Distribution Of Partition Function Zeros Research Articles

Partition function zeros of zeta-urns

We discuss the distribution of partition function zeros for the grand-canonical ensemble of the zeta-urn model, where tuning a single parameter can give a first or any higher order condensation transition. We compute the locus of zeros for finite-size systems and test scaling relations describing the accumulation of zeros near the critical point against theoretical predictions for both the first and higher order transition regimes.

Distribution of partition function zeros of the ±J model on the Bethe lattice

The distribution of partition function zeros is studied for the ±J model of spin glasses on the Bethe lattice. We find a relation between the distribution of complex cavity fields and the density of zeros, which enables us to obtain the density of zeros for the infinite system size by using the cavity method. The phase boundaries thus derived from the location of the zeros are consistent with the results of direct analytical calculations. This is the first example in which the spin-glass transition is related to the distribution of zeros directly in the thermodynamical limit. We clarify how the spin-glass transition is characterized by the zeros of the partition function. It is also shown that in the spin-glass phase a continuous distribution of singularities touches the axes of real field and temperature.

Evolution and structure formation of the distribution of partition function zeros: triangular type Ising lattices with cell decoration.

The distribution of partition function zeros of the two-dimensional Ising model in the complex temperature plane is studied within the context of triangular decorated lattices and their triangle-star transformations. Exact recursion relations for the zeros are deduced for the description of the evolution of the distribution of the zeros subject to the change of decoration level. In the limit of infinite decoration level, the decorated lattices essentially possess the Sierpiński gasket or its triangle-star transformation as the inherent structure. The positions of the zeros for the infinite decorated lattices are shown to coincide with the ones for the Sierpiński gasket or its triangle-star transformation, and the distributions of zeros all appear to be a union of infinite scattered points and a Jordan curve, which is the limit of the scattered points.

On spin and matrix models in the complex plane

We describe various aspects of statistical mechanics defined in the complex temperature or coupling-constant plane. Using exactly solvable models, we analyse such aspects as renormalization group flows in the complex plane, the distribution of partition function zeros, and the question of new coupling-constant symmetries of complex-plane spin models. The double-scaling form of matrix models is shown to be exactly equivalent to finite-size scaling of two-dimensional spin systems. This is used to show that the string susceptibility exponents derived from matrix models can be obtained numerically with very high accuracy from the scaling of finite- N partition function zeros in the complex plane.

The exact location of partition function zeros, a new method for statistical mechanics

A new mathematical mechanism is proposed for locating exactly sections of the locus of the limiting distribution of partition function zeros for model systems in statistical mechanics. Illustrations and applications to the two-dimensional Ising model and the hard square lattice gas are given; the exact location of the branch point singularity for the hard square lattice gas on the negative activity z axis is conjectured to be the negative root of the polynomial (2z6+9z5+42z4+90z3+96z2+27z+2).