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arXiv:hep-th/0611142 · Journal of Statistical Mechanics Theory and Experiment

On correlation functions of integrable models associated with the six-vertexR-matrix

N Kitanine, Karol Kajetan Kozlowski, Jean Michel Maillet, N. A. Slavnov, V. Terras

Published Jan 30, 2007• 59 citations
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Abstract

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.

DOI
10.1088/1742-5468/2007/01/p01022

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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.
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 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.
We consider the interaction-round-a-face version of the isotropic six-vertex model. The associated spin chain is made of two coupled Heisenberg spin chains with different boundary twists. The phase diagram of the model and the long distance correlations were studied in Tavares and Ribeiro (2023) [43] . Here, we compute the short-distance correlation functions of the model in the ground state for finite system sizes via non-linear integral equations and in the thermodynamic limit. This was possible since the model satisfies the face version of the discrete quantum Knizhnik-Zamolodchikov (qKZ) equation. A suitable ansatz for the density matrix is proposed in the form of a direct sum of two Heisenberg density matrices, which allows us to obtain the discrete functional equation for the two-site function ω ( λ 1 , λ 2 ) . Thanks to the known results on the factorization of correlation functions of the Heisenberg chain, we are able to compute the density matrix of the IRF model for up to four sites and its associated spin chain for up to three sites.
A general approach to the computation of exact correlation functions of lattice quantum integrable models proposed recently is reviewed. It is based on the resolution of the quantum inverse scattering problem. The application of this method to theXXZ Heisenberg spin- 12\tfrac{1}{2}21​ in a magnetic field leads to multiple integral representations of its n-point correlation functions. For zero magnetic field, it gives a proof, in both the massless and massive (anti-ferromagnetic) regimes, of the formulas obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra Uq(s^l2){\mathcal{U}_q}(\widehat s{l_2})Uq​(sl2​) together with the corner transfer matrix approach (massive regime). We also discuss further problems, related in particular to long distance asymptotics of these correlation functions.
We introduce and solve a special family of integrable interacting vertex models that generalizes the well known six-vertex model. In addition to the usual nearest neighbour interactions among the vertices, there exist extra hard-core interactions among pairs of vertices at larger distances. The associated row-to-row transfer matrices are diagonalized by using the recently introduced matrix product ansatz. Like for the relation of the six-vertex model with the XXZ quantum chain, the row-to-row transfer matrices of these new models are also the generating functions of an infinite set of commuting conserved charges. Among these charges we identify the integrable generalization of the XXZ chain that contains hard-core exclusion interactions among the spins. These quantum chains have already appeared in the literature. The present paper explains their integrability.