Exact Critical Casimir Amplitude of Anisotropic Systems from Conformal Field Theory and Self-Similarity of Finite-Size Scaling Functions in $d\geq 2$ Dimensions

TL;DR

This paper derives the exact critical Casimir amplitude for anisotropic systems in two dimensions using conformal field theory, revealing complex self-similarity and universality properties of finite-size scaling functions.

Contribution

It combines CFT with anisotropic $$ theory to obtain explicit results for anisotropic models, demonstrating multiparameter universality and nonuniversality of the Casimir amplitude.

Findings

Exact Casimir amplitude derived for anisotropic 2D systems.

Revealed complex self-similarity near anisotropy instability.

Predicted self-similar structures in higher dimensions for $O(n)$ systems.

Abstract

The exact critical Casimir amplitude is derived for anisotropic systems within the $d=2$ Ising universality class by combining conformal field theory (CFT) with anisotropic $\varphi^4$ theory. Explicit results are presented for the general anisotropic scalar $\varphi^4$ model and for the fully anisotropic triangular-lattice Ising model in finite rectangular and infinite strip geometries with periodic boundary conditions (PBC). These results demonstrate the validity of multiparameter universality for confined anisotropic systems and the nonuniversality of the critical Casimir amplitude. We find an unexpected complex form of self-similarity of the anisotropy effects near the instability where weak anisotropy breaks down. This can be traced back to the property of modular invariance of isotropic CFT for $d=2$. More generally, for $d>2$ we predict the existence of self-similar structures of the finite-size scaling functions of $O(n)$-symmetric systems with planar anisotropies and PBC both in the critical region for $ n \geq 1$ as well as in the Goldstone-dominated low-temperature region for $ n \geq 2$.