Nonmonotonic External Field Dependence of the Magnetization in a Finite Ising Model: Theory and MC Simulation

TL;DR

This paper combines field theory and Monte Carlo simulations to analyze the nonmonotonic behavior of magnetization in finite 3D Ising models under external fields, revealing detailed finite-size effects and scaling properties.

Contribution

It provides a theoretical and numerical investigation of finite-size effects and nonmonotonic magnetization behavior in the 3D Ising model with external fields, confirming predictions with simulations.

Findings

Nonmonotonic dependence of magnetization on external field at criticality.

Good quantitative agreement between theory and Monte Carlo data.

Identification of crossover from nonmonotonic to monotonic behavior.

Abstract

Using $\phi^4$ field theory and Monte Carlo (MC) simulation we investigate the finite-size effects of the magnetization $M$ for the three-dimensional Ising model in a finite cubic geometry with periodic boundary conditions. The field theory with infinite cutoff gives a scaling form of the equation of state $h/M^\delta = f(hL^{\beta\delta/\nu}, t/h^{1/\beta\delta})$ where $t=(T-T_c)/T_c$ is the reduced temperature, $h$ is the external field and $L$ is the size of system. Below $T_c$ and at $T_c$ the theory predicts a nonmonotonic dependence of $f(x,y)$ with respect to $x \equiv hL^{\beta\delta/\nu}$ at fixed $y \equiv t/h^{1/\beta \delta}$ and a crossover from nonmonotonic to monotonic behaviour when $y$ is further increased. These results are confirmed by MC simulation. The scaling function $f(x,y)$ obtained from the field theory is in good quantitative agreement with the finite-size MC data. Good agreement is also found for the bulk value $f(\infty,0)$ at $T_c$.