Functional renormalization-group approach to interacting bosons at zero temperature

Abstract

We investigate the single-particle spectral density of interacting bosons within the non-perturbative functional renormalization group technique. The flow equations for a Bose gas are derived in a scheme which treats the two-particle density-density correlations exactly but neglects irreducible correlations among three and more particles. These flow equations are solved within a truncation which allows to extract the complete frequency and momentum structure of the normal and anomalous self-energies. Both the asymptotic small momentum regime, where perturbation regime fails, as well as the perturbative regime at larger momenta are well described within a single unified approach. The self-energies do not exhibit any infrared divergences, satisfy the U (1) symmetry constraints, and are in accordance with the Nepomnyashchy relation which states that the anomalous self-energy vanishes at zero momentum and zero frequency. From the self-energies we extract the single-particle spectral density of the two-dimensional Bose gas. The dispersion is found to be of the Bogoliubov form and shows the crossover from linear Goldstone modes to the quadratic behavior of quasi-free bosons. The damping of the quasiparticles is found to be in accordance with the standard Beliaev damping. We furthermore recover the exact asymptotic limit of the propagators derived by Gavoret and Nozières and discuss the nature of the non-analyticities of the self-energies in the very small momentum regime.

Infrared behavior of interacting bosons at zero temperature

“…7,20,11 It shows that the Bogoliubov theory, where the linear spectrum and the superfluidity rely on a finite value of the anomalous self-energy, breaks down at low energy in agreement with the conclusions drawn from perturbation theory (Sec. 2.2).…”

“…The suppression of Z C,k , together with a finite value of V A,k=0 shows that the average effective action (17) exhibits a "relativistic" invariance in the infrared limit and therefore becomes equivalent to that of the classical O(2) model in dimensions d + 1. In the ordered phase, the coupling constant of this model vanishes as λ k ∼ k 4−(d+1) , 11 which agrees with (20). For k → 0, the existence of a linear spectrum is due to the relativistic form of the average effective action (rather than a non-zero value of λ k n 0,k as in the Bogoliubov approximation).…”

“…12,13,16,17,18,14,15,19,20 The strategy of the NPRG is to build a family of theories indexed by a momentum scale k such that fluctuations are smoothly taken into account as k is lowered from the microscopic scale Λ down to 0. 25,26 This is achieved by adding to the action (1) an infrared regulator term…”

“…3, we discuss the non-perturbative renormalization group (NPRG) and show how it enables to obtain the exact infrared behavior of the normal and anomalous single-particle propagators without encountering infrared divergences. 12,13,14,15,16,17,18,19,20 …”

Green's-function formalism for a condensed Bose gas consistent with infrared-divergent longitudinal susceptibility and Nepomnyashchii-Nepomnyashchii identity

“…The infrared singularity directly appears in some quantities such as the correlation functions of the phase and amplitude fluctuations of a BEC order parameter [11][12][13][14][15][16][17][18][19][20] (that are also referred to as the transverse and longitudinal response functions in the literature, respectively).…”