Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Solid state

Bogoliubov theory for a many-boson system

Gary S. Grest, A. K. Rajagopal

Published Dec 1, 1975• 2 citations
Publisher

Abstract

Presently there are two distinct classes of microscopic theories which describe a system of many identical spinless bosons. One is the Bogoliubov theory, which places special emphasis on the occupation of the zero-momentum state and as such depends explicitly on the number of particles in this state, ${N}_{0}$. The other depends on density fluctuations to describe the fluid. In this work, we derive a modified version of the Bogoliubov theory which results in a number-conserving Hamiltonian with no explicit dependence on ${N}_{0}$. The ground- and first-excited-state energies evaluated through second order in $\frac{1}{{N}^{\frac{1}{2}}}$ are equivalent to the results of a variational-perturbative procedure based on the method of correlated basis functions in the uniform limit.

DOI
10.1103/physrevb.12.4847

Related papers

Ground-State Energy of a Many-Particle Boson System

Deok Kyo Lee

The ground state of a many-particle boson system is studied for two closely related limits: the uniform limit and the weak coupling limit. The former is defined by $\ensuremath{\alpha}=1\ensuremath{-}g(0)\ensuremath{\ll}1$ and the latter by $\ensuremath{\beta}=\frac{(N\ensuremath{-}{N}_{0})}{N}\ensuremath{\ll}1$, where $g(r)$ is the radial distribution function and ${N}_{0}$ is the mean occupation number of the zero-momentum state. In the uniform limit the variation-perturbation approach based on (a) the method of correlated basis functions and (b) the series expansion in powers of $\ensuremath{\alpha}$ is found to be equivalent to the field-theoretic treatment given by Brueckner (for the charged-boson gas) in the weak coupling limit. In particular, it is shown that the variation-perturbation energy obtained for the uniform limit in the momentum representation is identical through second order to the ground-state energy evaluated for $\ensuremath{\beta}\ensuremath{\ll}1$ by summing one- and two-ring diagrams in the Bogoliubov occupation-number representation. The charged-boson gas and the one-dimensional boson system with a $\ensuremath{\delta}$-function interaction are considered to examine some of the interesting features of the uniform-limit procedure.

Physical Review A•Oct 1, 1971•29 citations

Theory of Many-Boson Systems: Pair Theory

Marvin D. Girardeau, R. Arnowitt

The many-boson system with repulsive interactions is treated by a variational method based on a variational trial state of an exponential pair-excitation type obtained by a generalization of that of Bogoliubov; the treatment is closely related to an intermediate-coupling approximation with respect to pairs. The nonlinear integral equation which determines the variational ground state is derived, and various properties of this ground state and the associated excited states are examined. The resultant low-lying spectrum lies below that of Bogoliubov by an amount proportional to the total number of particles. The variational principle is shown to produce rigorous energy eigenvalue differences for the pair part of the Hamiltonian. The variational states, however, still exhibit unphysical features characteristic of pair-excitation states: The pair correlation function does not go to zero at zero particle separation and the phonon spectrum exhibits a gap above the ground state. It is suggested that these features can be removed by using states which take into account excitation of momentum-conserving groups of more than two particles.

Physical Review•Feb 1, 1959•325 citations

Perturbation Theory Based on Currents for Interacting Bosons at Absolute Zero

B. Tsu-Shen Chang

As a continuation of a recent paper, using density and current density as coordinates for interacting bosons at absolute zero, this work deals with the perturbation theory. The zerothorder Hamiltonian is the one shown to yield the Bogoliubov energy spectrum. First- or secondorder perturbation calculations have been made of the three-phonon and the four-phonon vertices for the energy corrections of the low-lying excited states. The results in the long-wavelength limit differ, although only slightly, from the quantum hydrodynamic calculations, and indicate that the perturbed excitation energy ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ is an odd-power series in $k$ if the Fourier coefficients of the two-body potential concerned can be approximated by a constant or an evenpower series in $k$. However, whether this remains so in all higher orders of various perturbations is an open question. Two methods were used in the calculation. The first involves functional representations of the coordinates in the ${\ensuremath{\rho}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ representation recently reported. The second method is new, and employs "occupation-number" representations of the current algebra by introducing creation and annihilation operators for "phonons" from density fluctuations. These latter representations enable us to benefit from the advantages of the canonical field theory.

Physical Review A•Mar 1, 1972•3 citations

Ground State of an Interacting Bose Gas

Paul Berdahl, Deok Kyo Lee

The ground-state energy of an interacting many-boson system obtained by Bogoliubov and Zubarev using the collective-coordinate method is shown to be equivalent through second order to the variation-perturbation energy evaluated in the uniform limit by means of the method of correlated basis functions. It is also shown that the wave function derived with a slight modification of the Bogoliubov-Zubarev approach is equivalent to that determined formally from the Rayleigh-Schr\"odinger perturbation theory in the uniform limit.

Physical Review A•Apr 1, 1973•9 citations

Re-Examining Bogoliubov's Theory of an Interacting Bose Gas

A. M. Ettouhami

As is well-known, in the conventional formulation of Bogoliubov's theory of an interacting Bose gas, the Hamiltonian is written as a decoupled sum of contributions from different momenta of the form . Then, each of the single-mode Hamiltonians is diagonalized separately, and the resulting ground state wavefunction of the total Hamiltonian is written as a simple product of the ground state wavefunctions of each of the single-mode Hamiltonians . While this way of diagonalizing the total Hamiltonian may seem to be valid from the perspective of the standard, number non-conserving Bogoliubov's method, where the k = 0 state is removed from the Hilbert space and hence the individual Hilbert spaces where the Hamiltonians {} are diagonalized are disjoint from one another, we argue that from a number-conserving perspective this diagonalization method may not be adequate since the true Hilbert spaces where the Hamiltonians {} should be diagonalized all have the k = 0 state in common, and hence the ground state wavefunction of the total Hamiltonian may not be written as a simple product of the ground state wavefunctions of the 's. In this paper, we give a thorough review of Bogoliubov's method, and discuss a variational and number-conserving formulation of this theory in which the k = 0 state is restored to the Hilbert space of the interacting gas, and where, instead of diagonalizing the Hamiltonians separately, we diagonalize the total Hamiltonian . When this is done, we find that the ground state energy is lowered below the Bogoliubov result, and the depletion of bosons is significantly reduced with respect to the one obtained in the number non-conserving treatment. We also find that the spectrum of the usual αk excitations of Bogoliubov's method changes from a gapless one, as predicted by the standard, number non-conserving formulation of this theory, to one which exhibits a finite gap in the k → 0 limit. We discuss the presence of a gap in the spectrum of the αk's in light of Goldstone's theorem, and show that there is no contradiction with the latter.

arXiv:cond-mat/0703498•Progress of Theoretical Physics•Mar 1, 2012•15 citations

Ground state and excited states of a confined condensed Bose gas

Alexander L. Fetter

The Bogoliubov approximation is used to study the ground state and low-lying excited states of a dilute gas of N atomic bosons held in an isotropic harmonic potential characterized by frequency \ensuremath{\omega} and oscillator length ${\mathit{d}}_{0}$=\ensuremath{\surd}\ensuremath{\Elzxh}/m\ensuremath{\omega}. By assumption, the self-consistent condensate has a macroscopic occupation number ${\mathit{N}}_{0}$\ensuremath{\gg}1, with N-${\mathit{N}}_{0}$\ensuremath{\ll}${\mathit{N}}_{0}$. A linearized hydrodynamic description yields operator forms of the particle-conservation law and Bernoulli's theorem, expressed in terms of the small density fluctuation operator \ensuremath{\rho}^\ensuremath{'} and velocity potential operator \ensuremath{\Phi}^\ensuremath{'}, along with the condensate density ${\mathit{n}}_{0}$ and velocity ${\mathbf{v}}_{0}$. For positive scattering length a and large stationary condensate (${\mathit{N}}_{0}$\ensuremath{\gg}${\mathit{d}}_{0}$/a and ${\mathbf{v}}_{0}$=0), the spherical condensate has a well-defined radius ${\mathit{R}}_{0}$\ensuremath{\gg}${\mathit{d}}_{0}$, and the low-lying excited states are irrotational compressional waves localized near the surface. Approximate variational energies ${\mathit{E}}_{0\mathit{l}}$ of the lowest radial modes (n=0) for successive values of orbital angular momentum l form a rotational band given by ${\mathit{E}}_{0\mathit{l}}$\ensuremath{\approxeq}${\mathit{E}}_{00}$+1/2${\mathrm{\ensuremath{\Elzxh}}}^{2}$l(l+1)/${\mathrm{mR}}_{0}^{2}$, with radial zero-point energy ${\mathit{E}}_{00}$\ensuremath{\propto}\ensuremath{\Elzxh}\ensuremath{\omega}(${\mathit{R}}_{0}$/${\mathit{d}}_{0}$${)}^{2/3}$=(${\mathrm{\ensuremath{\Elzxh}}}^{2}$ m${\mathrm{\ensuremath{\omega}}}^{4}$${\mathit{R}}_{0}^{2}$${)}^{1/3}$. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9510037•Physical Review A•Jun 1, 1996•138 citations
Presently there are two distinct classes of microscopic theories which describe a system of many identical spinless bosons. One is the Bogoliubov theory, which places special emphasis on the occupation of the zero-momentum state and as such depends explicitly on the number of particles in this state, N0{N}_{0}N0​. The other depends on density fluctuations to describe the fluid. In this work, we derive a modified version of the Bogoliubov theory which results in a number-conserving Hamiltonian with no explicit dependence on N0{N}_{0}N0​. The ground- and first-excited-state energies evaluated through second order in 1N12\frac{1}{{N}^{\frac{1}{2}}}N21​1​ are equivalent to the results of a variational-perturbative procedure based on the method of correlated basis functions in the uniform limit.
The ground state of a many-particle boson system is studied for two closely related limits: the uniform limit and the weak coupling limit. The former is defined by $\ensuremath{\alpha}=1\ensuremath{-}g(0)\ensuremath{\ll}1$ and the latter by $\ensuremath{\beta}=\frac{(N\ensuremath{-}{N}_{0})}{N}\ensuremath{\ll}1$, where g(r)g(r)g(r) is the radial distribution function and N0{N}_{0}N0​ is the mean occupation number of the zero-momentum state. In the uniform limit the variation-perturbation approach based on (a) the method of correlated basis functions and (b) the series expansion in powers of $\ensuremath{\alpha}$ is found to be equivalent to the field-theoretic treatment given by Brueckner (for the charged-boson gas) in the weak coupling limit. In particular, it is shown that the variation-perturbation energy obtained for the uniform limit in the momentum representation is identical through second order to the ground-state energy evaluated for $\ensuremath{\beta}\ensuremath{\ll}1$ by summing one- and two-ring diagrams in the Bogoliubov occupation-number representation. The charged-boson gas and the one-dimensional boson system with a $\ensuremath{\delta}$-function interaction are considered to examine some of the interesting features of the uniform-limit procedure.
The many-boson system with repulsive interactions is treated by a variational method based on a variational trial state of an exponential pair-excitation type obtained by a generalization of that of Bogoliubov; the treatment is closely related to an intermediate-coupling approximation with respect to pairs. The nonlinear integral equation which determines the variational ground state is derived, and various properties of this ground state and the associated excited states are examined. The resultant low-lying spectrum lies below that of Bogoliubov by an amount proportional to the total number of particles. The variational principle is shown to produce rigorous energy eigenvalue differences for the pair part of the Hamiltonian. The variational states, however, still exhibit unphysical features characteristic of pair-excitation states: The pair correlation function does not go to zero at zero particle separation and the phonon spectrum exhibits a gap above the ground state. It is suggested that these features can be removed by using states which take into account excitation of momentum-conserving groups of more than two particles.
As a continuation of a recent paper, using density and current density as coordinates for interacting bosons at absolute zero, this work deals with the perturbation theory. The zerothorder Hamiltonian is the one shown to yield the Bogoliubov energy spectrum. First- or secondorder perturbation calculations have been made of the three-phonon and the four-phonon vertices for the energy corrections of the low-lying excited states. The results in the long-wavelength limit differ, although only slightly, from the quantum hydrodynamic calculations, and indicate that the perturbed excitation energy ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ is an odd-power series in kkk if the Fourier coefficients of the two-body potential concerned can be approximated by a constant or an evenpower series in kkk. However, whether this remains so in all higher orders of various perturbations is an open question. Two methods were used in the calculation. The first involves functional representations of the coordinates in the ${\ensuremath{\rho}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ representation recently reported. The second method is new, and employs "occupation-number" representations of the current algebra by introducing creation and annihilation operators for "phonons" from density fluctuations. These latter representations enable us to benefit from the advantages of the canonical field theory.
The ground-state energy of an interacting many-boson system obtained by Bogoliubov and Zubarev using the collective-coordinate method is shown to be equivalent through second order to the variation-perturbation energy evaluated in the uniform limit by means of the method of correlated basis functions. It is also shown that the wave function derived with a slight modification of the Bogoliubov-Zubarev approach is equivalent to that determined formally from the Rayleigh-Schr\"odinger perturbation theory in the uniform limit.
As is well-known, in the conventional formulation of Bogoliubov's theory of an interacting Bose gas, the Hamiltonian is written as a decoupled sum of contributions from different momenta of the form . Then, each of the single-mode Hamiltonians is diagonalized separately, and the resulting ground state wavefunction of the total Hamiltonian is written as a simple product of the ground state wavefunctions of each of the single-mode Hamiltonians . While this way of diagonalizing the total Hamiltonian may seem to be valid from the perspective of the standard, number non-conserving Bogoliubov's method, where the k = 0 state is removed from the Hilbert space and hence the individual Hilbert spaces where the Hamiltonians {} are diagonalized are disjoint from one another, we argue that from a number-conserving perspective this diagonalization method may not be adequate since the true Hilbert spaces where the Hamiltonians {} should be diagonalized all have the k = 0 state in common, and hence the ground state wavefunction of the total Hamiltonian may not be written as a simple product of the ground state wavefunctions of the 's. In this paper, we give a thorough review of Bogoliubov's method, and discuss a variational and number-conserving formulation of this theory in which the k = 0 state is restored to the Hilbert space of the interacting gas, and where, instead of diagonalizing the Hamiltonians separately, we diagonalize the total Hamiltonian . When this is done, we find that the ground state energy is lowered below the Bogoliubov result, and the depletion of bosons is significantly reduced with respect to the one obtained in the number non-conserving treatment. We also find that the spectrum of the usual αk excitations of Bogoliubov's method changes from a gapless one, as predicted by the standard, number non-conserving formulation of this theory, to one which exhibits a finite gap in the k → 0 limit. We discuss the presence of a gap in the spectrum of the αk's in light of Goldstone's theorem, and show that there is no contradiction with the latter.
The Bogoliubov approximation is used to study the ground state and low-lying excited states of a dilute gas of N atomic bosons held in an isotropic harmonic potential characterized by frequency \ensuremath{\omega} and oscillator length d0{\mathit{d}}_{0}d0​=\ensuremath{\surd}\ensuremath{\Elzxh}/m\ensuremath{\omega}. By assumption, the self-consistent condensate has a macroscopic occupation number N0{\mathit{N}}_{0}N0​\ensuremath{\gg}1, with N-N0{\mathit{N}}_{0}N0​\ensuremath{\ll}N0{\mathit{N}}_{0}N0​. A linearized hydrodynamic description yields operator forms of the particle-conservation law and Bernoulli's theorem, expressed in terms of the small density fluctuation operator \ensuremath{\rho}^\ensuremath{'} and velocity potential operator \ensuremath{\Phi}^\ensuremath{'}, along with the condensate density n0{\mathit{n}}_{0}n0​ and velocity v0{\mathbf{v}}_{0}v0​. For positive scattering length a and large stationary condensate (N0{\mathit{N}}_{0}N0​\ensuremath{\gg}d0{\mathit{d}}_{0}d0​/a and v0{\mathbf{v}}_{0}v0​=0), the spherical condensate has a well-defined radius R0{\mathit{R}}_{0}R0​\ensuremath{\gg}d0{\mathit{d}}_{0}d0​, and the low-lying excited states are irrotational compressional waves localized near the surface. Approximate variational energies E0l{\mathit{E}}_{0\mathit{l}}E0l​ of the lowest radial modes (n=0) for successive values of orbital angular momentum l form a rotational band given by E0l{\mathit{E}}_{0\mathit{l}}E0l​\ensuremath{\approxeq}E00{\mathit{E}}_{00}E00​+1/2${\mathrm{\ensuremath{\Elzxh}}}^{2}$l(l+1)/mR02{\mathrm{mR}}_{0}^{2}mR02​, with radial zero-point energy E00{\mathit{E}}_{00}E00​\ensuremath{\propto}\ensuremath{\Elzxh}\ensuremath{\omega}(R0{\mathit{R}}_{0}R0​/d0{\mathit{d}}_{0}d0​)2/3{)}^{2/3})2/3=(${\mathrm{\ensuremath{\Elzxh}}}^{2}$ m${\mathrm{\ensuremath{\omega}}}^{4}$R02{\mathit{R}}_{0}^{2}R02​)1/3{)}^{1/3})1/3. \textcopyright{} 1996 The American Physical Society.