We consider the Bose gas on a $d$-dimensional anisotropic lattice employing the imperfect (mean-field) gas as a prototype example. We study the dimensional crossover arising as a result of varying the dispersion relation at finite temperature $T$. We analyze in particular situations where one of the relevant effective dimensionalities is located at or below the lower critical dimension, so that the Bose-Einstein condensate becomes expelled from the system by anisotropically modifying the lattice parameters controlling the kinetic term in the Hamiltonian. We clarify the mechanism governing this phenomenon. Subsequently we study the thermodynamic Casimir effect occurring in this system. We compute the exact profile of the scaling function for the Casimir energy. As an effect of strongly anisotropic scale invariance, the Casimir force below or at the critical temperature $T_c$ may be repulsive even for periodic boundary conditions. The corresponding Casimir amplitude is universal only in a restricted sense, and the power law governing the decay of the Casimir interaction becomes modified. We also demonstrate that, under certain circumstances, the scaling function is constant for sufficiently large values of the scaling variable, and in consequence is not an analytical function. At $T>T_c$ the Casimir-like interactions reflect the structure of the correlation function, and, for certain orientations of the confining walls, show exponentially damped oscillatory behavior so that the corresponding force is attractive or repulsive depending on the distance.

中文翻译：

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各向异性光学晶格中玻色气体的尺寸分界和卡西米尔力

我们认为 $d$以不完全（平均场）气体为原型的三维各向异性晶格。我们研究在有限温度下由于改变色散关系而产生的尺寸交叉$Ť$。我们在特定情况下分析了其中一个有效有效维数位于或低于较低临界维的情况，从而通过各向异性地修改控制哈密顿量的动力学项的晶格参数，使玻色-爱因斯坦凝聚物从系统中排出。我们阐明了控制这种现象的机制。随后，我们研究了该系统中发生的热力学卡西米尔效应。我们为卡西米尔能量计算缩放函数的精确轮廓。由于强烈的各向异性尺度不变性，低于或处于临界温度的卡西米尔力${Ť}_C$即使在周期性边界条件下也可能令人反感。相应的卡西米尔振幅仅在有限的意义上是通用的，并且控制卡西米尔相互作用衰减的幂定律也被修改。我们还证明，在某些情况下，缩放函数对于缩放变量的足够大的值是恒定的，因此不是解析函数。在$Ť>{Ť}_C$ 类似于卡西米尔的相互作用反映了相关函数的结构，并且对于约束壁的某些方向，表现出指数阻尼的振荡行为，因此相应的力取决于距离是吸引还是排斥。