Numerical study of the dimensionally reduced 3D Ising model

TL;DR

This paper numerically investigates the 3D Ising model and a dimensionally reduced version, analyzing critical temperatures and exponents, revealing that the reduced model shares universality with the 2D Ising model.

Contribution

It provides the first detailed numerical analysis of the dimensionally reduced 3D Ising model and its critical properties across different fixed $N_z$ values.

Findings

Critical temperature $T_c$ varies smoothly between 2D and 3D values.

Dimensionally reduced models share the same universality class as the 2D Ising model.

Critical exponents $eta,\, ext{and}\, u$ match 2D Ising values across $N_z$.

Abstract

We study the 3D Ising model in the infinite volume limit $N_{x,y,z}\to\infty$ by means of numerical simulations. We determine $T_c$ as well as the critical exponents $\beta,\gamma$ and $\nu$, based on finite-size scaling and histogram reweighting techniques. In addition, we study a ``dimensionally reduced'' scenario where $N_z$ is kept fixed (e.g. at 2, 4, 8), while the limit $N_{x,y}\to\infty$ is taken. For each fixed $N_z$ we determine $T_c$ as well as $\beta,\gamma,\nu$. For $T_c$ we find a smooth transition curve which connects the well known critical temperatures of the 2D and the 3D Ising model. Regarding $\beta,\gamma,\nu$ our data suggest that the ``dimensionally reduced'' Ising model is in the same universality class as the 2D Ising model, regardless of $N_z$.