Fluctuation-induced forces in periodic slabs: Breakdown of epsilon expansion at the bulk critical point and revised field theory

Abstract

Abstract. -Systems described by n-component φ 4 models in a ∞ d−1 × L slab geometry of finite thickness L are considered at and above their bulk critical temperature Tc,∞. The renormalization-group improved perturbation theory commonly employed to investigate the fluctuation-induced forces ("thermodynamic Casimir effect") in d = 4 − ǫ bulk dimensions is re-examined. It is found to be ill-defined beyond two-loop order because of infrared singularities when the boundary conditions are such that the free propagator in slab geometry involves a zero-energy mode at bulk criticality. This applies to periodic boundary conditions and the special-special ones corresponding to the critical enhancement of the surface interactions on both confining plates. The field theory is reorganized such that a small-ǫ expansion results which remains well behaved down to Tc,∞. The leading contributions to the critical Casimir amplitudes ∆per and ∆sp,sp beyond two-loop order are ∼ (u * ) (3−ǫ)/2 , where u * = O(ǫ) is the value of the renormalized φ 4 coupling at the infrared-stable fixed point. Besides integer powers of ǫ, the small-ǫ expansions of these amplitudes involve fractional powers ǫ k/2 , with k ≥ 3, and powers of ln ǫ. Explicit results to order ǫ Fluctuations associated with long wave-length, low-energy excitations play a crucial role in determining the physical properties of many macroscopic systems. When such fluctuations are confined by boundaries, walls, or size restrictions along one or several axes, important effective forces may result. In those cases where the continuous mode spectrum that emerges as the system becomes macroscopic in all directions is not separated from zero energy by a gap, these fluctuation-induced forces are longranged, decaying algebraically as a function of the relevant confinement length L (separation of walls, thickness of the system, etc).A prominent example of such forces are the Casimir forces [1] induced by vacuum fluctuations of the electromagnetic field between two metallic bodies a distance L apart [2][3][4]. Analogous long-range effective forces occur in condensed matter systems as the result of either (i) thermal fluctuations at continuous phase transitions or else (ii) Goldstone modes and similar "massless" excitations [5][6][7][8][9][10][11][12]. In particular the former ones, frequently called "critical Casimir forces," have attracted considerable theoretical and experimental attention recently. Beginning with the seminal paper by Fisher and de Gennes [5], they have been studied theoretically for more than a decade using renormalization group (RG) [6][7][8][9][10] and conformal field theory methods [13], exact solutions of models [12], as well as Monte c EDP Sciences

Critical Casimir forces forO(N)models from functional renormalization

“…Similarly to the case N = 1, the MC result is lower and outside our error bars. Our prediction is again closer to the MC results than the earlier perturbative RG studies, which yield 28,29 f (3,2) ≈ 0.84. The dependencies depicted in Figs.…”

“…As discussed in Ref. 28 (see also Ref. 29), the expansion is however problematic in the present case of periodic boundary conditions, yielding at three-loop level terms nonanalytic in .…”

Трехмерная $O(n)$-$\phi^4$-модель в плоско-параллельном слое со свободными граничными условиями: точные результаты для нетривиального размерного кроссовера в пределе $n\to\infty$

“…In the PBC case, the Landau theory erroneously predicts a critical point of the film right at T c . Hence, the -expansion fails at T c [41]. For FBCs, the film becomes critical in the Landau theory at an Landc-dependent temperature T c,L < T c .…”

“…Perturbative renormalization-group (RG) approaches based on the -expansion about the upper critical dimension d * = 4 were found to give acceptable results for the disordered phase x > 0 [14], [15], [39], [40] but normally cannot describe dimensional crossovers properly. Hence, the -expansion fails for L < ∞ near a critical or pseudo-critical point of the film [14], [41]- [43]. In the PBC case, the Landau theory erroneously predicts a critical point of the film right at T c .…”

Critical and near-critical phase behavior and interplay between the thermodynamic Casimir and van der Waals forces in a confined nonpolar fluid medium with competing surface and substrate potentials

“…[47] one has performed measurements of the Casimir force in thin wetting films of binary liquid mixture. One the theoretical side, the effect has been studied via exact calculations in the two-dimensional Ising model [48][49][50][51][52][53][54][55][56][57], the three dimensional spherical model [58][59][60][61][62][63][64][65][66], via conformal-theoretical methods [67][68][69][70][71][72][73], within mean-field type calculations on Ising type [74][75][76][77][78] and XY models [80], through renormalization-group studies via ε-expansion [81][82][83][84][85][86][87][88] and via fixed dimension d techniques [87][88][89] of O(n) models, as well as via MonteCarlo calculations [79,[90][91]…”