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.

中文翻译：

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$N\to \infty$ 类中强各向异性系统中的热力学卡西米尔力

我们分析了板几何中矢量 $N\to\infty$ 类强各向异性系统中的热力学卡西米尔效应。以不完美（平均场）Bose 气体为例，我们证明了空间维度 $d$ 在确定围墙之间有效波动介导的相互作用的特征中的关键作用。对于各向异性色散和周期性边界条件的特定的、物理上可以想象的选择，我们表明，在临界状态以及低温相内的 Casimir 力对于维数 $d\in (\frac{5}{2}, 4)\cup (6,8)\cup (10,12)\cup\dots$ 并且对 $d\in (4,6)\cup (8,10)\cup \dots$ 有吸引力。我们认为，对于 $d\in\{4,6,8\dots\}$，Casimir 相互作用在缩放限制中完全消失。