Tensor Renormalization Group Calculations of Partition-Function Ratios

TL;DR

This paper uses tensor renormalization group methods to compute partition-function ratios in 2D models, confirming their universal critical values predicted by conformal field theory and observing logarithmic corrections in the four-state Potts model.

Contribution

It introduces a numerical approach to evaluate universal partition-function ratios at criticality across different models, aligning with CFT predictions.

Findings

Partition-function ratios follow finite-size scaling similar to the Binder parameter.

Critical ratios match universal values from conformal field theory.

Logarithmic corrections are observed in the four-state Potts model.

Abstract

The behavior of dimensionless quantities defined as ratios of partition functions is analyzed to investigate phase transitions and critical phenomena. At criticality, the universal values of these ratios can be predicted from conformal field theory (CFT) through the modular-invariant partition functions on a torus. We perform numerical calculations using the bond-weighted tensor renormalization group for three two-dimensional models belonging to different universality classes: the Ising model, the three-state Potts model, and the four-state Potts model. The partition-function ratios obey the same finite-size scaling form as the Binder parameter, and their critical values agree well with the universal values predicted by CFT. In the four-state Potts model, we observe logarithmic corrections in the system-size dependence of these ratios.