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CRC Press eBooks

Green’s Function Method and Theory of Repulsive Bose Gas

Flordivino Basco, Hidetoshi Fukuyama

Published Nov 27, 2020• 0 citations
Publisher

Abstract

A unified theory of repulsive bose gas is constructed. The theory is based on the universal feature of the excitation spectra - linear and quadratic for small and large values of momenta, respectively. The ensemble of bosons is treated as a mixture of condensates and noncondensates. The measurable amount of condensates is determined by imposing that they coexist at equilibrium. This scenario of coexistence gives a natural explanation why the excitation spectra are gapless. Calculations made, based on this theory, yield the established results for one and three dimensions. For the first time, a new result is obtained in two dimensions indicating a strong dimensional crossover. The theory is consistent with vanishing of the anomalous self-energy for zero momentum and energy for dimensions less than or equal to three.

DOI
10.1201/9781003078296-10

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International Journal of Modern Physics B•Jul 16, 2018•4 citations
A unified theory of repulsive bose gas is constructed. The theory is based on the universal feature of the excitation spectra - linear and quadratic for small and large values of momenta, respectively. The ensemble of bosons is treated as a mixture of condensates and noncondensates. The measurable amount of condensates is determined by imposing that they coexist at equilibrium. This scenario of coexistence gives a natural explanation why the excitation spectra are gapless. Calculations made, based on this theory, yield the established results for one and three dimensions. For the first time, a new result is obtained in two dimensions indicating a strong dimensional crossover. The theory is consistent with vanishing of the anomalous self-energy for zero momentum and energy for dimensions less than or equal to three.
The properties of a Bose-Einstein gas with repulsive interparticle interactions are determined at low temperature. It is shown that the contribution to the energy arising from particle excitation with small momentum transfers computed in conventional perturbation theory is divergent, but that this difficulty can be avoided by an alternative procedure. An exact method is developed for dealing with that part of the Hamiltonian which gives rise to the perturbation divergence and an exact solution is obtained. The treatment of the region of large momentum transfers is then carried out by introducing the concept of the scattering length of the interaction. The energy is given by the well-known result of the optical theorem corrected by a series in powers of ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$, where $\ensuremath{\rho}$ is the density and aaa the scattering length. Examination of the processes contributing to the energy shows that 95.8% of the correction of order ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$ results from a simple alteration of the perturbation energy denominators which takes into account the interaction energy of excited pairs with the unexcited pairs of the medium. The remaining 4.2% arises from multiple particle excitation. These methods also predict the existence of phonon excitation with phonon energies that are linear in the phonon momentum for small excitation and approach q22M\frac{{q}^{2}}{2M}2Mq2​ for large momentum. The detailed consideration of this spectrum forms part of the following paper.The results are compared with the theory of fermion systems with strong interactions, where again the largest correction to the first-order "optical" energy arises from alteration of the two-particle propagator which is required to take into account interactions with the many unexcited particles of the medium. The propagator corrections in the fermion and boson systems are similar except for a characteristic difference arising from the statistics. The theory for bosons is shown to be essentially identical with that given by Brueckner and Levinson in their general formulation of the theory of many-body systems with strong interactions.Systems with attractive but saturating interactions are also considered and some new difficulties are shown to arise which cannot be treated by a straightforward application of the methods of this paper.
Theory of a weakly non-ideal Bose gas in the canonical ensemble is developed without assumption of the C-number representation of the creation and annihilation operators with zero momentum. Instead of this assumption, we use the assumption on the C-number nature of the density operator N 0 = a † 0 a 0 with zero momentum. It is shown that the pole of the "density-density" Green function (DDGF), as well as the pole of the single-particle Green function (SPGF), exactly coincide with the Bogoliubov phonon-roton spectrum of excitations for both assumptions. This spectrum, as is known confirmed by many neutron and x-ray experimental measurements of the dynamic structure factor in He II, is straightly related to the DDGF. At the same time, we show that in the other case under consideration, when neither N 0 nor a † 0 and a 0 are C-numbers, a gap can exist in SPGF. This gap in SPGF excitations is straightly related to the density of particles in the "condensate". Therefore, the spectra of excitations for the DDGF and SPGF in the last case under consideration are different, in contrast to the Bogoliubov-type theory where these spectra are identical.
Theory of a weakly non-ideal Bose gas in the canonical ensemble is developed without assumption of the C-number representation of the creation and annihilation operators with zero momentum. It is shown that the pole of the "density-density" Green's function exactly coincides with the Bogolybov's phonon-roton spectrum of excitations. At the same time, a gap exists in the one-particle excitation spectrum . This gap is related to the density of particles in the "condensate".
Because of the vast growth of the many-body level density with excitation energy, its smoothed form is of central relevance for spectral and thermodynamic properties of interacting quantum systems. We compute the cumulative of this level density for confined one-dimensional continuous systems with repulsive short-range interactions. We show that the crossover from an ideal Bose gas to the strongly correlated, fermionized gas, i.e., partial fermionization, exhibits universal behavior: Systems with very few and up to many particles share the same underlying spectral features. In our derivation we supplement quantum cluster expansions with short-time dynamical information. Our nonperturbative analytical results are in excellent agreement with numerics for systems of experimental relevance in cold atom physics, such as interacting bosons on a ring (Lieb-Liniger model) or subject to harmonic confinement. Our method provides predictions for excitation spectra that enable access to finite-temperature thermodynamics in large parameter ranges.
In Chapter 3, we introduced a simple but reasonable approximation for the nonequilibrium dynamics of a Bose-condensed gas based on a generalized GP equation coupled to a kinetic equation. In Chapters 4–7, we turn to the question of how to derive such coupled equations for the condensate and noncondensate components in a way that gives a deeper understanding of the ZNG theory. Chapters 4–7 involve an introduction to Green's functions and field theoretic techniques for nonequilibrium problems. These provide the natural language and formalism to deal with the many subtle aspects of a Bose-condensed gas at finite temperatures. These four chapters are fairly technical. This chapter is mainly based on Kadanoff and Baym (1962) and Imamović-Tomasović (2001). Readers who are not interested in these questions can go straight to Chapter 8, which begins the discussion of applications of the ZNG coupled equations given in Chapter 3. Overview of Green's function approach To derive a microscopic theory of the nonequilibrium behaviour of a dilute weakly interacting Bose-condensed gas at finite temperatures, there are several different approaches available in the literature. We will use the wellknown Kadanoff–Baym (KB) nonequilibrium Green's function method. The generalization of this formalism to a Bose-condensed system was first considered by Kane and Kadanoff (1965), whose goal was to derive the Landau two-fluid hydrodynamic equations for a system with a Bose broken symmetry. The general problem consists of how to calculate the nonequilibrium response of a system induced by an external (space- and time-dependent) perturbation. In response to such an external perturbation, many interesting physical phenomena appear, including the excitation of collective modes and various transport processes.
A one-dimensional gas mixture consisting of bosons and fermions without spin interacting via a repulsive [Formula: see text]-function potential is considered. The model is integrable and soluble via two nested Bethe ansatz, if all particles are assumed to have equal masses and the interaction strength between the bosons and among the bosons and fermions is the same. The low energy excitation spectrum is a two-component Luttinger liquid and can be parametrized by a conformal field theory with conformal charges c = 1. In the low-energy limit, where the band curvature terms in the dispersion can be neglected, the linear dispersion of a Luttinger liquid is asymptotically exact. The spectral function, however, displays deviations from the Luttinger behavior for higher energy excitations. In the neighborhood of the single-particle (hole) energy, the spectral function is represented by an effective X-ray edge type model. Expressions of the critical exponents for the single-hole Green’s function are obtained using the Bethe ansatz solution in the limit of the bosonic gas. The results could be of relevance in the context of ultracold atoms confined to an elongated optical trap.