Critical exponents from cluster coefficients

TL;DR

This paper develops a renormalization-based method to accurately compute critical exponents in repulsive interaction models, overcoming limitations of traditional series extrapolation techniques near phase transitions.

Contribution

It extends previous results by deriving a corrected formula for critical exponents applicable to more general asymptotic behaviors of cluster matrix elements.

Findings

Derived a generalized formula for critical exponents using RG analysis.

Validated the formula numerically on physical models.

Identified the limits of the formula's applicability.

Abstract

For a large class of repulsive interaction models, the Mayer cluster integrals can be transformed into a tridiagonal real symmetric matrix $R_{mn}$, whose elements converge to two constants. This allows for an effective extrapolation of the equation of state for these models. Due to a nearby (nonphysical) singularity on the negative real z axis, standard methods (e.g. Pad\`e approximants based on the cluster integrals expansion) fail to capture the behavior of these models near the ordering transition, and, in particular, do not detect the critical point. A recent work (Eisenberg and Baram, PNAS {\bf 104}, 5755 (2007)) has shown that the critical exponents $\sigma$ and $\sigma'$, characterizing the singularity of the density as a function of the activity, can be exactly calculated if the decay of the $R$ matrix elements to their asymptotic constant follows a $1/n^2$ law. Here we employ renormalization arguments to extend this result and analyze cases for which the asymptotic approach of the $R$ matrix elements towards their limiting value is of a more general form. The relevant asymptotic correction terms (in RG sense) are identified and we then provide a corrected exact formula for the critical exponents. We identify the limits of usage of the formula, and demonstrate one physical model which is beyond its range of validity. The new formula is validated numerically and then applied to analyze a number of concrete physical models.