Two-point correlation function in systems with van der Waals type interaction

The behavior of the bulk two-point correlation function $G({\bf r};T|d)$ in $d$-dimensional system with van der Waals type interactions is investigated and its consequences on the finite-size scaling properties of the susceptibility in such finite systems with periodic boundary conditions is discussed within mean-spherical model which is an example of Ornstein and Zernike type theory. The interaction is supposed to decay at large distances $r$ as $r^{-(d+\sigma)}$, with $2<d<4$, $2<\sigma<4$ and $d+\sigma \le 6$. It is shown that $G({\bf r};T|d)$ decays as $r^{-(d-2)}$ for $1\ll r\ll \xi$, exponentially for $\xi\ll r \ll r^*$, where $r^*=(\sigma-2)\xi \ln \xi$, and again in a power law as $r^{-(d+\sigma)}$ for $r\gg r^*$. The analytical form of the leading-order scaling function of $G({\bf r};T|d)$ in any of these regimes is derived.Comment: 12 pages, 3 figures, revtex. Two references added To be published in EPJ

Casimir force in the rotor model with twisted boundary conditions

We investigate the three dimensional lattice XY model with nearest neighbor interaction. The vector order parameter of this system lies on the vertices of a cubic lattice, which is embedded in a system with a film geometry. The orientations of the vectors are fixed at the two opposite sides of the film. The angle between the vectors at the two boundaries is $\alpha$ where $0 \le \alpha \le \pi$. We make use of the mean field approximation to study the mean length and orientation of the vector order parameter throughout the film---and the Casimir force it generates---as a function of the temperature $T$, the angle $\alpha$, and the thickness $L$ of the system. Among the results of that calculation are a Casimir force that depends in a continuous way on both the parameter $\alpha$ and the temperature and that can be attractive or repulsive. In particular, by varying $\alpha$ and/or $T$ one controls \underline{both} the sign \underline{and} the magnitude of the Casimir force in a reversible way. Furthermore, for the case $\alpha=\pi$, we discover an additional phase transition occurring only in the finite system associated with the variation of the orientations of the vectors.Comment: 14 pages, 9 figure

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

Casimir force in O(n) lattice models with a diffuse interface

On the example of the spherical model we study, as a function of the temperature $T$, the behavior of the Casimir force in O(n) systems with a diffuse interface and slab geometry $\infty^{d-1}\times L$, where $2<d<4$ is the dimensionality of the system. We consider a system with nearest-neighbor anisotropic interaction constants $J_\parallel$ parallel to the film and $J_\perp$ across it. The model represents the $n\to\infty$ limit of O(n) models with antiperiodic boundary conditions applied across the finite dimension $L$ of the film. We observe that the Casimir amplitude $\Delta_{\rm Casimir}(d|J_\perp,J_\parallel)$ of the anisotropic $d$-dimensional system is related to that one of the isotropic system $\Delta_{\rm Casimir}(d)$ via $\Delta_{\rm Casimir}(d|J_\perp,J_\parallel)=(J_\perp/J_\parallel)^{(d-1)/2} \Delta_{\rm Casimir}(d)$. For $d=3$ we find the exact Casimir amplitude $\Delta_{\rm Casimir}= [ {\rm Cl}_2 (\pi/3)/3-\zeta (3)/(6 \pi)](J_\perp/J_\parallel)$, as well as the exact scaling functions of the Casimir force and of the helicity modulus $\Upsilon(T,L)$. We obtain that $\beta_c\Upsilon(T_c,L)=(2/\pi^{2}) [{\rm Cl}_2(\pi/3)/3+7\zeta(3)/(30\pi)] (J_\perp/J_\parallel)L^{-1}$, where $T_c$ is the critical temperature of the bulk system. We find that the effect of the helicity is thus strong that the Casimir force is repulsive in the whole temperature region.Comment: 15 pages, 3 figure

Excess free energy and Casimir forces in systems with long-range interactions of van-der-Waals type: General considerations and exact spherical-model results

We consider systems confined to a $d$-dimensional slab of macroscopic lateral extension and finite thickness $L$ that undergo a continuous bulk phase transition in the limit $L\to\infty$ and are describable by an O(n) symmetrical Hamiltonian. Periodic boundary conditions are applied across the slab. We study the effects of long-range pair interactions whose potential decays as $b x^{-(d+\sigma)}$ as $x\to\infty$, with $2<\sigma<4$ and $2<d+\sigma\leq 6$, on the Casimir effect at and near the bulk critical temperature $T_{c,\infty}$, for $2<d<4$. For the scaled reduced Casimir force per unit cross-sectional area, we obtain the form L^{d} {\mathcal F}_C/k_BT \approx \Xi_0(L/\xi_\infty) + g_\omega L^{-\omega}\Xi\omega(L/\xi_\infty) + g_\sigma L^{-\omega_\sigm a} \Xi_\sigma(L \xi_\infty). The contribution $\propto g_\sigma$ decays for $T\neq T_{c,\infty}$ algebraically in $L$ rather than exponentially, and hence becomes dominant in an appropriate regime of temperatures and $L$. We derive exact results for spherical and Gaussian models which confirm these findings. In the case $d+\sigma =6$, which includes that of nonretarded van-der-Waals interactions in $d=3$ dimensions, the power laws of the corrections to scaling $\propto b$ of the spherical model are found to get modified by logarithms. Using general RG ideas, we show that these logarithmic singularities originate from the degeneracy $\omega=\omega_\sigma=4-d$ that occurs for the spherical model when $d+\sigma=6$, in conjunction with the $b$ dependence of $g_\omega$.Comment: 28 RevTeX pages, 12 eps figures, submitted to PR