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

arXiv:1208.3651

Generalized Wick's theorem at finite temperature for a quadratic Hamiltonian

M. O. C. Pires

Published Aug 17, 2012• 0 citations
arXivHTMLPDF

Abstract

In Gaudin (1960), Michel Gaudin showed the Wick's theorem at finite temperature using a diagonal Hamiltonian. We extend the Gaudin's prove for a statistical density operator which depend on a quadratic Hamiltonian. To illustrate the utility of the theorem, we evaluate the ratio $\sqrt{[-^2]/}$ of a homogeneous weakly interacting Bose gas at temperature below the Bose-Einstein condensation temperature. At this condition, the quadratic Hamiltonian approximation is valued and, in this evaluation, we show the sub-Poissonian behaviour of the fundamental state distribution at zero temperature.

DOI
10.48550/arxiv.1208.3651

Related papers

Generalized Wick’s Theorem at Finite Temperature and Non-Poissonian Fluctuations in Weakly Interacting Bose Gases

M. O. C. Pires

Abstract In Ref. Gaudin (Nuclear Physics 15 , 89 1960), Gaudin established a generalized version of Wick’s theorem at finite temperature for systems described by a diagonal quadratic Hamiltonian. In this work, we extend Gaudin’s proof to statistical density operators generated by a general quadratic Hamiltonian, prior to diagonalization, as commonly encountered in interacting Bose systems. We show explicitly that the resulting thermal contractions depend on the Bogoliubov transformation parameters and on the quasiparticle spectrum, providing a transparent framework for evaluating thermal averages of arbitrary operator products. As an illustration of the usefulness of this generalized formulation, we evaluate particle-number fluctuations in a homogeneous, weakly interacting Bose gas below the Bose–Einstein condensation temperature. We show that particle-number fluctuations are generically non-Poissonian as a consequence of interaction-induced correlations and quantum depletion, with the Poissonian limit recovered only for an ideal Bose gas at zero temperature. Our results clarify the role of quadratic Hamiltonians and Bogoliubov quasiparticles in shaping fluctuation statistics and highlight the generalized Wick’s theorem as a controlled tool for computing correlation functions in interacting quantum gases.

Brazilian Journal of Physics•Apr 26, 2026•0 citations

A gapless theory of Bose-Einstein condensation in dilute gases at finite temperature

S. A. Morgan

In this paper we develop a gapless theory of Bose-Einstein condensation (BEC) which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number-conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher-order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first- and second-order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infrared divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultraviolet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T -matrix leads naturally to a high-energy renormalization. The theory developed in this paper is therefore well defined at both low and high energy and provides a systematic description of BEC in dilute gases. It can therefore be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition.

arXiv:cond-mat/9911278•Journal of Physics B Atomic Molecular and Optical Physics•Sep 14, 2000•116 citations

Generalized Wick's theorem for multiquasiparticle overlaps as a limit of Gaudin's theorem

S. Pérez-Martín, Luis Miguel Robledo

By using the extension of the statistical Wick's theorem (Gaudin's theorem) to deal with generalized statistical density operators (those which can be expressed as the product of and operator carrying out a canonical transformations times a density operator) and using the appropriate limits we are able to rederive in a very simple way the standard generalized Wick's theorem for overlaps of mean-field wave functions. Due to the simplicity of the derivation it is now straightforward to consider more involved cases and some of them are discussed. The present derivation also allows us to obtain general and compact formulas for other particular cases of the generalized Wick theorem involving overlaps of multiquasiparticle excitations of product wave functions. The new expressions allow us to reduce the combinatorial complexity of the standard calculation of the above overlaps.

arXiv:0707.3365•Physical Review C•Dec 17, 2007•28 citations

Deviations from Wick's theorem in the canonical ensemble

Kurt Schönhammer

Wick's theorem for the expectation values of products of field operators for a system of noninteracting fermions or bosons plays an important role in the perturbative approach to the quantum many-body problem. A finite-temperature version holds in the framework of the grand canonical ensemble, but not for the canonical ensemble appropriate for systems with fixed particle number such as ultracold quantum gases in optical lattices. Here we present formulas for expectation values of products of field operators in the canonical ensemble using a method in the spirit of Gaudin's proof of Wick's theorem for the grand canonical case. The deviations from Wick's theorem are examined quantitatively for two simple models of noninteracting fermions.

arXiv:1707.05111•Physical Review A•Jul 5, 2017•15 citations

Perturbation Theory: Wick’s Theorem for Strings of Operators Ordered along a Contour

Giancarlo Calvanese Strinati

This chapter considers a general form of the Wick’s theorem, which leads to a perturbation expansion of the (contour) single- and two-particle Green’s functions, which are expressed in terms of the contour time-ordering operator. The strategy for proving the Wick’s theorem is similar to that adopted within the Matsubara formalism for the Green’s functions at finite temperature and relies on the Gibbs form of the statistical operator in the interaction picture. An extension to superfluid Bose and Fermi systems is also considered.

Cambridge University Press eBooks•Feb 6, 2026•0 citations

A gapless theory of Bose-Einstein condensation in dilute gases at finite temperature

S. A. Morgan

In this thesis we develop a gapless theory of BEC which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first and second order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infra-red divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultra-violet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T-matrix leads naturally to a high-energy renormalization. The theory developed in this thesis is therefore well-defined at both low and high energy and provides a systematic description of Bose-Einstein condensation in dilute gases. It can be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition. We present analytical results for these quantities in the homogeneous limit and describe how they can be calculated numerically for trapped gases. (author)

•Jan 1, 1999•73 citations
In Gaudin (1960), Michel Gaudin showed the Wick's theorem at finite temperature using a diagonal Hamiltonian. We extend the Gaudin's prove for a statistical density operator which depend on a quadratic Hamiltonian. To illustrate the utility of the theorem, we evaluate the ratio [−2]/\sqrt{[-^2]/}[−2]/​ of a homogeneous weakly interacting Bose gas at temperature below the Bose-Einstein condensation temperature. At this condition, the quadratic Hamiltonian approximation is valued and, in this evaluation, we show the sub-Poissonian behaviour of the fundamental state distribution at zero temperature.
Abstract In Ref. Gaudin (Nuclear Physics 15 , 89 1960), Gaudin established a generalized version of Wick’s theorem at finite temperature for systems described by a diagonal quadratic Hamiltonian. In this work, we extend Gaudin’s proof to statistical density operators generated by a general quadratic Hamiltonian, prior to diagonalization, as commonly encountered in interacting Bose systems. We show explicitly that the resulting thermal contractions depend on the Bogoliubov transformation parameters and on the quasiparticle spectrum, providing a transparent framework for evaluating thermal averages of arbitrary operator products. As an illustration of the usefulness of this generalized formulation, we evaluate particle-number fluctuations in a homogeneous, weakly interacting Bose gas below the Bose–Einstein condensation temperature. We show that particle-number fluctuations are generically non-Poissonian as a consequence of interaction-induced correlations and quantum depletion, with the Poissonian limit recovered only for an ideal Bose gas at zero temperature. Our results clarify the role of quadratic Hamiltonians and Bogoliubov quasiparticles in shaping fluctuation statistics and highlight the generalized Wick’s theorem as a controlled tool for computing correlation functions in interacting quantum gases.
In this paper we develop a gapless theory of Bose-Einstein condensation (BEC) which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number-conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher-order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first- and second-order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infrared divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultraviolet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T -matrix leads naturally to a high-energy renormalization. The theory developed in this paper is therefore well defined at both low and high energy and provides a systematic description of BEC in dilute gases. It can therefore be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition.
By using the extension of the statistical Wick's theorem (Gaudin's theorem) to deal with generalized statistical density operators (those which can be expressed as the product of and operator carrying out a canonical transformations times a density operator) and using the appropriate limits we are able to rederive in a very simple way the standard generalized Wick's theorem for overlaps of mean-field wave functions. Due to the simplicity of the derivation it is now straightforward to consider more involved cases and some of them are discussed. The present derivation also allows us to obtain general and compact formulas for other particular cases of the generalized Wick theorem involving overlaps of multiquasiparticle excitations of product wave functions. The new expressions allow us to reduce the combinatorial complexity of the standard calculation of the above overlaps.
Wick's theorem for the expectation values of products of field operators for a system of noninteracting fermions or bosons plays an important role in the perturbative approach to the quantum many-body problem. A finite-temperature version holds in the framework of the grand canonical ensemble, but not for the canonical ensemble appropriate for systems with fixed particle number such as ultracold quantum gases in optical lattices. Here we present formulas for expectation values of products of field operators in the canonical ensemble using a method in the spirit of Gaudin's proof of Wick's theorem for the grand canonical case. The deviations from Wick's theorem are examined quantitatively for two simple models of noninteracting fermions.
This chapter considers a general form of the Wick’s theorem, which leads to a perturbation expansion of the (contour) single- and two-particle Green’s functions, which are expressed in terms of the contour time-ordering operator. The strategy for proving the Wick’s theorem is similar to that adopted within the Matsubara formalism for the Green’s functions at finite temperature and relies on the Gibbs form of the statistical operator in the interaction picture. An extension to superfluid Bose and Fermi systems is also considered.
In this thesis we develop a gapless theory of BEC which can be applied to both trapped and homogeneous gases at zero and finite temperature. The starting point for the theory is the second quantized, many-body Hamiltonian for a system of structureless bosons with pairwise interactions. A number conserving approach is used to rewrite this Hamiltonian in a form which is approximately quadratic with higher order cubic and quartic terms. The quadratic part of the Hamiltonian can be diagonalized exactly by transforming to a quasiparticle basis, while requiring that the condensate satisfy the Gross-Pitaevskii equation. The non-quadratic terms are assumed to have a small effect and are dealt with using first and second order perturbation theory. The conventional treatment of these terms, based on factorization approximations, is shown to be inconsistent. Infra-red divergences can appear in individual terms of the perturbation expansion, but we show analytically that the total contribution beyond quadratic order is finite. The resulting excitation spectrum is gapless and the energy shifts are small for a dilute gas away from the critical region, justifying the use of perturbation theory. Ultra-violet divergences can appear if a contact potential is used to describe particle interactions. We show that the use of this potential as an approximation to the two-body T-matrix leads naturally to a high-energy renormalization. The theory developed in this thesis is therefore well-defined at both low and high energy and provides a systematic description of Bose-Einstein condensation in dilute gases. It can be used to calculate the energies and decay rates of the excitations of the system at temperatures approaching the phase transition. We present analytical results for these quantities in the homogeneous limit and describe how they can be calculated numerically for trapped gases. (author)