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Physical review. B, Solid state

Nonequivalence of N -fixed and μ -fixed formalisms adopting a0†=a0=N012 for condensed Bose systems

J.C. Lee

Published Nov 1, 1975• 0 citations
Publisher

Abstract

The idea of replacing the condensate operators by the $c$ number ${N}_{0}^{\frac{1}{2}}$ in treating condensed Bose systems is known as Bogoliubov prescription (BP). Once BP is adopted, there are two ways to proceed. The first is the popular method of Hugenholtz and Pines, which we call $\ensuremath{\mu}$-fixed formalism. The second may be found in the well-known works of Foldy and Brueckner on a dense charged boson gas. Unlike the $\ensuremath{\mu}$-fixed formalism, the latter method does not require the introduction of the chemical potential, and thus it is called the $N$-fixed formalism. We argue that BP is rigorous in the $\ensuremath{\mu}$-fixed formalism but is not in the $N$-fixed one. BP in the $N$-fixed one gives correct results for terms of leading orders (Foldy and Brueckner took just these terms) but incorrect results for terms of higher orders.

DOI
10.1103/physrevb.12.3749

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Nonequivalence of N-fixed and $mu$-fixed formalisms adopting a$sub 0$ = a$sub 0$ = N$sub 0$/sup 1/2/ for condensed Bose systems

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The idea of replacing the condensate operators by the c number N$sub 0$$sup 1$/$sup 2$ in treating condensed Bose systems is known as Bogoliubov prescription (BP). Once BP is adopted, there are two ways to proceed. The first is the popular method of Hugenholtz and Pines, which we call $mu$-fixed formalism. The second may be found in the well-known works of Foldy and Brueckner on a dense charged boson gas. Unlike the $mu$-fixed formalism, the latter method does not require the introduction of the chemical potential, and thus it is called the N- fixed formalism. We argue that BP is rigorous in the $mu$-fixed formalism but is not in the N-fixed one. BP in the N-fixed one gives correct results for terms of leading orders (Foldy and Brueckner took just these terms) but incorrect results for terms of higher orders. (auth)

•Nov 1, 1975•0 citations

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arXiv:quant-ph/9703005•Physical Review A•Aug 1, 1997•256 citations
The idea of replacing the condensate operators by the ccc number N012{N}_{0}^{\frac{1}{2}}N021​​ in treating condensed Bose systems is known as Bogoliubov prescription (BP). Once BP is adopted, there are two ways to proceed. The first is the popular method of Hugenholtz and Pines, which we call \ensuremathμ\ensuremath{\mu}\ensuremathμ-fixed formalism. The second may be found in the well-known works of Foldy and Brueckner on a dense charged boson gas. Unlike the \ensuremathμ\ensuremath{\mu}\ensuremathμ-fixed formalism, the latter method does not require the introduction of the chemical potential, and thus it is called the NNN-fixed formalism. We argue that BP is rigorous in the \ensuremathμ\ensuremath{\mu}\ensuremathμ-fixed formalism but is not in the NNN-fixed one. BP in the NNN-fixed one gives correct results for terms of leading orders (Foldy and Brueckner took just these terms) but incorrect results for terms of higher orders.
Nonequivalence of N-fixed and mumumu-fixed formalisms adopting asub0sub 0sub0 = asub0sub 0sub0 = Nsub0sub 0sub0/sup 1/2/ for condensed Bose systems
The idea of replacing the condensate operators by the c number Nsub0sub 0sub0sup1sup 1sup1/sup2sup 2sup2 in treating condensed Bose systems is known as Bogoliubov prescription (BP). Once BP is adopted, there are two ways to proceed. The first is the popular method of Hugenholtz and Pines, which we call mumumu-fixed formalism. The second may be found in the well-known works of Foldy and Brueckner on a dense charged boson gas. Unlike the mumumu-fixed formalism, the latter method does not require the introduction of the chemical potential, and thus it is called the N- fixed formalism. We argue that BP is rigorous in the mumumu-fixed formalism but is not in the N-fixed one. BP in the N-fixed one gives correct results for terms of leading orders (Foldy and Brueckner took just these terms) but incorrect results for terms of higher orders. (auth)
The usual symmetry-breaking procedures for Bose condensed systems, namely, the Bogoliubov prescription, the symmetry-breaking term added to the Hamiltonian, and the canonical shift transformation are unified into a single formalism. By taking into account the condensate reservoir as a source and sink of excited particles, exact Ward identities are solved in the shielded-potential approximation. A relationship between the condensate density n0{\mathit{n}}_{0}n0​ and the superfluid density nS{\mathit{n}}_{\mathit{S}}nS​ is obtained in closed form. It is shown that the Bogoliubov prescription yields n0{\mathit{n}}_{0}n0​\ensuremath{\simeq}nS{\mathit{n}}_{\mathit{S}}nS​ and nU0{\mathit{nU}}_{0}nU0​\ensuremath{\ll}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}, where n is the total density, U0{\mathit{U}}_{0}U0​ the interaction constant, and \ensuremath{\mu} the chemical potential. On the other hand, for the canonical shift transformation one has n0{\mathit{n}}_{0}n0​\ensuremath{\ll}nS{\mathit{n}}_{\mathit{S}}nS​ and nU0{\mathit{nU}}_{0}nU0​\ensuremath{\gg}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}. The latter, applied to superfluid 4He^{4}\mathrm{He}4He at saturated vapor pressure, gives excellent agreement between theory and experiment, without any adjustable parameter. The condensate density turns out to be strongly dependent on pressure as observed experimentally. The formalism provides in a natural way a consistent description of Bose systems in arbitrary D-dimensional space.
In treating system of bosons localized in a trapping potential, having a macroscopic number N0 of them condensing at the lowest single-particle level v0, Bogoliubov approximation is to replace the creation/annihilation operators [Formula: see text] of the state v0 by [Formula: see text]. We show that this approximation is justified if the inter-particle potential is repulsive in the sense specified. In fact, we show, by using [Formula: see text], that [Formula: see text] is effectively of the order [Formula: see text] under the condition stated.
We present an extension of the well-known Bogoliubov theory to treat low-dimensional degenerate Bose gases in the limit of weak interactions and low density fluctuations. We use a density-phase representation and show that a precise definition of the phase operator requires a space discretization in cells of size l. We perform a systematic expansion of the Hamiltonian in terms of two small parameters, the relative density fluctuations inside a cell and the phase change over a cell. The resulting macroscopic observables can be computed in one, two, and three dimensions with no ultraviolet or infrared divergence. Furthermore, this approach exactly matches Bogoliubov's approach when there is a true condensate. We give the resulting expressions for the equation of state of the gas, the ground state energy, and the first order and second order correlation functions of the field. Explicit calculations are done for homogeneous systems.
The simultaneous effects of the condensate and of the long-range nature of the interaction have made the high-density charged Bose gas an interesting theoretical problem. In this paper we present a straight-forward procedure which takes these effects into account. It is based on the perturbation theory in the Bogoliubov representation and the principle of the compensation of dangerous diagrams. Both the ground-state energy and the excitation spectrum are studied. Numerical results are given for the first two orders in rs34{{r}_{s}}^{\frac{3}{4}}rs​43​. The ground-state energy reduces to that calculated correctly by Brueckner, and earlier by Lee and Feenberg in a variational treatment. The excitation spectrum agrees with that calculated in our preceding paper using an entirely different method.
The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number NNN of particles. This generalization yields a description of the Schr\"odinger picture field operators as the product of an annihilation operator AAA for the total number of particles and the sum of a ``condensate wave function'' $\ensuremath{\xi}(\mathbf{x})$ and a phonon field operator $\ensuremath{\chi}(\mathbf{x})$ in the form $\ensuremath{\psi}(\mathbf{x})\ensuremath{\approx}A{\ensuremath{\xi}(\mathbf{x})+\ensuremath{\chi}(\mathbf{x})/\sqrt{N}}$ when the field operator acts on the NNN particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of N\sqrt{N}N​, and thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different NNN. The excitation spectrum can be obtained by essentially the same method as Bogoliubov only if $\ensuremath{\xi}(\mathbf{x})$ is a solution of the time-independent Gross-Pitaevskii equation for NNN particles and any chemical potential $\ensuremath{\mu}$, which yields a valid and stable solution of the Gross-Pitaevskii equation. The treatment within a subspace of fixed NNN is identical in form to that usually used, but the interpretation of the operators is slightly different. A time-dependent generalization is then made, yielding an asymptotic expansion in decreasing powers of N\sqrt{N}N​ for the equations of motion. In this expansion the condensate wave function has the time-dependent form $\ensuremath{\xi}(\mathbf{x},t)$, and the condition for the validity of the expansion is that $\ensuremath{\xi}(\mathbf{x},t)$ satisfies the time-dependent Gross-Pitaevskii equation $\ensuremath{\partial}\ensuremath{\xi}/\ensuremath{\partial}t=\ensuremath{-}({\ensuremath{\Elzxh}}^{2}/2m){\ensuremath{\nabla}}^{2}\ensuremath{\xi}+V\ensuremath{\xi}+Nu|\ensuremath{\xi}{|}^{2}\ensuremath{\xi}$. The physics is then described in a kind of interaction picture, called the condensate picture, in which the phonon operator can be expressed as $\ensuremath{\chi}(\mathbf{x},t)={\ensuremath{\sum}}_{k}{\ensuremath{\xi}}_{k}(\mathbf{x},t){\ensuremath{\alpha}}_{k},$ where the operators ${\ensuremath{\alpha}}_{k}$ are time-independent annihilation operators, and the state vector has a time evolution described by a Schr\"odinger equation in which the Hamiltonian is a time-dependent quadratic form in the phonon creation and annihilation operators, whose coefficients are explicitly determined in terms of the time-dependent condensate wave function $\ensuremath{\xi}(\mathbf{x},t)$.