arXiv:1401.1357 · Journal of Physics A Mathematical and Theoretical
TheO(n) ϕ4model with free surfaces in the large-nlimit: some exact results for boundary critical behaviour, fluctuation-induced forces and distant-wall corrections
The O ( n ) ϕ 4 model on a slab bounded by free surfaces is studied for 2 < d < 4 in the limit n → ∞ . The self-consistent potential V ( z ) which the exact n → ∞ solution of the model involves is analysed by means of boundary operator expansions. Building on the known exact n → ∞ solution for V ( z ) in the semi-infinite case L = ∞ at the bulk critical point, we exactly determine two types of corrections to this potential: (i) those linear in the temperature scaling field t at L = ∞ , and (ii) the leading L -dependent (distant-wall) corrections at the critical point. From (i) exact analytical results at d = 3 are obtained for the leading temperature singularity of the excess surface free energy and the implied asymptotic behaviours of the scaling functions Θ 3 ( x ) and ϑ 3 ( x ) of the residual free energy f res = L 1 − d Θ d ( tL ) and the critical Casimir force in the limit x → 0 ±. The second derivative is computed exactly.
The O ( n ) ϕ 4 model on a slab bounded by free surfaces is studied for 2 < d < 4 in the limit n → ∞ . The self-consistent potential V ( z ) which the exact n → ∞ solution of the model involves is analysed by means of boundary operator expansions. Building on the known exact n → ∞ solution for V ( z ) in the semi-infinite case L = ∞ at the bulk critical point, we exactly determine two types of corrections to this potential: (i) those linear in the temperature scaling field t at L = ∞ , and (ii) the leading L -dependent (distant-wall) corrections at the critical point. From (i) exact analytical results at d = 3 are obtained for the leading temperature singularity of the excess surface free energy and the implied asymptotic behaviours of the scaling functions Θ 3 ( x ) and ϑ 3 ( x ) of the residual free energy f res = L 1 − d Θ d ( tL ) and the critical Casimir force in the limit x → 0 ±. The second derivative is computed exactly.
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.
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
Semi-infinite O ( n ) systems are considered at a bulk tricritical point. A continuum model with a |ϕ| 6 bulk nonlinearity and a |ϕ| 4 surface nonlinearity is used to analyse the critical behaviour at the special transition in d = 3 and d = 3−ε dimensions. For n = 0 the model describes the statistics of polymers at the θ-point, in the presence of a wall. It is shown that the surface critical exponents have a systematic expansion in powers of ε 1/2 . For d = 3, logarithmic corrections of an unusual form are found for surface quantities. In the special case of vanishing bulk nonlinearity the surface exponents can be expressed in terms of known bulk exponents for systems with long-range interactions.
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.
Abstract Based on the results published recently (Hucht 2017 J. Phys. A: Math. Theor . 50 065201), the universal finite-size contributions to the free energy of the square lattice Ising model on the L × M rectangle, with open boundary conditions in both directions, are calculated exactly in the finite-size scaling limit L , M → ∞ , T → T c , with fixed temperature scaling variable x ∝ ( T / T c − 1 ) M and fixed aspect ratio ρ ∝ L / M . We derive exponentially fast converging series for the related Casimir potential and Casimir force scaling functions. At the critical point T = T c we confirm predictions from conformal field theory (Cardy and Peschel 1988 Nucl. Phys . B 300 377, Kleban and Vassileva 1991 J. Phys. A: Math. Gen . 24 3407). The presence of corners and the related corner free energy has dramatic impact on the Casimir scaling functions and leads to a logarithmic divergence of the Casimir potential scaling function at criticality.