The critical Casimir force and its fluctuations in lattice spin models: exact and Monte Carlo results

TL;DR

This paper develops a stress tensor operator for lattice spin models to analyze the critical Casimir force and its fluctuations, providing exact results and Monte Carlo simulations across various models and dimensions.

Contribution

It introduces a new stress tensor operator for lattice models and characterizes the fluctuations and distribution of the critical Casimir force near $T_c$.

Findings

Exact results for 2D Ising, Gaussian, and spherical models.

Monte Carlo simulations for 3D Ising, XY, and Heisenberg models.

Force fluctuations follow a Gaussian distribution at $T_c$.

Abstract

We present general arguments and construct a stress tensor operator for finite lattice spin models. The average value of this operator gives the Casimir force of the system close to the bulk critical temperature $T_c$. We verify our arguments via exact results for the force in the two-dimensional Ising model, $d$-dimensional Gaussian and mean spherical model with $2<d<4$. On the basis of these exact results and by Monte Carlo simulations for three-dimensional Ising, XY and Heisenberg models we demonstrate that the standard deviation of the Casimir force $F_C$ in a slab geometry confining a critical substance in-between is $k_b T D(T)(A/a^{d-1})^{1/2}$, where $A$ is the surface area of the plates, $a$ is the lattice spacing and $D(T)$ is a slowly varying nonuniversal function of the temperature $T$. The numerical calculations demonstrate that at the critical temperature $T_c$ the force possesses a Gaussian distribution centered at the mean value of the force $<F_C>=k_b T_c (d-1)\Delta/(L/a)^{d}$, where $L$ is the distance between the plates and $\Delta$ is the (universal) Casimir amplitude.