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arXiv:0901.4631 · Physical Review Letters

Unified Picture of Superfluidity: From Bogoliubov’s Approximation to Popov’s Hydrodynamic Theory

N. Dupuis

Published May 14, 2009• 43 citations
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Abstract

Using a nonperturbative renormalization-group technique, we compute the momentum and frequency dependence of the anomalous self-energy and the one-particle spectral function of two-dimensional interacting bosons at zero temperature. Below a characteristic momentum scale k_{G}, where the Bogoliubov approximation breaks down, the anomalous self-energy develops a square-root singularity and the Goldstone mode of the superfluid phase (Bogoliubov sound mode) coexists with a continuum of excitations, in agreement with the predictions of Popov's hydrodynamic theory. Thus our results provide a unified picture of superfluidity in interacting boson systems and connect Bogoliubov's theory (valid for momenta larger than k_{G}) to Popov's hydrodynamic approach.

DOI
10.1103/physrevlett.102.190401

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Using a nonperturbative renormalization-group technique, we compute the momentum and frequency dependence of the anomalous self-energy and the one-particle spectral function of two-dimensional interacting bosons at zero temperature. Below a characteristic momentum scale k_{G}, where the Bogoliubov approximation breaks down, the anomalous self-energy develops a square-root singularity and the Goldstone mode of the superfluid phase (Bogoliubov sound mode) coexists with a continuum of excitations, in agreement with the predictions of Popov's hydrodynamic theory. Thus our results provide a unified picture of superfluidity in interacting boson systems and connect Bogoliubov's theory (valid for momenta larger than k_{G}) to Popov's hydrodynamic approach.
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 nonperturbative renormalization-group calculation of the one-particle Green's function of an interacting boson system at zero temperature. We find two regimes separated by a characteristic momentum scale kG{k}_{G}kG​ (``Ginzburg'' scale). While the Bogoliubov approximation is valid at large momenta and energies, $|\mathbf{p}|,\text{ }|\ensuremath{\omega}|/c⪢{k}_{G}$ (with ccc as the velocity of the Bogoliubov sound mode), in the infrared (hydrodynamic) regime, $|\mathbf{p}|,\text{ }|\ensuremath{\omega}|/c⪡{k}_{G}$, 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 ${\ensuremath{\Sigma}}_{\text{an}}(\mathbf{p},\ensuremath{\omega})\ensuremath{\simeq}\text{const}$ at high energies and behaves as ${\ensuremath{\Sigma}}_{\text{an}}(\mathbf{p},\ensuremath{\omega})\ensuremath{\sim}{({c}^{2}{\mathbf{p}}^{2}\ensuremath{-}{\ensuremath{\omega}}^{2})}^{(d\ensuremath{-}3)/2}$ in the infrared regime (with ddd as the space dimension), in agreement with the Nepomnyashchii identity ${\ensuremath{\Sigma}}_{\text{an}}(0,0)=0$ and the predictions of Popov's hydrodynamic theory. We argue that the hydrodynamic limit of the one-particle Green's function is fully determined by the knowledge of the exponent $3\ensuremath{-}d$ characterizing the divergence of the longitudinal susceptibility and the Ward identities associated to gauge and Galilean invariances. The infrared singularity of ${\ensuremath{\Sigma}}_{\text{an}}(\mathbf{p},\ensuremath{\omega})$ leads to a continuum of excitations (coexisting with the sound mode) which shows up in the one-particle spectral function.
We investigate the single-particle spectral density of interacting bosons within the nonperturbative 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 the 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 quasifree 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 Nozi\`eres and discuss the nature of the nonanalyticities of the self-energies in the very small momentum regime.
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We review the infrared behavior of interacting bosons at zero temperature. After a brief discussion of the Bogoliubov approximation and the breakdown of perturbation theory due to infrared divergences, we present two approaches that are free of infrared divergences—Popov’s hydrodynamic theory and the nonperturbative renormalization group—and allow us to obtain the exact infrared behavior of the correlation functions. We also point out the connection between the infrared behavior in the superfluid phase and the critical behavior at the superfluid-Mott-insulator transition in the Bose-Hubbard model.
We investigate Bose-Einstein condensation for interacting bosons at zero and nonzero temperature. Functional renormalization provides us with a consistent method to compute the effect of fluctuations beyond the Bogoliubov approximation. For three-dimensional dilute gases, we find an upper bound on the scattering length aaa which is of the order of the microphysical scale---typically the range of the van der Waals interaction. In contrast to fermions near the unitary bound, no strong interactions occur for bosons with approximately pointlike interactions, thus explaining the high quantitative reliability of perturbation theory for most quantities. For zero temperature we compute the quantum phase diagram for bosonic quasiparticles with a general dispersion relation, corresponding to an inverse microphysical propagator with terms linear and quadratic in the frequency. We compute the temperature dependence of the condensate and particle density nnn, and find for the critical temperature Tc{T}_{c}Tc​ a deviation from the free theory, $\ensuremath{\Delta}{T}_{c}/{T}_{c}=2.1a{n}^{1/3}$. For the sound velocity at zero temperature we find very good agreement with the Bogoliubov result, such that it may be used to determine the particle density accurately.
We investigate superfluidity of bosons in gapped topological bands and discover a new phase that has no counterparts in the previous literature. This phase is characterized by a highly unconventional modulation of the order parameter, breaking the crystallographic symmetry, and for which the condensation momentum is neither zero nor any other high-symmetry vector of the Brillouin zone. This unconventional structure impacts the spectrum of Bogoliubov excitations and, consequently, the speed of sound in the system. Even in the case of perfectly flat bands, the speed of sound and Bogoliubov excitations remain nonvanishing, provided that the underlying topology and quantum geometry are nontrivial. Furthermore, we derive detailed expressions for the superfluid weight using the Popov hydrodynamic formalism for superfluidity and provide estimates for the Berezinskii-Kosterlitz-Thouless temperature, which is significantly enhanced by the nontriviality of the underlying quantum metric. These results are applicable to generic topological bosonic bands, with or without dispersion. To illustrate our findings, we employ the Haldane model with a tunable bandwidth, including the narrow lowest-band case. Within this model, we also observe a re-entrant superfluid behavior: As the Haldane's magnetic flux is varied, the Berezinskii-Kosterlitz-Thouless transition temperature initially decreases to almost zero, only to resurface with renewed vigor.