Classification of the phases of 1D spin chains with commuting Hamiltonians

TL;DR

This paper classifies phases of 1D spin chains with translationally invariant, commuting, and scale-invariant Hamiltonians by analyzing the structure of their ground states through associated directed graphs.

Contribution

It introduces a graph-based framework to determine ground state degeneracy and phase distinctions in a specific class of 1D spin Hamiltonians, extending previous algebraic methods.

Findings

Ground state degeneracy corresponds to graph cycles.

Ground state degeneracy uniquely characterizes phases.

The structure of ground space can be inferred from the associated graph.

Abstract

We consider the class of spin Hamiltonians on a 1D chain with periodic boundary conditions that are (i) translational invariant, (ii) commuting and (iii) scale invariant, where by the latter we mean that the ground state degeneracy is independent of the system size. We correspond a directed graph to a Hamiltonian of this form and show that the structure of its ground space can be read from the cycles of the graph. We show that the ground state degeneracy is the only parameter that distinguishes the phases of these Hamiltonians. Our main tool in this paper is the idea of Bravyi and Vyalyi (2005) in using the representation theory of finite dimensional C^*-algebras to study commuting Hamiltonians.