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

## 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 -algebrasReference 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 of -cocycle if

- (1)
for all , ,

- (2)
for all .

Define the product of two by their point-wise product. The set of all -cocycles of -cocycles 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 The set of all 2-cocycles of this type forms a subgroup -cocycle. 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 , of -cocycle 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 subspaces of -invariant 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

### 1.3. The split property and projective representations

Next let us introduce the split property.

Recall that a type I factor is