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arXiv:cond-mat/0002219 · Journal of Physics A Mathematical and General

Asymptotic form of the two-point correlation function of theXXZspin chain

Yasuhiro Fujii, Miki Wadati

Published Sep 6, 2000• 0 citations
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Abstract

Correlation functions of the XXZ spin chain in the critical regime are studied at zero temperature. They are exactly represented in the Fredholm determinant form and are related to an operator-valued Riemann-Hilbert problem. Analysing this problem we prove that a two-point correlation function is asymptotically expressed in terms of power functions of the distance between spin operators.

DOI
10.1088/0305-4470/33/37/303

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Asymptotic form of the two-point correlation function of theXXZspin chain

Y Fujii and M Wadati

In equation (3.23), we introduced the function f ( m , α) satisfying the condition f ( m + 1, α) / f ( m , α) = 1 + O( m -1 ). For the calculation of correlation functions, one needs the derivative of this function with respect to α. There is no guarantee that the derivative f '( m , α) = ∂ f ( m , α)/∂α satisfies f '( m + 1, α) / f '( m , α) = 1 + O( m -1 ). Therefore, equation (3.24) is incorrect. This error is crucial because a main result of this paper is incorrect. However, the asymptotic analysis of the Riemann-Hilbert problem is new and correct.

Journal of Physics A Mathematical and General•Mar 19, 2001•0 citations
Correlation functions of the XXZ spin chain in the critical regime are studied at zero temperature. They are exactly represented in the Fredholm determinant form and are related to an operator-valued Riemann-Hilbert problem. Analysing this problem we prove that a two-point correlation function is asymptotically expressed in terms of power functions of the distance between spin operators.
We analyze the transverse dynamical two-point correlation function of the XX chain by means of a thermal form factor series. The series is rewritten in terms of the resolvent and the Fredholm determinant of an integrable integral operator. This connects it with a matrix Riemann-Hilbert problem. We express the correlation function in terms of the solution of the matrix Riemann-Hilbert problem. The matrix Riemann-Hilbert problem is then solved asymptotically in the high-temperature limit. This allows us to obtain the leading high-temperature contribution to the two-point correlation function at any fixed space-time separation.
We study zero temperature correlation functions of the spin-121\over 221​ Heisenberg XXZ model in the critical regime −1<Δ≤1-1< Δ\leq 1−1<Δ≤1 in a magnetic field by means of the {\tenit Dual Field Approach}. We show for one particular example how to derive determinant representations for correlation functions and how to use these to embed the correlation functions in integrable systems of integro-difference equations (IDE). These IDE are associated with a Riemann-Hilbert problem.
In equation (3.23), we introduced the function f ( m , α) satisfying the condition f ( m + 1, α) / f ( m , α) = 1 + O( m -1 ). For the calculation of correlation functions, one needs the derivative of this function with respect to α. There is no guarantee that the derivative f '( m , α) = ∂ f ( m , α)/∂α satisfies f '( m + 1, α) / f '( m , α) = 1 + O( m -1 ). Therefore, equation (3.24) is incorrect. This error is crucial because a main result of this paper is incorrect. However, the asymptotic analysis of the Riemann-Hilbert problem is new and correct.