Abstract
We analyze the thermodynamic Casimir effect in strongly anisotropic systems from the vectorial N\to\inftyN→∞ class in a slab geometry. Employing the imperfect (mean-field) Bose gas as a representative example, we demonstrate the key role of spatial dimensionality dd in determining the character of the effective fluctuation-mediated interaction between the confining walls. For a particular, physically conceivable choice of anisotropic dispersion relation and periodic boundary conditions, we show that the Casimir force at criticality as well as within the low-temperature phase is repulsive for dimensionality d\in (\frac{5}{2},4)\cup (6,8)\cup (10,12)\cup\dotsd∈(52,4)∪(6,8)∪(10,12)∪… and attractive for d\in (4,6)\cup (8,10)\cup \dotsd∈(4,6)∪(8,10)∪…. We argue, that for d\in\{4,6,8\dots\}d∈{4,6,8…} the Casimir interaction entirely vanishes in the scaling limit. We discuss implications of our results for systems characterized by 1/N>01/N>0 and possible realizations in the contexts of optical lattice systems and quantum phase transitions.
Highlights
The thermodynamic Casimir effect received substantial interest over the last years [1,2,3,4,5,6,7,8] both from theoretical and experimental points of view
That for d ∈ {4, 6, 8 . . . } the Casimir interaction entirely vanishes in the scaling limit
According to exact theorems [12, 13] formulated in the context of the electrodynamic Casimir effect, the fluctuation-induced Casimir force acting between bodies related by a reflection must be attractive
Summary
The thermodynamic Casimir effect received substantial interest over the last years [1,2,3,4,5,6,7,8] both from theoretical and experimental points of view. Our analysis indicates that it leads to a far-going deviation from the usual situation, and, in particular, yields the Casimir interaction attractive, repulsive, or zero depending on the system dimensionality This is completely opposite to the usual cases extensively studied before, where dimensionality has no impact on the force sign. . This is in stark contrast to the case of isotropic systems with quadratic dispersion, where (for periodic boundary conditions) the Casimir force is always attractive (in any dimensionality and for the entire family of O(N ) universality classes), as guaranteed by the exact statements of Refs. [35] for the isotropic case, the imperfect Bose gas is equivalent to the O(2N ) model in the limit N → ∞ and the corresponding scaling functions [36, 37] for Casimir energy differ by a global factor of two. V contains a summary and a portion of technical details of the analysis is postponed to appendices A and B
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