries of the Hamil-tonian in (35) and the Dirac operator in (37), this leads to 64 index pairings, each leadingto a Fredholm operator which inherits symmetry properties from (35) and (37). The complexpairings (8) allow to pair unitaries with projections and this covers all the Z and 2 Z entries inTable 1. Following [9], the pairing of two projections P and E is given by T = E ( − P ) E + − E , (38)the pairing of two unitaries U and F by T = 12 (cid:18) UU ∗ (cid:19) (cid:18) F F (cid:19) (cid:18) UU ∗ (cid:19) + − (cid:18) UU ∗ (cid:19) . (39)Due to the locality estimates (1), these are also Fredholm operators. Moreover, these indexpairings as well as those given in (8) inherit symmetries. These symmetries may imply thatthe index Ind( T ) is even or vanishes. In the latter case, a secondary Z -valued indexInd ( T ) = dim (cid:0) Ker( T ) (cid:1) mod 2 ∈ Z may be well-defined. In (38) and (39), one can exchange the roles of E with P and U with F respectively, without changing the numerical index Ind ( T ) [9].15 heorem 2 (Theorem 2 in [9]) With P and U having the symmetries from (35) and thedimension d leading to the symmetries (37) , the index pairings given by (8) , (38) and (39) arewell-defined and take values as stated in Table 1 . Let us stress that all these Z -indices are generalizations of the standard Z -indices used inthe literature (like the Kane-Mele index for CAZ class AII for d = 2). For the pairings of unitaries with projections the spectral localizer is defined by (9) and (10),respectively for even or odd pairings. Then the symmetries (35) and (37) lead to symmetriesof the spectral localizer which in turn allow to show how Z - or Z -valued invariants can beextracted from the finite volume spectral localizer. For the odd index pairings, this is discussedin Section 1.7 of [15]. The proof that these invariants are indeed equal the entries of Table 1 isnot given in [15]. This as well as an exhaustive treatment of the other 32 cases involving thepairings (38) and (39) is the object of another publication. The main objective here is to exhibitapproximate laws that allow to find Z -valued invariants which determine the Z -invariants ofTable 1. In each column and line of the real CAZ classes of Table 1 there is one Z -valued invariant andone 2 Z -valued invariant. Both can be computed using the spectral localizer from the complextheory as described in Section 2.3. Here the focus is on the Z -entries of Table 1 and, morespecifically, to present an approximate law for each case that allows to introduce a Z -valuedinvariant that modulo 2 determines the Z -invariant described in Section 4.3. Furthermore,for each case a twisted spectral localizer is presented that allows to compute the new Z -valuedinvariant. In principle, the approach is algebraically the same as in the complex cases describedin Section 3. However, we will attempt to argue in each case that the approximate symmetry orconservation law can be a physically reasonable assumption. In the subsections below, we willgo through all cases by following the staircase of Z -invariants in Table 1 downward. To givethe reader some traction, we start out with zero-dimensional systems of CAZ class BDI and D ,then follow the one-dimensional systems, etc. , up to d = 8. The proof that the Z -invariant givesmodulo 2 the Z -invariant is carried out in detail only in the low-dimensional cases d = 1 , , ( j, d ) = (1 , and ( j, d ) = (2 , As a warm-up, let us begin with the zero-dimensional case which appears in Table 1 as the d = 8 mod 8 column. The Hamiltonian is a finite dimensional matrix in this case as thephysical space consists of merely one point. According to Table 1, there are two Z -invariantscorresponding to j = 1 and j = 2. In both cases, the Hamiltonian is of the BdG form (33).The Z -invariant is given by the sign of the Pfaffian of ı times the Majorana representation16 Maj of the Hamiltonian, as given in (34). If now the Hamiltonian has an approximate chargeconservation, this Z -invariant can be calculated as the parity of an integer valued signature.The extra symmetry for j = 1 w.r.t. j = 2 is irrelevant for the following. Proposition 3
Suppose that η = k [ H, τ ] k satisfies η < k H − k − and let h ∈ C L × L be theupper diagonal entry of the BdG Hamiltonian H ∈ C L × L , see (33) . Then sgn (cid:0) Pf( ı H
Maj ) (cid:1) = ( − L h )2 . Proof:
First of all, the bound on η implies that t ∈ [0 , H ( t ) = (cid:18) h t ∆ t ∆ ∗ − h (cid:19) is a path within the invertible matrices along which hence both sides of the equality remainunchanged. Thus it is sufficient to consider the case η = 0 so that ∆ = 0 and H = diag( h, − h )with h invertible. Let h = udu ∗ be the spectral decomposition of h , namely d is diagonal and u is unitary. There is a path t ∈ [0 , u ( t ) of unitaries connecting u (0) = u to u (1) = . Then t ∈ [0 , ı ( τ C ) ∗ diag( u ( t ) du ( t ) ∗ , − u ( t ) du ( t ) ∗ ) τ C is a path of real skew-adjoint invertiblesconnecting ı ( τ C ) ∗ Hτ C to ı ( τ C ) ∗ diag( d, − d ) τ C . Thereforesgn (cid:0) Pf( ı ( τ C ) ∗ Hτ C ) (cid:1) = sgn (cid:0) Pf( ı ( τ C ) ∗ diag( d, − d ) τ C ) (cid:1) = sgn (cid:18) Pf (cid:18) − dd (cid:19)(cid:19) = ( − L ( L − sgn(det( − d )) = ( − L ( L − ( − Tr ( − p ) , where p = χ ( h < − p ) = Sig( h )+ L the claim now follows. ✷ ( j, d ) = (2 , This section deals with one-dimensional BdG operators of CAZ class D. This includes a Kitaevchain without TRS. Hence H = − ( τ ) ∗ Hτ is of the BdG form (33). Moreover, the Hamiltonianis supposed to have a supplementary approximate chiral symmetry implemented by a Paulimatrix ν commuting with the τ i , namely η = k ν H ν + H k is required to be small. Thus oneis in the setting of Section 3.1 and, provided that η < g , one has an integer valued invariantgiven by (18). In particular, the spectral localizer is given by (15) and Proposition 1 appliesfor η < g and admissible ( κ, ρ ). This invariant then allows to compute the Z -invariant: Proposition 4
In the above situation,
Ind (cid:0) E ( − P ) E + − E (cid:1) = Ind (cid:0) E A | A | − E + − E (cid:1) mod 2 . Proof.
For H of the form (14), let us consider the path of BdG Hamiltonians t ∈ [0 , H ( t ) = (cid:18) t H + AA ∗ t H − (cid:19) . ν i . In particular, one has ( τ ) ∗ A τ = − A . If η < g , thisis a path of invertibles so that t ∈ [0 , P ( t ) = χ ( H ( t ) <
0) is continuous. Moreover, andas ( τ ) ∗ H ( t ) τ = − H ( t ), t ∈ [0 , Ind (cid:0) E ( − P ( t )) E + − E (cid:1) is constant by Proposition 7 in [9]. Thus it is sufficient to showInd (cid:0) E ( − P (0)) E + − E (cid:1) = Ind (cid:0) E A | A | − E + − E (cid:1) mod 2 . Let U = A | A | − denote the unitary phase of A . Modulo 2, one hasInd (cid:0) E ( − P (0)) E + − E (cid:1) = dim (cid:18) Ker (cid:18) E (cid:18) UU ∗ (cid:19) E + − E (cid:19)(cid:19) = dim (cid:0) Ker(
EU E + − E ) (cid:1) + dim (cid:0) Ker( EU ∗ E + − E ) (cid:1) = dim (cid:0) Ker(
EU E + − E ) (cid:1) − dim (cid:0) Ker( EU ∗ E + − E ) (cid:1) = Ind (cid:0) EU E + − E (cid:1) , which concludes the proof. ✷ ( j, d ) = (3 , Here j = 3 so that the Hamiltonian H lies in the CAZ class DIII, namely( τ ) ∗ H τ = − H , ( s ) ∗ H s = H . (40)Now the 3-component of the spin is implemented in the BdG representation by τ ⊗ s (see [6]).Therefore approximate spin conservation means that η = k [ H, τ ⊗ s ] k is small. Let us bring all this data into a normal form. The chiral symmetry inherited from(40) is with τ ⊗ s (which is imaginary here). The suitable basis change from τ i ⊗ s j to tensorproducts σ i ⊗ ν j of two new sets of Pauli matrices is M = 1 √ − ı ı ı − ı , because one then checks M ∗ τ ⊗ s M = σ ⊗ , M ∗ τ ⊗ s M = − ⊗ ν , M T τ ⊗ M = − ıσ ⊗ ν . In this basis the Hamiltonian therefore verifies( σ ) ∗ M ∗ HM σ = − M ∗ HM , ( σ ⊗ ν ) ∗ M ∗ H M σ ⊗ ν = − M ∗ H M , (41)18nd η = k [ M ∗ HM, ν ] k . In particular, M ∗ HM is precisely in the case of item (ii) in Section 3.4, with the secondequation in (41) as a supplementary property. Thus, if η < g , there is an integer index pairingInd( EA + | A + | − E + − E ) with A + given by M ∗ HM = (cid:18) AA ∗ (cid:19) , A = (cid:18) A + BC A − (cid:19) , (42)in the gradings of the σ i and ν i respectively. Then η = k [ A, ν ] k = 2 max {k B k , k C k} . Thereis a corresponding twisted spectral localizer given in (30). Let us also note that the secondequation in (41) implies ( ν ) ∗ A T ν = A and hence ( A + ) T = A − . Therefore again the sum ofthe indices of A ± vanishes. The strong Z -index is given in terms of this invariant: Proposition 5
In the above situation,
Ind (cid:0) EU E + − E (cid:1) = Ind (cid:0) EA + | A + | − E + − E (cid:1) mod 2 . Proof. As η < g , the path t ∈ [0 , A ( t ) = (cid:18) A + t Bt C ( A + ) T (cid:19) , lies in the invertible operators. Thus t ∈ [0 , U ( t ) = A ( t ) | A ( t ) | − is continuous. As( ν ) ∗ U ( t ) T ν = U ( t ) and therefore T ( t ) = EU ( t ) E + − E satisfies ( ν ) ∗ T ( t ) T ν = T ( t ), itfollows that t ∈ [0 , Ind ( T ( t )) is constant by Proposition 5 in [9], see also [24]. To completethe argument let us show thatInd (cid:0) EU (0) E + − E (cid:1) = Ind (cid:0) EU + E + − E (cid:1) mod 2 , where U + = A + | A + | − denotes the unitary phase of A + . Indeed, modulo 2 one hasInd (cid:0) EU (0) E + − E (cid:1) = dim (cid:0) Ker( EU + E + − E ) (cid:1) + dim (cid:0) Ker( EU T + E + − E ) (cid:1) = dim (cid:0) Ker( EU + E + − E ) (cid:1) + dim (cid:0) Ker( EU ∗ + E + − E ) (cid:1) = dim (cid:0) Ker( EU + E + − E ) (cid:1) − dim (cid:0) Ker( EU ∗ + E + − E ) (cid:1) = Ind (cid:0) EU + E + − E (cid:1) , completing the proof. ✷ ( j, d ) = (3 , The symmetry is the same as in the last Section 5.3. Moreover, H is supposed to have anapproximate conservation law expressed in terms of η = k [ H, τ ] k . τ of this approximate conservation law anticommutes with the symmetry τ ⊗ s implementing the chiral symmetry. To bring this in a normal form, let us again apply the basischange M from Section 5.3 so that (41) holds and M ∗ HM is off-diagonal as in (42) withoffdiagonal entry A satisfying ( ν ) ∗ A T ν = A . Moreover, due to M ∗ τ ⊗ M = σ ⊗ ν , one has η = k [ M ∗ HM, σ ⊗ ν ] k = k ( ıν A ) ∗ − ıν A k = k A + A k . (43)Hence this case of anticommuting (approximate) laws was already dealt with for the complexCAZ classes in Section 3.3, with a supplementary property given by the second equation in (41).Let us perform another basis change so that ( M N ) ∗ H ( M N ) satisfies (25). This is attained by N = 1 √ − − , because indeed N ∗ σ ⊗ N = σ ⊗ , N ∗ σ ⊗ ν N = σ ⊗ , N T σ ⊗ ν N = σ ⊗ . As in Section 3.3, the diagonal entry H + of ( M N ) ∗ H ( M N ) is invertible and for P + = χ ( H + < Z -valued invariant Ind( P + F P + + − P + ). For η < g and ( κ, ρ ) admissible, it canbe calculated via the spectral localizer given in (27). This invariant allows to compute the Z -invariant. Proposition 6
In the above set-up,
Ind (cid:0) EU E + − E (cid:1) = Ind (cid:0) P + F P + + − P + (cid:1) mod 2 . Proof.
Carrying out the basis change N , one directly checks H + = ı ( A − A ) ν . As η < g ,the path t ∈ [0 , A ( t ) = A − t ( A + A ) is within the invertibles. Thus t ∈ [0 , U ( t ) = A ( t ) | A ( t ) | − is continous. Moreover, ( ν ) ∗ U ( t ) T ν = U ( t ) so that t ∈ [0 , Ind (cid:0) EU ( t ) E + − E (cid:1) is constant by Proposition 5 in [9], see also [24]. Thus it is sufficient to showInd (cid:0) EU (1) E + − E (cid:1) = Ind (cid:0) P + F P + + − P + (cid:1) mod 2 . As − P + = ı ( A − A ) ν | ı ( A − A ) ν | − = ı ( A − A ) | ı ( A − A ) | − ν = ı U (1) ν , one has Ind (cid:0) EU (1) E + − E (cid:1) = Ind (cid:0) E ı U (1) ν E + − E (cid:1) = dim (cid:0) Ker(( − P + ) F + F ( − P + )) (cid:1) mod 2= dim (cid:0) Ker( F ∗ ( − P + ) F + ( − P + )) (cid:1) mod 2= Ind (cid:0) P + F P + + − P + (cid:1) mod 2 , where the second equality holds by Proposition 2 in [9]. ✷ .5 Approximately conserved odd spin: case ( j, d ) = (4 , This section deals with two-dimensional systems with odd TRS ( s ) ∗ Hs = H . Such systemshave a Z -invariant, allowing to distinguish a trivial system from a quantum spin Hall system.Let us suppose that H has an approximate spin conservation in the sense that η = k [ H, s ] k issmall. By the results of Section 3.2, the bound η < g = 2 k H − k − then allows to consider twoinvariants Ind( P ± F P ± + − P ± ) which, moreover, satisfy (22). This actually does not pend onthe odd TRS which does, on the other hand, imply ( s ) ∗ P s = P so that combined with F T = F one deduces Ind( P F P + − P ) = 0. Thus the sum of the invariants Ind( P ± F P ± + − P ± )vanishes and Proposition 2 implies that for η < g and admissible ( κ, ρ )Ind (cid:0) P + F P + + − P + (cid:1) = 14 Sig( L κ,ρ ) , where L κ,ρ is given by (23). The invariants Ind( P ± F P ± + − P ± ) are called the spin Chernnumbers because for covariant systems there is an index theorem showing that these invariantsare equal to the Chern numbers Ch( P ± ) [22]. They were initially introduced by Sheng et al. [26] for periodic systems with twisted boundary conditions and then by Prodan [20] in the formdiscussed at the end of Section 3.2. As already stressed before, the condition η < g is lessstringent than the one used in [20]. The physical implication of non-vanishing Chern numberson boundary currents is discussed in [23]. That the parity of the spin Chern numbers is equalto the Z -invariant was already proved in [24], but the following argument is more direct. Proposition 7
For η < g , Ind (cid:0) P F P + − P (cid:1) = Ind (cid:0) P + F P + + − P + (cid:1) mod 2 . Proof.
Recall that in the grading of s , H = (cid:18) H + AA ∗ H + (cid:19) , η = 2 k A k . Moreover, P = χ ( H ≤ P + = χ ( H + <
0) and F = ( X + ıX ) | X + ıX | − . By the boundon η , t ∈ [0 , H ( t ) = (cid:18) H + tAtA ∗ H + (cid:19) is a path of invertibles with odd TRS. Hence t ∈ [0 , P ( t ) = χ ( H ( t ) <
0) is continuousand by Proposition 5 in [9] t ∈ [0 , Ind (cid:0) P ( t ) F P ( t ) + − P ( t ) (cid:1) is constant, see also [24]. Thus the claim follows fromInd (cid:0) P (0) F P (0) + − P (0) (cid:1) = Ind (cid:0) P + F P + + − P + (cid:1) mod 2 . P (0) = diag( P + , P + ), one has modulo 2Ind (cid:0) P (0) F P (0) + − P (0) (cid:1) = dim (cid:0) Ker( P + F P + + − P + ) (cid:1) + dim (cid:0) Ker( P + F P + + − P + ) (cid:1) = dim (cid:0) Ker( P + F P + + − P + ) (cid:1) + dim (cid:0) Ker( P + F ∗ P + + − P + ) (cid:1) = Ind (cid:0) P + F P + + − P + (cid:1) , concluding the proof. ✷ ( j, d ) = (4 , The symmetry is the same as in Section 5.5, namely H satisfies s Hs = H . Now approximatespin inversion symmetry means that η = k s Hs + H k . is small. Thus one is in the setting of Section 3.1 and provided that η < g , one has an integervalued invariant given by (18). For η < g and ( κ, ρ ) admissible, it can be calculated via thespectral localizer given in (18). This invariant allows to compute the Z -invariant: Proposition 8
In the above situation,
Ind (cid:0) E ( − P ) E + − E (cid:1) = Ind (cid:0) EA | A | − E + − E (cid:1) mod 2 . Proof.
For H of the form (14), let us consider the path of Hamiltonians t ∈ [0 , H ( t ) = (cid:18) tH + AA ∗ tH − (cid:19) , in the grading of the s i . In particular, one has A = − A T . For η < g this is a path of invertibles,so t ∈ [0 , P ( t ) = χ ( H ( t ) <
0) is continuous. Moreover, and as s H ( t ) s = H ( t ) t ∈ [0 , Ind (cid:0) E ( − P ( t )) E + − E (cid:1) is constant by Proposition 5 in [9], see also [24]. Thus it is sufficient to showInd (cid:0) E ( − P (0)) E + − E (cid:1) = Ind (cid:0) EA | A | − E + − E (cid:1) mod 2 . Let U = A | A | − denote the unitary phase of A , so − P (0) = (cid:0) UU ∗ (cid:1) . Modulo 2 one hasInd (cid:0) E ( − P (0)) E + − E (cid:1) = dim (cid:0) Ker(
EU E + − E ) (cid:1) + dim (cid:0) Ker( EU ∗ E + − E ) (cid:1) = dim (cid:0) Ker(
EU E + − E ) (cid:1) − dim (cid:0) Ker( EU ∗ E + − E ) (cid:1) = Ind (cid:0) EU E + − E (cid:1) , completing the argument. ✷ .7 Approximate conservation law: case ( j, d ) = (5 , As j = 5, one has( ıτ ⊗ ) ∗ H ıτ ⊗ = − H , ( ⊗ ıs ) ∗ H ⊗ ıs = H . (44)Suppose that η = k [ H, τ ⊗ s ] k is small. One way to interprete this is as a particle-hole exchange togehter with a spin inversionalong the 1-direction. Another way to look at it is as an approximate conservation law thatcommutes with the chiral symmetry τ ⊗ s . This case is covered as item (ii) in Section 3.4.Let us bring this into a normal form. The suitable basis change from τ i ⊗ s j to tensor poducts σ i ⊗ ν j of two new sets of Pauli matrices is M = 1 √ −
11 0 0 10 − . This leads to M ∗ τ ⊗ s M = σ ⊗ , M ∗ τ ⊗ s M = ⊗ ν . In this basis one hence has( σ ) ∗ M ∗ HM σ = − M ∗ HM , η = k [ M ∗ HM, ν ] k , So M ∗ HM is in the case of item (ii) in Section 3.4 M ∗ H M = (cid:18) AA ∗ (cid:19) , A = (cid:18) A + BC A − (cid:19) , in the gradings of σ i and ν i respectively. The TRS in (44) leads, due to M ∗ ⊗ ıs M = − ⊗ ıν ,to ( ⊗ ıν ) ∗ M ∗ H M ⊗ ıν = M ∗ H M . This implies ( ν ) ∗ Aν = A and hence A + = A − . Theinteger valued invariant and the corresponding spectral localizer are given in (30). The strong Z -index is given in terms of this invariant: Proposition 9
In the above situation,
Ind (cid:0) EU E + − E (cid:1) = Ind (cid:0) EA + | A + | − E + − E (cid:1) mod 2 . Proof. As η < g t ∈ [0 , A ( t ) = (cid:18) A + tBtC A + (cid:19) , lies in the invertible operators. Thus t ∈ [0 , U ( t ) = A ( t ) | A ( t ) | − is continuous and as( ν ) ∗ U ( t ) ν = U ( t ) for all t ∈ [0 , t ∈ [0 , Ind (cid:0) EU ( t ) E + − E (cid:1) ,
23s constant by Proposition 7 in [9]. The claim follows fromInd (cid:0) EU (0) E + − E (cid:1) = Ind (cid:0) EU + E + − E (cid:1) mod 2 , where U + = A + | A + | − denotes the unitary phase of A + . Modulo 2 one hasInd (cid:0) EU (0) E + − E (cid:1) = dim (cid:0) Ker( EU + E + − E ) (cid:1) + dim (cid:0) Ker( EU + E + − E ) (cid:1) = dim (cid:0) Ker( EU + E + − E ) (cid:1) + dim (cid:0) Ker(( − E ) U + ( − E ) + E ) (cid:1) = dim (cid:0) Ker( EU + E + − E ) (cid:1) − dim (cid:0) Ker( EU ∗ + E + − E ) (cid:1) = Ind (cid:0) EU + E + − E (cid:1) , where the second step follows as Σ ∗ E Σ = − E for some odd symmetry operator Σ commutingwith U + by (37) and the third step follows from Lemma 1 in [9]. ✷ Let us note that there is another way to attain the case ( j, d ) = (5 , z -component of the spin. ( j, d ) = (5 , The symmetry is the same as in Section 5.7, that is, H satisfies (44). Moreover, the BdGHamiltonian H is supposed to have an approximate charge conservation law expressed in termsof η = k [ H, τ ] k . There is a basis change M passing from tensor products τ i ⊗ s j to tensor products σ i ⊗ ν j oftwo new sets of Pauli matrices such that M ∗ τ ⊗ s M = − σ ⊗ , M ∗ τ ⊗ M = σ ⊗ , this leads to ( σ ) ∗ M ∗ H M σ = − M ∗ H M , η = k [ M ∗ H M, σ ] k . Hence one is in the situation of Section 3.3, that is, M ∗ HM = (cid:18) AA ∗ (cid:19) in the grading of σ with η = k A − A ∗ k . As in Section 3.3, let us conjugate by the eigenbasis N of σ and then set b H = N ∗ M ∗ HM N . This leads to( σ ) ∗ b H σ = − b H , η = k [ b H, σ ] k . Hence b H is of the form (26). Moreover the basis change M can be chosen such that M ∗ ( ıτ ⊗ ) M = ıσ ⊗ ν , then the TRS in (44) leads to ( σ ⊗ ν ) ∗ b Hσ ⊗ ν = − b H . As in Section 3.3, the24iagonal entry H + = ( A ∗ + A ) of b H is invertible and for P + = χ ( H + <
0) there is a Z -valuedinvariant Ind( P + F P + + − P + ). For η < g and ( κ, ρ ) admissible, it can be calculated via thespectral localizer given in (27). As for ( j, d ) = (3 , Z -invariant:Ind (cid:0) EU E + − E (cid:1) = Ind (cid:0) P + F P + + − P + (cid:1) mod 2 . ( j, d ) = (6 , As j = 6, the Hamiltonian satisfies ( τ ) ∗ Hτ = − H . Moreover, let us again suppose thatcharge is approximately conserved, namely η = k [ H, τ ] k is small. Thus one is in the settingof Section 3.2. Moreover, the TRS implies H − = − H + . For η < g one has an integer valuedinvariant given by Ind( P + F P + + − P + ). For η < g one can calculate this invariant via thespectral localizer given in (23). As for ( j, d ) = (4 , Z -index isgiven in terms of this invariant:Ind (cid:0) P F P + − P (cid:1) = Ind (cid:0) P + F P + + − P + (cid:1) mod 2 . ( j, d ) = (6 , The symmetry is the same as in Section 5.9, namely H satisfies ( τ ) ∗ Hτ = − H . Now approx-imately spin inversion symmetry means that η = k s Hs + H k is small. Thus one is in the setting of Section 3.1 and provided that η < g , one has an integervalued invariant given by (18). For η < g and ( κ, ρ ) admissible, it can be calculated via thespectral localizer given in (18). As for ( j, d ) = (2 , Z -invariant:Ind (cid:0) E ( − P ) E + − E (cid:1) = Ind (cid:0) EA | A | − E + − E (cid:1) mod 2 . ( j, d ) = (7 , After reducing out the SU(2) invariance of a BdG-Hamiltonian with spin (see [2, 6]) one gets H = h h − ∆ 00 − ∆ − h
0∆ 0 0 − h , with ∆ T = ∆. Thus one has two reduced operators H red = (cid:18) h ± ∆ ± ∆ − h (cid:19) . ıτ ) ∗ H red ıτ = − H red . (45)The odd TRS (due to spin ) is ( ıs ) ∗ Hıs = H . Writing this out leads for the reducedoperators to H red = H red . (46)Summing up, a class CI system is specified by (45) and (46). It is now useful to pass to arepresentation that diagonalizes the associated chiral symmetry ( ıτ ) ∗ H red ıτ = − H red . Withthe Cayley transform τ C defined as in (34) one has τ C ( ıτ ) τ ∗ C = ıτ . Thus τ C H red τ ∗ C = (cid:18) h ∓ ı ∆( h ∓ ı ∆) ∗ (cid:19) , is indeed off-diagonal, and the off-diagonal entry A = h ∓ ı ∆ satisfies A T = A . For d = 5, thisimplies that Ind ( EA | A | − E + − E ) = 0 [9]. Now let us suppose that the Hamiltonian has anapproximate chiral symmetry implemented in an extra sublattice degree of freedom by a Paulimatrix ν , namely η = k ν Hν + H k is small. This case is covered as item (i) in Section 3.4. Hence let the entries of A in the gradingof ν i be given as in (28), namely A = (cid:18) A + BC A − (cid:19) . Then A T = A implies C = B T . The integervalued invariant and the correspondig spectral localizer are given in (29). As for ( j, d ) = (3 , Z -index is given in terms of this invariant:Ind (cid:0) EU E + − E (cid:1) = Ind (cid:0) EB | B | − E + − E (cid:1) mod 2 . ( j, d ) = (7 , As again j = 7, let us work in the framework of Section 5.11. We identify H with CH red C ∗ .Moreover, we assume that there is again a sublattice degree of freedom with associated Paulimatrices ν , ν , ν , and that η = k ( σ ⊗ ν ) ∗ Hσ ⊗ ν + H k = k [ H, σ ⊗ ν ] k = k ν ∗ A ∗ ν + A k is small. Hence one can proceed similarly as in the case ( j, d ) = (3 , N as in Section 5.4. Because N ∗ σ ⊗ N = σ ⊗ , N ∗ σ ⊗ ν N = σ ⊗ , N T σ ⊗ N = σ ⊗ ν , ( M N ) ∗ H ( M N ) satisfies (25). As in Section 3.3, the diagonal entry H + of ( M N ) ∗ H ( M N ) isinvertible and for P + = χ ( H + <
0) there is a Z -valued invariant Ind( P + F P + + − P + ). For η < g and ( κ, ρ ) admissible, it can be calculated via the spectral localizer given in (27). Asfor ( j, d ) = (3 , Z -invariant:Ind (cid:0) EU E + − E (cid:1) = Ind (cid:0) P + F P + + − P + (cid:1) mod 2 . .13 Approximately conserved even spin: case ( j, d ) = (0 , As j = 0, the Hamiltonian fulfills an even TRS which is supposed to be of the form s Hs = H .This may not come directly from spinless particles, but after reducing out other internal degreesof freedom. Then suppose that η = k [ H, s ] k is small. Thus one is in the setting of Section 3.2. Moreover, the TRS implies H − = H + . For η < g , one has an integer valued invariant given by Ind( P + F P + + − P + ). For η < g and( κ, ρ ) admissible, one can calculate this invariant via the spectral localizer given in (23). Asfor ( j, d ) = (4 , Z -index is given in terms of this invariant:Ind (cid:0) P F P + − P (cid:1) = Ind (cid:0) P + F P + + − P + (cid:1) mod 2 . ( j, d ) = (0 , The symmetry is the same as in Section 5.13, namely s Hs = H . Then suppose that η = k s Hs + H k is small. Thus one is in the setting of Section 3.1. Moreover, the TRS implies A = A T . Providedthat η < g , one has an integer valued invariant given by (18). For η < g and ( κ, ρ ) admissible,it can be calculated via the spectral localizer given in (18). As for ( j, d ) = (2 ,
1) one can showthat this invariant allows to compute the Z -invariant:Ind (cid:0) E ( − P ) E + − E (cid:1) = Ind (cid:0) EA | A | − E + − E (cid:1) mod 2 . ( j, d ) = (1 , As j = 1, one has ( τ ⊗ ) ∗ H τ ⊗ = − H , ( ⊗ s ) ∗ H ⊗ s = H . (47)Suppose that η = k [ H, τ ⊗ s ] k is small. Let us bring this into a normal form. The chiral symmetry inherited from (47) is with τ ⊗ s . There is a basis change from τ i ⊗ s j to tensor products σ i ⊗ ν j of two new sets of Paulimatices such that M ∗ τ ⊗ s M = σ ⊗ , M ∗ τ ⊗ s M = − ⊗ ν . In this basis one hence has( σ ) ∗ M ∗ HM σ = − M ∗ HM , η = k [ M ∗ HM, ν ] k . M ∗ HM is in the case of item (ii) in Section 3.4 M ∗ H M = (cid:18) AA ∗ (cid:19) , A = (cid:18) A + BC A − (cid:19) , in the gradings of σ i and ν i respectively. Moreover the basis change M can be chosen such that M ∗ ⊗ s M = ⊗ ν , thus the TRS in (47) leads to ( ⊗ ν ) ∗ M ∗ H M ⊗ ν = M ∗ H M . Thisimplies ( ν ) ∗ Aν = A and hence A + = A − . The integer valued invariant and the correspondigspectral localizer are given in (30). As for ( j, d ) = (3 ,
1) one can show that the strong Z -indexis given in terms of this invariant:Ind (cid:0) EU E + − E (cid:1) = Ind (cid:0) EA + | A + | − E + − E (cid:1) mod 2 . ( j, d ) = (1 , One again has an even spin, but here we have to suppose that the spin is not
0. For sakeof concreteness, let us suppose that the spin is 1 so that a three-dimensional representation s , s , s of su(2) is relevant which we choose (as in the appendix of [6]) to be s = − , e ıπs = − − − . Let H satisfy ( τ ⊗ ) ∗ H τ ⊗ = − H , ( ⊗ e ıπs ) ∗ H ⊗ e ıπs = H . (48)Suppose that η = k [ H, τ ⊗ e ıπs ] k , is small. The chiral symmetry is with τ ⊗ e ıπs . There is a basis change M such that M ∗ τ ⊗ e ıπs M = − σ ⊗ , M ∗ τ ⊗ e ıπs M ∗ = σ ⊗ . This leads to ( σ ) ∗ M ∗ H M σ = M ∗ H M , η = k [ M ∗ H M, σ ] k . Hence one is in the situation of Section 3.3, and thus, in the grading of σ , M ∗ HM = (cid:18) AA ∗ (cid:19) . As in Section 3.3, let us conjugated by the eigenbasis N of σ and then set b H = N ∗ M ∗ HM N .This leads to ( σ ) ∗ b H σ = − b H , η = k [ b H, σ ] k . Hence b H is of the form (25). One can chose M such that M ∗ ⊗ e ıπs M = ⊗ e ıπs , so theTRS in (48) leads to ( ⊗ e ıπs ) ∗ b H ⊗ e ıπs = − b H . As in Section 3.3, H + = ( A ∗ + A ) isinvertible and for P + = χ ( H + <
0) there is a Z -valued invariant Ind( P + F P + + − P + ). For η < g and ( κ, ρ ) admissible, it can be calculated via the spectral localizer given in (27). Asfor ( j, d ) = (3 ,
2) one can show that this invariant allows to compute the Z -invariant:Ind (cid:0) EU E + − E (cid:1) = Ind (cid:0) P + F P + + − P + (cid:1) mod 2 . .17 Approximately conserved charge: case ( j, d ) = (2 , As j = 6 the Hamiltonian satisfies τ Hτ = − H . Moreover charge is approximately conserved,namely η = k [ H, τ ] k is supposed to be small. This is exactly as in the zero-dimensional casediscussed in Section 5.1. Thus one is in the setting of Section 3.2. Moreover the TRS implies H − = − H + . For η < g one has an integer valued invariant given by Ind( P + F P + + − P + ).For η < g one can calculate this invariant via the spectral localizer given in (23). As for( j, d ) = (4 , Z -index is given in terms of this invariant:Ind (cid:0) P F P + − P (cid:1) = Ind (cid:0) P + F P + + − P + (cid:1) mod 2 . A Proofs of properties of the twisted spectral localizers
Proof of Proposition 1.
Let us begin by verifying that t ∈ [0 , L κ,ρ ( t ) = (cid:18) κ σ D H ( t ) H ( t ) κ σ D (cid:19) ρ with H ( t ) = (cid:18) tH + AA ∗ tH − (cid:19) is a path of invertibles. By the bound on η , one has k H − H (0) k < g . Thus g (0) = k H (0) − k − satisfies 2 g < g (0) < g . Therefore (16) implies κ ≤ g k H kk [ D, A ] k < g (0)) k H (0) kk [ D, A ] k = g (0) k H (0) kk [ D, A ] k . By (17) ρ > g κ > g (0) κ . As L κ,ρ (0) is unitary equivalent to b L κ,ρ (0) = (cid:18) κD AA ∗ − κD (cid:19) ρ ⊕ (cid:18) − κD A ∗ A κD (cid:19) ρ , Theorem 1 implies that L κ,ρ (0) is invertible and L κ,ρ (0) ≥ g (0) ≥ ( g ) g . Therefore k L κ,ρ (0) − k − ≥ g . As k L κ,ρ (0) − L κ,ρ ( t ) k < g by the bound on η , this implies that L κ,ρ ( t ) is invertible for all t ∈ [0 , L κ,ρ ) = 12 Sig ( L κ,ρ (0)) = 2 Ind( EA | A | − E + − E ) , ✷ Proof of Proposition 2.
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