Critical Casimir forces forO(n)systems with long-range interaction in the spherical limit

Abstract

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−σ , where σ < d < 2σ and 0 < σ ≤ 2. For a film geometry and under periodic boundary conditions we consider the behavior of these quantities near the bulk critical temperature Tc, as well as for T > Tc and T < Tc. 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 ofwhere 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 < σ, as well as the case with d − 1 > σ, 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 < Tc 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 σ ≥ 1 it is everywhere negative, i.e. an attraction between the surfaces bounding the system is to be observed. At T = Tc the force is an increasing function of T for σ > 1 and a decreasing one for σ < 1. For any d and σ the minimum of the force at T = Tc is always achieved at some H = 0.

Thermodynamic Casimir effect in films: The exchange cluster algorithm

“…For a discussion of this point see for example the introduction of [39]. Exact results can be obtained in the large N limit for periodic and free boundary conditions [39][40][41][42][43][44][45][46]. Also for the two-dimensional Ising model with various boundary conditions exact results were obtained [47][48][49][50][51][52].…”

Casimir force in theO(n→∞)model with free boundary conditions

“…Theoretically, the effect has been studies via exact calculations in the two dimensional Ising model [16][17][18][19][20][21][22][23], the three dimensional spherical model [24][25][26][27][28], with the use of conformal-theoretical methods [29][30][31][32][33], via mean-field type calculations on Ising type [34][35][36][37][38][39] and XY models [40], through renormalization-group studies via εexpansion of O(n) models [41][42][43][44][45][46][47][48], and via Monte-Carlo calculations [49][50][51][52][53][54][55][56][57][58][59][60]. In the models envisaged above nonzero critical temperature exists and the thermal fluctuations play the essential role.…”

Thermodynamics from a scaling Hamiltonian

“…However, several theoretical and practical aspects, which appeared in the physical properties of one-dimensional systems, caused specialists to reverse those beliefs. [1,2,3,4,5,6,7,8,9,10,11] Recently, ferromagnetism in one dimension has been reported in various systems. Transitions between two different magnetic ordering phases are obtained using different approaches and considerations, and we will discuss some of these.…”