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arXiv:cond-mat/0112310

Violation of finite-size scaling for the free energy near criticality

X. S. Chen, V. Dohm

Published Dec 17, 2001• 1 citation
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Abstract

The singular part of the finite-size free energy density $f_s$ of the O($n$) symmetric $ϕ^4$ field theory is calculated for confined geometries of linear size L with periodic boundary conditions in the large-N limit and with Dirichlet boundary conditions in one-loop order. We find that both a sharp cutoff and a subleading long-range interaction cause a non-universal L dependence of $f_s$ near $T_c$. For film geometry this implies a non-universal critical Casimir force with an algebraic L dependence that dominates the exponential finite-size scaling behavior above $T_c$ for both periodic and Dirichlet boundary conditions.

DOI
10.48550/arxiv.cond-mat/0112310

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The singular part of the finite-size free energy density fsf_sfs​ of the O(nnn) symmetric ϕ4ϕ^4ϕ4 field theory is calculated for confined geometries of linear size L with periodic boundary conditions in the large-N limit and with Dirichlet boundary conditions in one-loop order. We find that both a sharp cutoff and a subleading long-range interaction cause a non-universal L dependence of fsf_sfs​ near TcT_cTc​. For film geometry this implies a non-universal critical Casimir force with an algebraic L dependence that dominates the exponential finite-size scaling behavior above TcT_cTc​ for both periodic and Dirichlet boundary conditions.
The singular part of the finite-size free-energy density f(s) of the O(n) symmetric phi(4) field theory is calculated for confined geometries of linear size L with periodic boundary conditions in the large-n limit and with Dirichlet boundary conditions in one-loop order. We find that both a sharp cutoff and a subleading long-range interaction cause a leading nonuniversal L dependence of f(s) near T(c). This implies a significant restriction for the validity of universal finite-size scaling for model systems and real systems. For film geometry we predict a leading nonuniversal contribution to the critical Casimir force above the superfluid transition of (4)He.
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.
A detailed analysis of the finite-size effects on the bulk critical behaviour of the d -dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different kinds of boundary conditions are applied: periodic ( p ), antiperiodic ( a ) and free surfaces with Dirichlet ( D ), Neumann ( N ) and a combination of Neumann and Dirichlet ( ND ) on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d . It is found, for 2 < d < 4, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for ( p ) and ( a ). For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At d = 3, the critical amplitude of the singular part of the free energy (related to the so-called Casimir amplitude) is estimated. We obtain Δ ( p ) = −2ζ(3)/(5π) = −0.153 051..., Δ ( a ) = 0.274 543... and Δ ( ND ) = 0.019 22..., implying a fluctuation-induced attraction between the surfaces for ( p ) and repulsion in the other two cases. For ( D ) and ( N ) we find a logarithmic dependence on L .
A detailed analysis of the finite–size effects on the bulk critical behaviour of the d–dimensional mean spherical model confined to a film geometry with finite thickness L is reported. Along the finite direction different boundary conditions are applied: periodic (p), antiperiodic (a) and free surfaces with Dirichlet (D), Neumann (N) and a combination of Neumann and Dirichlet (ND) on both surfaces. A systematic method for the evaluation of the finite–size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary d. It is found, for 2 < d < 4, that the singular part of the free energy has the required finite–size scaling form at the bulk critical temperature only for (p) and (a). For the remaining boundary conditions the standard finite–size scaling hypothesis is not valid. At d = 3, the critical amplitude (related to the so called Casimir amplitude) is estimated. We obtain
We briefly review recent results of exact calculations of critical Casimir forces of the O(n) ϕ 4 model as n→∞ on a three-dimensional strip bounded by two planar free surfaces at a distance L. This model has long-range order below the critical temperature Tc of the bulk phase transition only in the limit L→∞ but remains disordered for all T > 0 for an arbitrary finite strip width L < ∞. A proper description of the system scaling behavior near T c turns out to be a quite challenging problem because in addition to bulk, boundary, and finite-size critical behaviors, a nontrivial dimensional crossover must be handled. The model admits an exact solution in the limit n → ∞ in terms of the eigenvalues and eigenenergies of a self-consistent Schrödinger equation. This solution contains a potential v(z) with the near-boundary singular behavior v(z → 0+) ≈ −1/(4z 2)+ 4m/(π 2 z), where m = 1/ξ+(|t|) is the inverse bulk correlation length and t ~ (T − T c )/T c , and a corresponding singularity at the second boundary plane. In recent joint work with colleagues, the potential v(z), the excess free energy, and the Casimir force were obtained numerically with high precision. We explain how these numerical results can be complemented by exact analytic ones for several quantities (series expansion coefficients of v(z), the scattering data of v(z) in the semi-infinite case L = ∞ for all m >/< 0, and the low-temperature asymptotic behavior of the residual free energy and the Casimir force) by a combination of boundary-operator and short-distance expansions, proper extensions of the inverse scattering theory, new trace formulas, and semiclassical expansions.
Scaling density profiles of critical O(N)-symmetric systems confined to a film geometry are analyzed by field-theoretic renormalization-group techniques in d=4-\ensuremath{\varepsilon} spatial dimensions. At bulk criticality we explicitly calculate the energy density profile for five different boundary conditions and determine their corresponding universal scaling functions. In order to discuss the influence of the boundary conditions at the two surfaces of the film on the global shape of the profiles, a suitable exponentiation scheme is constructed. Using these exponentiated scaling functions as well as results from conformal field theory, the shapes of the energy density profiles in d=2,3, and 4 are compared. Finally, we discuss the short-distance behavior of the energy density profiles close to the surfaces of the film as it follows from the knowledge of the full profiles in first order \ensuremath{\varepsilon}.