A Quasi-exact Formula for Ising critical temperatures on Hypercubic Lattices

TL;DR

This paper presents a quasi-exact power law formula for estimating the critical temperatures of the Ising model on hypercubic lattices, achieving high accuracy across multiple dimensions.

Contribution

The authors introduce a new quasi-exact formula for Ising critical temperatures on hypercubes, with minimal errors compared to known exact estimates.

Findings

Formula accurately predicts critical temperatures with errors less than 0.0005.

The power law behavior holds across dimensions 2 to 7.

Extension to other lattice types is discussed.

Abstract

We report a quasi-exact power law behavior for Ising critical temperatures on hypercubes. It reads $J/k_BT_c=K_0[(1-1/d)(q-1)]^a$ where $K_0=0.6260356$, $a=0.8633747$, $d$ is the space dimension, $q$ the coordination number ($q=2d$), $J$ the coupling constant, $k_B$ the Boltzman constant and $T_c$ the critical temperature. Absolute errors from available exact estimates ($d=2$ up to $d=7$) are always less than $0.0005$. Extension to other lattices is discussed.