Crossover from Goldstone to critical fluctuations: Casimir forces in confined O${\bf(n)}$ symmetric systems

TL;DR

This paper analyzes the crossover of Casimir forces in confined O(n) systems, bridging Goldstone and critical fluctuation regimes, with analytical and Monte Carlo validation for the XY model.

Contribution

It provides a finite-size renormalization-group framework describing the entire crossover of Casimir forces in O(n) systems, including analytical results and Monte Carlo comparisons.

Findings

Analytic expression for Casimir force crossover in O(n) systems.

Good agreement with Monte Carlo data for 3D XY model.

Quantitative predictions for Heisenberg universality class.

Abstract

We study the crossover between thermodynamic Casimir forces arising from long-range fluctuations due to Goldstone modes and those arising from critical fluctuations. Both types of forces exist in the low-temperature phase of O$(n)$ symmetric systems for $n>1$ in a $d$-dimensional ${L_\parallel^{d-1} \times L}$ slab geometry with a finite aspect ratio $\rho = L/L_\parallel$. Our finite-size renormalization-group treatment for periodic boundary conditions describes the entire crossover from the Goldstone regime with a nonvanishing constant tail of the finite-size scaling function far below $T_c$ up to the region far above $T_c$ including the critical regime with a minimum of the scaling function slightly below $T_c$. Our analytic result for $\rho \ll 1$ agrees well with Monte Carlo data for the three-dimensional XY model. A quantitative prediction is given for the crossover of systems in the Heisenberg universality class.