On fermion grading symmetry for quasi-local systems

We discuss fermion grading symmetry for quasi-local systems with graded commutation relations. We introduce a criterion of spontaneously symmetry breaking (SSB) for general quasi-local systems. 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 complementary outside system of a local region specified by the given quasi-local structure. Under a completely model independent setting, we show the absence of SSB for fermion grading symmetry in the above sense. We obtain some structural results for equilibrium states of lattice systems. If there would exist an even KMS state for some even dynamics that is decomposed into noneven KMS states, then those noneven states inevitably violate our local thermal stability condition.