Crossover from low-temperature to high-temperature fluctuations. II. Nonuniversal thermodynamic Casimir forces of anisotropic systems

TL;DR

This paper extends finite-size renormalization-group theory to anisotropic systems, deriving exact results for thermodynamic Casimir forces and correlation functions, revealing nonuniversal behavior influenced by anisotropy.

Contribution

It introduces a multiparameter universality framework for anisotropic systems, generalizing previous isotropic models and providing explicit calculations for Casimir forces and correlation functions.

Findings

Anisotropy affects the sign and magnitude of Casimir forces.

Exact large-$n$ results for free energy and Casimir force.

Validation of multiparameter universality in anisotropic systems.

Abstract

The finite-size renormalization-group approach for isotropic O$(n)$-symmetric systems introduced previously [V. Dohm, Phys. Rev. Lett. {\bf 110}, 107207 (2013)] is extended to weakly anisotropic O$(n)$-symmetric systems. Our theory is formulated within the $\varphi^4$ model with lattice anisotropy in a $d$-dimensional block geometry with periodic boundary conditions. It describes the crossover from low- to high-temperature fluctuations including Goldstone-dominated and critical fluctuations for $1\leq n \leq \infty$ in $2<d<4$ dimensions. An exact representation is derived for the large-distance behavior of the bulk correlation function of anisotropic systems in terms of the principal correlation lengths and an anisotropy matrix ${\bf \bar A}$. This includes the long-ranged correlations with an anisotropic algebraic decay at low temperatures due to the Goldstone modes for $n>1$. We calculate the finite-size scaling functions of the excess free energy and thermodynamic Casimir force. Exact results are derived in the large-$n$ limit. Applications are given for $L_\parallel^{d-1} \times L$ slab geometries with a finite aspect ratio $\rho=L/L_\parallel$ as well as for the film limit $\rho \to 0$. For weakly anisotropic systems two-scale-factor universality is replaced by multiparameter universality. This implies a substantial reduction of the predictive power of bulk and finite-size theory for anisotropic systems as compared to isotropic systems. The validity of multiparameter universality is confirmed analytically for a nontrivial example of the $d=2,n=1$ universality class. Anisotropy-dependent minima of the Casimir force scaling function are found below $T_c$. Both the sign and magnitude of the Casimir amplitude in the Goldstone and critical regimes are affected by the lattice anisotropy. Quantitative predictions are made that can be tested by Monte Carlo simulations.