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arXiv:0707.1995 · Journal of Statistical Mechanics Theory and Experiment

Correlation functions of the openXXZchain: I

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

Published Oct 19, 2007• 106 citations
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Abstract

We consider the XXZ spin chain with diagonal boundary conditions in the framework of the algebraic Bethe ansatz. Using the explicit computation of the scalar products of Bethe states and a revisited version of the bulk quantum inverse problem, we calculate the elementary building blocks for the correlation functions. In the limit of a half-infinite chain, they are obtained as multiple integrals of usual functions, similar to the case of periodic boundary conditions.

DOI
10.1088/1742-5468/2007/10/p10009

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We consider the XXZ spin chain with diagonal boundary conditions in the framework of the algebraic Bethe ansatz. Using the explicit computation of the scalar products of Bethe states and a revisited version of the bulk quantum inverse problem, we calculate the elementary building blocks for the correlation functions. In the limit of a half-infinite chain, they are obtained as multiple integrals of usual functions, similar to the case of periodic boundary conditions.
In this paper, we consider the quantum XYZ open spin-1/2 chain with boundary fields. We focus on the particular case in which the six boundary parameters are related by a single constraint enabling us to describe part of the spectrum by standard Bethe equations. We derive for this model exact representations for a set of elementary blocks of correlation functions, hence generalising to XYZ the results obtained in the XXZ open case in [1]. Our approach is also similar to the approach proposed in the XXZ case [1] : we solve the model by Sklyanin’s version of the quantum Separation of Variables, using Baxter’s Vertex-IRF transformation; in this framework, we identify a basis of local operators with a relatively simple action on the transfer matrix eigenstates; we then use the solution of the quantum inverse problem and our recent formulae on scalar products of separate states [2] to compute some of the corresponding matrix elements, which can therefore be considered as elementary building blocks for the correlation functions. The latter are expressed in terms of multiple sums in the finite chain, and as multiple integrals in the thermodynamic limit. Our results evidence that, once the basis of local operators is properly chosen, the corresponding building blocks for correlation functions have a similar structure in the XXX/XXZ/XYZ open chains, and do not require any insertion of non-local ”tail operators”.
The scalar products, form factors, and correlation functions of the XXZ spin chain with a twisted (or antiperiodic) boundary condition are obtained based on the inhomogeneous T–Q relation and the Bethe states constructed via the off-diagonal Bethe ansatz. It is shown that the scalar product of two off-shell Bethe states, the form factors, and the two-point correlation functions can be expressed as the summation of certain determinants. The corresponding homogeneous limits are studied. The results are also checked by the numerical calculations.
We derive compact multiple integral formulae for several physical spin correlation functions in the semi-infinite XXZ chain with a longitudinal boundary magnetic field. Our formulae follow from several effective resummations of the multiple integral representation for the elementary blocks obtained in our previous paper (I). In the free fermion point we compute the local magnetization as well as the density of energy profiles. These quantities, in addition to their bulk behavior, exhibit Friedel-type oscillations induced by the boundary; their amplitudes depend on the boundary magnetic field and decay algebraically in terms of the distance to the boundary.
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.
Using the algebraic Bethe ansatz, we consider the correlation functions of the integrable higher spin chains. We apply a method recently developed for the spin ½ Heisenberg chain, based on the solution of the quantum inverse problem. We construct a representation for the correlation functions on a finite chain for arbitrary spin. Then we show how the string solutions of the Bethe equations can be considered in the framework of this approach in the thermodynamic limit. Finally, a multiple integral representation for the spin 1 zero-temperature correlation functions is obtained in the thermodynamic limit.