Multiparameter universality for confined weakly anisotropic systems with realistic boundary conditions: anisotropy parameters and the critical Casimir force

Near a continuous phase transition, the finite-size behavior of a confined isotropic system is governed by two-scale-factor universality, and the fluctuation-induced critical Casimir force is universal within a universality class. In weakly anisotropic systems — systems with direction-dependent couplings, as in non-cubic magnets, superconductors, and liquid crystals — up to $d(d+1)/2+1$ nonuniversal parameters enter instead (multiparameter universality): in two dimensions the ratio of the principal correlation lengths and the orientation angle of the correlation ellipse, in three dimensions five anisotropy parameters. This framework has been derived and tested primarily for periodic boundary conditions; for non-periodic boundaries, exact multiparameter results are so far restricted to special settings such as corner free energies with free boundaries at criticality and axes-aligned orthorhombic geometries. Dohm's review (2023/24) poses the verification of multiparameter universality for confined anisotropic systems with realistic boundary conditions (Dirichlet, Neumann, open) as a major open challenge: no analytic prediction, no Monte Carlo data, and no experimental measurement exists for the critical Casimir force of anisotropic systems with realistic boundary conditions, and the anisotropy parameters are known exactly only for the triangular-lattice Ising model and the $\varphi^4$ model, not for other anisotropic Ising models (e.g. the Z-invariant models) nor for three-dimensional systems. These gaps remain open: adjacent partial results exist — exact corner free energies with free boundaries at criticality (2024), axes-aligned orthorhombic strips and cylinders (2020–2024), and a free-surface propagator for strongly anisotropic Lifshitz-point systems (2025) — but none determines the required anisotropy parameters for further anisotropic models, and none computes or verifies the anisotropic critical Casimir force with realistic boundary conditions.

Background

Near a continuous phase transition the correlation length $\xi$ diverges, and when a system is confined to a linear extent $L$ comparable to $\xi$, its free energy acquires universal finite-size contributions. For isotropic systems (identical correlations in all directions) the principle of two-scale-factor universality of Privman and Fisher states that all finite-size scaling functions depend on only two nonuniversal metric factors, such as the amplitudes of the correlation length above and below the critical temperature $T_c$. In particular, the critical Casimir force — the fluctuation-induced force between the confining boundaries, the thermodynamic analogue of the electromagnetic Casimir effect, predicted in 1978 for critical binary mixtures — is described in isotropic systems by a single universal scaling function $X(x)$ of the scaling variable $x=t(L/\xi_{0+})^{1/\nu}$, where $t=(T-T_c)/T_c$, $\nu$ is the correlation-length exponent, and $\xi_{0+}$ is a nonuniversal amplitude.

Weakly anisotropic systems have direction-dependent couplings (different exchange strengths along different lattice axes, or additional diagonal bonds), as in most real materials — non-cubic magnets, superconductors, liquid crystals, solids with non-cubic symmetry. Weak anisotropy leaves the bulk critical exponents unchanged but destroys two-scale-factor universality. A linear shear transformation maps the anisotropic system onto an isotropic one, and the finite-size scaling form then involves up to $d(d+1)/2+1$ nonuniversal parameters in $d$ dimensions — multiparameter universality. In two dimensions these are, besides two amplitude scales, the anisotropy ratio $q=\xi_{0+}^{(1)}/\xi_{0+}^{(2)}$ of the two principal correlation-length amplitudes and the orientation angle $\Omega$ of the principal axes of the correlation ellipse relative to the lattice; in three dimensions five anisotropy parameters enter (three orientation angles and two independent correlation-length ratios) besides two amplitude scales, seven nonuniversal parameters in all. As a consequence, quantities that are universal in isotropic systems — the critical Binder cumulant (a dimensionless ratio of moments of the order-parameter distribution used in finite-size analyses), the critical Casimir amplitude (the amplitude of the critical contribution to the confined free energy at $t=0$), and the critical Casimir force scaling function — become nonuniversal, with an intrinsic diversity fixed by the anisotropy parameters.

This framework has been established in full for periodic boundary conditions (BC), where the shear transformation exactly maps the finite anisotropic system onto an isotropic one. Dohm (2019) derived the exact anisotropic critical two-point function of the two-dimensional Ising universality class and confirmed it for the fully anisotropic triangular-lattice Ising model, whose anisotropy parameters $q$ and $\Omega$ are known exactly as functions of the microscopic couplings from Vaidya's exact triangular-lattice spin-spin correlation functions (1976). Dohm and Wessel (2021) obtained the exact critical Casimir amplitude of anisotropic systems in $d\geq2$ from conformal field theory (the continuum field theory governing two-dimensional critical points, whose modular-invariant torus partition functions generate complex self-similar finite-size scaling functions — functions that reproduce themselves, up to rescaling, under complex dilations of the scaling variable), and this was confirmed by Monte Carlo simulations of the anisotropic triangular-lattice Ising model on a torus (Dohm, Wessel, Kalthoff, and Selke, 2021). For three-dimensional anisotropic films with periodic BC, Casimir force scaling functions are available approximately from renormalization-group theory at fixed dimension (perturbative renormalization-group calculations carried out directly in $d=3$ rather than in an expansion about the upper critical dimension) and exactly in the large-$n$ limit (the solvable limit of the $O(n)$ spin model with infinitely many components) (Dohm, 2018; Dohm, 2023). Beyond the triangular-lattice Ising model and the $\varphi^4$ model, however, the anisotropy parameters are unknown even in two dimensions: for the general anisotropic Ising model they are generically undetermined, and in particular they have not been extracted for the Z-invariant Ising models of Au-Yang and Perk — exactly solvable inhomogeneous Ising models on isoradial graphs (graphs whose faces can each be circumscribed by a circle), constructed via the star-triangle integrability relation, whose correlation functions have been known from integrability theory since the 1980s. In three dimensions, the determination of the five anisotropy parameters for specific systems is, in the source's words, a nontrivial task.

For realistic boundary conditions the general framework is still open: exact multiparameter results beyond periodic BC exist only in restricted settings, such as corner free energies at criticality with free boundaries and axes-aligned (orthorhombic) geometries. Dirichlet BC (order parameter suppressed at the walls, appropriate for superconducting films), Neumann BC (normal derivative suppressed), and open BC (free boundary spins) all break the shear covariance that underlies the periodic-BC results: a shear that maps an anisotropic periodic torus onto an isotropic one tilts boundary-aligned surfaces. At present no analytic prediction, no Monte Carlo data, and no experimental measurement exist for the critical Casimir force in anisotropic systems with realistic BC; experiments on the critical Casimir effect exist only for isotropic media (superfluid $^4$He films and colloidal suspensions near critical demixing). This is the open challenge posed by Dohm's review of the multiparameter-universality program.

Problem Statement

Dohm's review poses the verification of multiparameter universality for confined anisotropic systems with realistic boundary conditions as a major open challenge. The problem is to extend the multiparameter-universality description of weakly anisotropic critical systems from periodic to realistic (Dirichlet, Neumann, or open) boundary conditions and to make it predictive and verifiable for concrete systems, with the measurable critical Casimir force as the central target. The challenge comprises three interconnected components.

(1) Nonuniversal inputs: the anisotropy parameters — in two dimensions the ratio $q$ of the principal correlation-length amplitudes and the orientation angle $\Omega$ of the correlation ellipse (together with two amplitude scales), and in three dimensions the five anisotropy parameters (three orientation angles and two independent correlation-length ratios) — enter the scaling description as functions of the microscopic couplings and must be known for the concrete system under study. The source also identifies the determination of these parameters for further anisotropic models as an open direction in its own right: for anisotropic two-dimensional Ising models beyond the triangular lattice whose correlation functions are already known exactly, such as the Z-invariant Ising models of Au-Yang and Perk, and for concrete three-dimensional systems.

(2) Universal structure with realistic boundary conditions: compute the critical Casimir force scaling function of confined weakly anisotropic systems with realistic BC — in particular the Dirichlet BC appropriate to anisotropic superconducting films near $T_c$, for which a measurable electrical potential difference between film and connected bulk has been proposed in the literature as the observable counterpart of the critical Casimir force — including its full dependence on the anisotropy parameters, by suitable methods — exact two-dimensional lattice techniques, boundary conformal field theory (conformal field theory adapted to domains with boundaries), renormalization-group theory at fixed dimension, the functional renormalization group (a formulation of the renormalization group as an exact flow equation for the full effective action), or large-$n$ limits.

(3) Verification: test the resulting predictions of multiparameter universality with realistic BC by Monte Carlo simulation of anisotropic lattice models and/or by experimental measurement of the critical Casimir force in an anisotropic critical medium (superconductors, liquid crystals, anisotropic magnets).

The components are interrelated: the anisotropy parameters are the nonuniversal inputs consumed by the universal scaling structure, and verification distinguishes an established extension from a conjectural one. In the framing of the source, the challenge is to predict the critical Casimir force of a weakly anisotropic confined system with realistic boundary conditions quantitatively — with the anisotropy parameters determined for the concrete system under study — and to verify the prediction against simulation or experiment. A resolution is the conjunction of components (2) and (3) for at least one concrete system: a quantitatively predictive critical Casimir force under realistic BC, with that system's anisotropy parameters determined, confirmed by Monte Carlo simulation (or, where available, experimental measurement). Determining the anisotropy parameters (1) for an additional model, or supplying a prediction that has not been verified, is a partial result rather than a resolution.

References

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