Quantum-field-theoretic descriptions of interacting condensed bosons have suffered from the lack of self-consistent approximation schemes satisfying Goldstone's theorem and dynamical conservation laws simultaneously. We present a procedure to construct such approximations systematically by using either an exact relation for the interaction energy or the Hugenholtz-Pines relation to express the thermodynamic potential in a Luttinger-Ward form. Inspection of the self-consistent perturbation expansion up to the third order with respect to the interaction shows that the two relations yield a unique identical result at each order, reproducing the conserving-gapless mean-field theory [T. Kita, J. Phys. Soc. Jpn. 74, 1891 (2005)] as the lowest-order approximation. The uniqueness implies that the series becomes exact when infinite terms are retained. We also derive useful expressions for the entropy and superfluid density in terms of Green's function and a set of real-time dynamical equations to describe thermalization of the condensate.
We formulate a conserving gapless mean-field theory for Bose-Einstein condensates on the basis of a Luttinger-Ward thermodynamic functional. It is applied to a weakly interacting uniform gas with density
n and s-wave scattering length
a to clarify its fundamental thermodynamic properties. It is found that the condensation here occurs as a first-order transition. The shift of the transition temperature
ΔTc from the ideal-gas result
T0 is positive and given to the leading order by
ΔTc=2.33an1/3T0, in agreement with a couple of previous estimates. The theory is expected to form a new theoretical basis for trapped Bose-Einstein condensates at finite temperatures.
Quantum field theory of equilibrium and nonequilibrium Bose-Einstein condensates is formulated so as to satisfy three basic requirements: the Hugenholtz-Pines relation; conservation laws; identities among vertices originating from Goldstone's theorem I. The key inputs are irreducible four-point vertices, in terms of which we derive a closed system of equations for Green's functions, three- and four-point vertices, and two-particle Green's functions. It enables us to study correlated Bose-Einstein condensates with a gapless branch of single-particle excitations without encountering any infrared divergence. The single- and two-particle Green's functions are found to share poles, i.e., the structure of the two-particle Green's functions predicted by Gavoret and Nozières for a homogeneous condensate at
T=0 is also shown to persist at finite temperatures, in the presence of inhomogeneity, and also in nonequilibrium situations.
This paper discusses exact relations in the Bose-Einstein condensates, starting from basic properties of an ideal Bose gas.In particular, focused on are the Hugenholtz-Pines relation, Nepomnyashchii-Nepomnyashchii identity, and identities for the density response function.After introducing these exact relations, a few approaches of many-body approximations are discussed, which satisfy the exact relations in the Bose-Einstein condensates.This paper will serve as a bridge between theories on exact relations and those on approximations in the Bose-Einstein condensates.
This paper presents a conserving gapless mean-field theory for weakly interacting Bose gases. We first construct a mean-field Luttinger-Ward thermodynamic functional in terms of the condensate wave function
Ψ and the Nambu Green's function
G^ for the quasiparticle field. Imposing its stationarity respect to
Ψ and
G^ yields a set of equations to determine the equilibrium for general non-uniform systems. They have a plausible property of satisfying the Hugenholtz-Pines theorem to provide a gapless excitation spectrum. Also, the corresponding dynamical equations of motion obey various conservation laws. Thus, the present mean-field theory shares two important properties with the exact theory: ``conserving'' and ``gapless.'' The theory is then applied to a homogeneous weakly interacting Bose gas with s-wave scattering length
a and particle mass
m to clarify its basic thermodynamic properties under two complementary conditions of constant density
n and constant pressure
p. The superfluid transition is predicted to be first-order because of the non-analytic nature of the order-parameter expansion near
Tc inherent in Bose systems, i.e., the Landau-Ginzburg expansion is not possible here. The transition temperature
Tc shows quite a different interaction dependence between the
n-fixed and
p-fixed cases. In the former case
Tc increases from the ideal gas value
T0 as
Tc/T0=1+2.33an1/3, whereas it decreases in the latter as
Tc/T0=1−3.84a(mp/2πℏ2)1/5. Temperature dependences of basic thermodynamic quantities are clarified explicitly.
Bose–Einstein condensation can be understood as a macroscopic occupation of a single-particle state. But an alternative definition, which requires only a macroscopic occupation of an arbitrarily small energy band of single-particle states, is also known, referred to as generalized Bose–Einstein condensation, and considered to be thermodynamically more stable. Depending on the number of macroscopically occupied single-particle states in the condensate, three types are then distin-guished. If the number of macroscopically occupied single-particle states is finite but at least one, then the condensation is said to be of type-I. A type-II condensate occurs if infinitely many single-particle states are macroscopically occupied. If a generalized Bose–Einstein condensation without any macroscopically occupied single-particle states is present, then one speaks of a type-III con-densation. In noninteracting Bose gases, it is generally easier to prove the occurrence of generalized Bose–Einstein condensation than to determine its type. While a main step for the former task is to show that the critical density is finite, quite detailed knowledge of the energy gaps of the cor-responding single-particle Hamiltonian at the lower edge of the spectrum is needed for the latter one. In this dissertation, we study the occurrence of a Bose–Einstein condensate in Bose gases that are placed in metrically transitive random potentials. We mainly consider the Poisson random potential. The thermodynamic limit and Dirichlet boundary conditions are used throughout this work. We explore noninteracting Bose gases in arbitrary dimension at positive temperatures and a one-dimensional interacting Bose gas at zero temperature. We provide fairly general sufficient conditions for the random potential that imply the almost sure occurrence of generalized Bose–Einstein condensation in noninteracting Bose gases. These conditions are fulfilled by a Poisson random potential, depending on its single-impurity potential. They are also met by the one-dimensional Luttinger–Sy model, which is a central system in the research area of Bose gases in random potentials, and a generalization thereof. Moreover, we state sufficient energy gap conditions such that one can conclude a type-I condensation in Poisson random potentials. We consider different probabilistic notions here, such as convergence in the rth mean, r P N, in probability, and with probability almost one. In particular, we prove that in the Luttinger–Sy model with infinitely strong random potential a type-I condensation occurs in the rth mean, r P N, if the particle density is greater than the critical one. For the Luttinger–Sy model with a finitely strong random potential, we show the same statement, but with probability almost one. Finally, we discuss the Luttinger–Sy model with a repulsive contact interaction between the particles at zero temperature. We provide sufficient requirements such that a generalized Bose–Einstein condensate almost surely occurs. We also show that the type of the condensate is altered by the interparticle interaction. In particular, we demonstrate a transition from a type-I to a type-III condensate when the strength of the contact interaction is increased. Furthermore, we prove that the interparticle interaction reduces the local particle density.