Parity flow as Z 2 -valued spectral flow
aa r X i v : . [ m a t h - ph ] J u l Parity as Z -valued spectral flow Nora Doll , Hermann Schulz-Baldes and Nils Waterstraat Department Mathematik, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg,Cauerstr. 11, 91058 Erlangen, Germany Institut f¨ur Mathematik, Martin-Luther-Universit¨at Halle-Wittenberg,Theodor-Lieser-Str. 5, 06120 Halle, Germany
Abstract
This note is about the topology of the path space of linear Fredholm operators on areal Hilbert space. Fitzpatrick and Pejsachowicz introduced the parity of such a path,based on the Leray-Schauder degree of a path of parametrices. Here an alternative ana-lytic approach is presented which reduces the parity to the Z -valued spectral flow of anassociated path of chiral skew-adjoints. Furthermore the related notion of Z -index of aFredholm pair of chiral complex structures is introduced and connected to the parity of asuitable path. Several non-trivial examples are provided. One of them concerns topologi-cal insulators, another an application to the bifurcation of a non-linear partial differentialequation. MSC2010: 47A53, 58J30 The spectral flow for paths of self-adjoint Fredholm operators on a complex Hilbert space isa well-known homotopy invariant [1, 19, 11, 17]. It plays a role in numerous other fields, e.g. index theory [1, 20, 6, 8] and bifurcation theory [11, 13]. For R -linear operators on a realHilbert space H R , spectral flow is still a well-defined and useful object. Moreover, for paths[0 , ∋ t B t of arbitrary (not necessarily self-adjoint) Fredholm operators on H R a Z -valuedparity σ has been introduced by Fitzpatrick and Pejsachowicz [11], and for paths [0 , ∋ t T t of skew-adjoint real Fredholm operators a Z -valued spectral flow Sf has also been studied [7].This note presents the parity of a path [0 , ∋ t B t of real Fredholm operators as the Z -valued spectral flow of an associated path of chiral skew-adjoint Fredholm operators on H R ⊕ H R : σ (cid:0) [0 , ∋ t B t (cid:1) = Sf (cid:18) [0 , ∋ t (cid:18) B t − B ∗ t (cid:19)(cid:19) . (1)This provides a new perspective on parity and also allows to deduce its main properties directlyfrom known facts on the Z -valued spectral flow. We also believe that the presented approach1akes the parity more accessible for computations. A new result for the parity is an indexformula for paths between conjugate Fredholm pairs of complex structures, see Section 6. Thiscorresponds to analogous results for the spectral flow between conjugate Fredholm pairs ofprojections [20] as well as the Z -valued spectral flow [7].To further stress the similarities between spectral flow, Z -valued spectral flow and parity,let us consider the classifying spaces for real K -theory as introduced by Atiyah and Singer [2].Let F k = F k ( H R ) denote the space of skew-adjoint Fredholm operators on a real separableHilbert space H R which anticommute with representations I , . . . , I k − of the generators of areal Clifford algebra of signature (0 , k −
1) [2]. By reducing out these relations in a concreterepresentation, it is possible (but tedious) to identify each F k with a set of Fredholm operatorson H R having certain supplementary symmetry relations. Relevant for the following is that F ∼ = F is isomorphic to the set of all Fredholm operators on H R , F is isomorphic to theset of skew-adjoint Fredholm operators while F is isomorphic to the self-adjoint Fredholmoperators on H R with positive and negative essential spectrum. Furthermore, F is isomorphicto the set of those elements of F that are linear over the quaternions. Atiyah and Singer [2]found that the homotopy groups of these spaces satisfy π j ( F i ) = π ( F i + j ) = π j + i ( F ) , and are given explicitly by i π ( F i ) Z Z Z Z π ( F i ) Z Z Z Z (2)The components π ( F i ) in the second row are labelled by the index (for i = 0 ,
4) and the Z -index is given by the nullity modulo 2 (for i = 1 , π ( F ) to Z , and also from π ( F ) to 2 Z . Here the factor 2 merely stressesthat eigenvalues of self-adjoint quaternionic operators are always of even multiplicity. Moreprecisely, if a self-adjoint quaternionic matrix is written as a complex matrix of double size,then this complex matrix has a symmetry leading to even dimensional eigenspaces (just liketime reversal for fermions with half-integer spin leads to Kramers’ degeneracy). Therefore alsothe spectral flow along paths of quaternionic operators is even. Furthermore the parity givesthe isomorphism π ( F ) ∼ = Z [13] and the Z -valued spectral flow provides the isomorphism π ( F ) ∼ = Z [7]. Hence the spectral flow, parity and Z -valued spectral flow allow to detect thetopology in the last row of (2). Furthermore, in view of table (2), one does not expect there tobe any other flow of interest. Let us also note that (1) results from realizing F as those elementsof F that anticommute with the representation J of the generator of a real Clifford algebra ofsignature (1 , J = diag( , − ) in its spectral representation and elements T ∈ F with the so-called chiral symmetry J T J = − T are off-diagonal as on the right hand-side of(1). This reduction is in the opposite direction than the one considered in [2]. Moreover,chiral skew-adjoints often also appear in different guise in applications. An example are chiralself-adjoints, see Section 4, which are naturally associated to chiral topological insulators, see2ection 8. Finally, let us stress that while table (2) only concerns closed loops, the definitionof spectral flow, Z -valued spectral flow and parity apply to arbitrary (open) paths.In Section 5 a Z -index of a Fredholm pair of chiral complex structures is introduced. Thisis the parity version of Kato’s index of a Fredholm pair of projections [15] as further studied byAvron, Seiler and Simon [3]. This is closely tied to the parity, as explained in Section 5 and ofparticular interest and importance for Fredholm pairs given by unitary conjugates. This leadsto an index formula proved in Section 6. Finally Sections 8 and 9 give two applications of theparity. The characterizing features of the parity can best be understood in finite dimension. Hencelet us consider a (continuous) path [0 , ∋ t B t of real N × N matrices acting on the realHilbert space H R = R N . Furthermore, let the path be admissible in the sense that its endpoints B and B are invertible, namely are in the general linear group Gl( N, R ). This group has twocomponents, specified by either a positive or a negative determinant. The parity of the path[0 , ∋ t B t is simply 1 if the endpoints are in the same component and − Definition 1
For an admissible path [0 , ∋ t B t of real N × N matrices, the parity isdefined as σ ([0 , ∋ t B t ) = sgn(det( B )) sgn(det( B )) ∈ Z , (3) where Z is viewed as the multiplicative group Z = {− , } . As this only depends on theendpoints, we will also simply write σ ( B , B ) . After rescaling, all of this also applies to paths [ a, b ] ∋ t B t with arbitrary endpoints a < b . The definition directly implies that the parity σ of admissible paths of real matrices isa homotopy invariant (under homotopies of the path keeping the endpoints fixed), it has aconcatenation property and it is normalized in the sense that the parity of a path in theinvertibles is 1. Furthermore, one has a multiplicativity property under direct sums, namely foranother admissible path [0 , ∋ t B ′ t of real L × L matrices, the definition directly impliesthat σ ([0 , ∋ t B t ⊕ B ′ t ) = σ ([0 , ∋ t B t ) · σ ([0 , ∋ t B ′ t ) , with multiplication in Z .For the generalization to infinite dimension there are several possibilities [17]. The routetaken by Fitzpatrick and Pejsachowicz [13] uses the fact that sgn(det( B )) can, under suitableconditions, be extended to infinite dimensions as the Leray-Schauder degree, for details seeSection 3 below. In this note we elaborate on another possibility which consists in first rewritingDefinition 1 in terms of skew-adjoint matrices on a doubled Hilbert space, just as suggestedby Atiyah and Singer [2]. This has the advantage that tools from the spectral analysis ofskew-adjoint operators can be used and the connection to the Z -valued spectral flow from37] is uncovered. Hence let us use the real Hilbert space H ′ R = H R ⊕ H R equipped with the Z -grading J = diag( , − ). Set: T t = (cid:18) B t − B ∗ t (cid:19) . (4)These operators have a so-called chiral symmetry: J T t J = − T t . (5)Conversely, if one has a real Hilbert space H ′ R equipped with the Z -grading given by a self-adjoint unitary J = J ∗ = J − and a path [0 , ∋ t T t of real chiral skew-adjoints, thengoing to the spectral representation of J in which J = diag( , − ) leads to the representationof T t in the form (4). Hence (4) provides a bijection between the set of paths of operatorson H R and the set of paths of chiral skew-adjoints on H ′ R . The chiral symmetry (5) impliesthat the spectrum always satisfies spec( T t ) = − spec( T t ) ⊂ ı R . A non-trivial topology in thepath is detected by the Z -valued spectral flow [7], the definition of which we recall next. Forthis purpose, let us note that the endpoints T and T are invertible (because the initial pathwas admissible) and therefore there exists an invertible A such that T = A ∗ T A . Then, bydefinition [7], Sf ([0 , ∋ t T t ) = sgn(det( A )) ∈ Z . (6)As the definition of Sf ([0 , ∋ t T t ) only depends on the endpoints we will also writeSf ( T , T ). For T and T in the form (4) one has T = A ∗ T A for A = diag(( B ∗ ) − B ∗ , ). Thisdirectly implies σ ([0 , ∋ t B t ) = Sf ([0 , ∋ t T t ) (7)whenever the identification (4) holds. This explains why (1) holds in finite dimension. The Z -valued spectral flow given by (6) has an invariance property under conjugation, namely if[0 , ∋ t O t is a path of orthogonals commuting with J , thenSf ([0 , ∋ t O t T t O ∗ t ) = Sf ([0 , ∋ t T t ) . This holds because O T O ∗ = O A ∗ O ∗ ( O T O ∗ ) O AO ∗ and det( O AO ∗ ) = det( A ) since O and O are in the same component of the orthogonal group. This transposes to an invarianceproperty for the parity. Similarly, other properties of parity result from properties of the Z -valued spectral flow.Let us next provide some examples that illustrate the topological stability associated to theparity. For N = 1 we first consider two paths T t = (cid:18) t − t (cid:19) , e T t = (cid:18) | t |−| t | (cid:19) , t ∈ [ − , . (8)Clearly these two paths are isospectral spec( T t ) = spec( e T t ) for all t ∈ [ − , ([ − , ∋ t T t ) = − ([ − , ∋ t e T t ) = 1. This has spectral consequences.4he latter can be perturbed to e T t ( s ) (within the class of real chiral skew-adjoints) in such away that 0 is not an eigenvalue for any t : e T t ( s ) = (cid:18) | t | + s − ( | t | + s ) 0 (cid:19) . Indeed, the eigenvalues are then ± ı ( | t | + s ) which both never vanish for positive s >
0. It isnot possible to construct such a perturbation for T t , namely any real skew-adjoint perturbationconserving the chiral symmetry can merely shift the eigenvalue crossing at 0.Furthermore, let us double the non-trivial example in (8) via a direct sum to T ′ t = T t ⊕ T t which is chiral with respect to J ⊕ J = diag(1 , − , , − T ′ t is block diagonal ratherthan in the off-diagonal form (4). However, using the permutation U of the second and thirdcomponent one obtains the spectral representation U ( J ⊕ J ) U ∗ = diag( , − ) and then U T ′ t U ∗ is of the form (4) with off-diagonal entry B ′ t = diag( t, t ). Then by the multiplicativity of the Z -valued spectral flow, Sf ([ − , ∋ t T ′ t ) = ( − −
1) = 1. Again it is then possibleto lift the kernel along the whole path by a real chiral skew-adjoint perturbation. One suchperturbation is
U T ′ t ( s ) U ∗ = t − s s t − t − s s − t . Indeed, the spectrum of T ′ t ( s ) is { ı ( t + s ) , − ı ( t + s ) } with a double degeneracy. Inparticular, for s = 0, T ′ t ( s ) is invertible for all t ∈ [ − , In this section, the separable real Hilbert space H R is now of infinite dimension and the contin-uous path [0 , ∋ t B t ∈ F is within the Fredholm operators. For the sake of simplicity, letus first suppose that it lies in the component of Fredholm operators with vanishing index. (Ina large part of the literature these are called Fredholm indices even though it was actually F.Noether who first exhibited a Fredholm operator with non-vanishing index [10].) The generalcase will then be dealt with towards the end of the section. In [11, 13], the parity of an admissi-ble path (namely with invertible endpoints) uses the Leray-Schauder degree which is defined asfollows. One first proves that there exists a second path of real invertibles [0 , ∋ t M t suchthat M t B t = + K t with a real compact operator K t . Then, if n t denotes the number of nega-tive eigenvalues of + K t counted with multiplicity, n t coincides with the number of eigenvaluesless than − K t and is therefore finite. The (linear) Leray-Schauderdegree is deg ( B t ) = ( − n t ∈ Z . (9)Let us explain how this fits together with Definition 1. If B t is a matrix, one can choose M t = ;the spectrum of B t is symmetric with respect to the reflection on the real axis; now non-realeigenvalues of B t come in complex conjugate pairs which do not contribute to sgn(det( B t ));5ence analyzing the real eigenvalues immediately leads to deg ( B t ) = sgn(det( B t )). For thepath, the parity is then as in Definition 1 given by σ ([0 , ∋ t B t ) = deg ( B )deg ( B ) ∈ Z [13]. One of the difficulties with this approach is that, in general, it is very hard to determinethe path M t and therefore also the parity by this procedure.This work provides an alternative approach in which the parity is defined as the Z -valuedspectral flow studied in [7] of a path of skew-adjoint operators on the doubled Hilbert space.This is based on the passage (4) to chiral skew-adjoint operators. Hence let H ′ R = H R ⊕ H R be a real Hilbert space equipped with the Z -grading J = diag( , − ). Then (4) identifies F = F ( H R ) with ˆ F = { T ∈ B ( H ′ R ) : T = − T ∗ = − J T J
Fredholm } . Hence ˆ F is a subspace of F = { T ∈ B ( H ′ R ) : T = − T ∗ Fredholm } , and any path in F can be viewed as a path [0 , ∋ t T t ∈ ˆ F in F . This path has a supplementary chiralsymmetry J T t J = − T t , but this is irrelevant for the definition of its Z -valued spectral flow thatwe review next. Hence let now [0 , ∋ t T t ∈ F . As already mentioned, we will first dealwith an admissible path with invertible endpoints T and T . Roughly, the idea is to reduce thedefinition of the Z -valued spectral flow to the finite dimensional definition by extracting from T t ∈ F only the finite-dimensional subspace corresponding to eigenvalues in a small intervalaround 0, just as in [19]. Thus, for a > Q a ( t ) = χ ( − a,a ) ( ı T t ) , where χ I denotes the characteristic function on I ⊂ R . The projection Q a ( t ) is of finitedimensional range for a sufficiently small by the Fredholm property of T t . Associated to theseprojections, one has the restrictions Q a ( t ) T t Q a ( t ) which are viewed as skew-adjoint matriceson E a ( t ) = Ran( Q a ( t )). By compactness (see the first Lemma in [19]), it is possible to choosea finite partition 0 = t < t < . . . < t N − < t N = 1 of [0 ,
1] and a n > n = 1 , . . . , N , suchthat each piece [ t n − , t n ] ∋ t Q a n ( t ) is continuous and hence of constant finite rank, and,moreover, for some ǫ , k Q a n ( t ) − Q a n ( t ′ ) k < ǫ , ∀ t, t ′ ∈ [ t n − , t n ] . (10)Let V n : E a n ( t n − ) → E a n ( t n ) be the orthogonal projection of E a n ( t n − ) onto E a n ( t n ), namely V n v = Q a n ( t n ) v . Then V n is a bijection allowing to identify E a n ( t n − ) with E a n ( t n ). Noweach interval [ t n − , t n ] leads to a path [ t n − , t n ] ∋ t Q a n ( t ) T t Q a n ( t ) of chiral, skew-adjointmatrices on E a n ( t ) = Ran( Q a n ( t )), but this path may not be admissible. To lift the (even-dimensional) kernel at the endpoint t n , one can add a skew-adjoint perturbation R n on thekernel of Q a n ( t ) T t n Q a n ( t n ) so that T ( a n ) t n = Q a n ( t n ) T t n Q a n ( t n ) + R n , (11)are skew-adjoint invertible operators on E a ( t ). Clearly the choice of the R n is largely arbitrary,but it is part of Theorem 1 below that the following definition is independent of the choice ofthe R n . 6 efinition 2 For an admissable path [0 , ∋ t T t ∈ F , let t n and a n as well as T ( a ) t and V n be as above. Then the Z -valued spectral flow is defined by Sf ([0 , ∋ t T t ) = Y n =1 ,...,N Sf (cid:0) T ( a n ) t n − , V ∗ n T ( a n ) t n V n (cid:1) , (12) where on the right hand side the Sf is the finite dimensional Z -valued spectral flow on E a n ( t n − ) as given in (6) , and the product is in the multiplicative group ( Z , · ) . Let us stress that in infinite dimension, it is in general not possible to write Sf ( T , T ) forSf ([0 , ∋ t T t ) because the Z -valued spectral flow depends on the choice of the path. Thebasic result on the Z -valued spectral flow is that it is well-defined by the above procedure. Theorem 1 (Theorem 4.2 in [7])
Let [0 , ∋ t T t ∈ F be an admissible path. Thedefinition of Sf ([0 , ∋ t T t ) is independent of the choice of the partition t < t <. . . < t N − < t N = 1 of [0 , and the values a n > such that [ t n − , t n ] ∋ t Q a n ( t ) iscontinuous and satisfies (10) , and also the choice of the R n in (11) . As ˆ F ⊂ F , one can now use the Z -valued spectral flow to define the parity. Definition 3
Let [0 , ∋ t B t ∈ F be an admissible path and [0 , ∋ t T t ∈ ˆ F be thepath associated by (4) . Then the parity is defined by σ ([0 , ∋ t B t ) = Sf ([0 , ∋ t T t ) ∈ Z . (13)Let us stress again that T t ∈ ˆ F implies the chiral symmetry J T t J = − T t , but this is not ofimportance for the definition of the Z -valued spectral flow on the right-hand side of (13). Onecan, however, make more specific choices in the construction of Sf above, notably the spectralprojections satisfy due to the symmetry of [ − a, a ] Q a ( t ) = J Q a ( t ) J , and one can choose the skew-adjoint perturbations R n to be chiral. Due to Definition 3, theparity inherits from the Z -valued spectral flow all of the properties stated in [7]. They arecollected in the following result. Most of these properties are already stated in Chapter 6 of[13]. Theorem 2
Let [0 , ∋ t B t ∈ F be an admissible path. (i) The parity is homotopy invariant under homotopies in the paths of Fredholm operatorskeeping the endpoints fixed. (ii) If B t is invertible for all t ∈ [0 , , then σ ([0 , ∋ t B t ) = 1 . (iii) The parity has a concatenation property, namely if [0 , ∋ t B t ∈ F is a path suchthat B is invertible, then σ ([0 , ∋ t B t ) · σ ([1 , ∋ t B t ) = σ ([0 , ∋ t B t ) . The parity is independent of the orientation of the path: σ ([0 , ∋ t B t ) = σ ([0 , ∋ t B − t ) . (v) The parity has a multiplicativity property under direct sums, namely if [0 , ∋ t B ′ t ∈F is a second admissible path, σ ([0 , ∋ t B t ⊕ B ′ t ) = σ ([0 , ∋ t B t ) · σ ([0 , ∋ t B ′ t ) . (vi) The parity is invariant under the conjugation by a path [0 , ∋ t O t of orthogonals: σ ([0 , ∋ t O t B t O ∗ t ) = σ ([0 , ∋ t B t ) . In particular, the parity is independent under reflection of the path: σ ([0 , ∋ t B t ) = σ ([0 , ∋ t
7→ − B t ) . The following result is already stated in [13].
Theorem 3
The map σ on loops in F is a homotopy invariant and induces an isomorphismof π ( F ) with Z . Proof. As π ( F ) ∼ = Z is already known [2] and σ is homotopy invariant, one only has tocheck that σ takes two different values on the two different components of the based loop spacein F . For constant paths (and thus all contractible ones) the parity vanishes. An examplewith a parity equal to − ✷ Up to now, only paths [0 , ∋ t B t in the component of F with vanishing index wereconsidered. For general paths, one hasdim (cid:0) Ker( T t ) (cid:1) − (cid:12)(cid:12) Ind( B t ) (cid:12)(cid:12) ∈ N . In particular, for non-vanishing Ind( B t ) the dimension of the kernel of T t is positive for all t andthus there are no admissible paths. However, there are several possibilities to reduce this case tothe prior one. For that purpose, let us now call a path admissible if dim (cid:0) Ker( T i ) (cid:1) = (cid:12)(cid:12) Ind( B i ) (cid:12)(cid:12) for i = 0 ,
1. Recall that Ker( T t ) is J -invariant. Let now [0 , ∋ t P t be a continuous path of J -invariant orthogonal projections onto parts of the kernel of T t , and being of the dimension ofKer( T i ) for i = 0 ,
1. Then follow the constructions and arguments from above for T t restrictedto the range of − P t . This construction is independent of the choice of [0 , ∋ t P t for,if [0 , ∋ t P ′ t is another projection with the above properties, then P ′ t = O t P t O ∗ t for apath [0 , ∋ t O t of orthogonals commuting with J . Hence it follows from property (vi) ofTheorem 2 that one obtains the same parity. All properties of Theorem 2 transpose directly,except for (ii) which now states that paths with constant nullity have a parity equal to 1.8 Reformulation with chiral self-adjoints
Given an admissible path [0 , ∋ t B t of Fredholm operators on H R , it is possible to associateself-adjoint real operators on H ′ R = H R ⊕ H R via H t = (cid:18) B t B ∗ t (cid:19) . (14)This identifies F with the set ˜ F of chiral self-adjoint Fredholm operators˜ F = { H ∈ B ( H ′ R ) : H = H ∗ = − J HJ
Fredholm } . A bijection between ˆ F and ˜ F is given byˆ F = ı J ˜ F ( J ) ∗ , where J = diag( , ı ) is the square root of J . In some applications (as in Section 8) onerather finds admissible paths [0 , ∋ t H t ∈ ˜ F of chiral self-adjoint real operators. Suchpaths then have a parity given by σ ([0 , ∋ t H t ) = Sf (cid:0) [0 , ∋ t ı J H t ( J ) ∗ (cid:1) . A further modification concerns a setting with complex Hilbert spaces and a reality conditioninvolving another symmetry. Suppose thus that one has a complex Hilbert space H C with areal structure given by a (anti-linear involutive) complex conjugation C : H C → H C , naturallyextended to H ′ C = H C ⊕ H C . For any linear operator A on H C or H ′ C let us set A = C A C .Further suppose given a real self-adjoint involution K on H ′ C , namely K = K ∗ = K and K = which, moreover, commutes with J . Then an operator A is called K -real if K ∗ AK = A . Nowone considers admissible paths [0 , ∋ t H t of K -real self-adjoint chiral operators, namely K ∗ H t K = H t , H ∗ t = H t , J ∗ H t J = − H t . Also for such paths one can define the parity. Indeed, let L be the root of K with spectrum { , ı } . It commutes with J . Then set b H t = L ∗ H t L .
It can be checked that b H t is real, self-adjoint and chiral with respect to J . Consequently, b H t can be restricted to an R -linear operator on H ′ R = Ker( C − ) ⊂ H ′ C . Thus it is within theclass of paths considered above and the parity of [0 , ∋ t H t can be defined as that of[0 , ∋ t b H t . The aim of this section is to construct an alternative formula for the parity. This will first bedone for special paths between complex structures that are close in the Calkin algebra, then9ater on it will also be extended to general paths. Recall that a complex structure I on H ′ R is alinear, skew-adjoint and unitary operator on H ′ R . It is called chiral if, moreover, J IJ = − I fora symmetry J , namely J = J ∗ = J − . Hence any chiral complex structure is an element of ˆ F .The following definition is motivated by [5, 7], as well as Kato’s Fredholm pair of projectionsand its index [15, 3]. Definition 4
A pair ( I , I ) of chiral complex structures on H ′ R is called a Fredholm pair ofchiral complex structures if k π ( I − I ) k Q < . The Z -index of ( I , I ) is then defined by Ind ( I , I ) = (cid:16) dim R (Ker R ( I + I )) (cid:17) mod 2 ∈ Z . (15)The index is indeed well-defined because I + I = 2 I + ( I − I ) has no essential spectrumat 0 and it will be shown in the proof of Theorem 4 that the kernel of I + I is even dimensional.On the right hand side of (15) the additive version of Z was used and is tacitly identified withthe multiplicative one. The following justifies Definition 4. Theorem 4
The map ( I , I ) Ind ( I , I ) ∈ Z is a homotopy invariant on the set of Fred-holm pairs of chiral symmetries. Moreover, for both signs one has Ind ( I , I ) = dim R (Ker R ( I − I ± ı )) mod 2 . (16)Before going into the proof, let us elaborate on the connection to the index of a Fredholm pairof projections [3]. Here there are two projections P = ( ıI + ) and P = ( ıI + ) associatedto the complex structures. The property k π ( I − I ) k Q < k π ( P − P ) k Q < P , P ) being a Fredholm pair. Furthermore, these two projections satisfy P j = − P j and J P j J = P j . In the terminology of [14] this means that the P j are even real and evenLagrangian projections. These symmetries imply that the index of the Fredholm pair ( P , P )vanishes, namely the two signs on the right hand side of (16) lead to the same dimension(compare with eq. (3.1) in [3]). Hence one sees that the Z -index Ind ( I , I ) is a secondaryinvariant associated to the Fredholm pair ( P , P ) which is well-defined due to Theorem 4.The proof of Theorem 4 will be based on the following lemma in which the chiral symmetryand reality are irrelevant. The lemma can be traced back to [5] and is stated as Lemma 5.3in [7]. An equivalent algebraic fact has also been used for pairs of orthogonal projections [3,Theorem 2.1]. Lemma 1
Let I and I be complex structures. Set T = ( I + I ) , T = ( I − I ) . Then the following identities hold: T ∗ T + T ∗ T = = T T ∗ + T T ∗ , T ∗ T + T ∗ T = 0 = T T ∗ + T T ∗ , as well as T I = I T , T I = I T , T I = − I T , T I = − I T . roof. Everything is verified by straightforward computations. ✷ Proof of Theorem 4. First of all, as noted above T = I + ( I − I ) is a skew-adjointFredholm operator by the assumption that ( I , I ) is a Fredholm pair so that there is onlydiscrete spectrum in a neighborhood of 0. The idea of the proof is to show that every (small)eigenvalue of T of finite multiplicity has even multiplicity. As T is chiral and its spectrumsatisfies spec( T ) = − spec( T ), this then implies that the nullity of T only changes by multiplesof 4 under homotopic changes of T (induced by a homotopy of I and I ). The main tool isto view I as a complex structure on H ′ R . Now ( T ∗ T ) I = − T I T = I ( T ∗ T ) so that T ∗ T is a complex linear operator on H ′ R viewed as complex Hilbert space (using the complexstructure I ). Consequently the real multiplicity of all eigenvalues of T ∗ T is even. In particular,Ind ( I , I ) given by (15) indeed takes values in { , } . Next let us show that for λ ∈ (0 , T ∗ T is a multiple of 2 (then the real multiplicity is amultiple of 4). Suppose that T ∗ T v = λv for some non-vanishing vector v . Then set w = T ∗ T v .First of all, its norm does not vanish: k w k = v ∗ T ∗ T T ∗ T v = v ∗ T ∗ ( − T T ∗ ) T v = λ (1 − λ ) k v k . It is also an eigenvector of T ∗ T : T ∗ T w = T ∗ T T ∗ T v = − T ∗ T T ∗ T v = T ∗ T T ∗ T v = λw . Moreover, it is complex linearly independent of v . In fact, suppose the contrary, namely that w = ( µ + µ I ) v for some µ , µ ∈ R . Multiplying this with T ∗ T leads to λ (1 − λ ) v = T ∗ T T ∗ T v = T ∗ T w = T ∗ T ( µ + µ I ) v = − ( µ − µ I ) w , where in the last equality the identity T ∗ T I = − I T ∗ T was used. Multiplying now by( µ + µ I ) shows λ (1 − λ ) w = − ( µ + µ ) w , that is, a contradiction. If there are further eigenvectors of T ∗ T with eigenvalue λ , one canrestrict to the orthogonal complement and iterate the above argument.As to the alternative formula for Ind ( I , I ), let us note that the kernel of T ∗ T coincideswith the eigenspace of T ∗ T to the eigenvalue 1, which in turn is given by the direct sum ofthe eigenspaces of T for the eigenvalues ı and − ı . This proves the formula. Let us commentthat another proof of the homotopy invariance uses the chiral symmetry of T and checks thedouble degeneracy of all eigenvalues of T in (0 , ✷ The following result establishes the link of Ind ( I , I ) with the Z -valued spectral flow ofthe straight line connecting I and I , which indeed lies in ˆ F ⊂ F . Proposition 1
For any Fredholm pair of chiral complex structures ( I , I ) on H ′ R , one has Sf (cid:0) [0 , ∋ t (1 − t ) I + t I (cid:1) = Ind ( I , I ) . roof. This is essentially identical to the argument leading to Proposition 6.2 in [7], so let usjust give a sketch. The operators T t = (1 − t ) I + tI are indeed Fredholm because, for t ∈ [0 , ], T t = I + t ( I − I ) is a perturbation of an operator I with spectrum {− ı, ı } by an operatorwith bound 1 in the Calkin algebra so that T t has its essential spectrum bounded away from0. For t ∈ [ , T t = I + (1 − t )( I − I ). Moreover, T t is invertible except possibly at t = . Hence in Definition 2 it is sufficient to work with threeintervals [0 , − ǫ ], [ − ǫ, + ǫ ] and [ + ǫ,
1] for some ǫ >
0. Only the middle interval has apossibly non-vanishing contribution coming from the parity of the nullity of T = ( I + I ).But this is precisely the definition (15) of the Z -index. ✷ Further following [20] or [7], one can go on and rewrite the definition of the parity.
Proposition 2
Let [0 , ∋ t B t ∈ F be an admissible path and associated T t ∈ ˆ F to B t by (4) . Let I t be chiral complex structures obtained by completing the phase T t | T t | − on the kernel.Then, for a sufficiently fine partition t < t · · · < t N = 1 satisfying k π ( I n − I n − ) k < ,one has for the parity σ ([0 , ∋ t B t ) = (cid:16) X n =1 ,...,N Ind ( I t n − , I t n ) (cid:17) mod 2 . Proof.
Let us begin by rewriting Definition 2. One can choose R n in (11) sufficiently smalland the partition t = 0 < t < · · · < t N = 1 sufficient fine such that a n from Definition 2 isnot in the spectrum of (1 − t ) ı ( T t n − + R n − ) + t ı ( T t n + R n ) for any t ∈ [0 , (cid:0) T ( a n ) t n − , V ∗ n T ( a n ) t n V n (cid:1) = Sf (cid:0) [0 , ∋ t (1 − t ) ( T t n − + R n − ) + t ( T t n + R n ) (cid:1) and Sf ([0 , ∋ t T t ) = Y n =1 ,...,N Sf (cid:0) [0 , ∋ t (1 − t ) ( T t n − + R n − ) + t ( T t n + R n ) (cid:1) . By identifying T t n + R n with T t n , we can from now on assume that T t n is invertible. Next letus claim that for each n = 1 , . . . , N one hasSf (cid:0) [0 , ∋ t (1 − t ) T t n − + t T t n (cid:1) = Sf (cid:0) [0 , ∋ t (1 − t ) I t n − + t I t n (cid:1) . Indeed, as T t n and T t n − are both invertible,[0 , ∋ s (1 − t ) T t n − | T t n − | − s + t T t n | T t n | − s deforms the initial path into the path [0 , ∋ t (1 − t ) I t n − + t I t n . During this homotopythe endpoints remain invertible so that the Z -valued spectral flow is unchanged. Now theassertion follows from Proposition 1. ✷ Parity of paths between unitary conjugates
Let H ′ R = H R ⊕ H R be equipped with the Z -grading J = diag( , − ). The orthogonal grouppreserving J is O ( H ′ R , J ) = { O ∈ O ( H ′ R ) : O ∗ J O = J } . This is a subgroup of O ( H ′ R ) naturally identified with O ( H R ) × O ( H R ) because O ∗ J O = J isequivalent to J OJ = O which requires O to be diagonal in the grading of J . For any real chiralcomplex structure I , let us set O I ( H ′ R , J ) = { O ∈ O ( H ′ R , J ) : [ O, I ] ∈ K ( H ′ R ) } , (17)where K ( H ′ R ) denotes the compact operators on H ′ R . This is a subgroup of O ( H ′ R , J ). Let usnote that for O ∈ O I ( H ′ R , J ) one has π ( O ∗ IO ) = π ( I ) in the Calkin algebra. Furthermore,recall the definition of the based loop space Ω I ˆ F of ˆ F based at I :Ω I ˆ F = (cid:8) [0 , ∋ t T t ∈ ˆ F : T = T = I (cid:9) . Theorem 5
For any chiral complex structure I on H ′ R , the group O I ( H ′ R , J ) is homotopyequivalent to Ω I ˆ F . In particular, π ( O I ( H ′ R , J )) ∼ = Z . As a preparatory result for the proof, let us state the following.
Proposition 3
The space F ∼ = ˆ F is homotopy equivalent to the space C ( H ′ R ) = { π ( I ) ∈ Q : π ( I ) chiral complex structure } . Proof.
We closely follow the proof of Theorem 7.1 of [7], which in turn is based on [2, 20].Let ρ : ˆ F → ˆ F be the (non-linear and discontinuous) map sending T to the partial isometry I = T | T | − in the polar decomposition. If π denotes as before the quotient map onto theCalkin algebra Q = Q ( H ′ R ) over H ′ R , then the map ˆ ρ = π ◦ ρ sends ˆ F surjectively onto C ( H ′ R ). Indeed, any chiral complex structure π ( I ) ∈ C ( H ′ R ) has a chiral and skew-adjoint lift I ′ for which ( I ′ ) ∗ I ′ − is compact; then the Riez projections P ′± on the positive and negativespectral projections of − ıI ′ lead to a lift I = ıP ′ + − ıP ′− for which I ∗ I − is a finite dimensionalprojection (on the kernel of I ′ ). The Bartle-Graves selection theorem [4] now provides a rightinverse θ : C ( H ′ R ) → ˆ F to ˆ ρ , namely ˆ ρ ◦ θ = . Moreover, θ ◦ ˆ ρ is homotopic to the identityvia [0 , ∋ t t T + (1 − t ) θ ( ˆ ρ ( T )). As θ ( ˆ ρ ( T )) = T | T | − + K for some chiral skew-adjointcompact K , this is a homotopy in ˆ F . Thus ˆ ρ is actually a homotopy equivalence so that ˆ F and C ( H ′ R ) are homotopy equivalent. ✷ Proof of Theorem 5. Due to Proposition 3 it is sufficient to show the homotopy equivalenceof O I ( H ′ R , J ) and Ω ˆ ρ ( I ) C ( H ′ R ). Here, the chiral complex structure I on H ′ R also specifies abase point ˆ ρ ( I ) in C ( H ′ R ). Associated to I , one can define a map β I : O ( H ′ R , J ) → C ( H ′ R )via β I ( O ) = ˆ ρ ( OIO ∗ ) = π ( OIO ∗ ). This map is actually a Serre fibration by the argument inTheorem 3.9 of [18]. The fiber over the base point ˆ ρ ( I ) = π ( I ) is precisely the set O I ( H ′ R , J )from (17). Hence one can use the long exact sequence of homotopy groups, which due to the13riviality of the homotopy groups of O ( H ′ R , J ) implies that the set Ω ˆ ρ ( I ) C ( H ′ R ) of based loops inthe base space is homotopy equivalent to the fiber over the base point which here is O I ( H ′ R , J ).Because the loop functor respects homotopy, we conclude from the above that the based loopspace Ω I ˆ F is homotopy equivalent to O I ( H ′ R , J ). The last claim follows from π ( ˆ F ) ∼ = Z . ✷ It is possible to use the index map j I : O I ( H ′ R , J ) → Z defined by j I ( O ) = Ind ( I, OIO ∗ )to distinguish the two components of O I ( H ′ R , J ). Furthermore, applying Theorem 4 and Propo-sition 1 to I = I and I = OIO ∗ leads to the following. Corollary 1
For any chiral complex structure I , j I is a homotopy invariant homomorphismlabelling the two components of O I ( H ′ R , J ) . One has j I ( O ) = Sf (cid:0) [0 , ∋ t (1 − t ) I + t OIO ∗ (cid:1) . The (Noether) index of a Toeplitz operator associated to a given index pairing can alwaysbe expressed as a spectral flow [19, 20, 8]. The following result is the parity version of thisresult, similar to [7] which contains a corresponding result for the Z -valued spectral flow. Theorem 6
Let I be a real chiral complex structure on H ′ R and O ∈ O I ( H ′ R , J ) . If P isthe spectral projection onto the positive imaginary spectrum of I where I was extended to askew-adjoint operator on H ′ R ⊗ C , then Sf (cid:0) [0 , ∋ t (1 − t ) I + tOIO ∗ (cid:1) = dim C (cid:0) Ker C ( P OP + − P ) (cid:1) mod 2 . Proof.
First of all, the Z -index on the right hand side is of the type ( j, d ) = (1 ,
8) inTheorem 1 of [14]. Indeed, P satisfies P = − P and J P J = − P (namely, P is even realand even Lagrangian in the terminology of [14]) as well as J OJ = O . In particular, the indexpairing on the right hand side is a homotopy invariant under variations of O and P respectingall the properties mentioned above. Now given I , the set O I ( H ′ R , J ) has two components byTheorem 5. The proof of Theorem 6 is thus remarkably simple. Both sides of the equality arehomotopy invariants and lie in Z . Hence it is sufficient to verify equality on both components.For O = , both sides vanish. For the other component, the equality is verified for a non-trivialexample in the next section. ✷ Let p be a one-dimensional projection on an infinite-dimensional Hilbert space H R . We consider I = (cid:18) − (cid:19) , O = (cid:18) − p (cid:19) . Then all conditions in Theorem 6 are satisfied. One has P = 12 (cid:18) − ı ı (cid:19) , P OP + ( − P ) = − (cid:18) p − ıpıp p (cid:19) . In particular, dim(Ker(
P OP )) = dim( p ) = 1. Hence the index on the right hand side ofTheorem 6 is equal to 1. On the other hand, the straight-line path is I t = (1 − t ) I + t O ∗ IO = (cid:18) − t p − + 2 t p (cid:19) . Hence this contains exactly one copy of the example (8), and so Sf ([0 , ∋ t I t ) = − , ∋ t I t = O ∗ t IO t where [1 , ∋ t O t is a Kuipers path connecting O to . As this second path is in the invertibles it has trivial Z -valued spectral flow. Therefore [0 , ∋ t I t is a loop with non-trivial Z -valued spectralflow. This provides the example needed in the proof of Theorem 3.Let us also calculate the parity of [0 , ∋ t I t as in [13]. One needs to look at the off-diagonal entry B t as in (4), and determine an invertible operator M t such that M t B t − = K t is compact. Clearly, M t = will do, and then K = 0 and K = − p so that deg ( T ) = 1 anddeg ( T ) = −
1. Thus one finds again that the parity of the path is − In the following the reformulation with chiral self-adjoints from Chapter 4 is used. Let H C = ℓ ( Z ) ⊗ C N and consider the following operator on H ′ C = H C ⊕ H C : H t = (cid:18) S t ) k ⊗ N ( S ∗ t ) k ⊗ N (cid:19) , where k ∈ Z and S t is the bilateral shift perturbed on one link from 1 to cos( πt ), namely inDirac notation S t = X n =0 | n ih n + 1 | + cos( πt ) | ih | . The Hamiltonian has the chiral symmetry (5) and is real as well as self-adjoint for all t : H t = H ∗ t = H t = − J H t J .
Thus it is possible to consider the parity of the path [0 , ∋ t H t . One finds σ ([0 , ∋ t H t ) = ( − kN . This property is now stable under any kind of perturbations not closing the spectral gap of H , such as a chiral disordered potential V ω = − J V ω J of moderate strength. Here ω is apoint in a compact W ∗ -dynamical system (Ω , T, Z , P ) given by the shift action T of Z and an15nvariant and ergodic probability measure P . Let us comment that the non-triviality of thepath [0 , ∋ t H t has nothing to do with the strong invariants appearing in the periodictable of topological insulators. The Hamiltonian has an even time-reversal symmetry and achiral symmetry. Hence it lies in the so-called BDI class. As such, in dimension d = 1 thereare infinitely many distinct phases labelled by the strong invariant, which in the above exampleis the number kN specifying the winding of the off-diagonal entry of H . For each k , theHamiltonian is then in the corresponding component of Fredholm operators and stays withinit along the path [0 , ∋ t H t , because it resulted from a merely local perturbation of H . Itis now a fact that such paths can be topologically non-trivial because the fundamental groupof F is Z . The parity detects this topology.Let us now come to the physical implications of the non-trivial parity. One can directlyconclude that H t has to have an eigenvalue crossing through 0 at some t ∈ [0 , Theorem 7 If kN is odd, then H has an odd number of evenly degenerate zero modes, namelythe multiplicity of as eigenvalue is modulo . Proof.
Let us introduce the gauge transformation G = X n> | n ih n | − X n ≤ | n ih n | . Then GS t G = S − t so that GH t G = H − t . Consequently the zero eigenvalue crossings for t leadto zero eigenvalue crossings for 1 − t . As the parity is invariant under a change of orientationand also unitary conjugations, these eigenvalue crossing cancel and do not lead to a net parity,except at t = . This implies that at t = one has to have an odd number of eigenvaluecrossings. Consequently, the multiplicity of the zero eigenvalue is 2 modulo 4. ✷ Note that at half-flux, the shift does not connect left and right half-space so that H is adiret sum of a left and a right half-space Hamiltonian. Each has to have a zero mode, leadingto the two-fold symmetry. Let us further add a few comments on how to interpret Theorem 7against the background of the periodic table of topological insulators. As already stated, allthe above Hamiltonians are from the BDI class of chiral Hamiltonians with an even time-reversal symmetry (integer spin). Also the Hamiltonian H is within this class. If one extractsonly the low lying spectrum (eigenvalues in the vicinity of 0), this reduced Hamiltonian is afinite dimensional matrix and hence represents a system of dimension d = 0 (correspondingto the local defect induced by a half-flux). The set of 0-dimensional BDI Hamiltonians hastwo components which are distinguished by the parity of the zero modes (of each half-sidedHamiltonian). Theorem 7 states that H is in the non-trivial component of the 0-dimensionalBDI Hamiltonians always having a zero mode. The aim of this final section is to apply the parity in the bifurcation theory of solutions tononlinear operator equations depending on a real parameter. It was precisely for this purpose16hat the parity was originally introduced and put to work [11, 12, 13]. The treatment given inthis note suggests to construct examples with a skew-adjoint linearization which has a chiralsymmetry built in. This is essentially what is done below.Let us begin by exposing the theoretical framework of bifurcation theory and the main resultused later on. Given two real Banach spaces X and Y and an interval I ⊂ R , one considerscontinuous maps F : I × X → Y for which we assume throughout that F ( t,
0) = 0 for all t ∈ I .In this context, one then calls the set I × { } the trivial branch of solutions of F ( t, u ) = 0. Abifurcation point for the family of equations F ( t, u ) = 0, t ∈ I , is a parameter value t ∗ where anew branch of solutions appears. Definition 5
A parameter value t ∗ ∈ I is a bifurcation point for the family of equations F ( t, u ) = 0 if in every neighborhood of ( t ∗ , ∈ I × X there is some ( t, u ) such that u = 0 and F ( t, u ) = 0 . Let us now assume that the map F is continuously differentiable in u . The implicit functiontheorem then implies that if the linear map D u F ( t ∗ ,
0) : X → Y is invertible, there is aneighborhood of ( t ∗ ,
0) in I × X for which there is a unique solution of the equation F ( t, u ) = 0.As F ( t,
0) = 0 by assumption, we see that t ∗ cannot be a bifurcation point in this case.Consequently, D u F ( t ∗ ,
0) must be singular if t ∗ is a bifurcation point. Let us stress, however,that not every t ∗ for which D u F ( t ∗ ,
0) is singular, is necessarily a bifurcation point. The aim ofbifurcation theory is to find sufficient conditions under which a singular point t ∗ is a bifurcationpoint. While such problems have been considered for centuries, topological criteria for existenceof bifurcations in an infinite dimensional set-up were only made by Krasnoselskii in the sixties[16].One extension of his ideas is the work of Fitzpatrick and Pejsachowicz [12] which uses theparity and is described next. For a continuously differentiable F let us consider the boundedlinear operator B t = D u F ( t,
0) and suppose that their index vanishes. As I ∋ t B t is acontinuous path by assumption, its parity is defined. Theorem 8 ([12])
Suppose that I ∋ t B t is an admissible path. If σ ( I ∋ t B t ) = − ,then there is a bifurcation point t ∗ ∈ I for the family of equations F ( t, u ) = 0 . In the following, we provide an example of a parameter dependent system of partial differ-ential equations for which the parity can be calculated explicitly. On Ω = (0 , π ) × (0 , π ) let usconsider the family of elliptic systems parametrized by t ∈ R − ∆ u = tv + f ( t, x, u, v ) , in Ω , − ∆ v = tu + g ( t, x, u, v ) , in Ω ,u = v = 0 , on ∂ Ω , (18)where u, v : Ω → R and f, g : R × Ω × R → R are continuously differentiable. We assume that f ( t, x, ,
0) = g ( t, x, ,
0) = 0 for all ( t, x ) ∈ R × Ω so that ( u, v ) = (0 ,
0) is a solution of (18)17or all t ∈ R . Moreover, all partial derivatives of f and g with respect to u and v are supposedto be bounded and satisfy D ( u,v ) f ( t, x, ,
0) = D ( u,v ) g ( t, x, ,
0) = 0 , ( t, x ) ∈ R × Ω . (19)As the Laplacian as operator on L (Ω) with domain H (Ω) ∩ H (Ω) is invertible with compactresolvent, one can transform the first two equations of (18) to the system F ( t, u, v ) = (cid:18) uv (cid:19) + t (cid:18) KvKu (cid:19) + (cid:18) Kf ( t, x, u, v ) Kg ( t, x, u, v ) (cid:19) = 0 , where K = ∆ − : L (Ω) → L (Ω) is compact. The assumptions on f , g and the compactnessof K imply that F : R × L (Ω) × L (Ω) → L (Ω , R ) is differentiable. Moreover, the derivativeat (0 ,
0) of the nonlinear part vanishes by (19) and therefore B t ( u, v ) = D ( u,v ) F ( t, , u, v ) = (cid:18) uv (cid:19) + t (cid:18) KvKu (cid:19) . By applying ∆ to each component of the equation B t ( u, v ) = 0, one checks that( u ( x ) , v ( x )) = (sin( x ) sin( x ) , sin( x ) sin( x )) , x = ( x , x ) ∈ Ω , is in the kernel of B . To find out if t ∗ = 2 is a bifurcation point of F ( t, u, v ) = 0, let us nowcompute σ ([2 − δ, δ ] ∋ t B t ). First of all, one needs to consider the eigenvalue problem (cid:18) B t − B ∗ t (cid:19) (cid:18) zw (cid:19) = µ (cid:18) zw (cid:19) . By setting z = ( u , v ) and w = ( u , v ) and applying the Laplace operator in each component,this amounts to solve the system of equations ∆ u + tv = µ ∆ u , in Ω , ∆ v + tu = µ ∆ v , in Ω , − ∆ u − tv = µ ∆ u , in Ω , − ∆ v − tu = µ ∆ v , in Ω ,u = u = v = v = 0 , on ∂ Ω . (20)Setting for integer k j , m j , l j , n j where j = 1 , u ( x ) = sin( k x ) sin( k x ) , u ( x ) = sin( m x ) sin( m x ) ,v ( x ) = sin( l x ) sin( l x ) , v ( x ) = sin( n x ) sin( n x ) , the equations (20) are equivalent to − ( m + m ) u + tv = − µ ( k + k ) u , in Ω , − ( n + n ) v + tu = − µ ( l + l ) v , in Ω , ( k + k ) u − tv = − µ ( m + m ) u , in Ω , ( l + l ) v − tu = − µ ( n + n ) v , in Ω .
18t is readily seen that for t close to 2, one can only have an eigenvalue crossing zero in thesubspace of L (Ω , C ) ⊕ L (Ω , C ) spanned by ( u , , , , v , , , , u ,
0) and (0 , , , v )when k = k = m = m = l = l = n = n = 1. The four eigenvalues in this subspace are λ = − ı ( t − , λ = ı ( t − , λ = − ı ( t + 2) , λ = ı ( t + 2) . The eigenvalue crossing is simple and analytic, and of the type of the first example in (8).Consequently, σ ([2 − δ, δ ] ∋ t B t ) = − δ > t ∗ = 2 is indeed abifurcation point for (18) by Theorem 8. Acknowledgements:
This work was partially supported by the DFG. N.W. thanks theFriedrich-Alexander-Universtit¨at for a Visiting Professorship.
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