Self-energy and critical temperature of weakly interacting bosons

Abstract

Using the exact renormalization group we calculate the momentum-dependent self-energy Sigma (k) at zero frequency of weakly interacting bosons at the critical temperature T_c of Bose-Einstein condensation in dimensions 3 <= D < 4. We obtain the complete crossover function interpolating between the critical regime k << k_c, where Sigma (k) propto k^{2 - eta}, and the short-wavelength regime k >> k_c, where Sigma (k) propto k^{2 (D-3)} in D> 3 and Sigma (k) \propto ln (k/k_c) in D=3. Our approach yields the crossover scale k_c on the same footing with a reasonable estimate for the critical exponent eta in D=3. From our Sigma (k) we find for the interaction-induced shift of T_c in three dimensions Delta T_c / T_c approx 1.23 a n^{1/3}, where a is the s-wave scattering length and n is the density.Comment: 4 pages,1 figur

Functional renormalization group in the broken symmetry phase: momentum dependence and two-parameter scaling of the self-energy

“…While in the symmetric phase the sharp cutoff is very convenient [3,17], it leads to technical complications in the broken symmetry phase (see the discussion after (36) below). These can be avoided if we use a smooth cutoff procedure, which we implement via an additive regulator R Λ (k) in the inverse propagator [5].…”

“…Different strategies to calculate the k-dependence of the self-energy (and, more generally, the momentum dependence of the higher order vertices) has recently been developed in [17] and in [18]. On the other hand, we present in this work an approximate calculation of the two-parameter scaling function σ − (kξ, k/k c ) describing the scaling of the self-energy of the system slightly below the critical temperature T c .…”

“…For models with weak interactions, a universal regime, covering the complete crossover from the vicinity of the Gaussian fixed point to the vicinity of the Wilson-Fisher-fixed point, exists which can be described completely within a two parameter scaling theory [21,22,4]. For the weakly interacting Bose gas at criticality the one-parameter scaling function σ * (k/k c ) = σ − (∞, k/k c ) has been calculated in [26,27,17,18]. However, the full implications of a finite scale k c at criticality and the resulting two-parameter scaling theory of the correlation function away from criticality has only recently been examined [20].…”

Critical behavior of weakly interacting bosons: A functional renormalization-group approach

“…(3.9) whereΓ (6) l andΓ (8) l involve various combinations of the two-, six-, four-, eightand ten-point vertices. All terms entering the inhomogeneity of the six-point vertex and the terms needed for a calculation up to second order in the relevant and marginal parameters of the inhomogeneity of the eightpoint can be found in Appendix A, see Eqs.…”

“…In this work we present a detailed study of the momentumdependence of the self-energy at the critical point of BoseEinstein condensation at zero frequency, using the functional renormalization group formalism in the form introduced by Wetterich [6] and by Morris [7]. Some results of this manuscript were already presented in a brief form [8]. Here we give a detailed account of the calculation and further include an extensive treatment of marginal terms.…”