Infrared behavior in systems with a broken continuous symmetry: classical O(N) model vs interacting bosons

In systems with a spontaneously broken continuous symmetry, the perturbative loop expansion is plagued with infrared divergences due to the coupling between transverse and longitudinal fluctuations. As a result the longitudinal susceptibility diverges and the self-energy becomes singular at low energy. We study the crossover from the high-energy Gaussian regime, where perturbation theory remains valid, to the low-energy Goldstone regime characterized by a diverging longitudinal susceptibility. We consider both the classical linear O($N$) model and interacting bosons at zero temperature, using a variety of techniques: perturbation theory, hydrodynamic approach (i.e., for bosons, Popov's theory), large-$N$ limit and non-perturbative renormalization group. We emphasize the essential role of the Ginzburg momentum scale $p_G$ below which the perturbative approach breaks down. Even though the action of (non-relativistic) bosons includes a first-order time derivative term, we find remarkable similarities in the weak-coupling limit between the classical O($N$) model and interacting bosons at zero temperature.Comment: v2) 19 pages, 8 figure

Superfluid equation of state of dilute composite bosons

We present an exact theory of the BEC-BCS crossover in the BEC regime, which treats explicitely dimers as made of two fermions. We apply our framework, at zero temperature, to the calculation of the equation of state. We find that, when expanding the chemical potential in powers of the density n up to the Lee-Huang-Yang order, proportional to n^3/2, the result is identical to the one of elementary bosons in terms of the dimer-dimer scattering length a_M, the composite nature of the dimers appearing only in the next order term proportional to n^2 .Comment: 5 pages, 3 figure

Packing dimension of mean porous measures

We prove that the packing dimension of any mean porous Radon measure on $\mathbb R^d$ may be estimated from above by a function which depends on mean porosity. The upper bound tends to $d-1$ as mean porosity tends to its maximum value. This result was stated in \cite{BS}, and in a weaker form in \cite{JJ1}, but the proofs are not correct. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure $\mu$ on $\mathbb R$ such that $\mu(A)=0$ for all mean porous sets $A\subset\mathbb R$.Comment: Revised versio

Damping in 2D and 3D dilute Bose gases

Damping in 2D and 3D dilute gases is investigated using both the hydrodynamical approach and the Hartree-Fock-Bogoliubov (HFB) approximation . We found that the both methods are good for the Beliaev damping at zero temperature and Landau damping at very low temperature, however, at high temperature, the hydrodynamical approach overestimates the Landau damping and the HFB gives a better approximation. This result shows that the comparison of the theoretical calculation using the hydrodynamical approach and the experimental data for high temperature done by Vincent Liu (PRL {\bf21} 4056 (1997)) is not proper. For two-dimensional systems, we show that the Beliaev damping rate is proportional to $k^3$ and the Landau damping rate is proportional to $T^2$ for low temperature and to $T$ for high temperature. We also show that in two dimensions the hydrodynamical approach gives the same result for zero temperature and for low temperature as HFB, but overestimates the Landau damping for high temperature.Comment: 11 pages, 4 figure

Effective field theory and dispersion law of the phonons of a non-relativistic superfluid

We study the recently proposed effective field theory for the phonon of an arbitrary non-relativistic superfluid. After computing the one-loop phonon self-energy, we obtain the low temperature T contributions to the phonon dispersion law at low momentum, and see that the real part of those can be parametrized as a thermal correction to the phonon velocity. Because the phonons are the quanta of the sound waves, at low momentum their velocity should agree with the speed of sound. We find that our results match at order T^4ln(T) with those predicted by Andreev and Khalatnikov for the speed of sound, derived from the superfluid hydrodynamical equations and the phonon kinetic theory. We get also higher order corrections of order T^4, which are not reproduced pushing naively the kinetic theory computation. Finally, as an application, we consider the cold Fermi gas in the unitarity limit, and find a universal expression for the low T relative correction to the speed of sound for these systems.Comment: 14 pages, 2 figures. References adde

Thermodynamics of a Bose-Einstein Condensate with Weak Disorder

We consider the thermodynamics of a homogeneous superfluid dilute Bose gas in the presence of weak quenched disorder. Following the zero-temperature approach of Huang and Meng, we diagonalize the Hamiltonian of a dilute Bose gas in an external random delta-correlated potential by means of a Bogoliubov transformation. We extend this approach to finite temperature by combining the Popov and the many-body T-matrix approximations. This approach permits us to include the quasi-particle interactions within this temperature range. We derive the disorder-induced shifts of the Bose-Einstein critical temperature and of the temperature for the onset of superfluidity by approaching the transition points from below, i.e., from the superfluid phase. Our results lead to a phase diagram consistent with that of the finite-temperature theory of Lopatin and Vinokur which was based on the replica method, and in which the transition points were approached from above.Comment: 11 pages, 5 figure

Thermodynamics of the superfluid dilute Bose gas with disorder

We generalize the Beliaev-Popov diagrammatic technique for the problem of interacting dilute Bose gas with weak disorder. Averaging over disorder is implemented by the replica method. Low energy asymptotic form of the Green function confirms that the low energy excitations of the superfluid dirty Boson system are sound waves with velocity renormalized by the disorder and additional dissipation due to the impurity scattering. We find the thermodynamic potential and the superfluid density at any temperature below the superfluid transition temperature and derive the phase diagram in temperature vs. disorder plane.Comment: 4 page

Behavior of the anomalous correlation function in uniform 2D Bose gas

We investigate the behavior of the anomalous correlation function in two dimensional Bose gas. In the local case, we find that this quantity has a finite value in the limit of weak interactions at zero temperature. The effects of the anomalous density on some thermodynamic quantities are also considered. These effects can modify in particular the chemical potential, the ground sate energy, the depletion and the superfluid fraction. Our predictions are in good agreement with recent analytical and numerical calculations. We show also that the anomalous density presents a significant importance compared to the non-condensed one at zero temperature. The single-particle anomalous correlation function is expressed in two dimensional homogenous Bose gases by using the density-phase fluctuation. We then confirm that the anomalous average accompanies in analogous manner the true condensate at zero temperature while it does not exist at finite temperature.Comment: 15 pages, 3 figure

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

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 $k_G$ ("Ginzburg" scale). While the Bogoliubov approximation is valid at large momenta and energies, |\p|,|\w|/c\gg k_G (with $c$ the velocity of the Bogoliubov sound mode), in the infrared (hydrodynamic) regime |\p|,|\w|/c\ll 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 \Sigan(\p,\w)\simeq\const at high-energies and behaves as \Sigan(\p,\w)\sim (c^2\p^2-\w^2)^{(d-3)/2} in the infrared regime (with $d$ the space dimension), in agreement with the Nepomnyashchii identity \Sigan(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 \Sigan(\p,\w) leads to a continuum of excitations (coexisting with the sound mode) which shows up in the one-particle spectral function.Comment: v1) 23 pages, 11 figures. v2) Changes following referee's comments. To appear in Phys. Rev.A. v3) Typos correcte

BCS-BEC crossover in a system of microcavity polaritons

We investigate the thermodynamics and signatures of a polariton condensate over a range of densities, using a model of microcavity polaritons with internal structure. We determine a phase diagram for this system including fluctuation corrections to the mean-field theory. At low densities the condensation temperature, T_c, behaves like that for point bosons. At higher densities, when T_c approaches the Rabi splitting, T_c deviates from the form for point bosons, and instead approaches the result of a BCS-like mean-field theory. This crossover occurs at densities much less than the Mott density. We show that current experiments are in a density range where the phase boundary is described by the BCS-like mean-field boundary. We investigate the influence of inhomogeneous broadening and detuning of excitons on the phase diagram.Comment: 20 pages, 6 figure