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:1905.04907

Riemann-Hilbert approach to a generalised sine kernel

Roozbeh Gharakhloo, Alexander Rudol'fovich Its, Karol Kajetan Kozlowski

Published May 13, 2019• 2 citations
arXivHTMLPDF

Abstract

We derive the large distance asymptotics of the Fredholm determinant of the so-called generalised sine kernel at the critical point. This kernel corresponds to a generalisation of the pure sine kernel arising in the theory of random matrices and has potential applications to the analysis of the large-distance asymptotic behaviour of the so-called emptiness formation probability for various quantum integrable models away from their free fermion point.

DOI
10.48550/arxiv.1905.04907

Related papers

Riemann-Hilbert approach to the time-dependent generalized sine kernel

Karol Kajetan Kozlowski

We derive the leading asymptotic behavior and build a new series representation for the Fredholm determinant of integrable integral operators appearing in the representation of the time and distance-dependent correlation functions of integrable models described by a six-vertex R-matrix.This series representation opens a systematic way for the computation of the long-time, long-distance asymptotic expansion for the correlation functions of the aforementioned integrable models away from their free fermion point.Our method builds on a Riemann-Hilbert based analysis.

arXiv:1011.5897•Advances in Theoretical and Mathematical Physics•Jan 1, 2011•25 citations

Asymptotic analysis of a Fredholm determinant occurring in the description of the dynamical correlation functions of the Lieb--Liniger Bose gas

Frank Göhmann, Karol Kajetan Kozlowski, Mikhail D. Minin

We perform a large-$x$ asymptotic analysis of the Fredholm determinant of an integrable integral operator with generalized sine kernel, where $x$ controls the strength of the oscillations along the integration contour of the operator and will play the role of the distance variable in applications to the correlation functions of integrable quantum systems. Our generalized sine kernel involves a number of functional parameters that will allow us to adapt it to the analysis of the dynamical correlation functions of the Lieb--Liniger model at finite temperature and for all positive values of the coupling constant. It will also allow us to consider a class of equilibrium correlation functions that are governed by generalized Gibbs ensembles. Our work is based on the analysis of a matrix Riemann--Hilbert problem that is canonically connected with our integrable integral operator.

arXiv:2608.16727•Aug 17, 2026•0 citations

FREDHOLM DETERMINANT REPRESENTATION OF QUANTUM CORRELATION FUNCTION FOR SINE-GORDON AT SPECIAL VALUE OF COUPLING CONSTANT

H. Itoyama, V. E. Korepin, H. B. Thacker

Correlation functions of the Sine-Gordon model (which is equivalent to the Massive-Thirring model) are considered at the free fermion point. We derive a determinant formula for local correlation functions of the Sine-Gordon model, starting from Bethe ansatz wave function. Kernel of integral operator is trigonometric version of the one for Impenetrable Bosons.

Modern Physics Letters B•Sep 20, 1992•3 citations

Exact solution of Riemann–Hilbert problem for a correlation function of the XY spin chain

Yasuhiro Fujii

A correlation function of the XY spin chain is studied at zero temperature. This is called the Emptiness Formation Probability (EFP) and is expressed by the Fredholm determinant in the thermodynamic limit. We formulate the associated Riemann–Hilbert problem and solve it exactly. The EFP is shown to decay in Gaussian. In this letter we study the Riemann–Hilbert problem associated with a correlation function of the XY spin chain. The Hamiltonian of the model is given by

•Jan 1, 2000•0 citations

The large time asymptotics of the temperature correlation functions of the XXO Heisenberg ferromagnetic: The Reimann-Hilbert approach

Xiaojun Jie

The finite temperature correlation functions of the time-dependent local spin operators of the Heisenberg XXO ferromagnet are considered. The Fredholm determinant representation for these correlation functions were obtained in 1992 by Colomo, Izergin, Korepin, and Tognetti. The problem addressed in this dissertation consists of the asymptotic analysis of the XXO correlation functions based on the determinant representations mentioned. The integral kernel involved in the Colomo-Izergin-Korepin-Tognetti determinant representations belongs to the class of “integrable kernels” which usually appear in the theory of exactly solvable statistical mechanics models, and also in the theory of random matrices. This makes it possible to analyze the large time behavior of the correlation functions in the framework of the Riemann-Hilbert approach. In the dissertation, the associated matrix Riemann-Hilbert problem is solved asymptotically via properly modified Deift-Zhou nonlinear steepest descent method. This in turn allows us to determine the structure of the long-time asymptotic series for the correlation functions and to evaluate explicitly the exponential, power-like pre-exponential, and the oscillatory factors of the asymptotics. A recursive procedure for evaluation of the corrections of any order, which is based on the relevant zero-curvature equation, is suggested as well.

•Jan 1, 1998•0 citations

Exact Solution of Riemann–Hilbert Problem for a Correlation Function of the XY Spin Chain

Yasuhiro Fujii

A correlation function of the XY spin chain is studied at zero temperature. This is called the Emptiness Formation Probability (EFP) and is expressed by the Fredholm determinant in the thermodynamic limit. We formulate the associated Riemann--Hilbert problem and solve it exactly. The EFP is shown to decay in Gaussian.

arXiv:cond-mat/0009458•Journal of the Physical Society of Japan•Oct 15, 2000•1 citation
We derive the large distance asymptotics of the Fredholm determinant of the so-called generalised sine kernel at the critical point. This kernel corresponds to a generalisation of the pure sine kernel arising in the theory of random matrices and has potential applications to the analysis of the large-distance asymptotic behaviour of the so-called emptiness formation probability for various quantum integrable models away from their free fermion point.
We derive the leading asymptotic behavior and build a new series representation for the Fredholm determinant of integrable integral operators appearing in the representation of the time and distance-dependent correlation functions of integrable models described by a six-vertex R-matrix.This series representation opens a systematic way for the computation of the long-time, long-distance asymptotic expansion for the correlation functions of the aforementioned integrable models away from their free fermion point.Our method builds on a Riemann-Hilbert based analysis.
We perform a large-xxx asymptotic analysis of the Fredholm determinant of an integrable integral operator with generalized sine kernel, where xxx controls the strength of the oscillations along the integration contour of the operator and will play the role of the distance variable in applications to the correlation functions of integrable quantum systems. Our generalized sine kernel involves a number of functional parameters that will allow us to adapt it to the analysis of the dynamical correlation functions of the Lieb--Liniger model at finite temperature and for all positive values of the coupling constant. It will also allow us to consider a class of equilibrium correlation functions that are governed by generalized Gibbs ensembles. Our work is based on the analysis of a matrix Riemann--Hilbert problem that is canonically connected with our integrable integral operator.
The finite temperature correlation functions of the time-dependent local spin operators of the Heisenberg XXO ferromagnet are considered. The Fredholm determinant representation for these correlation functions were obtained in 1992 by Colomo, Izergin, Korepin, and Tognetti. The problem addressed in this dissertation consists of the asymptotic analysis of the XXO correlation functions based on the determinant representations mentioned. The integral kernel involved in the Colomo-Izergin-Korepin-Tognetti determinant representations belongs to the class of “integrable kernels” which usually appear in the theory of exactly solvable statistical mechanics models, and also in the theory of random matrices. This makes it possible to analyze the large time behavior of the correlation functions in the framework of the Riemann-Hilbert approach. In the dissertation, the associated matrix Riemann-Hilbert problem is solved asymptotically via properly modified Deift-Zhou nonlinear steepest descent method. This in turn allows us to determine the structure of the long-time asymptotic series for the correlation functions and to evaluate explicitly the exponential, power-like pre-exponential, and the oscillatory factors of the asymptotics. A recursive procedure for evaluation of the corrections of any order, which is based on the relevant zero-curvature equation, is suggested as well.