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:1105.4722 · Journal of Statistical Mechanics Theory and Experiment

Reduction formula of form factors for the integrable spin-sXXZ chains and application to correlation functions

Tetsuo Deguchi

Published Apr 2, 2012• 3 citations
arXivHTMLPDF

Abstract

For the integrable spin- s XXZ chain we express explicitly any given spin- s form factor in terms of a sum over the scalar products of the spin-1/2 fundamental operators of the algebraic Bethe ansatz. Here, they are given by the operator-valued matrix elements of the monodromy matrix of the spin-1/2 XXZ spin chain. We derive the reduction formula of higher-spin form factors by the fusion method in detail. We call an arbitrary matrix element of a local operator between two Bethe eigenstates a form factor of the operator in this paper. We thus revise the derivation of the higher-spin XXZ form factors given in a previous paper. The revised method has several interesting applications in mathematical physics. For instance, we express the spin- s XXZ correlation function of an arbitrary entry at zero temperature in terms of a sum of multiple integrals.

DOI
10.1088/1742-5468/2012/04/p04001

Related papers

On the evaluation of form factors and correlation functions for the integrable spin-s XXZ chains via the fusion method

Tetsuo Deguchi, Chihiro Matsui

Revising the derivation of the previous papers, for the integrable spin-$s$ XXZ chain we express any form factor in terms of a single sum over scalar products of the spin-1/2 XXZ chain. With the revised method we express the spin-$s$ XXZ correlation function of any given entry at zero temperature in terms of a single sum of multiple integrals.

arXiv:1103.4206•Mar 22, 2011•3 citations

Correlation functions of the integrable higher-spin XXX and XXZ spin chains through the fusion method

Tetsuo Deguchi, Chihiro Matsui

For the integrable higher-spin XXX and XXZ spin chains we present multiple-integral representations for the correlation function of an arbitrary product of Hermitian elementary matrices in the massless ground state. We give a formula expressing it by a single term of multiple integrals. In particular, we explicitly derive the emptiness formation probability (EFP). We assume 2s-strings for the ground-state solution of the Bethe-ansatz equations for the spin-s XXZ chain, and solve the integral equations for the spin-s Gaudin matrix. In terms of the XXZ coupling Δ we define ζ by Δ=cosζ, and put it in a region 0⩽ζ<π/2s of the gapless regime: −1<Δ⩽1 (0⩽ζ<π), where Δ=1 (ζ=0) corresponds to the antiferromagnetic point. We calculate the zero-temperature correlation functions by the algebraic Bethe-ansatz, introducing the Hermitian elementary matrices in the massless regime, and taking advantage of the fusion construction of the R-matrix of the higher-spin representations of the affine quantum group.

arXiv:0907.0582•Nuclear Physics B•Jan 5, 2010•21 citations

Construction of Monodromy Matrix in the F- basis and scalar products in Spin chains.

А. А. Овчинников

We present in a simple terms the theory of the factorizing operator introduced recently by Maillet and Sanches de Santos for the spin- 1/2 chains. We obtain the explicit expressions for the matrix elements of the factorizing operator in terms of the elements of the Monodromy matrix. We use this results to derive the expression for the general scalar product for the quantum spin chain. We comment on the previous determination of the scalar product of Bethe eigenstate with an arbitrary dual state. We also establish the direct correspondence between the calculations of scalar products in the F- basis and the usual basis. 1. Introduction. One of the most important open problems in the theory of quantum integrable models is the calculation of the correlation functions. In the framework of the Algebraic Bethe Ansatz method [1] the problem is the combinatorial complexity of calculations due to the structure of Bethe eigenstates. Only the very limited number of physical results for the correlation

•Jan 1, 2000•3 citations

On the algebraic Bethe Ansatz approach to the correlation functions of the XXZ spin-1/2 Heisenberg chain(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

N Kitanine, Jean Michel Maillet, N. A. Slavnov +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 bamework, and leads to a mul- tiple 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.

数理解析研究所講究録•Apr 1, 2006•5 citations

ALGEBRAIC ASPECTS OF THE CORRELATION FUNCTIONS OF THE INTEGRABLE HIGHER-SPIN XXZ SPIN CHAINS WITH ARBITRARY ENTRIES

TETSUO DEGUCHI, Chihiro Matsui

We discuss some fundamental properties of the XXZ spin chain, which are important in the algebraic Bethe-ansatz derivation for the multiple-integral representations of the spin-s XXZ correlation function with an arbitrary product of elementary matrices. For instance, we construct Hermitian conjugate vectors in the massless regime and introduce the spin-s Hermitian elementary matrices.

arXiv:1005.0888•New Trends in Quantum Integrable Systems•Oct 1, 2010•3 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
For the integrable spin- s XXZ chain we express explicitly any given spin- s form factor in terms of a sum over the scalar products of the spin-1/2 fundamental operators of the algebraic Bethe ansatz. Here, they are given by the operator-valued matrix elements of the monodromy matrix of the spin-1/2 XXZ spin chain. We derive the reduction formula of higher-spin form factors by the fusion method in detail. We call an arbitrary matrix element of a local operator between two Bethe eigenstates a form factor of the operator in this paper. We thus revise the derivation of the higher-spin XXZ form factors given in a previous paper. The revised method has several interesting applications in mathematical physics. For instance, we express the spin- s XXZ correlation function of an arbitrary entry at zero temperature in terms of a sum of multiple integrals.
Revising the derivation of the previous papers, for the integrable spin-sss XXZ chain we express any form factor in terms of a single sum over scalar products of the spin-1/2 XXZ chain. With the revised method we express the spin-sss XXZ correlation function of any given entry at zero temperature in terms of a single sum of multiple integrals.
For the integrable higher-spin XXX and XXZ spin chains we present multiple-integral representations for the correlation function of an arbitrary product of Hermitian elementary matrices in the massless ground state. We give a formula expressing it by a single term of multiple integrals. In particular, we explicitly derive the emptiness formation probability (EFP). We assume 2s-strings for the ground-state solution of the Bethe-ansatz equations for the spin-s XXZ chain, and solve the integral equations for the spin-s Gaudin matrix. In terms of the XXZ coupling Δ we define ζ by Δ=cosζ, and put it in a region 0⩽ζ<π/2s of the gapless regime: −1<Δ⩽1 (0⩽ζ<π), where Δ=1 (ζ=0) corresponds to the antiferromagnetic point. We calculate the zero-temperature correlation functions by the algebraic Bethe-ansatz, introducing the Hermitian elementary matrices in the massless regime, and taking advantage of the fusion construction of the R-matrix of the higher-spin representations of the affine quantum group.
We present in a simple terms the theory of the factorizing operator introduced recently by Maillet and Sanches de Santos for the spin- 1/2 chains. We obtain the explicit expressions for the matrix elements of the factorizing operator in terms of the elements of the Monodromy matrix. We use this results to derive the expression for the general scalar product for the quantum spin chain. We comment on the previous determination of the scalar product of Bethe eigenstate with an arbitrary dual state. We also establish the direct correspondence between the calculations of scalar products in the F- basis and the usual basis. 1. Introduction. One of the most important open problems in the theory of quantum integrable models is the calculation of the correlation functions. In the framework of the Algebraic Bethe Ansatz method [1] the problem is the combinatorial complexity of calculations due to the structure of Bethe eigenstates. Only the very limited number of physical results for the correlation
We present a review of the method we have elaborated to compute the correlation functions of the XXZXXZXXZ spin-1/2 Heisenberg chain.This method is based on the resolution of the quantum inverse scattering problem in the algebraic Bethe Ansatz bamework, and leads to a mul- tiple 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.
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.