We study the critical behavior of the free energy and the thermodynamic Casimir force in a $L_\parallel^{d-1} \times L$ block geometry in $2<d<4$ dimensions with aspect ratio $ρ=L/L_\parallel$ above, at, and below $T_c$ on the basis of the O$(n)$ symmetric $ϕ^4$ lattice model with periodic boundary conditions (b.c.). We consider a simple-cubic lattice with isotropic short-range interactions. Exact results are derived in the large - $n$ limit describing the geometric crossover from film ($ρ=0$) over cubic $ρ=1$ to cylindrical ($ρ= \infty$) geometries. For $n=1$, three perturbation approaches are presented that cover both the central finite-size regime near $T_c$ for $1/4 \lesssim ρ\lesssim 3$ and the region outside the central finite-size regime well above and below $T_c$ for arbitrary $ρ$. At bulk $T_c$ of isotropic systems with periodic b.c., we predict the critical Casimir force in the vertical $(L)$ direction to be negative (attractive) for a slab ($ρ< 1$), positive (repulsive) for a rod ($ρ> 1$), and zero for a cube $(ρ=1)$. We also present extrapolations to the cylinder limit ($ρ=\infty$) and to the film limit ($ρ=0$) for $n=1$ and $d=3$. Our analytic results for finite-size scaling functions in the minimal renormalization scheme at fixed dimension $d=3$ agree well with Monte Carlo data for the three-dimensional Ising model by Hasenbusch for $ρ=1$ and by Vasilyev et al. for $ρ=1/6$ above, at, and below $T_c$.
"Screening" of universal van der Waals - Casimir terms by Coulomb gases in a fully-finite two-dimensional geometry
本文是前一篇论文的延续[Jancovici and Samaj，2004 J. Stat. Mech.P08006]处理半无限壁几何形状中的经典Casimir现象。在那篇论文中，使用微观库仑系统，长程Casimir力，由于热波动在导电壁被屏蔽的存在下，电解质之间的墙壁到一些残余的短程力。在这里，我们的目标是扩展的屏蔽（取消）现象的研究普遍Casimir条款中出现的大尺寸扩展的微观库仑系统的大电位局限于完全有限的二维几何形状，特别是盘几何。两种情况下得到了精确的解决：高温（德拜-休克尔）极限和Thirring自由费米子点。完全有限和半无限的几何形状之间的相似性和根本区别指出。
The exact critical Casimir amplitude is derived for anisotropic systems within the $d=2$ Ising universality class by combining conformal field theory (CFT) with anisotropic $φ^4$ theory. Explicit results are presented for the general anisotropic scalar $φ^4$ model and for the fully anisotropic triangular-lattice Ising model in finite rectangular and infinite strip geometries with periodic boundary conditions (PBC). These results demonstrate the validity of multiparameter universality for confined anisotropic systems and the nonuniversality of the critical Casimir amplitude. We find an unexpect…
We calculate formation probabilities of the ground state of the finite size quantum critical chains using conformal field theory (CFT) techniques. In particular, we calculate the formation probability of one interval in the finite open chain and also formation probability of two disjoint intervals in a finite periodic system. The presented formulas can be also interpreted as the Casimir energy of needles in particular geometries. We numerically check the validity of the exact CFT results in the case of the transverse field Ising chain.
