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We analyze the thermodynamic Casimir effect in strongly anizotropic systemsfrom the vectorial $N\to\infty$ class in a slab geometry. Employing theimperfect (mean-field) Bose gas as a representative example, we demonstrate thekey role of spatial dimensionality $d$ in determining the character of theeffective fluctuation-mediated interaction between the confining walls. For aparticular, physically conceivable choice of anisotropic dispersion andperiodic boundary conditions, we show that the Casimir force at criticality aswell 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 $d\in\{4,6,8\dots\}$ theCasimir interaction entirely vanishes in the scaling limit. We discussimplications of our results for systems characterized by $1/N>0$ and possiblerealizations in the context of quantum phase transitions.

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