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

Abstract

We analyze the thermodynamic Casimir effect in strongly anizotropic systems from the vectorial $N\to \infty$ class in a slab geometry. Employing the imperfect (mean-field) Bose gas as a representative example, we demonstrate the key role of spatial dimensionality $d$ in determining the character of the effective fluctuation-mediated interaction between the confining walls. For a particular, physically conceivable choice of anisotropic dispersion and periodic boundary conditions, we show that the Casimir force at criticality as well as within the low-temperature phase is repulsive for dimensionality $d\in (\frac{5}{2},4)\cup (6,8)\cup (10,12)\cup \dots$ and attractive for $d\in (4,6)\cup (8,10)\cup \dots$. We argue, that for $d\in \{4,6,8\dots \}$ the Casimir interaction entirely vanishes in the scaling limit. We discuss implications of our results for systems characterized by $1/N>0$ and possible realizations in the context of quantum phase transitions.

18 pages, 2 figures

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• Casimir forces for the ideal Bose gas in anisotropic optical lattices: the effect of alternating sign upon varying dimensionality
• Effective binding potential from Casimir interactions: the case of the Bose gas
• Thermal Casimir effect in the spin-orbit coupled Bose gas