Field theory of finite-size effects in Ising-like systems

Crossover from low-temperature to high-temperature fluctuations: Universal and nonuniversal Casimir forces of isotropic and anisotropic systems

We study the crossover from low-temperature to high-temperature fluctuations including Goldstone-dominated and critical fluctuations in confined isotropic and weakly anisotropic O(n)-symmetric systems on the basis of a finite-size renormalization-group approach at fixed dimension d introduced previously [V. Dohm, Phys. Rev. Lett. 110, 107207 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.107207]. Our theory is formulated within the φ^{4} lattice model in a d-dimensional block geometry with periodic boundary conditions. We calculate the finite-size scaling functions F^{ex} and X of the excess free-energy density and the thermodynamic Casimir force, respectively, for 1≤n≤∞, 2<d<4. Exact results are derived for n→∞. Applications are given for L_{∥}^{d-1}×L slab geometry with an aspect ratio ρ=L/L_{∥}>0 and for film geometry (ρ=0). Good overall agreement is found with Monte Carlo (MC) data for isotropic spin models with n=1,2,3. For ρ=0, the low-temperature limits of F^{ex} and X vanish for n=1, whereas they are finite for n≥2. For ρ>0 and n=1, we find a finite low-temperature limit of F^{ex}, which deviates from that of the Ising model. We attribute this deviation to the nonuniversal difference between the φ^{4} model with continuous variables and the Ising model with discrete variables. For n≥2 and ρ>0, a logarithmic divergence of F^{ex} in the low-temperature limit is predicted, in excellent agreement with MC data. For 2≤n≤∞ and ρ<ρ_{0}=0.8567 the Goldstone modes generate a negative low-temperature Casimir force that vanishes for ρ=ρ_{0} and becomes positive for ρ>ρ_{0}. For anisotropic systems a unified hypothesis of multiparameter universality is introduced for both bulk and confined systems. The dependence of their scaling functions on d(d+1)/2-1 microscopic anisotropy parameters implies a substantial reduction of the predictive power of the theory for anisotropic systems as compared to isotropic systems. An exact representation is derived for the nonuniversal large-distance behavior of the bulk correlation function of anisotropic systems and quantitative predictions are made. The validity of multiparameter universality is proven analytically for the d=2,n=1 universality class. A nonuniversal anisotropy-dependent minimum of the Casimir force scaling function X is found. Both the sign and magnitude of X and the shift of the film critical temperature are affected by the lattice anisotropy.

Critical free energy and Casimir forces in rectangular geometries

We study the critical behavior of the free energy and the thermodynamic Casimir force in a L(∥)(d-1) × L block geometry in 2<d<4 dimensions with aspect ratio ρ=L/L(∥) on the basis of the O(n) symmetric ϕ4 lattice model with periodic boundary conditions and with isotropic short-range interactions. Exact results are derived in the large-n limit describing the geometric crossover from film (ρ=0) over cubic (ρ=1) to cylindrical (ρ=∞) geometries. For n=1, three perturbation approaches in the minimal renormalization scheme at fixed d are presented that cover both the central finite-size regime near T(c) for 1/4≲ρ≲3 and the region well above and below T(c). At bulk T(c), we predict the critical Casimir force in the vertical (L) direction to be negative (attractive) for a slab (ρ<1), positive (repulsive) for a rod (ρ>1), and zero for a cube (ρ=1). Our results for finite-size scaling functions agree well with Monte Carlo data for the three-dimensional Ising model by Hasenbusch for ρ=1 and by Vasilyev et al. for ρ=1/6 above, at, and below T(c).

Diversity of critical behavior within a universality class

We study spatial anisotropy effects on the bulk and finite-size critical behavior of the O(n) symmetric anisotropic phi;{4} lattice model with periodic boundary conditions in a d -dimensional hypercubic geometry above, at, and below Tc. The absence of two-scale factor universality is discussed for the bulk order-parameter correlation function, the bulk scattering intensity, and for several universal bulk amplitude relations. The anisotropy parameters are observable by scattering experiments at Tc. For the confined system, renormalization-group theory within the minimal subtraction scheme at fixed dimension d for 2<d<4 is employed. In contrast to the epsilon=4-d expansion, the fixed- d finite-size approach keeps the exponential form of the order-parameter distribution function unexpanded. For the case of cubic symmetry and for n=1 , our perturbation approach yields excellent agreement with the Monte Carlo (MC) data for the finite-size amplitude of the free energy of the three-dimensional Ising model at Tc by Mon [Phys. Rev. Lett. 54, 2671 (1985)]. The epsilon expansion result is in less good agreement. Below Tc, a minimum of the scaling function of the excess free energy is found. We predict a measurable dependence of this minimum on the anisotropy parameters. The relative anisotropy effect on the free energy is predicted to be significantly larger than that on the Binder cumulant. Our theory agrees quantitatively with the nonmonotonic dependence of the Binder cumulant on the ferromagnetic next-nearest-neighbor (NNN) coupling of the two-dimensional Ising model found by MC simulations of Selke and Shchur [J. Phys. A 38, L739 (2005)]. Our theory also predicts a nonmonotonic dependence for small values of the antiferromagnetic NNN coupling and the existence of a Lifshitz point at a larger value of this coupling. The nonuniversal anisotropy effects in the finite-size scaling regime are predicted to satisfy a kind of restricted universality. The tails of the large- L behavior at T++Tc violate both finite-size scaling and universality even for isotropic systems as they depend on the bare four-point coupling of the phi4 theory, on the cutoff procedure, and on subleading long-range interactions.

Multicanonical Monte Carlo study and analysis of tails for the order-parameter distribution of the two-dimensional Ising model

The tails of the critical order-parameter distribution of the two-dimensional Ising model are investigated through extensive multicanonical Monte Carlo simulations. Results for fixed boundary conditions are reported here, and compared with known results for periodic boundary conditions. Clear numerical evidence for "fat" stretched exponential tails exists below the critical temperature, indicating the possible presence of fat tails at the critical temperature. Our work suggests that the true order-parameter distribution at the critical temperature must be considered to be unknown at present.

Test of renormalization predictions for universal finite-size scaling functions

We calculate universal finite-size scaling functions for systems with an n-component order parameter and algebraically decaying interactions. Just as previously found for short-range interactions, this leads to a singular epsilon expansion, where epsilon is the distance to the upper critical dimension. Subsequently, we check the results by numerical simulations of spin models in the same universality class. Our systems offer the essential advantage that epsilon can be varied continuously, allowing an accurate examination of the region where epsilon is small. The numerical calculations turn out to be in striking disagreement with the predicted singularity.