Renormalization group treatment of the scaling properties of finite systems with subleading long-range interaction

The finite size behavior of the susceptibility, Binder cumulant and some even moments of the magnetization of a fully finite O(n) cubic system of size L are analyzed and the corresponding scaling functions are derived within a field-theoretic $\epsilon$-expansion scheme under periodic boundary conditions. We suppose a van der Waals type long-range interaction falling apart with the distance r as $r^{-(d+\sigma)}$, where $2<\sigma<4$, which does not change the short-range critical exponents of the system. Despite that the system belongs to the short-range universality class it is shown that above the bulk critical temperature $T_c$ the finite-size corrections decay in a power-in-L, and not in an exponential-in-L law, which is normally believed to be a characteristic feature for such systems.Comment: 14 pages, revte

Fluctuation-Induced Interactions Between Ellipsoidal Particle and Planar Substrate Immersed in Critical Medium

In our study we investigate the behaviour of the net force (NF) emerging between an ellipsoidal particle and a thick plate (slab), when the interaction takes place in a near critical fluid medium with account for the omnipresent van der Waals forces (vdWF). Here we consider the case of complete wetting of the objects surfaces by the medium, due to strong adsorbing local surface potentials, exerted by thin solid coating films. The influence of the bulk inner regions of the particle and the slab on the constituents of the fluid results in long-ranged competing dispersion potentials. As a consequence from the critical fluctuations of the medium, the system experiences an additional effective interaction, traditionally termed critical Casimir force (CCF). The forces of interest are evaluated numerically from integral expressions obtained utilizing general scaling arguments and mean-field type calculations in combination with the so-called "surface integration approach" (SIA). Within the scenario considered here, this technique is applicable if one has knowledge of the forces between two parallel semi-infinite plates, confining in between some fluctuating fluid medium characterized by its temperature $T$ and chemical potential $\mu$. It is demonstrated that for a suitable set of particle-fluid, slab-fluid, and fluid-fluid coupling parameters the competition between the effects due to the coatings and the core regions of the objects result, when one changes $T$ or $\mu$, in {\it sign change} of the NF acting between the ellipsoid and the slab.Comment: 8 pages, 2 figues. arXiv admin note: text overlap with arXiv:1702.0491

Critical Casimir Effect: Exact Results

If a material body is immersed into a fluctuating medium, its shape and the properties of its constituents modify the fluctuations in the surrounding medium. If in the same medium there is a second body, modifications of the fluctuation due to the first one influence the modifications due to the second one. This mutual influence results in a force between these bodies. If the fluctuating medium consists of the confined electromagnetic field in vacuum, one speaks of the quantum mechanical Casimir effect. In the case that the order parameter of material fields fluctuates - such as differences of number densities or concentrations - and that the corresponding fluctuations of the order parameter are long-ranged, one speaks of the critical Casimir effect. This holds, e.g., in the case of systems which undergo a second-order phase transition and which are thermodynamically located near the corresponding critical point, or for systems with a continuous symmetry exhibiting Goldstone mode excitations. Here we review the currently available exact results concerning the critical Casimir effect in systems encompassing the one-dimensional Ising, XY, and Heisenberg models, the two-dimensional Ising model, the Gaussian and the spherical models, as well as the mean field results for the Ising and the XY model. Special attention is paid to the influence of the boundary conditions on the behavior of the Casimir force.Comment: 218 pages, 67 figure

Casimir versus Helmholtz forces: Exact results

Recently, attention has turned to the issue of the ensemble dependence of fluctuation induced forces. As a noteworthy example, in $O(n)$ systems the statistical mechanics underlying such forces can be shown to differ in the constant $\vec{M}$ magnetic canonical ensemble (CE) from those in the widely-studied constant $\vec{h}$ grand canonical ensemble (GCE). Here, the counterpart of the Casimir force in the GCE is the \textit{Helmholtz} force in the CE. Given the difference between the two ensembles for finite systems, it is reasonable to anticipate that these forces will have, in general, different behavior for the same geometry and boundary conditions. Here we present some exact results for both the Casimir and the Helmholtz force in the case of the one-dimensional Ising model subject to periodic and antiperiodic boundary conditions and compare their behavior. We note that the Ising model has recently being solved in Phys.Rev. E {\bf 106} L042103(2022), using a combinatorial approach, for the case of fixed value $M$ of its order parameter. Here we derive exact result for the partition function of the one-dimensional Ising model of $N$ spins and fixed value $M$ using the transfer matrix method (TMM); earlier results obtained via the TMM were limited to $M=0$ and $N$ even. As a byproduct, we derive several specific integral representations of the hypergeometric function of Gauss. Using those results, we rigorously derive that the free energies of the CE and grand GCE are related to each other via Legendre transformation in the thermodynamic limit, and establish the leading finite-size corrections for the canonical case, which turn out to be much more pronounced than the corresponding ones in the case of the GCE.Comment: 33 pages, 7 figures. The derivations in Appendix C are simplifie

Non-universal size dependence of the free energy of confined systems near criticality

The singular part of the finite-size free energy density $f_s$ of the O(n) symmetric $\phi^4$ field theory in the large-n limit is calculated at finite cutoff for confined geometries of linear size L with periodic boundary conditions in 2 < d < 4 dimensions. We find that a sharp cutoff $\Lambda$ causes a non-universal leading size dependence $f_s \sim \Lambda^{d-2} L^{-2}$ near $T_c$ which dominates the universal scaling term $\sim L^{-d}$. This implies a non-universal critical Casimir effect at $T_c$ and a leading non-scaling term $\sim L^{-2}$ of the finite-size specific heat above $T_c$.Comment: RevTex, 4 page

Finite-size effects in the spherical model of finite thickness

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 $\Delta^{(p)}=-2\zeta(3)/(5\pi)=-0.153051...$, $\Delta^{(a)}=0.274543...$ and $\Delta^{(ND)}=0.01922...$, 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 $L$.Comment: Version published in J. Phys. A: Math. Theo

The bulk correlation length and the range of thermodynamic Casimir forces at Bose-Einstein condensation

The relation between the bulk correlation length and the decay length of thermodynamic Casimir forces is investigated microscopically in two three-dimensional systems undergoing Bose-Einstein condensation: the perfect Bose gas and the imperfect mean-field Bose gas. For each of these systems, both lengths diverge upon approaching the corresponding condensation point from the one-phase side, and are proportional to each other. We determine the proportionality factors and discuss their dependence on the boundary conditions. The values of the corresponding critical exponents for the decay length and the correlation length are the same, equal to 1/2 for the perfect gas, and 1 for the imperfect gas

Universality of the thermodynamic Casimir effect

Recently a nonuniversal character of the leading spatial behavior of the thermodynamic Casimir force has been reported [X. S. Chen and V. Dohm, Phys. Rev. E {\bf 66}, 016102 (2002)]. We reconsider the arguments leading to this observation and show that there is no such leading nonuniversal term in systems with short-ranged interactions if one treats properly the effects generated by a sharp momentum cutoff in the Fourier transform of the interaction potential. We also conclude that lattice and continuum models then produce results in mutual agreement independent of the cutoff scheme, contrary to the aforementioned report. All results are consistent with the {\em universal} character of the Casimir force in systems with short-ranged interactions. The effects due to dispersion forces are discussed for systems with periodic or realistic boundary conditions. In contrast to systems with short-ranged interactions, for $L/\xi \gg 1$ one observes leading finite-size contributions governed by power laws in $L$ due to the subleading long-ranged character of the interaction, where $L$ is the finite system size and $\xi$ is the correlation length.Comment: 11 pages, revtex, to appear in Phys. Rev. E 68 (2003

Critical Casimir forces for O(n){\cal O}(n) systems with long-range interaction in the spherical limit

We present exact results on the behavior of the thermodynamic Casimir force and the excess free energy in the framework of the $d$-dimensional spherical model with a power law long-range interaction decaying at large distances $r$ as $r^{-d-\sigma}$, where $\sigma<d<2\sigma$ and $0<\sigma\leq2$. For a film geometry and under periodic boundary conditions we consider the behavior of these quantities near the bulk critical temperature $T_c$, as well as for $T>T_c$ and $T<T_c$. The universal finite-size scaling function governing the behavior of the force in the critical region is derived and its asymptotics are investigated. While in the critical and under critical region the force is of the order of $L^{-d}$, for $T>T_c$ it decays as $L^{-d-\sigma}$, where $L$ is the thickness of the film. We consider both the case of a finite system that has no phase transition of its own, when $d-1<\sigma$, as well as the case with $d-1>\sigma$, when one observes a dimensional crossover from $d$ to a $d-1$ dimensional critical behavior. The behavior of the force along the phase coexistence line for a magnetic field H=0 and $T<T_c$ is also derived. We have proven analytically that the excess free energy is always negative and monotonically increasing function of $T$ and $H$. For the Casimir force we have demonstrated that for any $\sigma \ge 1$ it is everywhere negative, i.e. an attraction between the surfaces bounding the system is to be observed. At $T=T_c$ the force is an increasing function of $T$ for $\sigma>1$ and a decreasing one for $\sigma<1$. For any $d$ and $\sigma$ the minimum of the force at $T=T_c$ is always achieved at some $H\ne 0$.Comment: 13 pages, revtex, 8 figure

On the finite-size behavior of systems with asymptotically large critical shift

Exact results of the finite-size behavior of the susceptibility in three-dimensional mean spherical model films under Dirichlet-Dirichlet, Dirichlet-Neumann and Neumann-Neumann boundary conditions are presented. The corresponding scaling functions are explicitly derived and their asymptotics close to, above and below the bulk critical temperature $T_c$ are obtained. The results can be incorporated in the framework of the finite-size scaling theory where the exponent $\lambda$ characterizing the shift of the finite-size critical temperature with respect to $T_c$ is smaller than $1/\nu$, with $\nu$ being the critical exponent of the bulk correlation length.Comment: 24 pages, late