Infrared behavior and spectral function of a Bose superfluid at zero temperature

Abstract

In a Bose superfluid, the coupling between transverse (phase) and longitudinal fluctuations leads to a divergence of the longitudinal correlation function, which is responsible for the occurrence of infrared divergences in the perturbation theory and the breakdown of the Bogoliubov approximation. We report a non-perturbative renormalization-group (NPRG) calculation of the one-particle Green function of an interacting boson system at zero temperature. We find two regimes separated by a characteristic momentum scale kG ("Ginzburg" scale). While the Bogoliubov approximation is valid at large momenta and energies, |p|, |ω|/c ≫ kG (with c the velocity of the Bogoliubov sound mode), in the infrared (hydrodynamic) regime |p|, |ω|/c ≪ kG the normal and anomalous self-energies exhibit singularities reflecting the divergence of the longitudinal correlation function. In particular, we find that the anomalous self-energy agrees with the Bogoliubov result Σan(p, ω) ≃ const at highenergies and behaves as Σan(p, ω) ∼ (c 2 p 2 − ω 2 ) (d−3)/2 in the infrared regime (with d the space dimension), in agreement with the Nepomnyashchii identity Σan(0, 0) = 0 and the predictions of Popov's hydrodynamic theory. We argue that the hydrodynamic limit of the one-particle Green function is fully determined by the knowledge of the exponent 3 − d characterizing the divergence of the longitudinal susceptibility and the Ward identities associated to gauge and Galilean invariances. The infrared singularity of Σan(p, ω) leads to a continuum of excitations (coexisting with the sound mode) which shows up in the one-particle spectral function.

Nonperturbative renormalization group approach to quantumXYspin models

“…Although not discussed here, it has been shown that the NPRG is a method of choice in studying superfluid phases. 29,35,[58][59][60][61][62] It is free of the infrared divergences usually encountered in perturbative approaches such as Bogoliubov or spin-wave theories, it satisfies the Hugenholtz-Pines theorem, 63 and recovers the hydrodynamic regime at low energy (described by Popov's hydrodynamic theory, 64 in the context of dilute superfluids). This is of particular importance for lattice systems away from the dilute limit.…”

“…Note that even though V A,k (n) is initially zero, it is important to include it in order to properly describe the infrared behavior in the superfluid phase. 12,13,29,35 Furthermore, it is crucial to keep the full lattice structure in the early stages of the RG flow (k Λ), and we have kept the full dispersion in…”

Analytic continuation of functional renormalization group equations

“…In refs. [39] and [40], the truncated renormalization group equations for a two-dimensional gas of non-relativistic bosons have been solved in the Matsubara formalism with imaginary frequencies. The result for the propagator at the macroscopic scale k = 0 has then been analytically continued to real frequencies using numerical Padé JHEP05(2012)021 approximant techniques.…”

“…Methods based for example on Padé approximants need information from many Matsubara frequencies and the numerical effort gets rather large. Nevertheless, this approach has been successfully followed in the past and allowed to calculate for example the spectral function of non-relativistic bosons in two spatial dimensions [39,40].…”

Critical behavior of the Higgs- and Goldstone-mass gaps for the two-dimensional S=1 XY model

“…The critical behaviors for the Higgs-and Goldstone-mass gaps are the main concern. smears out the Higgs-mode branch [16][17][18][19]. (For instance, the chemical-potential modulation for the bosonic system does not conflict with the symmetry.…”