String order and symmetries in quantum spin lattices

TL;DR

This paper establishes a fundamental link between string order and local symmetries in quantum spin lattices, providing a comprehensive characterization and implications for phase transitions and higher dimensions.

Contribution

It proves the equivalence of string order and local symmetries in finitely correlated states and offers a new framework for understanding their properties and generalizations.

Findings

String order is equivalent to local symmetry in finitely correlated states.

Characterization of local symmetries explains properties of string order parameters.

Extension of concepts to higher-dimensional lattices and phase transition detection.

Abstract

We show that the existence of string order in a given quantum state is intimately related to the presence of a local symmetry by proving that both concepts are equivalent within the framework of finitely correlated states. Once this connection is established, we provide a complete characterization of local symmetries in these states. The results allow to understand in a straightforward way many of the properties of string order parameters, like their robustness/fragility under perturbations and their typical disappearance beyond strictly one-dimensional lattices. We propose and discuss an alternative definition, ideally suited for detecting phase transitions, and generalizations to two and more spatial dimensions.