The   expansion in the symmetry-broken phase of an interacting Bose gas at finite temperature

Abstract

We discuss the application of the momentum-shell renormalization group method to the interacting homogeneous Bose gas in the symmetric and in the symmetry-broken phases. It is demonstrated that recently discussed discrepancies are artifacts of not taking proper care of infrared divergencies appearing at finite temperature. If these divergencies are taken into account and treated properly by means of the ε-expansion, the resulting renormalization group equations and the corresponding universal properties are identical in the symmetric and the symmetrybroken phases.The ε-expansion in the symmetry-broken phase of an interacting Bose gas 2 are sensitive to the way the resummation is performed and consequently somewhat ambiguous, see e.g. [26].An alternative method is to calculate the universal properties perturbatively as series of powers of g * (g * being the infrared stable fixed point for the interaction g) directly in D = 3, as first suggested in [27]. These series are then truncated to order g * L where L is the number of loops in which the calculation is performed. Though this method is fundamentally less satisfactory than the ε-expansion, see e.g. [15], it can be used in the regime of small but non-zero chemical potential. It has been employed for the calculation of critical exponents up to seventh order in g * for N = 0, 1, 2, 3 [28,29,26] and for arbitrary N [30, 31] where N is the number of components of the vector field. The series in g * are again asymptotic and have to be resummed. There is in general agreement with the corresponding ε-expansion results. We will be referring to this technique as the direct method.After the experimental realization of BEC in ultracold atomic gases [32,33,34], because of the renewed interest in these systems, a new generation of papers on the renormalization of Bose gases appeared. Starting with [35], a series of papers relied on the so-called momentum-shell approach [36,37,38,39]. In this method, momentum shells around the cutoff are successively integrated out directly at D = 3 according to Wilson's method, but unlike in the direct method no expansion of the critical exponents over g * is performed.This apparently new method, when applied in the symmetric (normal) phase, yields universal results which, when compared to experimental values, are worse than even the first-order ε-expansion results. However, when the momentum-shell method is applied to the symmetry-broken phase, it yields results which are far better than the firstorder ε-expansion and, in fact, as good as the results of the second-order ε-expansion. Based on this observation, it was assumed that the reliability of the momentum-shell method increases when it is used in the symmetry-broken phase, and a calculation of non-universal properties (for example transition temperature versus scattering length) from the symmetry-broken phase RG equations was attempted. For this reason, one may now wonder if applying the ε-expansion or the direct method in the symmetrybroken rather than in the symmetric phase as is us...

Quantum Effects in the Three-Dimensional Dilute Bose System at Finite Temperature: Renormalization Group Approach

“…27, the linear dependence of the critical temperature shift is essentially connected to the approximation of the Bose function by the classical expression used in Ref. 15 for the calculation of the critical exponent ν, but it gives also different values for the constant c. From Fig. 1, we can see that our result is close to the case from Ref.…”

“…The value ν = 0.6 obtained in the lowest order, 15 and which is the same as for the case of the classical Wilson theory, is due to the approximation of the Bose function f B (E) by f B (E) T /(E − µ). We mention that the same approximation has been made in Ref.…”

“…8 and 17, but identically with that in Ref. 15. The difference is in fact a constant, and it is connected to the presence of the two channels in the perturbation calculation.…”

“…A better approximation obtained in Ref. 15 is ν = 0.65, and this has improved the approximation, but the calculation still remains in the classical regime. The quantum effects have been captured by Andersen and Strickland, 20 keeping the quantum effects in a RG-calculation.…”

“…15 An important quantity of this system is the density of the particles in the critical region given by Ref. 17 as follows:…”