$\mathcal{PT}$ Symmetry in Statistical Mechanics and the Sign Problem

Abstract

Generalized \(\mathcal{PT}\) 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 \(\mathcal{PT}\)-symmetric Hamiltonian. There is a corresponding class of \(\mathcal{PT}\)-symmetric classical statistical mechanics models with non-Hermitian transfer matrices. We discuss a class of Z(N) spin models with explicit \(\mathcal{PT}\) symmetry and also the ANNNI model, which has a hidden \(\mathcal{PT}\) symmetry. For both quantum and classical models, the class of models with generalized \(\mathcal{PT}\) 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 \(\mathcal{PT}\) symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with \(\mathcal{PT}\) symmetry breaking of the ground state, and is generically a first-order transition. In the region where \(\mathcal{PT}\) 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 \(\mathcal{PT}\) symmetry which can be simulated for all parameter values, including cases where \(\mathcal{PT}\) symmetry is broken.