Some Numerical Results on the Block Spin Transformation for the 2D Ising Model at the Critical Point

TL;DR

This paper investigates the effects of block spin transformations on the 2D Ising model at criticality, providing numerical evidence that the critical temperature decreases after transformation, with implications for understanding phase transitions.

Contribution

It introduces a rigorous approach to analyze the transformed measure's Gibbs potential and presents numerical evidence that the critical temperature lowers after the block spin transformation.

Findings

Numerical evidence shows $T'_c < T_c$ after transformation.

The Gibbs potential for the transformed measure is well defined under certain conditions.

Study of the Dobrushin-Shlosman uniqueness condition related to the model.

Abstract

We study the block spin transformation for the 2D Ising model at the critical temperature $T_c$. We consider the model with the constraint that the total spin in each block is zero. An old argument by Cassandro and Gallavotti allows to show that the Gibbs potential for the transformed measure is well defined, provided that such model has a critical temperature $T'_c$ lower than $T_c$. After describing a possible rigorous approach to the problem, we present numerical evidence that indeed $T'_c<T_c$, and a study of the Dobrushin-Shlosman uniqueness condition.