Uniformizing Lee-Yang Singularities

TL;DR

This paper presents a method to reconstruct the critical behavior of thermodynamic systems near Lee-Yang singularities using limited local data, enabling analysis across different phase transition regions.

Contribution

It introduces a uniformizing map approach to extend the reconstruction of the equation of state beyond the convergence radius, connecting crossover and first-order transition regions.

Findings

Reconstruction of the equation of state from limited Taylor coefficients.

Extension of analysis from crossover to first-order transition regions.

Application demonstrated with the Chiral Random Matrix Model and Ising Model.

Abstract

Motivated by the search for the QCD critical point, we discuss how to obtain the singular behavior of a thermodynamic system near a critical point, namely the Lee-Yang singularities, from a limited amount of local data generated in a different region of the phase diagram. We show that by using a limited number of Taylor series coefficients, it is possible to reconstruct the equation of state past the radius of convergence, in particular in the critical region. Furthermore we also show that it is possible to extend this reconstruction to go from a crossover region to the first-order transition region in the phase diagram, using a uniformizing map to pass between Riemann sheets. We illustrate these ideas via the Chiral Random Matrix Model and the Ising Model.