The spectral gap for some spin chains with discrete symmetry breaking

TL;DR

This paper demonstrates that certain quantum spin chain models with broken discrete symmetries can be constructed with a spectral gap, using a general approach applicable to various systems, and provides explicit examples.

Contribution

It proves the existence of translation-invariant finite-range Hamiltonians with specified GVBS ground states and establishes spectral gaps for models satisfying natural conditions.

Findings

Existence of Hamiltonians with prescribed GVBS ground states.

All GVBS models with natural conditions have a spectral gap.

A general method for lower bounding spectral gaps in spin systems.

Abstract

We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and therefore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with an number of examples.