Critical behaviour of the Ising model on the 4-dimensional lattice

TL;DR

This paper examines the critical behavior of the 4D Ising model, finding that data supports either a bounded specific heat or a logarithmic divergence, with implications for future research methods.

Contribution

The study provides new insights into the critical singularity of the 4D Ising model, challenging the field-theory prediction of a logarithmic divergence.

Findings

Canonical ensemble data is consistent with both bounded and logarithmic singularity.

Micro-canonical ensemble data favors a bounded specific heat.

Larger system sizes may be necessary for definitive Monte Carlo studies.

Abstract

In this paper we investigate the nature of the singularity of the Ising model of the 4-dimensional cubic lattice. It is rigorously known that the specific heat has critical exponent $\alpha=0$ but a non-rigorous field-theory argument predicts an unbounded specific heat with a logarithmic singularity at $T_c$. We find that within the given accuracy the canonical ensemble data is consistent both with a logarithmic singularity and a bounded specific heat, but that the micro-canonical ensemble lends stronger support to a bounded specific heat. Our conclusion is that either much larger system sizes are needed for Monte Carlo studies of this model in four dimensions or the field theory prediction of a logarithmic singularity is wrong.