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arXiv:1003.1288 · New Trends in Quantum Integrable Systems

PROPERTIES OF LINEAR INTEGRAL EQUATIONS RELATED TO THE SIX-VERTEX MODEL WITH DISORDER PARAMETER

Hermann Boos, Frank Göhmann

Published Oct 1, 2010• 3 citations
arXivHTMLPDF

Abstract

One of the key steps in recent work on the correlation functions of the XXZ chain was to regularize the underlying six-vertex model by a disorder parameter $α$. For the regularized model it was shown that all static correlation functions are polynomials in only two functions. It was further shown that these two functions can be written as contour integrals involving the solutions of a certain type of linear and non-linear integral equations. The linear integral equations depend parametrically on $α$ and generalize linear integral equations known from the study of the bulk thermodynamic properties of the model. In this note we consider the generalized dressed charge and a generalized magnetization density. We express the generalized dressed charge as a linear combination of two quotients of $Q$-functions, the solutions of Baxter's $t$-$Q$-equation. With this result we give a new proof of a lemma on the asymptotics of the generalized magnetization density as a function of the spectral parameter.

DOI
10.1142/9789814324373_0001

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One of the key steps in recent work on the correlation functions of the XXZ chain was to regularize the underlying six-vertex model by a disorder parameter ααα. For the regularized model it was shown that all static correlation functions are polynomials in only two functions. It was further shown that these two functions can be written as contour integrals involving the solutions of a certain type of linear and non-linear integral equations. The linear integral equations depend parametrically on ααα and generalize linear integral equations known from the study of the bulk thermodynamic properties of the model. In this note we consider the generalized dressed charge and a generalized magnetization density. We express the generalized dressed charge as a linear combination of two quotients of QQQ-functions, the solutions of Baxter's ttt-QQQ-equation. With this result we give a new proof of a lemma on the asymptotics of the generalized magnetization density as a function of the spectral parameter.
We study certain functions arising in the context of the calculation of correlation functions of the XXZ spin chain and of integrable field theories related with various scaling limits of the underlying six-vertex model. We show that several of these functions that are related to linear integral equations can be obtained by acting with (deformed) difference operators on a master function ΦΦΦ. The latter is defined in terms of a functional equation and of its asymptotic behavior. Concentrating on the so-called temperature case we show that these conditions uniquely determine the high-temperature series expansions of the master function. This provides an efficient calculation scheme for the high-temperature expansions of the derived functions as well.
It was recently shown by Jimbo et al (2008 arXiv:0811.0439) that the correlation functions of a generalized XXZ chain associated with an inhomogeneous six-vertex model with a disorder parameter α and with arbitrary inhomogeneities on the horizontal lines factorize and can all be expressed in terms of only two functions ρ and ω. Here we approach the description of the same correlation functions and, in particular, of the function ω from a different direction. We start from a novel multiple integral representation for the density matrix of a finite chain segment of length m in the presence of a disorder field α. We explicitly factorize the integrals for m = 2. Based on this, we present an alternative description of the function ω in terms of the solutions of certain linear and nonlinear integral equations. We then prove directly that the two definitions of ω describe the same function. The definition in the work of Jimbo et al (2008 arXiv:0811.0439) was crucial for the proof of the factorization. The definition given here together with the known description of ρ in terms of the solutions of nonlinear integral equations is useful for performing, e.g., the Trotter limit in the finite temperature case, or for obtaining numerical results for the correlation functions at short distances. We also address the issue of the construction of an exponential form of the density matrix for finite α.
We derive an analogue of the master equation, obtained recently for correlation functions of the XXZ chain, for a wide class of quantum integrable systems described by the R -matrix of the six-vertex model, including in particular continuum models. This generalized master equation allows us to obtain multiple integral representations for the correlation functions of these models. We apply this method to derive the density–density correlation functions of the quantum non-linear Schrödinger model.
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 study exact solutions of the four-point correlation functions around a vertex of the anti-ferroelectric six-vertex model. After showing that an expression for the correlations is readily obtained by differentiating the known exact solution of the free energy, we present a new expression for the correlations derived through the recently developed vertex operator approach. It has the form of a double contour integral of a meromorphic function. To give a new verification of the vertex operator approach beyond the case of the one-point function, we prove the equality of the two expressions by calculating the contour integrals.