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

Journal of Physics A Mathematical and General

Temperature and magnetic field-dependent correlators of the exactly integrable (1+1)-dimensional gas of impenetrable fermions

Alexander Bérkovich

Published Apr 7, 1991• 31 citations
Publisher

Abstract

The author shows that the finite-temperature equal-time field correlators of the exactly integrable (1+1)-dimensional gas of impenetrable fermions in a magnetic field can be expressed as the first Fredholm minor of a completely integrable linear operator. This result enables the author to derive differential equations for quantum correlators and to obtain analytical formulae for long-distance asymptotics. In particular, a formula for correlation length as a function of temperature and magnetic field is presented. His analysis reveals the presence of two length scales, which separate short-distance, intermediate and extreme asymptotic domains. The importance of the cross-over region is emphasized.

DOI
10.1088/0305-4470/24/7/025

Related papers

Time and Temperature Dependent Correlation Functions of the One-Dimensional Impenetrable Electron Gas

Frank Göhmann, Anatoli G. Izergin, V. E. Korepin +1 more

We consider the one-dimensional delta-interacting electron gas in the case of infinite repulsion. We use determinant representations to study the long time, large distance asymptotics of correlation functions of local fields in the gas phase. We derive differential equations which drive the correlation functions. Using a related Riemann–Hilbert problem we obtain formulae for the asymptotics of the correlation functions, which are valid at all finite temperatures. At low temperatures these formulae lead to explicit asymptotic expressions for the correlation functions, which describe power law behavior and exponential decay as functions of temperature, magnetic field and chemical potential.

arXiv:cond-mat/9805192•International Journal of Modern Physics B•Sep 20, 1998•26 citations

Large-distance asymptotic behavior of the correlation functions of 1D impenetrable anyons at finite temperatures

O. I. Pâţu, V. E. Korepin, D. V. Averin

The large-distance asymptotic behavior of the field-field correlators has been computed for one-dimensional impenetrable anyons at finite temperatures. The asymptotic behavior agrees with the predictions of conformal field theory at low temperatures and reproduces the known results for impenetrable bosons and free fermions in appropriate limits. We have also obtained an integrable system of partial nonlinear differential equations which completely characterizes the 2-point correlation functions. The system is the same as for bosons but with different initial conditions.

arXiv:0811.2419•Europhysics Letters (EPL)•May 1, 2009•21 citations

Dynamical correlation functions in the Ising field theory

István Csépányi, Márton Kormos

We study finite temperature dynamical correlation functions of the magnetization operator in the one-dimensional Ising quantum field theory. Our approach is based on a finite temperature form factor series and on a Fredholm determinant representation of the correlators. While for space-like separations the Fredholm determinant can be efficiently evaluated numerically, for the time-like region it has convergence issues inherited from the form factor series. We develop a method to compute the correlation functions at time-like separations based on the analytic continuation of the space-time coordinates to complex values. Using this numerical technique, we explore all space-time and temperature regimes in both the ordered and disordered phases including short, large, and near-light-cone separations at low and high temperatures. We confirm the existing analytic predictions for the asymptotic behavior of the correlations except in the case of space-like correlations in the paramagnetic phase. For this case we derive a new closed form expression for the correlation length that has some unusual properties: it is a non-analytic function of both the space-time direction and the temperature, and its temperature dependence is non-monotonic.

arXiv:2406.05100•SciPost Physics•Dec 10, 2024•0 citations

One-dimensional impenetrable anyons in thermal equilibrium: III. Large distance asymptotics of the space correlations

Ovidiu I Pâţu, V. E. Korepin, D. V. Averin

Using the determinant representation for the field-field correlation functions of impenetrable anyons at finite temperature obtained in a previous paper, we derive a system of nonlinear partial differential equations completely characterizing the correlators. The system is the same as the one for impenetrable bosons but with different initial conditions. The large-distance asymptotic behavior of the correlation functions is obtained from the analysis of the Riemann-Hilbert problem associated with the system of differential equations. We calculate both the exponential and pre-exponential factors in the asymptotics of the field-field correlators. The asymptotics derived in this way agree with those of the free fermions and impenetrable bosons in the appropriate limits, $κ\to 1$ and $κ\to 0$, of the statistics parameter $κ$, and coincide with the predictions of the conformal field theory at low temperatures.

arXiv:0904.1835•Journal of Physics A Mathematical and Theoretical•Jun 17, 2009•29 citations

The Matrix Riemann-Hilbert Problem for Correlation Functions

V. E. Korepin, N. M. Bogoliubov, Anatoli G. Izergin

Introduction In the previous chapters correlation functions were represented as determinants of a special form of Fredholm integral operator. First, a differential equation was obtained for the time-independent zero temperature field correlation function for the impenetrable Bose gas (it depends only on one variable—distance). By means of an isomonodromic deformation technique A. Jimbo, T. Miwa, Y. Mori and M. Sato wrote down the ordinary differential equation (of Painlevè type). In the previous chapter differential equations for correlation functions in more general situations (with time, temperature and for finite coupling) were constructed. A.R. Its came up with the idea of applying the matrix Riemann problem for the description of these correlation functions (Fredholm determinants). This permits us to write down partial differential equations (completely integrable) for correlation functions and evaluate the asymptotics of correlation functions. Below we follow this line. Integrable nonlinear partial differential equations for correlation functions of the one-dimensional nonrelativistic Bose gas with point-like repulsion between gas particles were given in section XIV. So quantum correlation functions can be expressed in terms of the solutions of classical nonlinear integrable evolution systems. These classical systems have been investigated already by means of the classical inverse scattering method. Our approach allows one, in particular, to solve the particularly interesting problem of calculating the asymptotics of correlators at large time and distance; this is done in the next section.

Cambridge University Press eBooks•Aug 19, 1993•1 citation

DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS

Alexander Rudol'fovich Its, Anatoli G. Izergin, V. E. Korepin +1 more

The quantum nonlinear Schrödinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest weight vector permits us to represent correlation function as a determinant of a Fredholm integral operator. This integral operator can be treated as the Gelfand-Levitan operator for some new differential equation. These differential equations are written down in the paper. They generalize the fifth Painlève transcendent, which describe equal time, zero temperature correlation function of an impenetrable Bose gas. These differential equations drive the quantum correlation functions of the Bose gas. The Riemann problem, associated with these differential equations permits us to calculate asymp-totics of quantum correlation functions. Quantum correlation function (Fredholm determinant) plays the role of τ functions of these new differential equations. For the impenetrable Bose gas space and time dependent correlation function is equal to τ function of the nonlinear Schrödinger equation itself, For a penetrable Bose gas (finite coupling constant c) the correlator is τ-function of an integro-differentiation equation.

International Journal of Modern Physics B•Apr 1, 1990•386 citations
The author shows that the finite-temperature equal-time field correlators of the exactly integrable (1+1)-dimensional gas of impenetrable fermions in a magnetic field can be expressed as the first Fredholm minor of a completely integrable linear operator. This result enables the author to derive differential equations for quantum correlators and to obtain analytical formulae for long-distance asymptotics. In particular, a formula for correlation length as a function of temperature and magnetic field is presented. His analysis reveals the presence of two length scales, which separate short-distance, intermediate and extreme asymptotic domains. The importance of the cross-over region is emphasized.
We consider the one-dimensional delta-interacting electron gas in the case of infinite repulsion. We use determinant representations to study the long time, large distance asymptotics of correlation functions of local fields in the gas phase. We derive differential equations which drive the correlation functions. Using a related Riemann–Hilbert problem we obtain formulae for the asymptotics of the correlation functions, which are valid at all finite temperatures. At low temperatures these formulae lead to explicit asymptotic expressions for the correlation functions, which describe power law behavior and exponential decay as functions of temperature, magnetic field and chemical potential.
The large-distance asymptotic behavior of the field-field correlators has been computed for one-dimensional impenetrable anyons at finite temperatures. The asymptotic behavior agrees with the predictions of conformal field theory at low temperatures and reproduces the known results for impenetrable bosons and free fermions in appropriate limits. We have also obtained an integrable system of partial nonlinear differential equations which completely characterizes the 2-point correlation functions. The system is the same as for bosons but with different initial conditions.
We study finite temperature dynamical correlation functions of the magnetization operator in the one-dimensional Ising quantum field theory. Our approach is based on a finite temperature form factor series and on a Fredholm determinant representation of the correlators. While for space-like separations the Fredholm determinant can be efficiently evaluated numerically, for the time-like region it has convergence issues inherited from the form factor series. We develop a method to compute the correlation functions at time-like separations based on the analytic continuation of the space-time coordinates to complex values. Using this numerical technique, we explore all space-time and temperature regimes in both the ordered and disordered phases including short, large, and near-light-cone separations at low and high temperatures. We confirm the existing analytic predictions for the asymptotic behavior of the correlations except in the case of space-like correlations in the paramagnetic phase. For this case we derive a new closed form expression for the correlation length that has some unusual properties: it is a non-analytic function of both the space-time direction and the temperature, and its temperature dependence is non-monotonic.
Using the determinant representation for the field-field correlation functions of impenetrable anyons at finite temperature obtained in a previous paper, we derive a system of nonlinear partial differential equations completely characterizing the correlators. The system is the same as the one for impenetrable bosons but with different initial conditions. The large-distance asymptotic behavior of the correlation functions is obtained from the analysis of the Riemann-Hilbert problem associated with the system of differential equations. We calculate both the exponential and pre-exponential factors in the asymptotics of the field-field correlators. The asymptotics derived in this way agree with those of the free fermions and impenetrable bosons in the appropriate limits, κ→1κ\to 1κ→1 and κ→0κ\to 0κ→0, of the statistics parameter κκκ, and coincide with the predictions of the conformal field theory at low temperatures.
Introduction In the previous chapters correlation functions were represented as determinants of a special form of Fredholm integral operator. First, a differential equation was obtained for the time-independent zero temperature field correlation function for the impenetrable Bose gas (it depends only on one variable—distance). By means of an isomonodromic deformation technique A. Jimbo, T. Miwa, Y. Mori and M. Sato wrote down the ordinary differential equation (of Painlevè type). In the previous chapter differential equations for correlation functions in more general situations (with time, temperature and for finite coupling) were constructed. A.R. Its came up with the idea of applying the matrix Riemann problem for the description of these correlation functions (Fredholm determinants). This permits us to write down partial differential equations (completely integrable) for correlation functions and evaluate the asymptotics of correlation functions. Below we follow this line. Integrable nonlinear partial differential equations for correlation functions of the one-dimensional nonrelativistic Bose gas with point-like repulsion between gas particles were given in section XIV. So quantum correlation functions can be expressed in terms of the solutions of classical nonlinear integrable evolution systems. These classical systems have been investigated already by means of the classical inverse scattering method. Our approach allows one, in particular, to solve the particularly interesting problem of calculating the asymptotics of correlators at large time and distance; this is done in the next section.
The quantum nonlinear Schrödinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest weight vector permits us to represent correlation function as a determinant of a Fredholm integral operator. This integral operator can be treated as the Gelfand-Levitan operator for some new differential equation. These differential equations are written down in the paper. They generalize the fifth Painlève transcendent, which describe equal time, zero temperature correlation function of an impenetrable Bose gas. These differential equations drive the quantum correlation functions of the Bose gas. The Riemann problem, associated with these differential equations permits us to calculate asymp-totics of quantum correlation functions. Quantum correlation function (Fredholm determinant) plays the role of τ functions of these new differential equations. For the impenetrable Bose gas space and time dependent correlation function is equal to τ function of the nonlinear Schrödinger equation itself, For a penetrable Bose gas (finite coupling constant c) the correlator is τ-function of an integro-differentiation equation.