Partition function zeros for the one-dimensional ordered plasma in Dirichlet boundary conditions

Abstract

We consider the grand canonical partition function for the ordered one-dimensional, two-component plasma at fugacityζ in an applied electric fieldE with Dirichlet boundary conditions. The system has a phase transition from a low-coupling phase with equally spaced particles to a high-coupling phase with particles clustered into dipolar pairs. An exact expression for the partition function is developed. In zero applied field the zeros in theζ plane occupy the imaginary axis from −i∞ to −iζ_c and iζ_c to i∞ for some ζ_c. They also occupy the diamond shape of four straight lines from ±iζ_c to ζ_c and from ±iζ_c to −ζ_c. The fugacityζ acts like a temperature or coupling variable. The symmetry-breaking field is the applied electric fieldE. A finite-size scaling representation for the partition in scaled coupling and scaled electric field is developed. It has standard mean field form. When the scaled coupling is real, the zeros in the scaled field lie on the imaginary axis and pinch the real scaled field axis as the scaled coupling increases. The scaled partition function considered as a function of two complex variables, scaled coupling and scaled field, has zeros on a two-dimensional surface in a domain of four real variables. A numerical discussion of some of the properties of this surface is presented.