Classification of Gapped Symmetric Phases in 1D Spin Systems

TL;DR

This paper classifies gapped quantum phases in 1D spin systems, showing that without symmetry all states are trivial, but with symmetry, multiple topologically distinct phases can exist, using local unitary equivalence.

Contribution

It provides a classification scheme for 1D gapped quantum phases based on local unitary equivalence, including symmetry-protected topological orders.

Findings

All 1D gapped states are trivial without symmetry.

Presence of symmetry leads to multiple distinct topological phases.

Results extend to higher dimensions with symmetry considerations.

Abstract

Quantum many-body systems divide into a variety of phases with very different physical properties. The question of what kind of phases exist and how to identify them seems hard especially for strongly interacting systems. Here we make an attempt to answer this question for gapped interacting quantum spin systems whose ground states are short-range correlated. Based on the local unitary equivalence relation between short-range correlated states in the same phase, we classify possible quantum phases for 1D matrix product states, which represent well the class of 1D gapped ground states. We find that in the absence of any symmetry all states are equivalent to trivial product states, which means that there is no topological order in 1D. However, if certain symmetry is required, many phases exist with different symmetry protected topological orders. The symmetric local unitary equivalence relation also allows us to obtain some simple results for quantum phases in higher dimensions when some symmetries are present.