Abstract
We consider a set S P G ( A ) SPG(\mathcal {A}) of pure split states on a quantum spin chain A \mathcal {A} which are invariant under the on-site action τ \tau of a finite group G G . For each element ω \omega in S P G ( A ) SPG(\mathcal {A}) we can associate a second cohomology class c ω , R c_{\omega ,R} of G G . We consider a classification of S P G ( A ) SPG(\mathcal {A}) whose criterion is given as follows: ω 0 \omega _{0} and ω 1 \omega _{1} in S P G ( A ) SPG(\mathcal {A}) are equivalent if there are automorphisms Ξ R \Xi _{R} , Ξ L \Xi _L on A R \mathcal {A}_{R} , A L \mathcal {A}_{L} (right and left half infinite chains) preserving the symmetry τ \tau , such that ω 1 \omega _{1} and ω 0 ∘ ( Ξ L ⊗ Ξ R ) \omega _{0}\circ \left ( \Xi _{L}\otimes \Xi _{R}\right ) are quasi-equivalent. It means that we can move ω 0 \omega _{0} close to ω 1 \omega _{1} without changing the entanglement nor breaking the symmetry. We show that the second cohomology class c ω , R c_{\omega ,R} is the complete invariant of this classification.
Highlights
It is well-known that the pure state space P (A) of a quantum spin chain A (UHFalgebra, see subsection 1.1) is homogeneous under the action of the asymptotically inner automorphisms [P], [B], [FKK]
We focus on the subset SP (A) of P (A) consisting of pure states satisfying the split property. (See Definition 1.4.) One equivalent condition for a state ω ∈ P (A) to satisfy the split property is that ω is quasi-equivalent to ω|AL ⊗ ω|algebra πωR (AR) . (See Remark 1.5.) Here, ω|AL, ω|AR are restrictions of ω onto the left/right half-infinite chains. (See subsection 1.1.) Recall that two state being quasi-equivalent can be understood physically that they are “macroscopically same”, because it means that one state can be represented as a local perturbation of the other and vice versa
Using the result of [P], [B], [FKK],[KOS], one can see that for any ω0, ω1 ∈ SP (A), there exist asymptotically inner automorphisms ΞL, ΞR on AL, AR such that ω1|AL ∼q.e. ω0|AL ◦ ΞL and ω1|AR ∼q.e. ω0|AR ◦ ΞR. (Here ∼q.e. means quasi-equivalence.) From this and the split property of ω0, ω1, we see that ω1 and ω0 ◦(ΞL ⊗ ΞR) are quasi-equivalent
Summary
It is well-known that the pure state space P (A) of a quantum spin chain A (UHFalgebra, see subsection 1.1) is homogeneous under the action of the asymptotically inner automorphisms [P], [B], [FKK]. The proof is the same as that of the genuine representations (see [S] Theorem II.4.2 for example.) For σ ∈ Z2(G, T), we denote by Pσ, the set of all unitarily equivalence classes of irreducible projective representations with 2-cocycle σ. Let (H, u) be a projective unitary representation of G with 2-cocycle σ, α ∈ Pσ, and N N ≥ Nm(σ). Let (L, ρ, u, σ) be a quadruple such that (i): ρ is a ∗-representation of AΓ on a Hilbert space L, (ii): u is a projective unitary representation of G on L with a 2-cocycle σ, (iii): for any g ∈ G, we have (23). The projective unitary representation u has an irreducible decomposition given by some Hilbert spaces {Kγ | γ ∈ Pσ} (Lemma 1.1 and Notation 1.2). U contains all elements of Pσ with infinite multiplicity
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