Self-consistent calculation of the coupling constant in the Gross-Pitaevskii equation

A method is proposed for a self-consistent evaluation of the coupling constant in the Gross-Pitaevskii equation without involving a pseudopotential replacement. A renormalization of the coupling constant occurs due to medium effects and the trapping potential, e.g. in quasi-1D or quasi-2D systems. It is shown that a simplified version of the Hartree-Fock-Bogoliubov approximation leads to a variational problem for both the condensate and a two-body wave function describing the behaviour of a pair of bosons in the Bose-Einstein condensate. The resulting coupled equations are free of unphysical divergences. Particular cases of this scheme that admit analytical estimations are considered and compared to the literature. In addition to the well-known cases of low-dimensional trapping, cross-over regimes can be studied. The values of the kinetic, interaction, external, and release energies in low dimensions are also evaluated and contributions due to short-range correlations are found to be substantial.Comment: 15 pages, ReVTEX, no figure

Hidden Symmetries and their Consequences in t2gt_{2g} Cubic Perovskites

The five-band Hubbard model for a $d$ band with one electron per site is a model which has very interesting properties when the relevant ions are located at sites with high (e. g. cubic) symmetry. In that case, if the crystal field splitting is large one may consider excitations confined to the lowest threefold degenerate $t_{2g}$ orbital states. When the electron hopping matrix element ($t$) is much smaller than the on-site Coulomb interaction energy ($U$), the Hubbard model can be mapped onto the well-known effective Hamiltonian (at order $t^{2}/U$) derived by Kugel and Khomskii (KK). Recently we have shown that the KK Hamiltonian does not support long range spin order at any nonzero temperature due to several novel hidden symmetries that it possesses. Here we extend our theory to show that these symmetries also apply to the underlying three-band Hubbard model. Using these symmetries we develop a rigorous Mermin-Wagner construction, which shows that the three-band Hubbard model does not support spontaneous long-range spin order at any nonzero temperature and at any order in $t/U$ -- despite the three-dimensional lattice structure. Introduction of spin-orbit coupling does allow spin ordering, but even then the excitation spectrum is gapless due to a subtle continuous symmetry. Finally we showed that these hidden symmetries dramatically simplify the numerical exact diagonalization studies of finite clusters.Comment: 26 pages, 3 figures, 520 KB, submitted Phys. Rev.

The ground state energy of the weakly interacting Bose gas at high density

We prove the Lee-Huang-Yang formula for the ground state energy of the 3D Bose gas with repulsive interactions described by the exponential function, in a simultaneous limit of weak coupling and high density. In particular, we show that the Bogoliubov approximation is exact in an appropriate parameter regime, as far as the ground state energy is concerned.Comment: RevTeX4, 16 page

Non Local Theories: New Rules for Old Diagrams

We show that a general variant of the Wick theorems can be used to reduce the time ordered products in the Gell-Mann & Low formula for a certain class on non local quantum field theories, including the case where the interaction Lagrangian is defined in terms of twisted products. The only necessary modification is the replacement of the Stueckelberg-Feynman propagator by the general propagator (the ``contractor'' of Denk and Schweda) D(y-y';tau-tau')= - i (Delta_+(y-y')theta(tau-tau')+Delta_+(y'-y)theta(tau'-tau)), where the violations of locality and causality are represented by the dependence of tau,tau' on other points, besides those involved in the contraction. This leads naturally to a diagrammatic expansion of the Gell-Mann & Low formula, in terms of the same diagrams as in the local case, the only necessary modification concerning the Feynman rules. The ordinary local theory is easily recovered as a special case, and there is a one-to-one correspondence between the local and non local contributions corresponding to the same diagrams, which is preserved while performing the large scale limit of the theory.Comment: LaTeX, 14 pages, 1 figure. Uses hyperref. Symmetry factors added; minor changes in the expositio

Condensate Oscillations, Kinetic Equations and Two-Fluid Hydrodynamics in a Bose Gas

This is based on 4 lectures given at the 13th Australian Physics Summer School, Australia National University, Canberra, Jan 17-28, 2000. The main topic is the theory of collective modes in a trapped Bose gas at finite temperatures. A generalized Gross-Pitaevskii equation is derived at finite temperatures, which is used to discuss a new mechanism for damping in the collisionless region arising from interactions with a static thermal cloud of non-condensate atoms. Next, introducing a kinetic equation for the thermal cloud, we derive two-fluid equations of motion for the condensate and non-condensate components in the collision-dominated hydrodynamic region. We show that these are precisely the equivalent of the Landau two-fluid equations in the limit that the two components are in diffusive local equilibrium. However, our equations also predict the existence of a new zero frequency relaxational mode, in addition to the usual Landau hydrodynamic modes (such as first and second sound). The special importance and simplicity of two-fluid hydrodynamics is stressed.Comment: 50 pages, 7 figures; To appear in "Proceedings of the 13th Physics Summer S chool: Bose-Einstein Condensation", eds. C.M.Savage and M.Das (World Scientific, 2000

Ground state properties and excitation spectra of non-Galilean invariant interacting Bose systems

We study the ground state properties and the excitation spectrum of bosons which, in addition to a short-range repulsive two body potential, interact through the exchange of some dispersionless bosonic modes. The latter induces a time dependent (retarded) boson-boson interaction which is attractive in the static limit. Moreover the coupling with dispersionless modes introduces a reference frame for the moving boson system and hence breaks the Galilean invariance of this system. The ground state of such a system is depleted {\it linearly} in the boson density due to the zero point fluctuations driven by the retarded part of the interaction. Both quasiparticle (microscopic) and compressional (macroscopic) sound velocities of the system are studied. The microscopic sound velocity is calculated up the second order in the effective two body interaction in a perturbative treatment, similar to that of Beliaev for the dilute weakly interacting Bose gas. The hydrodynamic equations are used to obtain the macroscopic sound velocity. We show that these velocities are identical within our perturbative approach. We present analytical results for them in terms of two dimensional parameters -- an effective interaction strength and an adiabaticity parameter -- which characterize the system. We find that due the presence of several competing effects, which determine the speed of the sound of the system, three qualitatively different regimes can be in principle realized in the parameter space and discuss them on physical grounds.Comment: 6 pages, 2 figures, to appear in Phys. Rev.

The fate of phonons in freely expanding Bose-Einstein condensates

Phonon-like excitations can be imprinted into a trapped Bose-Einstein condensate of cold atoms using light scattering. If the condensate is suddenly let to freely expand, the initial phonons lose their collective character by transferring their energy and momentum to the motion of individual atoms. The basic mechanisms of this evaporation process are investigated by using the Gross-Pitaevskii theory and dynamically rescaled Bogoliubov equations. Different regimes of evaporation are shown to occur depending on the phonon wavelength. Distinctive signatures of the evaporated phonons are visible in the density distribution of the expanded gas, thus providing a new type of spectroscopy of Bogoliubov excitations.Comment: 13 pages, 16 figure

The generalized Fenyes-Nelson model for free scalar field theory

The generalized Fenyes--Nelson model of quantum mechanics is applied to the free scalar field. The resulting Markov field is equivalent to the Euclidean Markov field with the times scaled by a common factor which depends on the diffusion parameter. This result is consistent between Guerra's earlier work on stochastic quantization of scalar fields. It suggests a deep connection between Euclidean field theory and the stochastic interpretation of quantum mechanics. The question of Lorentz covariance is also discussed.Comment: 6 page

Time evolution of correlation functions and thermalization

We investigate the time evolution of a classical ensemble of isolated periodic chains of O(N)-symmetric anharmonic oscillators. Our method is based on an exact evolution equation for the time dependence of correlation functions. We discuss its solutions in an approximation which retains all contributions in next-to-leading order in a 1/N expansion and preserves time reflection symmetry. We observe effective irreversibility and approximate thermalization. At large time the system approaches stationary solutions in the vicinity of, but not identical to, thermal equilibrium. The ensemble therefore retains some memory of the initial condition beyond the conserved total energy. Such a behavior with incomplete thermalization is referred to as "mesoscopic dynamics". It is expected for systems in a small volume. Surprisingly, we find that the nonthermal asymptotic stationary solutions do not change for large volume. This raises questions on Boltzmann's conjecture that macroscopic isolated systems thermalize.Comment: 40 pages, 9 figure

The Second Order Upper Bound for the Ground Energy of a Bose Gas

Consider $N$ bosons in a finite box $\Lambda= [0,L]^3\subset \mathbf R^3$ interacting via a two-body smooth repulsive short range potential. We construct a variational state which gives the following upper bound on the ground state energy per particle $$\bar\lim_{\rho\to0} \bar \lim_{L \to \infty, N/L^3 \to \rho} (\frac{e_0(\rho)- 4 \pi a \rho}{(4 \pi a)^{5/2}(\rho)^{3/2}})\leq \frac{16}{15\pi^2},$$ where $a$ is the scattering length of the potential. Previously, an upper bound of the form $C 16/15\pi^2$ for some constant $C > 1$ was obtained in \cite{ESY}. Our result proves the upper bound of the the prediction by Lee-Yang \cite{LYang} and Lee-Huang-Yang \cite{LHY}.Comment: 62 pages, no figure