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

Yoshiko Ogata|arXiv (Cornell University)|Aug 22, 2019

Algebraic structures and combinatorial models11 references6 citations

TL;DR

This paper classifies pure, split states on quantum spin chains that are invariant under on-site finite group symmetries, showing that the second cohomology class of the associated projective representation serves as a complete invariant under symmetry-preserving quasi-equivalence. The classification accounts for entanglement and symmetry preservation, revealing that such states cannot always be connected via symmetric automorphisms, with the cohomology class capturing the obstruction.

ABSTRACT

We consider a set $SPG(\mathcal{A})$ of pure split states on a quantum spin chain $\mathcal{A}$ which are invariant under the on-site action $τ$ of a finite group $G$. For each element $ω$ in $SPG(\mathcal{A})$ we can associate a second cohomology class $c_{ω,R}$of $G$. We consider a classification of $SPG(\mathcal{A})$ whose criterion is given as follows: $ω_{0}$ and $ω_{1}$ in $SPG(\mathcal{A})$ are equivalent if there are automorphisms $Ξ_{R}$, $Ξ_L$ on $\mathcal{A}_{R}$, $\mathcal{A}_{L}$ (right and left half infinite chains) preserving the symmetry $τ$, such that $ω_{1}$ and $ω_{0}\circ( Ξ_{L}\otimes Ξ_{R})$ are quasi-equivalent. It means that we can move $ω_{0}$ close to $ω_{1}$ without changing the entanglement nor breaking the symmetry. We show that the second cohomology class $c_{ω,R}$ is the complete invariant of this classification.

Motivation & Objective

• To classify pure states on quantum spin chains that satisfy the split property and are invariant under on-site finite group symmetries.
• To determine whether such states can be connected via symmetry-preserving automorphisms on left and right half-chains without altering entanglement.
• To identify a complete invariant for this classification, capturing topological obstructions arising from group symmetry.
• To clarify the role of second cohomology classes in characterizing symmetry-protected topological phases in quantum spin chains.

Proposed method

• The paper defines a set $SPG({ ilde{A}})$ of pure, split states invariant under the on-site action of a finite group $G$.
• For each state $\omega \in SPG({\tilde{A}})$, it constructs a projective representation of $G$ and associates a second cohomology class $c_{\omega,R} \in H^2(G, \mathbb{C}^*)$.
• It introduces a notion of equivalence where $\omega_0$ and $\omega_1$ are equivalent if $\omega_1 \sim_{q.e.} \omega_0 \circ (\Xi_L \otimes \Xi_R)$ for symmetry-preserving automorphisms $\Xi_L$, $\Xi_R$ on the left and right half-chains.
• It uses results from asymptotically inner automorphisms and quasi-equivalence to relate the structure of states to cohomological invariants.
• It applies techniques from $C^*$-algebra theory, including Glimm's Lemma and the Kadison transitivity theorem, to establish approximation properties of states and unitaries.
• It proves that the second cohomology class $c_{\omega,R}$ is a complete invariant for the equivalence relation defined on $SPG({\tilde{A}})$.

Experimental results

Research questions

• RQ1 Can any two symmetric pure split states on a quantum spin chain be connected via symmetry-preserving automorphisms on the left and right half-chains without altering their entanglement structure?
• RQ2 What topological or algebraic invariant captures the obstruction to such a connection when symmetry is imposed?
• RQ3 How does the second cohomology class of the group $G$ arise naturally from the structure of symmetric pure states satisfying the split property?
• RQ4 Is the second cohomology class $c_{\omega,R}$ sufficient to classify all such symmetric pure split states up to the defined equivalence relation?
• RQ5 To what extent does the split property constrain the possible symmetry-protected topological phases in quantum spin chains?

Key findings

• The second cohomology class $c_{\omega,R} \in H^2(G, \mathbb{C}^*)$ is a complete invariant for the classification of symmetric pure split states under symmetry-preserving quasi-equivalence.
• Not all symmetric pure split states can be connected via symmetric automorphisms on the half-chains, indicating a non-trivial topological obstruction.
• The obstruction to connecting states via symmetric automorphisms is fully captured by the second cohomology class of the group $G$ associated with the projective representation of the state.
• The classification reveals that symmetry-protected topological phases in quantum spin chains are classified by $H^2(G, \mathbb{C}^*)$, generalizing the notion of group cohomology in condensed matter physics.
• The result establishes a precise link between the algebraic structure of $C^*$-algebras and the topological invariants of quantum many-body systems with global symmetries.
• The paper confirms that the split property ensures small entanglement between left and right chains, and that symmetry-preserving automorphisms preserve this entanglement structure.