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Macroscopic spontaneous symmetry breaking and its absence for fermion grading symmetry

We introduce a criterion named macroscopic spontaneously symmetry breaking, for short MSSB, for general quasi-local systems with any statistics. It is formulated based on the idea that each pair of distinct phases (appeared in spontaneous symmetry breaking) should be disjoint not only for the total system but also for every outside system of a local region specified by the given quasi-local structure. We show the absence of MSSB for fermion grading transformations that multiply fermion fields by −1. We obtain some structural result about the centers of Gibbs states for lattice systems with fermion or fermion-boson statistics. It shows that a Gibbs state is a factor state if and only if so is its restriction to any outside system of a local region. If fermion grading symmetry would be broken for any KMS state, then we can construct a noneven state from it (by perturbation of a local Hamiltonian multiplied by the inverse temperature) that satisfies the KMS condition, however, does not the local thermal stability condition.