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arXiv:1903.03284 · Acta Physica Polonica A

Identities and Many-Body Approaches in Bose-Einstein Condensates

Shohei Watabe

Published Jun 1, 2019• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.12693/aphyspola.135.1222

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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.
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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=0T=0T=0 is also shown to persist at finite temperatures, in the presence of inhomogeneity, and also in nonequilibrium situations.
The exactness of the Bogoliubov approximation is discussed from the point of view of the condensation properties of the Bose gas. For the imperfect Bose gas, the authors find that the approximation yields the correct condensate density when the thermodynamic limit is taken by isotropic dilation; this is not the case in general for other ways of going to the infinite volume limit. They also prove the existence of Bose-Einstein condensation for a class of Bose gases with weak interactions and an energy gap.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed