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arXiv:1309.4867 · Journal of Physics A Mathematical and Theoretical

Fisher–Hartwig expansion for the transverse correlation function in the XX spin-1/2 chain

Dmitri A. Ivanov, Alexander G Abanov

Published Dec 3, 2013• 1 citation
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Abstract

Motivated by the recent results on the asymptotic behavior of Toeplitz determinants with Fisher-Hartwig singularities, we develop an asymptotic expansion for transverse spin correlations in the XX spin-1/2 chain. The coefficients of the expansion can be calculated to any given order using the relation to discrete Painleve equations. We present explicit results up to eleventh order and compare them with a numerical example.

DOI
10.1088/1751-8113/47/1/015001

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Motivated by the recent results on the asymptotic behavior of Toeplitz determinants with Fisher-Hartwig singularities, we develop an asymptotic expansion for transverse spin correlations in the XX spin-1/2 chain. The coefficients of the expansion can be calculated to any given order using the relation to discrete Painleve equations. We present explicit results up to eleventh order and compare them with a numerical example.
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Abstract We derive an explicit expression for the leading term in the long-time, large-distance asymptotic expansion of a transverse dynamical two-point function of the XX chain in the spacelike regime. This expression is valid for all nonzero finite temperatures and for all magnetic fields below the saturation threshold. It is obtained here by means of a straightforward term-by-term analysis of a thermal form factor series, derived in previous work, and demonstrates the usefulness of the latter.
We present the new expressions for the formfactors of local operators for the XX- quantum spin chain as a Cauchy determinants. Using the known functional form of the correlator at large distances we propose the new expression for the constant for the asymptotics of the correlator as a Cauchy determinant. We calculate the momentum distribution for the general case of the XXZ- spin chain and point out that it is completely different from that for the Luttinger model (the system of fermions). For the XX chain we compare numerically the value of the lowest formfactor and the expectation value of momentum- zero operators which is determined by the functional form of the correlator. 1.Introduction. In the present paper we study the spin-spin equal-time correlator in the XX- spin chain
We present new expressions for the formfactors of local operators for the XX-quantum spin chain as Cauchy determinants. Using the known functional form of the correlator at large distances we propose a new expression for the constant for the asymptotics of the correlator as a Cauchy determinant. We calculate the momentum distribution for the general case of the XXZ-spin chain and point out that it is completely different from that for the Luttinger model (the system of fermions). For the XX chain we compare numerically the value of the lowest formfactor and the expectation value of momentum-zero operators which is determined by the functional form of the correlator.
We consider the Rényi entropies in the one-dimensional spin-1/2 Heisenberg XX chain in a magnetic field. The case n = 1 corresponds to the von Neumann ‘entanglement’ entropy. Using a combination of methods based on the generalized Fisher–Hartwig conjecture and a recurrence relation connected to the Painlevé VI differential equation we obtain the asymptotic behaviour, accurate to order , of the Rényi entropies for large block lengths . For n = 1, 2, 3, 10 this constitutes the 3, 6, 10, 48 leading terms respectively. The o(1) contributions are found to exhibit a rich structure of oscillatory behaviour, which we analyse in some detail both for finite n and in the limit .