The idea of replacing the condensate operators by the c number N$sub 0$$sup 1$/$sup 2$ in treating condensed Bose systems is known as Bogoliubov prescription (BP). Once BP is adopted, there are two ways to proceed. The first is the popular method of Hugenholtz and Pines, which we call $mu$-fixed formalism. The second may be found in the well-known works of Foldy and Brueckner on a dense charged boson gas. Unlike the $mu$-fixed formalism, the latter method does not require the introduction of the chemical potential, and thus it is called the N- fixed formalism. We argue that BP is rigorous in the $mu$-fixed formalism but is not in the N-fixed one. BP in the N-fixed one gives correct results for terms of leading orders (Foldy and Brueckner took just these terms) but incorrect results for terms of higher orders. (auth)
Nonequivalence of N-fixed and mu-fixed formalisms adopting asub0 = asub0 = Nsub0/sup 1/2/ for condensed Bose systems
The idea of replacing the condensate operators by the c number Nsub 0$$sup 1/sup2 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)
The idea of replacing the condensate operators by the c number 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 $\ensuremath{\mu}$-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.
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 and the superfluid density nS is obtained in closed form. It is shown that the Bogoliubov prescription yields n0\ensuremath{\simeq}nS and nU0\ensuremath{\ll}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}, where n is the total density, U0 the interaction constant, and \ensuremath{\mu} the chemical potential. On the other hand, for the canonical shift transformation one has n0\ensuremath{\ll}nS and nU0\ensuremath{\gg}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}. The latter, applied to superfluid 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.
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 rs43. 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.
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 Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number N of particles. This generalization yields a description of the Schr\"odinger picture field operators as the product of an annihilation operator A 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 N particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of 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 N. 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 N 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 N 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 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 $\ensuremath{\xi}(\mathbf{x},t)$.
We re-examine the way in which Bogoliubov's theory of a dilute Bose gas at T=0 has been extended to describe the statistical mechanics of interacting bosons at finite temperature. We show explicitly that the field-theoretic calculation of the grand partition function in this formulation amounts to a canonical trace over the eigenfunctions of the Bogoliubov Hamiltonian at fixed total number of bosons N, and that the additional trace over N that is required in the grand-canonical formalism is never carried out. This implies that what usually passes as the grand-canonical treatment of the Bogoliubov Hamiltonian is not quite grand-canonical, and is in fact a canonical one. We also show that the discontinuity in the condensate density predicted by previous formulations of this theory as the temperature T goes past the critical transition temperature Tc is a direct consequence of an inappropriate generalization of the Bogoliubov prescription to finite temperatures, and that this discontinuity disappears when this prescription is either used as a zero temperature approximation or avoided altogether. Armed with the above findings, we reformulate the statistical mechanics of interacting bosons in the canonical ensemble and derive the thermodynamics of the system. We then show how the canonical treatment can be used to setup a truly grand-canonical description of the statistical mechanics of a weakly interacting Bose gas where the average number of bosons in the system varies with temperature, unlike existing formulations where the total number of bosons N is taken to be a constant that does not depend on T. Consequences on the physics of interacting bosons are briefly discussed.