Thermodynamic Casimir forces in strongly anisotropic systems within the $N\to \infty$ class

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.