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.
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.
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.
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.
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.
Two-dimensional Ising strips subject to identical surface fields h(1)=h(2) > or =0 are studied for temperatures above and below the bulk critical temperature T(c) and a range of bulk fields h by means of the density-matrix renormalization-group method. In the case of nonvanishing surface fields, the near-critical behavior of the solvation force f(solv), total adsorption Gamma, inverse longitudinal correlation length xi(parallel)( -1) and specific heat C(H) is strongly influenced by the (pseudo) capillary condensation that occurs below T(c). We obtain scaling functions of f(solv), Gamma, and xi(parallel)(-1). C(H) exhibits a weakly rounded singularity on crossing the pseudocoexistence line. We contrast these results with those for the case of free boundaries where, for temperatures slightly below T(c), f(solv) and C(H) exhibit a sharp extremum away from h=0. Our results have direct repercussions for the properties of near-critical Ising films in three dimensions and we argue that the long-ranged solvation (Casimir) force in confined fluids should be more attractive in the neighborhood of the capillary critical point than exactly at the bulk critical point.
I use an exact variational formulation of Mikheev and Fisher to study the critical Ising strip with a grain boundary and confining surfaces characterized by arbitrary and different surface magnetic fields. Energy density profiles that serve as order parameters of the system within the used method show strong nonmonotonous behavior in the vicinity of confining surfaces. I consider short-distance expansion of energy density profiles. New universal amplitudes associated with distant-wall corrections exhibit nontrivial crossover behaviors from positive to negative values as the field's variables are continuously varied. Casimir amplitudes calculated in a self-contained manner are characterized by complex manifolds which may comprise two disconnected positive wings separated by an area of negative values. I define and determine the generalized de Gennes-Fisher amplitude, which strongly suggests that a stress tensor is present within one of the distant-wall corrections. This is an unanticipated result given that a similar discovery for standard extraordinary (E) and ordinary (O) surface universality classes was based on the conformal invariance symmetry, which is broken under the present boundary conditions. A grain boundary essentially influences both energy density profiles and Casimir amplitudes, besides surface fields with a number of accompanying features that are examined in detail. We present closed analytic forms of energy density profiles for standard (N) and (O) BCs: ɛNN(x,g),ɛOO(x,g),ɛNO(x,g) [(N) is the normal surface universality class, g is the grain boundary's strength, x is the relative distance x:=z/L with L as a film width], thus enabling us to deduce their important symmetry properties and to have a detailed insight into their general behaviors.
The quantum critical behavior of the (2 + 1)-dimensional Gross-Neveu model in the vicinity of its zero-temperature critical point is considered.The model is known to be renormalisable in the large-N limit, which offers the possibility to obtain expressions for various thermodynamic functions in closed form.We have used the concept of finite-size scaling to extract information about the leading temperature behavior of the free energy and the mass term, defined by the fermionic condensate and determined the crossover lines in the coupling (g)-temperature (T ) plane.These are given by T ∼ |g -gc|, where gc denotes the critical coupling at zero temperature.According to our analysis no spontaneous symmetry breaking survives at finite temperature.We have found that the leading temperature behavior of the fermionic condensate is proportional to the temperature with the critical amplitude √ 5 3 π.The scaling function of the singular part of the free energy is found to exhibit a maximum at ln 2 2π corresponding to one of the crossover lines.The critical amplitude of the singular part of the free energy is given by the universal number 1 3 1 2π ζ(3) -Cl2 π 3 = -0.274543 . .., where ζ(z) and Cl2(z) are the Riemann zeta and Clausen's functions, respectively.Interpreted in terms of the thermodynamic Casimir effect, this result implies an attractive Casimir "force".This study is expected to be useful in shedding light on a broader class of four fermionic models.