The Z 2 -phase label

The next aim is to distinguish different gapped ground states of fermionic Hamiltonians, ideally via a topological phase label. To this end, we again utilize the following decomposition obtained from the Jordan–Wigner transform, see Section 6.1:

Ab_Z = A^car_Z oγ−Z2 ∼= A^car_Z ⊕ T A^car_Z , A^P_Z ∼= (A^P_Z)⁰⊕(A^P_Z)¹ ∼= (A^car_Z )⁰ ⊕ T(A^car_Z )¹ . (51) Here γ− is the Z2-action from Equation (46). One can extend any state ω on A^car

Z to a state on Ab_Z and then restrict to a state ω^P on A^P_Z. If one starts with a Θ-invariant and pure state on A^car_Z , by Theorem 6.1 the purity of ω^P depends on the representations of ω and ω◦γ− on A^car

Z and (A^car

Z )⁰.

In the quasifree case, this obstruction can be expressed in terms of a Hilbert-Schmidt condition and a Z2-index on canonical transformations. Let us now consider this question for more general states.

The next results do not needω to be a ground state.

Lemma 6.8 Letωbe aΘ-invariant state onA^car_Z that satisfies the split property. Thenωis quasiequiva-lent to ω◦γ−.

Proof. If ω is Θ-invariant then so are the restrictions ω_L and ω_R to the subalgebras A^car_L and A^car_R . Furthermore, we observe that

γ−

A^car_L = Θ

A^car_L , γ−

A^car_R = Id_A^car

R , and so

ω_L⊗_F ω_R(γ−(a_La_R)) = ω_L(γ−(a_L))ω_R(γ−(a_R)) = ω_L(Θ(a_L))ω_R(a_R) = ω_L(a_L)ω_R(a_R). That is, ωL⊗_F ωR ◦ γ− = ωL ⊗_F ωR. Therefore by Corollary 2.3.17 of [17], there is a unitary W ∈ B(h_ωL⊗FωR) such thatWΩ_ωL⊗_FωR = Ω_ωL⊗_FωR and W π_ωL⊗_FωR(a)W^∗ =π_ωL⊗_FωR(γ−(a)).

Becauseω_L⊗_Fω_Ris quasiequivalent toω, there is an isomorphismϕ:π_ωL⊗_FωR(A^car

Z )⁰⁰→π_ω(A^car

Z )⁰⁰ such that ϕ(π_ωL⊗_FωR(a)) =π_ω(a) for all a∈A^car

Z . Let us now consider the map ϕ◦Ad_W which has the property that

ϕ(W π_ωL⊗_FωR(a)W^∗) = ϕ(π_ωL⊗_FωR(γ−(a))) = π_ω(γ−(a)) = πω◦γ−(a), a∈A^car_Z . Henceϕ◦Ad_W gives an isomorphism π_ωL⊗_FωR(A^car

Z )⁰⁰∼=πω◦γ−(A^car

Z )⁰⁰ that implements a quasiequiv-alence betweenωL⊗_F ωR and ω◦γ−. Because quasiequivalence is transitive, ω is quasiequivalent to

ω◦γ−. 2

Let us now assume thatω is pure and Θ-invariant. In particular,πω(A^car_Z )⁰⁰=B(h_ω) and the GNS space is graded by a self-adjoint unitary Σ. If, moreover, ω is equivalent to ω ◦γ−, there exists a unitary V ∈ B(h_ω) such that π_ω(γ−(a)) = V π_ω(a)V^∗. It turns out that this unitary can be either even or odd.

Proposition 6.9 Let ω be a pureΘ-invariant state on A^car

Z equivalent to ω◦γ−. (i) The states ω|_(A^car

Z )⁰ and ω|_(A^car

Z )⁰ ◦γ− are equivalent (that is, ω^P is pure) if and only if there is a self-adjoint unitaryV0∈πω((A^car_Z )⁰)⁰⁰ such that πω(γ−(a)) =V0πω(a)V₀^∗ for all a∈A^car_Z . (ii) If ω|_(Acar

Z )⁰ and ω|_(Acar

Z )⁰ ◦γ− are not equivalent (that is, ω^P is not pure), then there exists a unitaryV1 ∈πω((A^car_Z )¹)⁰⁰ such that πω(γ−(a)) =V1πω(a)V₁^∗ for all a∈A^car_Z . Furthermore, ω^P is a mixture of two inequivalent pure states.

We note that there is a large overlap between the above proposition and [48, Proposition 6.3].

Proof. (i) Given the state ω, one can identify the GNS space h_ω|

(Acar

Z )0 of its restriction to the even algebra withh⁰_ω=πω((A^car

Z )⁰)Ωω ∼= ¹₂(1+Σ)hω. Becauseωis Θ-invariant and pure,ω|_(A^car

Z )⁰ is pure [32, Lemma 6.23]. In particular, the states ω|_(A^car

Z )⁰ and ω|_(A^car

Z )⁰ ◦γ− on (A^car_Z )⁰ will be equivalent if and only if there is a self-adjoint unitary V =V₀ ∈π_ω((A^car

Z )⁰)⁰⁰ implementingγ− on h⁰_ω,i.e. ΣVΣ =V.

For part (ii), let us fix some j ∈ N and set Zj =a_j+a^∗_j which is an odd self-adjoint unitary in A^car

Z . By [32, Lemma 6.27] (applied with U = Zj and β = γ−), the pure state ω|_(A^car

Z )⁰ on (A^car

Z )⁰ is equivalent to ω|_(Acar

Z )⁰ ◦ γ− ◦Ad_Zj. Therefore there is some ˜W ∈ π_ω((A^car

Z )⁰)⁰⁰ such that Ad_W˜ implements γ− ◦AdZj on h⁰_ω ∼= ¹₂(1 + Σ)hω. Because (γ−◦AdZj)² = Id, for an appropriate phase we can takeW =e^iφW˜ self-adjoint with Ad_W implementing γ−◦Ad_Zj on the GNS space. We then compute that

π_ω(Z_j)W π_ω(a)W π_ω(Z_j) = π_ω(Ad_Zj◦γ−◦Ad_Zj(a)) = π_ω(γ−(a)), a∈(A^car_Z )⁰ .

Once again, because γ²₋ = Id, the operator π_ω(Z_j)W is self-adjoint up to a phase. In particular, π_ω(Z_j)W =e^iνW π_ω(Z_j) for some ν.

We now consider odd elements, where we compute that, for a₁ ∈(A^car

Z )¹,

π_ω(Z_j)W π_ω(a₁)W π_ω(Z_j) = e^iνW π_ω(Z_ja₁)W π_ω(Z_j) = e^iνπ_ω(γ−◦Ad_Zj(Z_ja₁))π_ω(Z_j)

= e^iνπω(γ−(a1)Zj)πω(Zj) = e^iνπω(γ−(a1)), (52) where we have used thatZja1 is even and our results on even elements. Because Equation (52) is true for all odd elements, we have that

πω(Zj)W πω(Zj)W πω(Zj) = e^iηπω(γ−(Zj)) = e^iηπω(Zj). (53) Because the left-hand side of Equation (53) is self-adjoint, so must be the right-hand side, which implies thate^iη =±1. Ife^iη = 1 we are done and can take the unitaryV1 =πω(Zj)W ∈πω((A^car_Z )¹)⁰⁰. Ife^iη =−1, then instead we consider π_ω(Z_j)WΣ, where for any a∈A^car

Z with homogeneous grading

|a| ∈ {0,1},

π_ω(Z_j)WΣπ_ω(a)ΣW π_ω(Z_j) = (−1)^|a|π_ω(Z_j)W π_ω(a)W π_ω(Z_j)

= (−1)^|a|(−1)^|a|πω(γ−(a)) = πω(γ−(a)).

ThusV1 =πω(Zj)WΣ∈πω((A^car_Z )¹)⁰⁰gives the required result. The last statement is Theorem6.1. 2 For completeness, let us now construct the corresponding states ω^P on A^P_Z in the two settings of Proposition 6.9. If for i = 0 or i = 1 there is an element Vi ∈ πω((A^car

Z )ⁱ)⁰⁰ such that πω(γ−(a)) = V_iπ_ω(a)V_i^∗, then recalling the decomposition (51) of A^P

Z, one can define a representation π : A^P

Z → B(h_ω) by

π(a0+T a1) = πω(a0) +Viπω(a1), aj ∈(A^car_Z )^j . We then set

ω^P(Q) = hΩ_ω, π(Q)Ωωi_hω = hΩ_ω, πω(a0)Ωωi_hω +hΩ_ω, Viπω(a1)Ωωi_hω , Q = a0+T a1 ∈A^P_Z . For the even unitary V₀, the second term in ω^P(Q) will vanish as Ω_ω is even and V₀π_ω(a₁)Ω_ω odd.

By [48, Proposition 6.3 (ii)], ω^P is the unique Θ-invariant pure state on A^P_Z coming from the state ω onA^car

Z . If the unitaryV₁ is odd, then the second term does not vanish andω^P is a sum of two states.

Let us now define a Z2-phase label for a class of pure Θ-invariant states on A^car

Z that are not necessarily quasifree. The definition distinguishes the two cases considered in Proposition6.9. Recall that Σ is the implementation of the parity Θ in the GNS representation.

Definition 6.10 Let ω be a pure Θ-invariant state on A^car

Z that is equivalent to ω◦γ−. Further let V ∈ π_ω((A^car

Z )ⁱ)⁰⁰ be a unitary such that π_ω(γ−(a)) = V π_ω(a)V^∗ for all a ∈ A^car

Z . Then a Z2-phase label of ω is assigned by j(ω) = (−1)ⁱ ∈Z2 with i= 0,1 as above, namelyΣVΣ = (−1)ⁱV.

Let us make some first comments on this definition. First, we note that anyV implementingγ−on h_ω has indeed homogeneous parity by Proposition 6.9. Such a unitaryV is determined up to unitary equivalence and, because π_ω is irreducible, any other operator U V U^∗ implementing γ− is the same as V up to a complex scalar of modulus one. Hence the parity of all unitaries implementing γ− is constant and thus the phase-label is well-defined. Moreover, Lemma 6.8 implies that the Z2-phase label is well-defined for pure and Θ-invariant states that satisfy the split property. In particular, the Z2-phase label is defined for any pure gapped ground state of a Hamiltonian for the form considered in Theorem 6.7. Moreover, for quasifree states the Z2-phase label is linked to a Z2-valued Fredholm index.

Proposition 6.11 Let E be a basis projection and ω_E the corresponding pure, Θ-invariant and quasifree state on A^car

Z . If ω is equivalent to ω ◦γ−, then for J = i(2E −1) and θ− the diagonal extension of (47),

j(ωE) = jJ(θ−).

Proof. By Theorem6.1,ω_E^P is pure in case (i) of Proposition6.9and not pure in case (ii). These cases correspond toj(ω_E) = 1 and j(ω_E) =−1 respectively. Therefore Corollary6.2implies the claim. 2 Recalling Example 5.18 in the quasifree setting, the automorphism γ− can be implemented by inserting a local half-flux through a Hamiltonian. Because the index j(ω) is a comparison between the state ω and the ‘half-flux-inserted state’ ω◦γ−, if j(ω) =−1, this indicates that a flux insertion induces a change in the ground state. In the quasifree setting, such a change of the ground state is detected by theZ2-valued spectral flow.

We now consider some basic stability properties of the phase label. The following is a simple application of standard properties of the GNS representation of pure states.

Proposition 6.12 Let ω₀ andω₁ be pureΘ-invariant states onA^car

Z equivalent toω₀◦γ− andω₁◦γ−

respectively. Suppose that there is an automorphism η ∈ Aut(A^car_Z ) commuting with Θ and γ− and such thatω1=ω0◦η. Then j(ω0) =j(ω1).

The hypothesis that η commutes with Θ and γ− is quite strong, though it is sufficient to assume that η commutes with Θ and leaves A^car_L and A^car_R invariant. Proposition 6.12 combined with the following remark shows that the Z2-phase label is perturbatively stable, for example, when weak interactions are added to a quasifree system.

Remark 6.13 Examples of such automorphisms η ofA^car_Z that satisfy the hypothesis of Proposition 6.12 can be constructed using the quasilocal structure ofA^car

Z and the quasiadiabatic evolution (also called the spectral flow) of uniformly gappedC¹-interactions [54]. In particular, let us consider a path of local Hamiltonians for allX ⊂Zfinite, where

H_X(s) = H_X + Φ_X(s)

and the path satisfies several assumptions. First, the ground state gap of HX(s) is required to be uniformly bounded for all s ∈ [0,1]. Furthermore, ΦX(s) ∈ B_F for all s ∈ [0,1] and X ∈ P₀(Z),

whereB_F is the space of stronglyC¹-interactions satisfying [54, Assumption 6.12] with the additional property that Θ(Φ_X(s)) = Φ_X(s) and γ−(Φ_X(s)) = Φ_X(s) for all s ∈ [0,1]. If these assumptions are satisfied, then the results in [54, Section 6-7] (adapted to the fermionic case, where the property Θ(ΦX(s)) = ΦX(s) is crucial) guarantee the existence of an automorphism η^Φ_s in the infinite-volume limit that maps between the ground states on A^car

Z with the property that Θ◦η^Φ_s = η^Φ_s ◦Θ and γ−◦η_s^Φ=η^Φ_s ◦γ− for all s∈[0,1].

To summarise, ifj(ω) is well-defined and comes from the thermodynamic limit of a finite-volume HamiltonianH_X(0) with gapped ground state, thenj(ω◦η_s^Φ) =j(ω) for alls∈[0,1]. While this result shows an important stability property of theZ2-phase label, the assumption thatγ−(ΦX(s)) = ΦX(s) is somewhat artificial. Given a Θ-invariant interaction Φ, one can consider ˜Φ = ¹₂ Φ +γ−(Φ)

which is γ−-invariant, but it is interesting to investigate to what degree the γ−-invariant assumption can be lessened. One may be able to use a construction similar to [57] in order to work with paths of

interactions that need not be γ−-invariant.

Proposition 6.14 Letω₀be a pure andΘ-invariant state onA^car

Z that is equivalent toω₀◦γ₋. Suppose that there is a path of states{ω_s}_s∈[0,1] with an associated family of Hilbert spaces{h_ωs}_s∈[0,1], as well as unitaries{U_s}_s∈[0,1] such that Us:h_ω0 →h_ωs. Then, j(ωs) =j(ω0) for alls∈[0,1].

Proof. Given such a path of unitaries, for anyA_s ∈ B(h_ωs) there is an operatorA₀∈ B(h_ω0) such that A_s=U_sA₀U_s^∗. We can therefore define a representationπ_ωs = Ad_Us◦π_ω0. Becauseπ_ω0 is irreducible, so isπωs. Furthermore, for Vs=UsV0U_s^∗, Σs=UsΣ0U_s^∗ one has

Vsπωs(a)Vs = πωs(γ−(a)), Σsπωs(a)Σs = πωs(Θ(a)), so that

ΣsVsΣs = UsΣ0V0Σ0U_s^∗ = (−1)^|V0|Vs.

Thus for alls∈[0,1],j(ωs) is well-defined withj(ωs) =j(ω0). 2 Results from [54] guarantee that our Z2-index is stable under strongly C¹-paths of interactions that are Θ-symmetric, γ−-symmetric and satisfy [54, Assumption 6.12]. In particular, if two pure gapped ground states ω₀ and ω₁ have different indices, j(ω₀) =−j(ω₁), these ground states cannot be connected by such a path. Similarly, by Proposition 6.14 there cannot be family of unitaries of unitaries connectingh_ω0 and h_ω1.

Let us now state a stability result of theZ2-phase label in the quasifree setting.

Proposition 6.15 Let(H,Γ)be a complex Hilbert space with real structure. Let H0 andH1 be gapped BdG Hamiltonians on H with quasifree ground states ω_E0 and ω_E1 such that j(ω_E0) and j(ω_E1) are well-defined. Suppose that H0 and H1 can be connected by a norm-continuous path of self-adjoint Fredholm operators [0,1]3t7→Ht such that ΓHtΓ =−H_t for allt∈[0,1]. Then j(ωE0) =j(ωE1).

Proof. By the assumptions on the pathH_t, theZ2-valued spectral flow Sf₂(iH_t) is well-defined. In par-ticular, there is a partition 0 =t₀ < t₁ < . . . < t_n= 1 such that Ind₂(J_tj, J_tj+1) = (−1)^{12Ker(Jtj+Jtj+1)} is well-defined forJtj =iHtj|H_tj|⁻¹(with an arbitrary complex structure on Ker(Htj) if needed). Now Proposition6.11implies that

j(ωE0) = jJt0(θ−) = Ind2(Jt0, θ−Jt0θ−)

withθ−the diagonal extension of (47). Recalling the concatenation and invariance properties of Ind2, in particular

Ind2(Jtj, Jtj+1) = Ind2(Jtj+1, Jtj) = Ind2(V JtjV^∗, V Jtj+1V^∗) for any unitaryV with ΓVΓ =V, we compute

j(ω_E0) = Ind₂(J_t0, θ−J_t0θ−)

= Ind₂(J_t0, J_t1)· · ·Ind₂(J_tn−1, J_tn) Ind₂(J_tn, θ−J_tnθ−) Ind₂(θ−J_tnθ−, θ−J_tn−1θ−)

× · · ·Ind2(θ−Jt1θ−, θ−Jt0θ−)

= Ind2(Jtn, θ−Jtnθ−)

= j(ωE1)

as all other terms cancel. 2

Proposition 6.15, in comparison with Proposition 6.14, shows that in special cases we can take paths of ground states such that the GNS spaces are not unitarily equivalent, but where theZ2-phase label remains constant. Furthermore, recalling Proposition5.17, if the pathiHtfrom Proposition6.15 has a non-trivial Z2-valued spectral flow, then the spectral gap of the GNS Hamiltonians will close.

Therefore, we see that in special cases the indexj(ω) is invariant on paths that can close the ground state gap.