Symmetry projection converts the thermally trivial 3D cluster model into a system with SPT, projected-paramagnetic, and disordered phases, distinguished by a quantized membrane invariant taking values -1, +1, and 0.
ParaToric 1.0: Continuous-time quantum Monte Carlo for the toric code in a parallel field
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We introduce ParaToric, a C++ package for simulating the toric code in a parallel field (i.e., $X$- and $Z$-fields) at finite temperature. We implement and extend the continuous-time quantum Monte Carlo algorithm of Wu, Deng, and Prokof'ev on the square, triangular, honeycomb, and cubic lattices with either periodic or smooth open boundaries. The package is expandable to arbitrary lattice geometries and custom observables diagonal in either the $X$- or $Z$-basis. ParaToric also supports snapshot extraction in both bases, making it ideal for generating training/benchmarking data for other methods, such as lattice gauge theories, cold atom or other quantum simulators, quantum spin liquids, artificial intelligence, and quantum error correction. The software provides bindings to C/C++ and Python, and is thus almost universally integrable into other software projects.
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Quantized topological invariant of symmetry-projected Gibbs states
Symmetry projection converts the thermally trivial 3D cluster model into a system with SPT, projected-paramagnetic, and disordered phases, distinguished by a quantized membrane invariant taking values -1, +1, and 0.