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XXX spin chain: From Bethe solution to open problems

V. E. Korepin, Ovidiu I. Pâţu

Published Jan 1, 2007• 1 citation
OpenAlex

Abstract

We present some open problems in the field of exactly solvable models. Two of the problems are related to the correlation functions of the XXX spin chain and the XXZ spin chain, one to the entropy of subsystems and one to the six vertex model with domain wall boundary conditions.

Related papers

XXX Spin Chain: from Bethe Solution to Open Problems

V. E. Korepin, Ovidiu I. Pâţu

We present some open problems in the field of exactly solvable models. Two of the problems are related to the correlation functions of the XXX spin chain and the XXZ spin chain, one to the entropy of subsystems and one to the six vertex model with domain wall boundary conditions.

arXiv:cond-mat/0701491•Jan 20, 2007•5 citations

Integrable quantum spin chains

Anastasia Doikou

Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...

•Jan 1, 1999•0 citations

On the spin–spin correlation functions of the XXZ spin- infinite chain

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

We obtain a new multiple integral representation for the spin–spin correlation functions of the XXZ spin- infinite chain. We show that this representation is closely related with the partition function of the six-vertex model with domain wall boundary conditions.

arXiv:hep-th/0407223•Journal of Physics A Mathematical and General•Aug 10, 2005•25 citations

Correlation Function and Simplified TBA Equations for XXZ Chain

Minoru Takahashi

The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem.Among these solvable models spin 1/2 XXX chain has been investigated for a long time.Even for this model only the nearest neighbor and the second neighbor correlations were known.In 1990's multiple integral formula for the general correlations is derived.But the integration of this formula is also very difficult problem.Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions are expressed by log 2 and Riemann's zeta functions with odd integer argument ζ(3), ζ(5), ζ(7), . . . .We can calculate density sub-matrix of successive seven sites.Entanglement entropy of seven sites is calculated.These methods can be extended to XXZ chain up to n = 4. Correlation functions are expressed by the generalized zeta functions.Several years ago I derived new thermodynamic Bethe ansatz equation for XXZ chain.This is quite different with Yang-Yang type TBA equations and contains only one unknown function.This equation is very useful to get the high temperature expansion.In this paper we get the analytic solution of this equation at ∆ = 0.

arXiv:1101.0035•Symmetry Integrability and Geometry Methods and Applications•Jan 8, 2011•2 citations

On solutions of Bethe equations for the XXZ model

V. Tarasov

It is shown that solutions of the Bethe ansatz equations for the inhomogeneous arbitrary spin XXX or XXZ model satisfy certain identites which generalize those, recently obtained by K.Fabricius and B.M.McCoy, for solutions of the Bethe ansatz equations for the six-vertex model. Even in the last case, the given proof is simpler than the original one.

arXiv:math-ph/0306002•Jun 2, 2003•0 citations

Bethe Ansatz and boundary energy of the open spin-1/2 XXZ chain

Rajan Murgan

We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{$\\alpha_-$,$\\alpha_+$,$\\beta_-$,$\\beta_+$}} are nonzero. A generalization of the Baxter $T-Q$ equation that involves more than one independent $Q$ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.

•Jan 1, 2006•0 citations
Integrable quantum spin chains are exactly solvable quantum mechanical models of N quantum spins, of which the Heisenberg model solved by Bethe is the prototype. In the antiferromagnetic regime, such spin chains can be regarded as lattice versions of corresponding integrable relativistic quantum field theories. For integrable spin chains, quantities of physical interest (spectrum, S matrix, etc.) can be calculated exactly by direct means, starting from the microscopic Hamiltonian, while for the corresponding field theories, such exact information has been primarily obtained by indirect means, such as the “ bootstrap ” approach and semiclassical approximations. We have been focused basically on the Heisenberg XXX and XXZ chains and their generalizations. Basic aims of this work were the study of the symmetries of these models and the computation, by means of Bethe ansatz equations, of the bulk and boundary S matrices for the closed and open chains respectively...
The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem.Among these solvable models spin 1/2 XXX chain has been investigated for a long time.Even for this model only the nearest neighbor and the second neighbor correlations were known.In 1990's multiple integral formula for the general correlations is derived.But the integration of this formula is also very difficult problem.Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions are expressed by log 2 and Riemann's zeta functions with odd integer argument ζ(3), ζ(5), ζ(7), . . . .We can calculate density sub-matrix of successive seven sites.Entanglement entropy of seven sites is calculated.These methods can be extended to XXZ chain up to n = 4. Correlation functions are expressed by the generalized zeta functions.Several years ago I derived new thermodynamic Bethe ansatz equation for XXZ chain.This is quite different with Yang-Yang type TBA equations and contains only one unknown function.This equation is very useful to get the high temperature expansion.In this paper we get the analytic solution of this equation at ∆ = 0.
We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.