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:1112.5768 · Physical Review A

Criteria of off-diagonal long-range order in Bose and Fermi systems based on the Lee-Yang cluster expansion method

Naoyuki Sakumichi, Norio Kawakami, Masahito Ueda

Published Apr 2, 2012• 3 citations
arXivHTMLPDF

Abstract

The quantum-statistical cluster expansion method of Lee and Yang is extended to investigate off-diagonal long-range order (ODLRO) in one-component and multicomponent mixtures of bosons or fermions. Our formulation is applicable to both a uniform system and a trapped system without local-density approximation and allows systematic expansions of one-particle and multiparticle reduced density matrices in terms of cluster functions, which are defined for the same system with Boltzmann statistics. Each term in this expansion can be associated with a Lee-Yang graph. We elucidate a physical meaning of each Lee-Yang graph; in particular, for a mixture of ultracold atoms and bound dimers, an infinite sum of the ladder-type Lee-Yang 0-graphs is shown to lead to Bose-Einstein condensation of dimers below the critical temperature. In the case of Bose statistics, an infinite series of Lee-Yang 1-graphs is shown to converge and gives the criteria of ODLRO at the one-particle level. Applications to a dilute Bose system of hard spheres are also made. In the case of Fermi statistics, an infinite series of Lee-Yang 2-graphs is shown to converge and gives the criteria of ODLRO at the two-particle level. Applications to a two-component Fermi gas in the tightly bound limit are also made.

DOI
10.1103/physreva.85.043601

Related papers

Quantum-statistical mechanics of an atom-dimer mixture: Lee-Yang cluster expansion approach

Takahiro Ohkuma, Masahito Ueda

We use the Lee-Yang cluster expansion method to study quantum-statistical properties of a mixture of interconvertible atoms and dimers, where the dimers form in a two-body bound state of the atoms. We point out an infinite series of cluster diagrams whose summation leads to the Bose-Einstein condensation of the dimers below a critical temperature. Our theory captures some important features of a cold atom-dimer mixture such as interconversion of atoms and dimers and properties of the mixture at the unitarity limit.

Physical Review A•Jun 7, 2006•5 citations

Deviations from off-diagonal long-range order in one-dimensional quantum systems

A. Colcelli, Giuseppe Mussardo, Andrea Trombettoni

A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue of the one-body-density matrix scales as , where N is the total number of particles. Putting to define the scaling exponent , then corresponds to ODLRO and to the single-particle occupation of the density matrix orbitals. When , can be used to quantify deviations from ODLRO. In this paper we study the exponent in a variety of one-dimensional bosonic and anyonic quantum systems at T = 0. For the 1D Lieb-Liniger Bose gas we find that for small interactions is close to 1, implying a mesoscopic condensation, i.e. , a value of the zero temperature "condensate" fraction appreciable at finite values of N (as the ones in experiments with 1D ultracold atoms). 1D anyons provide the possibility to fully interpolate between and 0. The behaviour of for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.

arXiv:1804.04084•Europhysics Letters (EPL)•Jul 18, 2018•18 citations

Scaling properties and Floquet engineering for low dimensional Bose gases

A. Colcelli

A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue λ 0 of the one-body-density matrix scales as λ 0 ∼ N , where N is the total number of particles.Putting λ 0 ∼ N C to define the scaling exponent C, then C = 1 corresponds to ODLRO and C = 0 to the single-particle occupation of the density matrix orbitals.When 0 < C < 1, C can be used to quantify deviations from ODLRO.In this paper we study the exponent C in a variety of one-dimensional bosonic and anyonic quantum systems at T = 0.For the 1D Lieb-Liniger Bose gas we find that for small interactions C is close to 1, implying a mesoscopic condensation, i.e. a value of the zero temperature "condensate" fraction λ 0 /N appreciable at finite values of N (as the ones in experiments with 1D ultracold atoms).1D anyons provide the possibility to fully interpolate between C = 1 and 0. The behaviour of C for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.[2] A. Colcelli, J. Viti, G. Mussardo, and A. Trombettoni, Universal off-diagonal longrange-order behavior for a trapped Tonks-Girardeau gas, Phys.Rev. A 98, 063633 (2018).

•Oct 7, 2020•0 citations

Reduced Density Matrices and the Hard-Sphere Bose Gas

Donald Wright

The thermodynamic properties of the hard-sphere Bose gas are calculated from the one- and two-particle density matrices. The one- and two-particle density matrices are evaluated in the Bogoliubov approximation. The off-diagonal elements of the one- and two-particle density matrices are studied for the presence or absence of off-diagonal long-range order in various temperature regions. Off-diagonal long-range order was found in the single-particle density matrix below the Bose-Einstein transition temperature. The two-particle density matrix also exhibited off-diagonal long-range order whenever the single-particle density matrix did.

Physical Review•Jun 14, 1965•0 citations

Off-Diagonal Long-Range Order and Generalized Bose Condensation

Marvin D. Girardeau

It is shown for boson systems with periodic boundary conditions that the existence of generalized Bose condensation, of which the simple type typified by the ideal Bose gas is a special case, is equivalent to the existence of off-diagonal long-range order (ODLRO) in the single-particle density matrix ρ1 provided that the two noncomuting limits involved in the criterion for ODLRO are taken in the proper order: first size of system → ∞, then interparticle separation → ∞. It is shown by means of an example that certain assumptions concerning the behavior of the single-particle momentum distribution function involved in proving this equivalence are actually satisfied in thermal equilibrium for some dynamical models. The analysis is generalized to box-enclosure boundary conditions by an extension of an argument due to Schafroth, according to which box-enclosure conditions are equivalent to homogeneous Neumann conditions except for a thermodynamically negligible surface effect, provided that one second-quantizes with respect to Hartree-Fock orbitals rather than free-particle orbitals. A general criterion for generalized Bose condensation in terms of eigenvalues of ρ1 is proposed. On the basis of the behavior of soluble models it is conjectured that for a boson system in thermal equilibrium subject to arbitrary boundary conditions, the existence of such Bose condensation is equivalent to the existence of ODLRO in ρ1. The two-particle density matrix ρ2 is discussed briefly. By means of a simplified model it is shown that for a Bose system generalized condensation implies large eigenvalues of ρ2 and ODLRO of ρ2, just as for ρ1. It is pointed out that the question of the existence or nonexistence of generalized Bose condensation of fermion pairs ought to be investigated.

Journal of Mathematical Physics•Jul 1, 1965•29 citations

Proof of off-diagonal long-range order in a mean-field trapped Bose gas via the Feynman–Kac formula

Tianyi Bai, Wolfgang König, Quirin Vogel

We consider the non-interacting Bose gas of N bosons in dimension d≥3 in a trap in a mean-field setting with a vanishing factor aN in front of the kinetic energy. The choice aN=N−2∕d is the semi-classical setting and was analysed in great detail in a special, interacting case in [DS21]. Using a version of the well-known Feynman–Kac representation and a further representation in terms of a Poisson point process, we derive precise asymptotics for the reduced one-particle density matrix, implying off-diagonal long-range order (ODLRO, a well-known criterion for Bose–Einstein condensation) for aN above a certain threshold and non-occurrence of ODLRO for aN below that threshold. In particular, we relate the condensate and its total mass to the amount of particles in long loops in the Feynman–Kac formula, the order parameter that Feynman suggested in [Fe53]. For aN≪N−2∕d, we prove that all loops have the minimal length one, and for aN≫N−2∕d we prove 100 percent condensation and identify the distribution of the long-loop lengths as the Poisson–Dirichlet distribution.

Electronic Journal of Probability•Jan 1, 2025•0 citations
The quantum-statistical cluster expansion method of Lee and Yang is extended to investigate off-diagonal long-range order (ODLRO) in one-component and multicomponent mixtures of bosons or fermions. Our formulation is applicable to both a uniform system and a trapped system without local-density approximation and allows systematic expansions of one-particle and multiparticle reduced density matrices in terms of cluster functions, which are defined for the same system with Boltzmann statistics. Each term in this expansion can be associated with a Lee-Yang graph. We elucidate a physical meaning of each Lee-Yang graph; in particular, for a mixture of ultracold atoms and bound dimers, an infinite sum of the ladder-type Lee-Yang 0-graphs is shown to lead to Bose-Einstein condensation of dimers below the critical temperature. In the case of Bose statistics, an infinite series of Lee-Yang 1-graphs is shown to converge and gives the criteria of ODLRO at the one-particle level. Applications to a dilute Bose system of hard spheres are also made. In the case of Fermi statistics, an infinite series of Lee-Yang 2-graphs is shown to converge and gives the criteria of ODLRO at the two-particle level. Applications to a two-component Fermi gas in the tightly bound limit are also made.
We use the Lee-Yang cluster expansion method to study quantum-statistical properties of a mixture of interconvertible atoms and dimers, where the dimers form in a two-body bound state of the atoms. We point out an infinite series of cluster diagrams whose summation leads to the Bose-Einstein condensation of the dimers below a critical temperature. Our theory captures some important features of a cold atom-dimer mixture such as interconversion of atoms and dimers and properties of the mixture at the unitarity limit.
A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue of the one-body-density matrix scales as , where N is the total number of particles. Putting to define the scaling exponent , then corresponds to ODLRO and to the single-particle occupation of the density matrix orbitals. When , can be used to quantify deviations from ODLRO. In this paper we study the exponent in a variety of one-dimensional bosonic and anyonic quantum systems at T = 0. For the 1D Lieb-Liniger Bose gas we find that for small interactions is close to 1, implying a mesoscopic condensation, i.e. , a value of the zero temperature "condensate" fraction appreciable at finite values of N (as the ones in experiments with 1D ultracold atoms). 1D anyons provide the possibility to fully interpolate between and 0. The behaviour of for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.
A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue λ 0 of the one-body-density matrix scales as λ 0 ∼ N , where N is the total number of particles.Putting λ 0 ∼ N C to define the scaling exponent C, then C = 1 corresponds to ODLRO and C = 0 to the single-particle occupation of the density matrix orbitals.When 0 < C < 1, C can be used to quantify deviations from ODLRO.In this paper we study the exponent C in a variety of one-dimensional bosonic and anyonic quantum systems at T = 0.For the 1D Lieb-Liniger Bose gas we find that for small interactions C is close to 1, implying a mesoscopic condensation, i.e. a value of the zero temperature "condensate" fraction λ 0 /N appreciable at finite values of N (as the ones in experiments with 1D ultracold atoms).1D anyons provide the possibility to fully interpolate between C = 1 and 0. The behaviour of C for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.[2] A. Colcelli, J. Viti, G. Mussardo, and A. Trombettoni, Universal off-diagonal longrange-order behavior for a trapped Tonks-Girardeau gas, Phys.Rev. A 98, 063633 (2018).
The thermodynamic properties of the hard-sphere Bose gas are calculated from the one- and two-particle density matrices. The one- and two-particle density matrices are evaluated in the Bogoliubov approximation. The off-diagonal elements of the one- and two-particle density matrices are studied for the presence or absence of off-diagonal long-range order in various temperature regions. Off-diagonal long-range order was found in the single-particle density matrix below the Bose-Einstein transition temperature. The two-particle density matrix also exhibited off-diagonal long-range order whenever the single-particle density matrix did.
It is shown for boson systems with periodic boundary conditions that the existence of generalized Bose condensation, of which the simple type typified by the ideal Bose gas is a special case, is equivalent to the existence of off-diagonal long-range order (ODLRO) in the single-particle density matrix ρ1 provided that the two noncomuting limits involved in the criterion for ODLRO are taken in the proper order: first size of system → ∞, then interparticle separation → ∞. It is shown by means of an example that certain assumptions concerning the behavior of the single-particle momentum distribution function involved in proving this equivalence are actually satisfied in thermal equilibrium for some dynamical models. The analysis is generalized to box-enclosure boundary conditions by an extension of an argument due to Schafroth, according to which box-enclosure conditions are equivalent to homogeneous Neumann conditions except for a thermodynamically negligible surface effect, provided that one second-quantizes with respect to Hartree-Fock orbitals rather than free-particle orbitals. A general criterion for generalized Bose condensation in terms of eigenvalues of ρ1 is proposed. On the basis of the behavior of soluble models it is conjectured that for a boson system in thermal equilibrium subject to arbitrary boundary conditions, the existence of such Bose condensation is equivalent to the existence of ODLRO in ρ1. The two-particle density matrix ρ2 is discussed briefly. By means of a simplified model it is shown that for a Bose system generalized condensation implies large eigenvalues of ρ2 and ODLRO of ρ2, just as for ρ1. It is pointed out that the question of the existence or nonexistence of generalized Bose condensation of fermion pairs ought to be investigated.
We consider the non-interacting Bose gas of N bosons in dimension d≥3 in a trap in a mean-field setting with a vanishing factor aN in front of the kinetic energy. The choice aN=N−2∕d is the semi-classical setting and was analysed in great detail in a special, interacting case in [DS21]. Using a version of the well-known Feynman–Kac representation and a further representation in terms of a Poisson point process, we derive precise asymptotics for the reduced one-particle density matrix, implying off-diagonal long-range order (ODLRO, a well-known criterion for Bose–Einstein condensation) for aN above a certain threshold and non-occurrence of ODLRO for aN below that threshold. In particular, we relate the condensate and its total mass to the amount of particles in long loops in the Feynman–Kac formula, the order parameter that Feynman suggested in [Fe53]. For aN≪N−2∕d, we prove that all loops have the minimal length one, and for aN≫N−2∕d we prove 100 percent condensation and identify the distribution of the long-loop lengths as the Poisson–Dirichlet distribution.