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arXiv:cond-mat/9604076 · Physical Review Letters

Infrared Behavior of Interacting Bosons at Zero Temperature

Claudio Castellani, C. Di Castro, F. Pistolesi, Giancarlo Calvanese Strinati

Published Mar 3, 1997• 61 citations
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Abstract

We exploit the symmetries associated with the stability of the superfluid phase to solve the long-standing problem of interacting bosons in the presence of a condensate at zero temperature. Implementation of these symmetries poses strong conditions on the renormalizations that heal the singularities of perturbation theory. The renormalized theory gives the following: For $d>3$ the Bogoliubov quasiparticles as an exact result; for $1<d\ensuremath{\le}3$ a nontrivial solution with the exact exponent for the singular longitudinal correlation function, with phonons again as low-lying excitations.

DOI
10.1103/physrevlett.78.1612

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We exploit the symmetries associated with the stability of the superfluid phase to solve the long-standing problem of interacting bosons in the presence of a condensate at zero temperature. Implementation of these symmetries poses strong conditions on the renormalizations that heal the singularities of perturbation theory. The renormalized theory gives the following: For d>3d>3d>3 the Bogoliubov quasiparticles as an exact result; for 1<d\ensuremath≤31<d\ensuremath{\le}31<d\ensuremath≤3 a nontrivial solution with the exact exponent for the singular longitudinal correlation function, with phonons again as low-lying excitations.
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 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 exploit the renormalization-group approach to establish the exact infrared behavior of an interacting Bose system at zero temperature. The local-gauge symmetry in the broken-symmetry phase is implemented through the associated Ward identities, which reduce the number of independent running couplings to a single one. For this coupling the \ensuremath{\epsilon} expansion can be controlled to all orders in \ensuremath{\epsilon} $(=3\ensuremath{-}d).$ For spatial dimensions 1<d< 31<d<~31<d< 3 the Bogoliubov fixed point is unstable towards a different fixed point characterized by the divergence of the longitudinal correlation function. The Bogoliubov linear spectrum, however, is found to be independent of the critical behavior of this correlation function, being exactly constrained by Ward identities. The new fixed point properly gives a finite value of the coupling among transverse fluctuations, but due to virtual intermediate longitudinal fluctuations, the effective coupling affecting the transverse correlation function flows to zero. As a result, no transverse anomalous dimension is present. This treatment allows us to recover known results for the quantum Bose gas in the context of a unifying framework and also to reveal the nontrivial skeleton structure of its perturbation theory.
We use a non-perturbative renormalization group technique to study interacting bosons at zero temperature. Our approach reveals the instability of the Bogoliubov fixed point when d ⩽3 and yields the exact infrared behavior in all dimensions d >1 within a rather simple theoretical framework. It also enables to compute the low-energy properties in terms of the parameters of a microscopic model. In one dimension and for not too strong interactions, it yields a good picture of the Luttinger-liquid behavior of the superfluid phase.
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.