Nonlocal Entanglement of 1D Thermal States Induced by Fermion Exchange Statistics

When two identical fermions exchange their positions, their wave function gains phase factor $-1$. We show that this distance-independent effect can induce nonlocal entanglement in one-dimensional (1D) electron systems having Majorana fermions at the ends. It occurs in the system bulk and has nontrivial temperature dependence. In a system having a single Majorana at each end, the nonlocal entanglement has a Bell-state form at zero temperature and decays as temperature increases, vanishing suddenly at certain finite temperature. In a system having two Majoranas at each end, it is in a cluster-state form and its nonlocality is more noticeable at finite temperature. By contrast, thermal states of corresponding 1D spins do not have nonlocal entanglement.