A recursion formula for the correlation functions of an inhomogeneous XXX model

Abstract

Abstract.A new recursion formula is presented for the correlation functions of the integrable spin 1/2 XXX chain with inhomogeneity. It links the correlators involving n consecutive lattice sites to those with n − 1 and n − 2 sites. In a series of papers by V. Korepin and two of the present authors, it was discovered that the correlators have a certain specific structure as functions of the inhomogeneity parameters. The formula mentioned above makes it possible to prove this structure directly, as well as to obtain an exact description of the rational functions that were left undetermined in the earlier work. §1. Introduction Consider the XXX antiferromagnet given by the HamiltonianThis model was solved in the famous paper by Bethe [3] already in 1931, by using what is now called the coordinate Bethe Ansatz. Nevertheless, it took some time before the physical content of this model in the thermodynamic limit was clarified completely. For the first time, the spectrum of excitations was described correctly in the paper [10] by Faddeev and Takhtajan; it was shown that the spectrum contains magnons of spin 1/2. These authors used the algebraic Bethe Ansatz formulated by Faddeev, Sklyanin, and Takhtajan (see [9]) on the basis of R-matrices and the Yang-Baxter equation. The origin of these new techniques goes back to the work of Baxter [1]. Restricting ourselves to our example, we recall the role of R-matrices in solvable models. The XXX model is related to the rational R-matrix that acts on C 2 ⊗ C 2 (see (2.7) for the explicit formula). We employ the usual notation R 1,2 (λ), where 1, 2 label the corresponding spaces and λ is the spectral parameter. The relationship between the R-matrix and the XXX Hamiltonian is as follows. Consider the transfer matrix

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“…As far as two-point correlation functions of the spin-1/2 chain are concerned, enormous progress has been made in the last decade to obtain exact expressions from Bethe Ansatz [11,12,13] for short distances [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] and from field-theory for both the amplitudes and the exponents of the leading terms in the asymptotic limit [29,30,31,32]. These results are not restricted to the isotropic point, but cover the critical anisotropic regime as well,…”

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“…It is desirable that this algebraic method will be generalized to the case with ∆ = 1. Actually, Boos, Jimbo, Miwa, Smirnov and Takeyama have derived an exponential formula for the density matrix elements of XXZ model, which does not contain multiple integrals [27,28,29,30,31]. It, however, seems still hard to evaluate the formula for general density matrix elements.…”

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“…1. The finding of [4] is striking, a two-point nearest neighbor function !ðx 1 ; x 2 Þ : ¼ 6 tr D 2 ðx 1 ; x 2 ÞS z 1 S z 2 and a set of ''structure constants'' f n;I;J determine the density matrix D n for arbitrary n…”

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