Symmetry protected topological order at nonzero temperature

TL;DR

This paper investigates the thermal stability of symmetry-protected topological (SPT) order, showing that certain SPT phases cannot survive at nonzero temperature, but identifying a 3D cluster state model with a nontrivial SPT thermal phase protected by a generalized symmetry.

Contribution

It proves the non-existence of nontrivial SPT order at nonzero temperature protected by on-site symmetry and identifies a 3D cluster state with a thermal SPT phase protected by a 1-form symmetry.

Findings

Nontrivial SPT order protected by on-site symmetry cannot persist at nonzero temperature.

The 3D cluster state model has a thermal SPT phase with a 1-form symmetry.

High error tolerance in the 3D cluster state is linked to quantum error correction enforcing a 1-form symmetry.

Abstract

We address the question of whether symmetry-protected topological (SPT) order can persist at nonzero temperature, with a focus on understanding the thermal stability of several models studied in the theory of quantum computation. We present three results in this direction. First, we prove that nontrivial SPT order protected by a global on-site symmetry cannot persist at nonzero temperature, demonstrating that several quantum computational structures protected by such on-site symmetries are not thermally stable. Second, we prove that the 3D cluster state model used in the formulation of topological measurement-based quantum computation possesses a nontrivial SPT-ordered thermal phase when protected by a global generalized (1-form) symmetry. The SPT order in this model is detected by long-range localizable entanglement in the thermal state, which compares with related results characterizing SPT order at zero temperature in spin chains using localizable entanglement as an order parameter. Our third result is to demonstrate that the high error tolerance of this 3D cluster state model for quantum computation, even without a protecting symmetry, can be understood as an application of quantum error correction to effectively enforce a 1-form symmetry.