A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries

By Yoshiko Ogata

Abstract

We consider a set of pure split states on a quantum spin chain which are invariant under the on-site action of a finite group . For each element in we can associate a second cohomology class of . We consider a classification of whose criterion is given as follows: and in are equivalent if there are automorphisms , on , (right and left half infinite chains) preserving the symmetry , such that and are quasi-equivalent. It means that we can move close to without changing the entanglement nor breaking the symmetry. We show that the second cohomology class is the complete invariant of this classification.

1. Introduction

It is well-known that the pure state space of a quantum spin chain (UHF-algebra, see subsection 1.1) is homogeneous under the action of the asymptotically inner automorphisms Reference P, Reference B, Reference FKK. In fact, the homogeneity is proven for much larger class, i.e., for all the separable simple -algebras Reference KOS.

In this paper, we focus on the subset of consisting of pure states satisfying the split property. (See Definition 1.4.) One equivalent condition for a state to satisfy the split property is that is quasi-equivalent to . (See Remark 1.5.) Here, , are restrictions of 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. On the other hand, a product state on has no entanglement between and by definition. Therefore, a state with the split property can be physically understood as a state without macroscopic entanglement between the left half and the right half of the chain. Using the result of Reference P, Reference B, Reference FKK,Reference KOS, one can easily see that for any , there exist asymptotically inner automorphisms , on , such that and . (Here means quasi-equivalence.) From this and the split property of , , we see that and are quasi-equivalent. The product of automorphisms clearly does not create/destroy any entanglement between the left half and the right half of the chain. Hence any can get “close to” any without changing the entanglement. In this sense, we may regard to be “homogeneous”.

What we would like to show in this paper is that the situation changes when symmetry comes into the game. This corresponds to the notion of symmetry protected topological phases in physics Reference O. Let be the set of all states in which are invariant under the onsite action of a finite group . (See Definition 1.4.) We now require that the automorphisms , above to preserve the symmetry i.e., and for all . (See Equation 3 for the definition of and .) For any , can we always find such automorphisms giving ? We show that the answer is no in general. The obstacle is given by the second cohomology class of the projective representation of associated to . We show that this second cohomology class is the complete invariant of this classification.

1.1. Setting

We consider the setting in this subsection throughout this paper. We use the basic notation in Appendix A freely. We start by summarizing standard setup of quantum spin chains on the infinite chain Reference BR1Reference BR2. Throughout this paper, we fix some . We denote the algebra of matrices by .

For each subset of , we denote the set of all finite subsets in by . We use the notation and .

For each , let be an isomorphic copy of , and for any finite subset , we set . For finite , the algebra can be regarded as the set of all bounded operators acting on the Hilbert space . We use this identification freely. If , the algebra is naturally embedded in by tensoring its elements with the identity. For an infinite subset , is given as the inductive limit of the algebras with . We call the quantum spin system on . In particular, we use notation , and . Occasionally, we call them quantum spin chain, right infinite chain, left infinite chain, respectively. Note that each of , can be regarded naturally as a subalgebra of . We also set , for any .

We denote the standard basis of by , and denote the standard matrix unit of by . Namely, is a matrix such that . For each finite , we denote the tensor product of along , by with and . We also use the notation

Furthermore, we set for .

Throughout this paper we fix a finite group and its unitary representation on satisfying

We denote the identity of by .

Let be a non-empty subset. For each , there exists a unique automorphism on such that

for any finite subset of . We call the group homomorphism , the on-site action of on given by . In particular, when , (resp. , ), we denote by (resp. , ). For , we denote by the fixed point subalgebra of with respect to . For simplicity, also use the notation and .

1.2. Projective representations of

Let . A map is called a -cocycle of if

(1)

, for all ,

(2)

for all .

Define the product of two -cocycles by their point-wise product. The set of all -cocycles of then becomes an abelian group. The resulting group we denote by . The identity of is given by , for . For an arbitrary function such that ,

defines a -cocycle. The set of all 2-cocycles of this type forms a subgroup of . (It is clearly normal because is abelian.) The quotient group is called the second cohomology group of . For each , we denote by the second cohomology class that belongs to.

A projective unitary representation of is a triple consisting of a Hilbert space , a map and a -cocycle of such that for all . Note that we get from the latter condition. We call , the -cocycle of associated to , and call the second cohomology class of associated to . We occasionally say is a projective unitary representation with -cocycle . The character of a finite dimensional projective unitary representation is given by , for .

We say a projective unitary representation of is irreducible if and are the only -invariant subspaces of . As is a finite group, for any irreducible projective unitary representation of , the Hilbert space is finite dimensional. Projective unitary representations and are said to be unitarily equivalent if there is a unitary such that , with . Clearly if and are unitarily equivalent, the -cocycles and coincides. Schur’s Lemma holds: let and be irreducible projective unitary representations of , and be a linear map such that for all . Then either or and are unitarily equivalent. The proof is the same as that of the genuine representations (see Reference S Theorem II.4.2 for example.)

For , we denote by , the set of all unitarily equivalence classes of irreducible projective representations with -cocycle . Note that is equal to , the dual of i.e. the set of equivalence classes of irreducible representations.

For each , we fix a representative . We denote the dimension of (which is finite) by and fix an orthonormal basis of . We introduce the matrix unit of given by

We will use the following vector later, in section 4

For each and , define a function on by

As in Theorem III.1.1 of Reference S, from Schur’s Lemma, we obtain the orthogonality relation:

for all and . Here denotes the number of elements in . In particular, is a finite set. We freely identify and . For example, , should be understood as , for , , and a projective unitary representation . We repeatedly use the following fact.

Lemma 1.1.

For any projective unitary representation , there are Hilbert spaces labeled by and a unitary such that

Furthermore, the commutant of is of the form

Proof.

For any -invariant subspace of , its orthogonal complement is -invariant as well. Therefore, from Zorn’s Lemma, we may decompose as an orthogonal sum of irreducible projective unitary representations with -cocycle . This proves Equation 9. The second statement Equation 10 follows from the orthogonality relation Equation 8.

Notation 1.2.

When Equation 9 holds, we say that (or ) has an irreducible decomposition given by Hilbert spaces . We say (or ) contains all elements of if for all . We say (or ) contains all elements of with infinite multiplicity if for all . We hence force omit in Equation 9 and identify and freely. The Hilbert space can be naturally regarded as a closed subspace of . We use this identification freely and call the -component of (or ).

Notation 1.3.

Let be a projective unitary representation. Let be a map such that . Setting as in Equation 4, we obtain . We also set , for . Then is a projective representation.

1.3. The split property and projective representations

Next let us introduce the split property.

Definition 1.4.

Let be a pure state on . Let be the restriction of to , and be the GNS triple of . We say satisfies the split property with respect to and , if the von Neumann algebra is a type I factor. We denote by the set of all pure states on which satisfy the split property with respect to and . We also denote by , the set of all states in , which are -invariant.

Recall that a type I factor is -isomorphic to