d → d ’ crossover in the anisotropic free bose gas

2014 The d ~ d’ crossover in the anisotropic free Bose gas is considered. The results agree with the renormalization group solution in the range of d’ ~ 2, where this solution exists and extend its validity to values of d’ 2. It is conjectured that for d’ = 1, n = 2 the crossover critical index equals two. LE JOURNAL DE PHYSIQUE TOME 36, AVRIL 1975, Classification Physics Abstracts 1.680 7.480 1. General. This paper treats the simple and exactly soluble model of a phase transition in the d-dimensional anisotropic Bose gas at a constant volume. The spectrum of the gas is assumed to be It was recognized somè’tùne ago [1, 2] that the BoseEinstein condensation of the isotropic a = 0 gas corresponds, by virtue of universality, to the n = oo vector order parameter phase transition in a d’-dimensional system with short range forces. E.g. in both cases the critical index of the susceptibility is It is then expected that the crossover a --> 0 in the system (1) corresponds to a d -+ d’ crossover of a n = oo vector model. The crossover occurs in strongly anisotropic systems. Above a certain crossover temperature, T* such a system behaves according to the laws appropriate to the lower d’ dimensionality, below the crossover temperature, it becomes d-dimensional and eventually undergoes a phase transition at Tc. Both T* and T, depend on the anisotropy parameter a in a way specified by the exponents gl and ç respectively. It is generally believed that these two exponents are equal. However, it should be mentioned that in this paper we are concemed only with the exponent ç, related to Tc. The renormalization group approach [3, 4] furnishes the solution of the crossover problem for an arbitrary n, provided that the renormalization group converges in d’ dimensions. It converges above the logarithmic line in the n, d’ diagram. This line is at d’ = 2 for n = oo. For n Go it bends to a lower d’. The renormalization group result is