The chemical potential for the inhomogeneous electron liquid in terms of its kinetic and potential parts with special consideration of the surface potential step and BCS-BEC crossover

The chemical potential μ of a many-body system is valuable since it carries fingerprints of phase changes. Here, we summarize results for μ for a three-dimensional electron liquid in terms of average kinetic and potential energies per particle. The difference between μ and the energy per particle is found to be exactly the electrostatic potential step at the surface. We also present calculations for an integrable one-dimensional many-body system with delta function interactions, exhibiting a BCS-BEC crossover. It is shown that in the BCS regime the chemical potential can be expressed solely in terms of the ground-state energy per particle. A brief discussion is also included of the strong coupling BEC limit.     

Exact results for the spectra of bosons and fermions with contact interaction

An N-body bosonic model with delta-contact interactions projected on the lowest Landau level is considered. For a given number of particles in a given angular momentum sector, any energy level can be obtained exactly by means of diagonalizing a finite matrix: they are roots of algebraic equations. A complete solution of the three-body problem is presented, some general properties of the N-body spectrum are pointed out, and a number of novel exact analytic eigenstates are obtained. The FQHE N-fermion model with Laplacian-delta interactions is also considered along the same lines of analysis. New exact eigenstates are proposed, along with the Slater determinant, whose eigenvalues are shown to be related to Catalan numbers.     

Flow equations for the spin-boson problem

Using continuous unitary transformations recently introduced by Wegner [1], we obtain flow equations for the parameters of the spin-boson Hamiltonian. Interactions not contained in the original Hamiltonian are generated by this unitary transformation. Within an approximation that neglects additional interactions quadratic in the bath operators, we can close the flow equations. Applying this formalism to the case of Ohmic dissipation at zero temperature, we calculate the renormalized tunneling frequency. We find a transition from an untrapped to trapped state at the critical coupling constant α _c =1. We also obtain the static susceptibility via the equilibrium spin correlation function. Our results are both consistent with results known from the Kondo problem and those obtained from mode-coupling theories. Using this formalism at finite temperature, we find a transition from coherent to incoherent tunneling atT ₂ ^* ≈2T ₁ ^* , whereT ₁ ^* is the crossover temperature of the dynamics known from the NIBA.     

Low frequency, low temperature properties of the spin-boson problem

Low temperature and low frequency properties of a spin-boson model are investigated within a super operator and Liouville space formulation. The leading contributions are identified with the help of projection operators projecting onto the equilibrium state. The quantities of interest are expressed in terms of weighted bath propagators and static linear and nonlinear susceptibilities. In particular the generalized Shiba relation and Wilson ratio are recovered. Received 7 July 2000 and Received in final form 8 August 2000     

Quantum macroscopic equations from Bohm potential and propagation of waves

Bohm’s interpretation of Quantum Mechanics leads to the derivation of a Quantum Kinetic Equation. In the present work, moments of this kinetic equation are taken, thus deriving conservation equations. These macroscopic equations are then applied to study the propagation of longitudinal density perturbations in neutral gases and plasmas, of either fermions or bosons. The dispersion relation is derived and the effect of the Bohm potential shown; the speed of propagation calculated and the difference between fermions and bosons investigated. Pseudosonic waves in quantum plasmas are obtained including the effect of the Bohm potential.     

Second quantized reduced Bloch equations and the exact solutions for pairing Hamiltonian

In this article, we present a set of hierarchy Bloch equations for the reduced density operators in either canonical or grand canonical ensembles in the occupation number representation. They provide a convenient tool for studying the equilibrium quantum statistical mechanics for some model systems. As an example of their applications, we solve the equations for the model system with a pairing Hamiltonian. With the aid of its symplectic group symmetry, we obtain the statistical reduced density matrices with different orders. As a special instance for the solutions, we also get the reduced density matrices of the ground state for a superconductor.     

Cumulant approach to dynamical correlation functions at finite temperatures

A new theoretical approach, based on the introduction of cumulants, to calculate thermodynamic averages and dynamical correlation functions at finite temperatures is developed. The method is formulated in Liouville instead of Hilbert space and can be applied to operators which do not require to satisfy fermion or boson commutation relations. The application of the partitioning and projection methods for the dynamical correlation functions is considered. The present method can be applied to weakly as well as to strongly correlated systems.     

New multiple histogram method for studying phase transitions

For temperature zero the effects of disorder for interacting bosons are considered. The disorder induced superfluid-insulator transition in thed-dimensional disordered Bogoliubov model is discussed. Results for a short-range and a long-range random potential are given. For short-range disorder we argue that ford<4 arbitrarily small disorder localizes the Bose condensate for vanishing interaction potential. Ford>4 a certain strength of the disorder potential is necessary in order to localize the condensate. For the three-dimensional Bogoliubov model our results are in agreement with a recent calculation. We compare our theoretical predictions with numerical experiments for a disordered boson Hubbard model.     

Impossibility to describe repulsion with contact interaction

Contact interactions always lead to attractive behavior. Arguments are presented to show why a repulsive interacting system, e.g. Bose gases, cannot be described by contact interactions and corresponding treatments are possibly obscured by the appearance of bound states. The usually used cut-offs are identified as finite range parameters.     

Self-Trapping of Two Bose-Einstein Condensates with a Coupling Drive

We present an approximate analytical solution to the population imbalance of two-component Bose-Einstein condensate with the coupling drive. The dependence of the time evolution of self-trapping upon the radio frequency wave, the Rabi coupling frequency, the initial atom number and relative phase between two condensates are investigated. The lower radio frequency wave, the same atom number and initial relative phase between condensates are beneficial to observe the self-trapping.     

Do bosons obey Bose-Einstein distribution: Two iterated limits of Gentile distribution

It is a common impression that by only setting the maximum occupation number to infinity, which is the demand of the indistinguishability of bosons, one can achieve the statistical distribution that bosons obey — the Bose-Einstein distribution. In this Letter, however, we show that only with an infinite maximum occupation number one cannot uniquely achieve the Bose-Einstein distribution, since in the derivation of the Bose-Einstein distribution, the problem of iterated limit is encountered. For achieving the Bose-Einstein distribution, one needs to take both the maximum occupation number and the total number of particles to infinities, and, then, the problem of the order of taking limits arises. Different orders of the limit operations will lead to different statistical distributions. For achieving the Bose-Einstein distribution, besides setting the maximum occupation number, we also need to state the order of the limit operations.     

First order in the plaquette expansion

The plaquette expansion, a general non-perturbative method for calculating the properties of lattice Hamiltonian systems, is analytically investigated at the first non-trivial order for an arbitrary system. At this level the approximation describes systems with either a bounded or an unbounded spectrum, depending on simple inequalities formed from the first four cumulants. The analysis yields analytic forms for the ground state energy, mass gaps and density of states in the thermodynamic limit. An exact form for the resolvent oeprator is given for a finite number of sites, as well as the asymptotic form in the thermodynamic limit.     

Single-particle self-energy and optical conductivity of the simplified Hubbard model

It is shown that the single-particle self-energy of the one and two-dimensional simplified Hubbard model exhibits different behavior characterized by Fermi-liquid, non-Fermi-liquid quasiparticle, or non-quasiparticle excitations, as a function of the strength of the on-site Coulomb repulsionU, temperature, and electron filling. For half-filled lattices, results for the optical conductivity indicate that the d.c. conductivity is zero for all temperatures andU>0.     

A new approach to electron-electron interactions in strongly correlated systems at arbitrary spin-polarization and temperature

The uniform electron fluid is the reference model for density functional calculations. Even for this system, many-body perturbation theory, and related methods become questionable when the density parameter r_s exceeds unity. Hence, quantum Monte Carlo (QMC) simulation has been almost the only applicable method. We review a new approach, which uses a mapping of the quantum fluid to a classical Coulomb fluid, based on density-functional concepts. It is applicable at finite temperatures and arbitrary spin polarizations as well, and correctly recovers even the logarithmic terms in the exchange and correlations energies close to T=0. We show by detailed comparison with available QMC data that the method yields accurate pair-distribution functions, spin-dependent energies, static local-field factors, Landau parameter-based quantities like m∗ and g∗, for strongly coupled electron fluids.     

Exchange-correlation and layer-thickness effects in quasi-two dimensional electron liquids

We demonstrate a replacement of the non-uniform sub-band density of quasi-2D electron layers by an effective uniform-slab density. Exchange, correlation and Fermi-liquid properties are determined via a mapping of the electron liquid to a classical fluid, using the hyper-netted-chain equation inclusive of bridge corrections, (i.e. the CHNC), as a function of the density, spin-polarization, layer width and the temperature. Our parameters-free theory is in good accord with quantum simulations, with effective-mass and spin-susceptibility results for 2D layers found in GaAs/AlGaAs structures.     

Plaquette expansion proof and interpretation

The plaquette expansion, a general non-perturbative method for calculating the properties of lattice Hamiltonian systems, is established up to the first two orders for an arbitrary system. This method employs an expansion of the Lanczos coefficients, the tridiagonal Hamiltonian matrix elements or equivalently the continued fraction coefficients of the resolvent, in a descending series in the size of the system. The coefficients of this series are formed from the low order cumulants or connected Hamiltonian moments. The lowest order approximation in the plaquette expansion corresponds to a gaussian model which is a consequence of the central limit theorem. The first nontrivial order yields a model with a spectrum on a bounded energy interval, becoming asymptotically uniform in the thermodynamic limit.     

Output from an atom laser: theory vs. experiment

Atom lasers based on rf-outcoupling can be described by a set of coupled generalized Gross–Pitaevskii equations (GPE). We compare the theoretical predictions obtained by numerically integrating the time-dependent GPE of an effective one-dimensional model with recently measured experimental data for the F=2 and F=1 states of Rb-87. We conclude that the output of a rf atom-laser can be described by this model in a satisfactory way. Received: 15 June 1999 / Revised version: 9 September 1999 / Published online: 10 November 1999     

Bose-Einstein condensation of bouncing balls

Microscopic bouncing balls, i.e., particles confined within a positive one-half-dimensional gravitational potential, display Bose-Einstein condensation (BEC) not only in the thermodynamic limit but also in the case of a finite number of particles, and the critical temperature with a finite number of particles is higher than that in the thermodynamic limit. This system is different from the one-dimensional harmonic potential one, for which the standard result indicates that the BEC is not possible unless the number of particles is finite.