Microcanonical finite-size scaling in specific heat diverging 2nd order phase transitions

TL;DR

This paper introduces a microcanonical finite-size scaling approach for second-order phase transitions with diverging specific heat, tested on models like the 3D Ising and 2D Potts, revealing precise critical exponent estimates.

Contribution

It extends phenomenological renormalization to the microcanonical ensemble using a new finite site Ansatz, validated through large-scale simulations and accounting for logarithmic corrections.

Findings

Accurate determination of Fisher-renormalized critical exponents.

Successful application of microcanonical cluster method to large systems.

Need for logarithmic corrections in Potts model analysis.

Abstract

A Microcanonical Finite Site Ansatz in terms of quantities measurable in a Finite Lattice allows to extend phenomenological renormalization (the so called quotients method) to the microcanonical ensemble. The Ansatz is tested numerically in two models where the canonical specific-heat diverges at criticality, thus implying Fisher-renormalization of the critical exponents: the 3D ferromagnetic Ising model and the 2D four-states Potts model (where large logarithmic corrections are known to occur in the canonical ensemble). A recently proposed microcanonical cluster method allows to simulate systems as large as L=1024 (Potts) or L=128 (Ising). The quotients method provides extremely accurate determinations of the anomalous dimension and of the (Fisher-renormalized) thermal $\nu$ exponent. While in the Ising model the numerical agreement with our theoretical expectations is impressive, in the Potts case we need to carefully incorporate logarithmic corrections to the microcanonical Ansatz in order to rationalize our data.