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:cond-mat/0005129 · Journal of Physics A Mathematical and General

Hellmann-Feynman theorem and correlation-fluctuation analysis for the Calogero-Sutherland model

Rudolf A. Römer, P. Ziesche

Published Feb 9, 2001• 10 citations
arXivHTMLPDF

Abstract

Exploiting the results of the exact solution for the ground state of the one-dimensional spinless quantum gas of fermions and impenetrable bosons with the µ/ x ij 2 particle-particle interaction, the Hellmann-Feynman theorem yields mutually compensating divergences of both the kinetic and the interaction energy in the limiting case µ→-¼. These divergences result from the peculiar behaviour of both the momentum distribution (for large momenta) and the pair density (for small inter-particle separation). The available analytical pair densities for µ = -¼, 0 and 2 allow one to analyse particle-number fluctuations. They are suppressed by repulsive interaction (µ>0), enhanced by attraction (µ<0), and may therefore measure the kind and strength of correlation. Other recently proposed purely quantum-kinematical measures of the correlation strength arise from the small-separation behaviour of the pair density or - for fermions - from the non-idempotency of the momentum distribution and its large-momenta behaviour. They are compared with each other and with reference-free, short-range correlation-measuring ratios of the kinetic and potential energies.

DOI
10.1088/0305-4470/34/7/320

Related papers

Fluctuation-correlation analysis of the Calogero-Sutherland model

Rudolf A. Römer, P. Ziesche

We exploit the exact solution for the ground state of the one-dimensional spinless quantum gas of fermions with $\ensuremath{\mu}{/x}_{\mathrm{ij}}^{2}$ interaction. The available analytical pair densities for $\ensuremath{\mu}=\ensuremath{-}1/4,$ 0, and 2 allow us to analyze particle-number fluctuations. They are suppressed by repulsive interaction $(\ensuremath{\mu}>0),$ enhanced by attraction $(\ensuremath{\mu}<0),$ and measure the kind and strength of correlation. Other recently proposed purely quantum-kinematical measures of the correlation strength arise from the small-separation behavior of the pair density or---for fermions---from the nonidempotency of the momentum distribution and its large-momenta behavior. They are compared with each other and with reference-free, short-range correlation-measuring ratios of the kinetic and potential energies.

Physical review. B, Condensed matter•Dec 15, 2000•7 citations

Off-diagonal correlations of the Calogero-Sutherland model

G. E. Astrakharchik, Dimitri M. Gangardt, Yu. E. Lozovik +1 more

We study correlation functions of the Calogero-Sutherland model in the whole range of the interaction parameter. Using the replica method we obtain analytical expressions for the long-distance asymptotics of the one-body density matrix in addition to the previously derived asymptotics of the pair-distribution function [D.M. Gangardt and A. Kamenev, Nucl. Phys. B 610, 578 (2001)]. The leading analytic and nonanalytic terms in the short-distance expansion of the one-body density matrix are discussed. Numerical results for these correlation functions are obtained using Monte Carlo techniques for all distances. The momentum distribution and static structure factor are calculated. The potential and kinetic energies are obtained using the Hellmann-Feynman theorem. Perfect agreement is found between the analytical expressions and numerical data. These results allow for the description of physical regimes of the Calogero-Sutherland model. The zero temperature phase diagram is found to be of a crossover type and includes quasicondensation, quasicrystallization and quasisupersolid regimes.

arXiv:cond-mat/0512470•Physical Review E•Aug 4, 2006•33 citations

Imperfect Fermi gas: Kinetic and interaction energies

Arkady A Shanenko

A uniform ground-state three-dimensional Fermi gas with short-range repulsive pairwise interaction is under consideration. Its kinetic and interaction energies are calculated up to the second order of the expansion in the gas parameter. Similar to recent results for an interacting Bose gas, the quantities in question are found to depend on the pairwise interaction through two characteristic lengths: the former, $a$, is the $s$-wave scattering length, and the latter, $b$, is related to $a$ by $b=a\ensuremath{-}m(\ensuremath{\partial}a∕\ensuremath{\partial}m)$, where $m$ stands for the fermion mass. To control the results, we proceed in two independent ways. The first involves the Hellmann-Feynman theorem applied to derive the kinetic and interaction energies from the total-energy expansion in the gas parameter first found by Huang and Yang. The second way operates with in-medium pair wave functions and allows one to calculate the quantities of interest ``from scratch.'' The results of the present investigation, taken together with those of the recent consideration of a dilute Bose gas, make it possible to conclude that the pairwise interaction in a quantum gas has an essential and nontrivial effect on the kinetic energy, which is not the case for a classical many-particle system.

arXiv:cond-mat/0401218•Physical Review A•Dec 21, 2004•0 citations

Short-range particle correlations in a dilute Bose gas

Alexander Yu. Cherny, Arkady A Shanenko

The thermodynamics of a homogeneous dilute Bose gas with an arbitrarily strong repulsion between particles is investigated on the basis of the exact relation connecting the pair correlation function with the in-medium pair wave functions and occupation numbers. It is shown that the effective-interaction scheme, which is reduced to the Bogoliubov model with the effective pairwise potential, is not acceptable for investigating the short-range particle correlations in a dilute strongly interacting Bose gas. In contrast to this scheme, our model is thermodynamically consistent and free of the ultraviolet divergences due to accurate treatment of the short-range boson correlations. An equation for the in-medium scattering amplitude is derived that makes it possible to find the in-medium renormalization for the pair wave functions at short boson separations. Low-density expansions for the main thermodynamic quantities are reinvestigated on the basis of this equation. In addition, the expansions are found for the interaction and kinetic energies per particle. It is demonstrated that for a many-boson system of hard spheres the interaction energy is equal to zero for any boson density. The exact relationship between the chemical potential and in-medium pair wave functions is also established.

arXiv:cond-mat/9909199•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 1, 2000•21 citations

Comparative study of one-dimensional Bose and Fermi gases with contact interactions from the viewpoint of universal relations for correlation functions

Yuta Sekino, Shina Tan, Yusuke Nishida

One-dimensional spinless Bose and Fermi gases with contact interactions have the close interrelation via Girardeau's Bose-Fermi mapping, leading to the correspondences in their energy spectra and thermodynamics. However, correlation functions are in general not identical between these systems. We derive in both systems the exact universal relations for correlation functions, which hold for any energy eigenstate and any statistical ensemble of the eigenstates with or without a trapping potential. These relations include the large-momentum behaviors of static structure factors and of momentum distributions as well as energy relations, which connect the sums of kinetic and interaction energies to the momentum distributions. The relations involve two- and three-body contacts, which are the integrals of local pair and triad correlations, respectively. We clarify how the relations for bosons and fermions differ and are connected with each other. In particular, we find that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one. In addition, we compute the exact momentum distribution for any number of fermions in the unitary limit.

arXiv:1710.01909•Physical Review A•Jan 17, 2018•26 citations

A Quantum Many-body Problem in Two Dimensions: Ground State

Avinash Khare, Koushik Ray

We obtain the exact ground state for the Calogero-Sutherland problem in arbitrary dimensions. In the special case of two dimensions, we show that the problem is connected to the random matrix problem for complex matrices, provided the strength of the inverse-square interaction g = 2. In the thermodynamic limit, we obtain the ground state energy and the pair-correlation function and show that in this case there is no long-range order. Calogero-Sutherland type models are a class of exactly solvable models. Starting from the inception [1], till today, these models continue to be of interest, especially due to their exact solvability [2]. Moreover, these models are related to (1 + 1)-dimensional conformal field theory, random matrices and a host of other things [3]. Some variants of the models have attracted attention in very recent past [3–5]. The original model of Calogero [1] described N particles in one dimension, interacting through an inverse square law and a two-body harmonic interaction. The latter interaction was added to the effect of complete discretization of the spectrum. This model gave in to an analytic treatment yielding an exact solution. Sutherland [6] considered a variant of this problem where the harmonic interaction was replaced by a harmonic well containing the N particles. It was shown that one can exactly solve for the ground state of the system. The norm of the ground state wave-function was shown to be of a form that coincides with the joint probability density function of the eigenvalues of random matrices from an orthogonal, unitary or symplectic Gaussian ensemble, provided, the

•Jan 1, 1997•31 citations
Exploiting the results of the exact solution for the ground state of the one-dimensional spinless quantum gas of fermions and impenetrable bosons with the µ/ x ij 2 particle-particle interaction, the Hellmann-Feynman theorem yields mutually compensating divergences of both the kinetic and the interaction energy in the limiting case µ→-¼. These divergences result from the peculiar behaviour of both the momentum distribution (for large momenta) and the pair density (for small inter-particle separation). The available analytical pair densities for µ = -¼, 0 and 2 allow one to analyse particle-number fluctuations. They are suppressed by repulsive interaction (µ>0), enhanced by attraction (µ<0), and may therefore measure the kind and strength of correlation. Other recently proposed purely quantum-kinematical measures of the correlation strength arise from the small-separation behaviour of the pair density or - for fermions - from the non-idempotency of the momentum distribution and its large-momenta behaviour. They are compared with each other and with reference-free, short-range correlation-measuring ratios of the kinetic and potential energies.
We exploit the exact solution for the ground state of the one-dimensional spinless quantum gas of fermions with $\ensuremath{\mu}{/x}_{\mathrm{ij}}^{2}$ interaction. The available analytical pair densities for $\ensuremath{\mu}=\ensuremath{-}1/4,$ 0, and 2 allow us to analyze particle-number fluctuations. They are suppressed by repulsive interaction $(\ensuremath{\mu}>0),$ enhanced by attraction $(\ensuremath{\mu}<0),$ and measure the kind and strength of correlation. Other recently proposed purely quantum-kinematical measures of the correlation strength arise from the small-separation behavior of the pair density or---for fermions---from the nonidempotency of the momentum distribution and its large-momenta behavior. They are compared with each other and with reference-free, short-range correlation-measuring ratios of the kinetic and potential energies.
We study correlation functions of the Calogero-Sutherland model in the whole range of the interaction parameter. Using the replica method we obtain analytical expressions for the long-distance asymptotics of the one-body density matrix in addition to the previously derived asymptotics of the pair-distribution function [D.M. Gangardt and A. Kamenev, Nucl. Phys. B 610, 578 (2001)]. The leading analytic and nonanalytic terms in the short-distance expansion of the one-body density matrix are discussed. Numerical results for these correlation functions are obtained using Monte Carlo techniques for all distances. The momentum distribution and static structure factor are calculated. The potential and kinetic energies are obtained using the Hellmann-Feynman theorem. Perfect agreement is found between the analytical expressions and numerical data. These results allow for the description of physical regimes of the Calogero-Sutherland model. The zero temperature phase diagram is found to be of a crossover type and includes quasicondensation, quasicrystallization and quasisupersolid regimes.
A uniform ground-state three-dimensional Fermi gas with short-range repulsive pairwise interaction is under consideration. Its kinetic and interaction energies are calculated up to the second order of the expansion in the gas parameter. Similar to recent results for an interacting Bose gas, the quantities in question are found to depend on the pairwise interaction through two characteristic lengths: the former, aaa, is the sss-wave scattering length, and the latter, bbb, is related to aaa by $b=a\ensuremath{-}m(\ensuremath{\partial}a∕\ensuremath{\partial}m)$, where mmm stands for the fermion mass. To control the results, we proceed in two independent ways. The first involves the Hellmann-Feynman theorem applied to derive the kinetic and interaction energies from the total-energy expansion in the gas parameter first found by Huang and Yang. The second way operates with in-medium pair wave functions and allows one to calculate the quantities of interest ``from scratch.'' The results of the present investigation, taken together with those of the recent consideration of a dilute Bose gas, make it possible to conclude that the pairwise interaction in a quantum gas has an essential and nontrivial effect on the kinetic energy, which is not the case for a classical many-particle system.
The thermodynamics of a homogeneous dilute Bose gas with an arbitrarily strong repulsion between particles is investigated on the basis of the exact relation connecting the pair correlation function with the in-medium pair wave functions and occupation numbers. It is shown that the effective-interaction scheme, which is reduced to the Bogoliubov model with the effective pairwise potential, is not acceptable for investigating the short-range particle correlations in a dilute strongly interacting Bose gas. In contrast to this scheme, our model is thermodynamically consistent and free of the ultraviolet divergences due to accurate treatment of the short-range boson correlations. An equation for the in-medium scattering amplitude is derived that makes it possible to find the in-medium renormalization for the pair wave functions at short boson separations. Low-density expansions for the main thermodynamic quantities are reinvestigated on the basis of this equation. In addition, the expansions are found for the interaction and kinetic energies per particle. It is demonstrated that for a many-boson system of hard spheres the interaction energy is equal to zero for any boson density. The exact relationship between the chemical potential and in-medium pair wave functions is also established.
One-dimensional spinless Bose and Fermi gases with contact interactions have the close interrelation via Girardeau's Bose-Fermi mapping, leading to the correspondences in their energy spectra and thermodynamics. However, correlation functions are in general not identical between these systems. We derive in both systems the exact universal relations for correlation functions, which hold for any energy eigenstate and any statistical ensemble of the eigenstates with or without a trapping potential. These relations include the large-momentum behaviors of static structure factors and of momentum distributions as well as energy relations, which connect the sums of kinetic and interaction energies to the momentum distributions. The relations involve two- and three-body contacts, which are the integrals of local pair and triad correlations, respectively. We clarify how the relations for bosons and fermions differ and are connected with each other. In particular, we find that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one. In addition, we compute the exact momentum distribution for any number of fermions in the unitary limit.
We obtain the exact ground state for the Calogero-Sutherland problem in arbitrary dimensions. In the special case of two dimensions, we show that the problem is connected to the random matrix problem for complex matrices, provided the strength of the inverse-square interaction g = 2. In the thermodynamic limit, we obtain the ground state energy and the pair-correlation function and show that in this case there is no long-range order. Calogero-Sutherland type models are a class of exactly solvable models. Starting from the inception [1], till today, these models continue to be of interest, especially due to their exact solvability [2]. Moreover, these models are related to (1 + 1)-dimensional conformal field theory, random matrices and a host of other things [3]. Some variants of the models have attracted attention in very recent past [3–5]. The original model of Calogero [1] described N particles in one dimension, interacting through an inverse square law and a two-body harmonic interaction. The latter interaction was added to the effect of complete discretization of the spectrum. This model gave in to an analytic treatment yielding an exact solution. Sutherland [6] considered a variant of this problem where the harmonic interaction was replaced by a harmonic well containing the N particles. It was shown that one can exactly solve for the ground state of the system. The norm of the ground state wave-function was shown to be of a form that coincides with the joint probability density function of the eigenvalues of random matrices from an orthogonal, unitary or symplectic Gaussian ensemble, provided, the