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/9702219 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Parametric Ward-Takahashi identity in disordered systems and the integral identity associated with the Calogero-Sutherland model

Nobuhiko Taniguchi

Published Apr 1, 1997• 2 citations
arXivHTMLPDF

Abstract

By utilizing the symmetric property known as the Ward-Takahashi identity in disordered systems, we explore the novel symmetry relations which hold in one-dimensional systems with inverse square interaction (the Calogero-Sutherland model). The identities emerge totally from the algebraic structure of the model. They show that the dynamical correlators are connected with one another, involving the higher-order integrals of motion. We obtain the result for the coupling strengths \ensuremath{\lambda}=1/2, 1, and 2, and conjecture that a similar relation may hold for arbitrary rational \ensuremath{\lambda}.

DOI
10.1103/physreve.55.4116

Related papers

Random Matrix Model and the Calogero-Sutherland Model: A Novel Current-Density Mapping

Nobuhiko Taniguchi, B. Sriram Shastry, B. L. Altshuler

We investigate the relation between the invariant correlators of random matrix theory and correlators of the integrable one-dimensional systems. Starting from the relation between correlators for the coupling strengths $\ensuremath{\lambda}=\frac{1}{2}, 1, \mathrm{and} 2$, we explore the local current-density mapping applicable to arbitrary $\ensuremath{\lambda}$ including irrational values, which results from the novel structure of the Calogero-Sutherland model. We find an interesting and novel relationship between equal time current and density correlations for any coupling, which exist in addition to the usual Ward identities for this class of systems.

arXiv:cond-mat/9510012•Physical Review Letters•Nov 13, 1995•28 citations

Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model

Edwin Langmann A

We present remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers N and M, and a particular function of N+M variables arising as anyon correlation function of N particles and M antiparticles. In addition to identities obtained from anyons with the same statistics parameter λ, we also obtain “dual” relations involving “mixed” correlation functions of anyons with two different statistics parameters λ and 1∕λ. We also give alternative, elementary proofs of these identities by direct computations.

arXiv:math-ph/0406061•Journal of Mathematical Physics•Feb 1, 2006•17 citations

Exact Dynamical Correlation Functions of Calogero-Sutherland Model and One-Dimensional Fractional Statistics

Zou Ha

One-dimensional model of nonrelativistic particles with inverse-square interaction potential known as the Calogero-Sutherland model is shown to possess fractional statistics. Using the theory of Jack polynomials the exact dynamical density-density correlation function and the hole propagator part of one-particle Green's function at any rational interaction coupling constant are obtained and used to show clear evidences of the fractional exclusion statistics in the sense of Haldane's "generalized Pauli exclusion principle." This model is also endowed with the corresponding natural exchange statistics.

arXiv:cond-mat/9405063•Physical Review Letters•Sep 19, 1994•188 citations

Infinite Symmetry of the Spin Systems with Inverse Square Interactions

Kazuhiro Hikami, Miki Wadati

One-dimensional quantum particle system with S U (ν) spins interacting through inverse square interactions is studied. We reveal algebraic structures of the system: hidden symmetry is the \(U(\nu )\cong SU(\nu )\otimes U(1)\) current algebra. This is consistent with the fact that the ground state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, which is the w 1+∞ algebra. With this W algebra we can clarify simultaneously the structures of the Calogero type (1/ x 2 -interactions) and Sutherland type (1/sin 2 x -interactions). The Yangian symmetry is briefly discussed.

Journal of the Physical Society of Japan•Dec 15, 1993•65 citations

Equivalence of the Calogero-Sutherland model to free harmonic oscillators

Nakka Gurappa, Prasanta Kumar Panigrahi

A similarity transformation is constructed, through which a system of particles interacting with inverse-square two-body and harmonic potentials in one dimension can be mapped identically to a set of free harmonic oscillators. This equivalence provides a straightforward method to find the complete set of eigenfunctions, the exact constants of motion, and a linear ${W}_{1+\ensuremath{\infty}}$ algebra associated with this model. It is also demonstrated that a large class of models with long-range interactions, both in one and higher dimensions can be made equivalent to decoupled oscillators.

arXiv:cond-mat/9710035•Physical review. B, Condensed matter•Jan 15, 1999•78 citations

Integrable Systems with Long-Range Interactions, W∞ Algebra, and Energy Spectrum

Kazuhiro Hikami, Miki Wadati

Quantum integrable particle systems with inverse square interactions in one-dimensional space are investigated by use of the quantum inverse scattering method. For the case of particles with $\mathrm{SU}(\ensuremath{\nu})$ spins, a hidden symmetry is the $\mathrm{SU}(\ensuremath{\nu})\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\simeq}\mathrm{U}(\ensuremath{\nu})$ current algebra. The system has a higher symmetry, the ${W}_{1+\ensuremath{\infty}}$ algebra. This $W$ algebra structure unifies the Calogero type ($\frac{1}{{x}^{2}}$ interactions) and the Sutherland type ($\frac{1}{{sin}^{2}x}$ interactions). The energy spectra of the systems are obtained in a simple manner based on the fermion representation of the algebra.

Physical Review Letters•Aug 29, 1994•39 citations
By utilizing the symmetric property known as the Ward-Takahashi identity in disordered systems, we explore the novel symmetry relations which hold in one-dimensional systems with inverse square interaction (the Calogero-Sutherland model). The identities emerge totally from the algebraic structure of the model. They show that the dynamical correlators are connected with one another, involving the higher-order integrals of motion. We obtain the result for the coupling strengths \ensuremath{\lambda}=1/2, 1, and 2, and conjecture that a similar relation may hold for arbitrary rational \ensuremath{\lambda}.
We investigate the relation between the invariant correlators of random matrix theory and correlators of the integrable one-dimensional systems. Starting from the relation between correlators for the coupling strengths $\ensuremath{\lambda}=\frac{1}{2}, 1, \mathrm{and} 2$, we explore the local current-density mapping applicable to arbitrary $\ensuremath{\lambda}$ including irrational values, which results from the novel structure of the Calogero-Sutherland model. We find an interesting and novel relationship between equal time current and density correlations for any coupling, which exist in addition to the usual Ward identities for this class of systems.
We present remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers N and M, and a particular function of N+M variables arising as anyon correlation function of N particles and M antiparticles. In addition to identities obtained from anyons with the same statistics parameter λ, we also obtain “dual” relations involving “mixed” correlation functions of anyons with two different statistics parameters λ and 1∕λ. We also give alternative, elementary proofs of these identities by direct computations.
One-dimensional model of nonrelativistic particles with inverse-square interaction potential known as the Calogero-Sutherland model is shown to possess fractional statistics. Using the theory of Jack polynomials the exact dynamical density-density correlation function and the hole propagator part of one-particle Green's function at any rational interaction coupling constant are obtained and used to show clear evidences of the fractional exclusion statistics in the sense of Haldane's "generalized Pauli exclusion principle." This model is also endowed with the corresponding natural exchange statistics.
One-dimensional quantum particle system with S U (ν) spins interacting through inverse square interactions is studied. We reveal algebraic structures of the system: hidden symmetry is the U(ν)≅SU(ν)⊗U(1)U(\nu )\cong SU(\nu )\otimes U(1)U(ν)≅SU(ν)⊗U(1) current algebra. This is consistent with the fact that the ground state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, which is the w 1+∞ algebra. With this W algebra we can clarify simultaneously the structures of the Calogero type (1/ x 2 -interactions) and Sutherland type (1/sin 2 x -interactions). The Yangian symmetry is briefly discussed.
A similarity transformation is constructed, through which a system of particles interacting with inverse-square two-body and harmonic potentials in one dimension can be mapped identically to a set of free harmonic oscillators. This equivalence provides a straightforward method to find the complete set of eigenfunctions, the exact constants of motion, and a linear ${W}_{1+\ensuremath{\infty}}$ algebra associated with this model. It is also demonstrated that a large class of models with long-range interactions, both in one and higher dimensions can be made equivalent to decoupled oscillators.
Quantum integrable particle systems with inverse square interactions in one-dimensional space are investigated by use of the quantum inverse scattering method. For the case of particles with $\mathrm{SU}(\ensuremath{\nu})$ spins, a hidden symmetry is the $\mathrm{SU}(\ensuremath{\nu})\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\simeq}\mathrm{U}(\ensuremath{\nu})$ current algebra. The system has a higher symmetry, the ${W}_{1+\ensuremath{\infty}}$ algebra. This WWW algebra structure unifies the Calogero type (1x2\frac{1}{{x}^{2}}x21​ interactions) and the Sutherland type (1sin2x\frac{1}{{sin}^{2}x}sin2x1​ interactions). The energy spectra of the systems are obtained in a simple manner based on the fermion representation of the algebra.