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arXiv:1212.2647 · Physical Review B

Critical Casimir forces for O(N) models from functional renormalization

Paweł Jakubczyk, Marcin Napiórkowski

Published Apr 29, 2013• 12 citations
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Abstract

We consider the classical $O(N)$-symmetric models confined in a $d$-dimensional slablike geometry and subject to periodic boundary conditions. Applying the one-particle-irreducible variant of functional renormalization group (RG), we compute the critical Casimir forces acting between the slab boundaries. The applied truncation of the exact functional RG flow equation retains interaction vertices of arbitrary order. We evaluate the critical Casimir amplitudes ${\ensuremath{\Delta}}_{f}(d,N)$ for continuously varying dimensionality between two and three and $N=1,2$. Our findings are in very good agreement with exact results for $d=2$ and $N=1$. For $d=3$, our results are closer to Monte Carlo predictions than earlier field-theoretic RG calculations. Inclusion of the wave function renormalization and the corresponding anomalous dimension in the calculation has negligible impact on the computed Casimir forces.

DOI
10.1103/physrevb.87.165439

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We consider the classical O(N)O(N)O(N)-symmetric models confined in a ddd-dimensional slablike geometry and subject to periodic boundary conditions. Applying the one-particle-irreducible variant of functional renormalization group (RG), we compute the critical Casimir forces acting between the slab boundaries. The applied truncation of the exact functional RG flow equation retains interaction vertices of arbitrary order. We evaluate the critical Casimir amplitudes \ensuremathΔf(d,N){\ensuremath{\Delta}}_{f}(d,N)\ensuremathΔf​(d,N) for continuously varying dimensionality between two and three and N=1,2N=1,2N=1,2. Our findings are in very good agreement with exact results for d=2d=2d=2 and N=1N=1N=1. For d=3d=3d=3, our results are closer to Monte Carlo predictions than earlier field-theoretic RG calculations. Inclusion of the wave function renormalization and the corresponding anomalous dimension in the calculation has negligible impact on the computed Casimir forces.
The mapping between a classical length and inverse temperature as imaginary time provides a direct equivalence between the Casimir force of a classical system in DDD dimensions and internal energy of a quantum system in $d=D\ensuremath{-}1$ dimensions. The scaling functions of the critical Casimir force of the classical system with periodic boundaries thus emerge from the analysis of the symmetry related quantum critical point. We show that both nonperturbative renormalization group and quantum Monte Carlo analysis of quantum critical points provide quantitative estimates for the critical Casimir force in the corresponding classical model, giving access to widely different aspect ratios for the geometry of confined systems. In light of these results, we propose protocols for the realization of critical Casimir forces for periodic boundaries through state-of-the-art cold-atom and solid-state experiments.
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.
A critical phase confined between two parallel plates, with symmetry-breaking fields h1{h}_{1}h1​ and h2{h}_{2}h2​ acting on each plate, is considered for (i) h1h2>0{h}_{1}{h}_{2}>0h1​h2​>0 (denoted ab=++ab\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}++ab=++) and (ii) h1h2<0{h}_{1}{h}_{2}<0h1​h2​<0 ( $ab\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+\ensuremath{-}$). Using local-functional methods, we calculate order-parameter scaling functions ${\ensuremath{\Psi}}_{\mathrm{ab}}(x)$ and Casimir amplitudes Aab{A}_{\mathrm{ab}}Aab​, for general dimension ddd. At d=2d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2d=2, our ${\ensuremath{\Psi}}_{++}(x)$ almost coincides with exact conformal predictions. For the Ising universality class, we obtain expansions for Aab{A}_{\mathrm{ab}}Aab​ in $\ensuremath{\epsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4\ensuremath{-}d\ensuremath{\downarrow}0$ in excellent agreement with those obtained from field theory and, in d=3d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3d=3, new results are presented for Aab{A}_{\mathrm{ab}}Aab​ and ${\ensuremath{\Psi}}_{\mathrm{ab}}(x)$.
We present a renormalization-group study of the order-parameter distribution function near the critical point of O(n) symmetric three-dimensional (3D) systems in a finite geometry. The distribution function is calculated within the φ4 field theory for a 3D cube with periodic boundary conditions by means of a novel approach that appropriately deals with the Goldstone modes below T c . Results are given for both vanishing and finite external field h. The results describe finite-size effects near the critical point in the h– T-plane including the first-order transition at the coexistence line at h = 0 below T c . Quantitative theoretical predictions of the finite-size scaling function are presented for the Ising (n=1), XY(n=2) and Heisenberg (n=3) models. Good agreement is found with recent Monte Carlo data.
Finite-size effects are investigated in the Gaussian model with isotropic and anisotropic short-range interactions in film geometry with nonperiodic boundary conditions (bc) above, at, and below the bulk critical temperature Tc. We have obtained exact results for the free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed bc in 1<d<4 dimensions. For the Casimir force, finite-size scaling is found to be valid for all bc. For the free energy, finite-size scaling is valid in 1<d<3 and 3<d<4 dimensions for antiperiodic, Neumann, and Dirichlet bc, but logarithmic deviations from finite-size scaling exist in d=3 dimensions for Neumann and Dirichlet bc. This is explained in terms of the borderline dimension d*=3 , where the critical exponent 1-α-ν=(d-3)∕2 of the Gaussian surface energy density vanishes. For Neumann-Dirichlet bc, finite-size scaling is strongly violated above Tc for 1<d<4 because of a cancelation of the leading scaling terms. For antiperiodic, Dirichlet, and Neumann-Dirichlet bc, a finite film critical temperature Tc,film(L)<Tc exists at finite film thickness L . Our results include an exact description of the dimensional crossover between the d -dimensional finite-size critical behavior near bulk Tc and the (d-1) -dimensional critical behavior near Tc,film(L). This dimensional crossover is illustrated for the critical behavior of the specific heat. Particular attention is paid to an appropriate representation of the free energy in the region Tc,film(L)≤T≤Tc. For 2<d<4 , the Gaussian results are renormalized and reformulated as one-loop contributions of the φ4 field theory at fixed dimension d and are then compared with the ε=4-d expansion results at ε=1 as well as with d=3 Monte Carlo data. For d=2 , the Gaussian results for the Casimir force scaling function are compared with those for the Ising model with periodic, antiperiodic, and free bc; unexpected exact relations are found between the Gaussian and Ising scaling functions. For both the d -dimensional Gaussian model and the two-dimensional Ising model it is shown that anisotropic couplings imply nonuniversal scaling functions of the Casimir force that depend explicitly on microscopic couplings. Our Gaussian results provide the basis for the investigation of finite-size effects of the mean spherical model in film geometry with nonperiodic bc above, at, and below the bulk critical temperature.
The Casimir force due to thermal fluctuations (or the pseudo-Casimir force) was previously calculated for a perfect Bose gas in the slab geometry for various boundary conditions. The Casimir pressure due to quantum fluctuations in a weakly-interacting dilute Bose–Einstein condensate (BEC) confined to a parallel plate geometry was recently calculated for Dirichlet boundary conditions. In this paper we calculate the Casimir energy and pressure due to quantum fluctuations in a zero-temperature homogeneous weakly-interacting dilute BEC confined to a parallel plate geometry with periodic boundary conditions and include higher-order corrections to the leading-order result which we refer to as Bogoliubov corrections. The leading-order term is identified as the Casimir energy of a massless scalar field moving with a wave velocity equal to the speed of sound in the BEC. We then obtain the leading-order Casimir pressure in a general three-dimensional rectangular cavity of arbitrary length and obtain the finite-size correction to the parallel plate scenario.