PT Symmetry in Statistical Mechanics and the Sign Problem

Generalized PT symmetry provides crucial insight into the sign problem for two classes of models. In the case of quantum statistical models at non-zero chemical potential, the free energy density is directly related to the ground state energy of a non-Hermitian, but generalized PT -symmetric Hamiltonian. There is a corresponding class of PT -symmetric classical statistical mechanics models with non-Hermitian transfer matrices. We discuss a class of Z(N) spin models with explicit PT symmetry and also the ANNNI model, which has a hidden PT symmetry. For both quantum and classical models, the class of models with generalized PT symmetry is precisely the class where the complex weight problem can be reduced to real weights, i.e., a sign problem. The spatial two-point functions of such models can exhibit three different behaviors: exponential decay, oscillatory decay, and periodic behavior. The latter two regions are associated with PT symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with PT symmetry breaking of the ground state, and is generically a first-order transition. In the region where PT symmetry is unbroken, the sign problem can always be solved in principle using the equivalence to a Hermitian theory in this region. The ANNNI model provides an example of a model with PT symmetry which can be simulated for all parameter values, including cases where PT symmetry is broken.