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/0402625 · Journal of Physics A Mathematical and General

Third-neighbour and other four-point correlation functions of spin-1/2XXZchain

Go Kato, Masahiro Shiroishi, Minoru Takahashi, Kazumitsu Sakai

Published Apr 28, 2004• 42 citations
arXivHTMLPDF

Abstract

The correlation functions of the spin-1/2 XXZ chain in the ground state were expressed in the form of multiple integrals for −1 < Δ ⩽ 1 and 1 < Δ. In particular, adjacent four-point correlation functions were given as certain four-dimensional integrals. We show that these integrals can be reduced to polynomials with respect to specific one-dimensional integrals. The results give the polynomial representation of the third-neighbour correlation functions.

DOI
10.1088/0305-4470/37/19/001

Related papers

Next-nearest-neighbour correlation functions of the spin-1/2XXZchain at the critical region

Go Kato, Masahiro Shiroishi, Minoru Takahashi +1 more

The correlation functions of the spin-1/2 XXZ spin chain in the ground state are expressed in the form of the multiple integrals. For -1< Delta <1, they were obtained by Jimbo and Miwa in 1996. Especially the next nearest-neighbour correlation functions are given as certain three-dimensional integrals. We shall show these integrals can be reduced to one-dimensional ones and thereby evaluate the values of the next nearest-neighbor correlation functions. We have also found that the remaining one-dimensinal integrals can be evaluated analytically, when nu = arccos(Delta)/pi is a rational number.

arXiv:cond-mat/0304475•Journal of Physics A Mathematical and General•May 28, 2003•45 citations

Next Nearest-Neighbor Correlation Functions of the Spin-1/2XXZChain at Massive Region

Minoru Takahashi, Go Kato, Masahiro Shiroishi

The second neighbor correlation functions of the spin-${1/2}$ $XXZ$ chain in the ground state are expressed in the form of three dimensional integrals. We show that these integrals can be reduced to one-dimensional ones and thereby evaluate the values of the next nearest-neighbor correlation functions for ${Δ>1}$.

arXiv:cond-mat/0308589•Journal of the Physical Society of Japan•Jan 15, 2004•31 citations

Fourth-neighbour two-point functions of the XXZ chain and the fermionic basis approach

Frank Göhmann, Raphael Kleinemühl, Alexander Weiße

Abstract We give a descriptive review of the fermionic basis approach to the theory of correlation functions of the XXZ quantum spin chain. The emphasis is on explicit formulae for short-range correlation functions which will be presented in a way that allows for their direct implementation on a computer. Within the fermionic basis approach a huge class of stationary reduced density matrices, compatible with the integrable structure of the model, assumes a factorized form. This means that all expectation values of local operators and all two-point functions, in particular, can be represented as multivariate polynomials in only two functions ρ and ω and their derivatives with coefficients that are rational in the deformation parameter q of the model. These coefficients are of ‘algebraic origin’. They do not depend on the choice of the density matrix, which only impacts the form of ρ and ω . As an example we work out in detail the case of the grand canonical ensemble at temperature T and magnetic field h for q in the critical regime. We compare our exact results for the fourth-neighbour two-point functions with asymptotic formulae for h , T = 0 and for finite h and T .

arXiv:2104.10071•Journal of Physics A Mathematical and Theoretical•Aug 23, 2021•4 citations

Exact Calculation of Correlation Functions for Spin-1/2 Heisenberg Chain

Masahiro Shiroishi, Minoru Takahashi

We review the recent progresses of the exact calculation of the correlation functions for the spin-1/2 Heisenberg XXX chain as well as the anisotropic XXZ chain. The spin-1/2 Heisenberg chain is exactly solvable by Bethe ansatz. Nevertheless the exact calculation of the correlation functions is still a very difficult problem in general. However, the recent theoretical developments have enabled us to derive the explicit analytical forms of some correlation functions in short range. For example, polynomial representations in terms of the one-dimensional integrals are obtained for the correlation functions of the XXZ chain up to four lattice sites. For the special case of the XXX chain, the correlation functions are further calculated up to five lattice sites.

Journal of the Physical Society of Japan•Jan 1, 2005•17 citations

Exact results for the σztwo-point function of theXXZchain at Δ = 1/2

N Kitanine, Jean Michel Maillet, N. A. Slavnov +1 more

We propose a new multiple-integral representation for the correlation function of the XXZ spin- Heisenberg chain in the disordered regime. We show that for Δ = 1/2 the integrals can be separated and computed exactly. As an example we give the explicit results up to the lattice distance m = 8. It turns out that the answer is given as integer numbers divided by 2 ( m +1) 2 .

arXiv:hep-th/0506114•Journal of Statistical Mechanics Theory and Experiment•Sep 6, 2005•14 citations

On the algebraic Bethe Ansatz approach to the correlation functions of the XXZ spin-1/2 Heisenberg chain

N Kitanine, Jean Michel Maillet, Slavnov, N. A. +1 more

We present a review of the method we have elaborated to compute the correlation functions of the XXZ spin-1/2 Heisenberg chain. This method is based on the resolution of the quantum inverse scattering problem in the algebraic Bethe Ansatz framework, and leads to a multiple integral representation of the dynamical correlation functions. We describe in particular some recent advances concerning the two-point functions: in the finite chain, they can be expressed in terms of a single multiple integral. Such a formula provides a direct analytic connection between the previously obtained multiple integral representations and the form factor expansions for the correlation functions.

arXiv:hep-th/0505006•Apr 30, 2005•26 citations
The correlation functions of the spin-1/2 XXZ chain in the ground state were expressed in the form of multiple integrals for −1 < Δ ⩽ 1 and 1 < Δ. In particular, adjacent four-point correlation functions were given as certain four-dimensional integrals. We show that these integrals can be reduced to polynomials with respect to specific one-dimensional integrals. The results give the polynomial representation of the third-neighbour correlation functions.
The correlation functions of the spin-1/2 XXZ spin chain in the ground state are expressed in the form of the multiple integrals. For -1< Delta <1, they were obtained by Jimbo and Miwa in 1996. Especially the next nearest-neighbour correlation functions are given as certain three-dimensional integrals. We shall show these integrals can be reduced to one-dimensional ones and thereby evaluate the values of the next nearest-neighbor correlation functions. We have also found that the remaining one-dimensinal integrals can be evaluated analytically, when nu = arccos(Delta)/pi is a rational number.
The second neighbor correlation functions of the spin-1/2{1/2}1/2 XXZXXZXXZ chain in the ground state are expressed in the form of three dimensional integrals. We show that these integrals can be reduced to one-dimensional ones and thereby evaluate the values of the next nearest-neighbor correlation functions for Δ>1{Δ>1}Δ>1.
Abstract We give a descriptive review of the fermionic basis approach to the theory of correlation functions of the XXZ quantum spin chain. The emphasis is on explicit formulae for short-range correlation functions which will be presented in a way that allows for their direct implementation on a computer. Within the fermionic basis approach a huge class of stationary reduced density matrices, compatible with the integrable structure of the model, assumes a factorized form. This means that all expectation values of local operators and all two-point functions, in particular, can be represented as multivariate polynomials in only two functions ρ and ω and their derivatives with coefficients that are rational in the deformation parameter q of the model. These coefficients are of ‘algebraic origin’. They do not depend on the choice of the density matrix, which only impacts the form of ρ and ω . As an example we work out in detail the case of the grand canonical ensemble at temperature T and magnetic field h for q in the critical regime. We compare our exact results for the fourth-neighbour two-point functions with asymptotic formulae for h , T = 0 and for finite h and T .
We review the recent progresses of the exact calculation of the correlation functions for the spin-1/2 Heisenberg XXX chain as well as the anisotropic XXZ chain. The spin-1/2 Heisenberg chain is exactly solvable by Bethe ansatz. Nevertheless the exact calculation of the correlation functions is still a very difficult problem in general. However, the recent theoretical developments have enabled us to derive the explicit analytical forms of some correlation functions in short range. For example, polynomial representations in terms of the one-dimensional integrals are obtained for the correlation functions of the XXZ chain up to four lattice sites. For the special case of the XXX chain, the correlation functions are further calculated up to five lattice sites.
We present a review of the method we have elaborated to compute the correlation functions of the XXZ spin-1/2 Heisenberg chain. This method is based on the resolution of the quantum inverse scattering problem in the algebraic Bethe Ansatz framework, and leads to a multiple integral representation of the dynamical correlation functions. We describe in particular some recent advances concerning the two-point functions: in the finite chain, they can be expressed in terms of a single multiple integral. Such a formula provides a direct analytic connection between the previously obtained multiple integral representations and the form factor expansions for the correlation functions.