Sam Roberts: Symmetry protected topological order at nonzero temperature
1We address the question of whether symmetry-protected topological (SPT) order can persist at nonzero temperature. 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. Second, we provide an explicit example of a three-dimensional spin lattice system protected by a global generalized (1-form) symmetry that possesses a nontrivial SPT ordered thermal phase. 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 model system is the 3D cluster state model, used in the formulation of topological measurement-based quantum computation. 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.