On Spin and Matrix Models in the Complex Plane

TL;DR

This paper explores the properties of statistical mechanics models in the complex plane, analyzing their renormalization flows, partition function zeros, and symmetries using exactly solvable models and matrix model techniques.

Contribution

It demonstrates the equivalence between matrix model double-scaling limits and finite-size scaling in 2D spin systems, providing a new numerical approach to determine string susceptibility exponents.

Findings

Renormalization group flows in the complex plane are characterized.

Partition function zeros distribution is analyzed for complex parameters.

Matrix model scaling relates to finite-size spin system behavior.

Abstract

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 2-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.