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< 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 investigate the phase diagram of a one-dimensional dissipative Bose-Hubbard model using the nonperturbative functional renormalization group (FRG). Each lattice site is coupled to an independent bath, generating long-range temporal interactions that encode non-Markovian dissipation. For a broad class of bath spectra - ohmic, sub-ohmic, and super-ohmic - we identify two competing low-energy regimes: a Luttinger-liquid line of fixed points and a dissipative fixed point characterized by finite compressibility, vanishing superfluid stiffness, and universal scaling exponents, separated by a Berezinskii-Kosterlitz-Thouless transition. The FRG framework is essential here, as it provides access to the complete renormalization group flow and all fixed points from a single microscopic action, beyond the reach of perturbative or variational methods. This work establishes a unified and systematically improvable framework for describing dissipative quantum phases in one dimension.
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.
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 (``Ginzburg'' scale). While the Bogoliubov approximation is valid at large momenta and energies, $|\mathbf{p}|,\text{ }|\ensuremath{\omega}|/c⪢{k}_{G}$ (with
c 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
d 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.