Distributions of zeros of ising spin-glass partition functions in two-dimensional lattices

Abstract

In this thesis the characteristics of zero distributions of Ising spin-glass partition functions in two-dimensional lattices, such as square, triangular, and kagome lattices, are shown in complex temperature z plane. It has been found that the zeros of partition function of the model on square lattice lie on a unit circle. Similarly, in the Ising spin-glass triangular lattice, the zeros are located on a unit circle and the real z axis. In the Ising spin-glass kagome lattice, however, the zeros are distributed in the complex z plane and some of them are located both on the real as well as imaginary z axes. It has been known that the locus of zeros of partition function divides the complex plane into holomorphic regions in which the free energy is analytic. Only those regions on the positive real axis correspond to physical phases of the system while the off-positive real axis regions are nonphysical. These holomorphic regions as well as transition temperatures for both isotropic Ising and Ising spin-glass models in two-dimensional lattices are discussed in detail.