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.
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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.