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arXiv:math-ph/0205045 · Journal of Physics Condensed Matter

Asymptotics and functional form of correlators in the XX-spin chain of finite length

А. А. Овчинников

Published Oct 18, 2002• 4 citations
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Abstract

We verify the functional form of the asymptotics of the spin–spin equal-time correlation function for the XX chain, predicted by the hypothesis of conformal invariance at large distances and by the bosonization procedure. We point out that the bosonization procedure also predicts the functional form of the correlators for the chains of finite length. We found the exact expression for the spin–spin equal-time correlator on a finite lattice. We find excellent agreement of the exact correlator with the prediction given by the leading asymptotics result up to very small distances. We also establish the correspondence between the value of the constant before the asymptotics for the XX chain with the expression for this constant proposed by Lukyanov and Zamolodchikov. We also evaluate the constant corresponding to the subleading term in the asymptotics in a way which is different from previous studies.

DOI
10.1088/0953-8984/14/43/316

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We verify the functional form of the asymptotics of the spin–spin equal-time correlation function for the XX chain, predicted by the hypothesis of conformal invariance at large distances and by the bosonization procedure. We point out that the bosonization procedure also predicts the functional form of the correlators for the chains of finite length. We found the exact expression for the spin–spin equal-time correlator on a finite lattice. We find excellent agreement of the exact correlator with the prediction given by the leading asymptotics result up to very small distances. We also establish the correspondence between the value of the constant before the asymptotics for the XX chain with the expression for this constant proposed by Lukyanov and Zamolodchikov. We also evaluate the constant corresponding to the subleading term in the asymptotics in a way which is different from previous studies.
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.
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 derive expressions for the form factors of the quantum transfer matrix of the spin- X X Z chain which are suitable for taking the infinite Trotter number limit. These form factors determine the finitely many amplitudes in the leading asymptotics of the finite-temperature correlation functions of the model. We consider form factor expansions of the longitudinal and transversal two-point functions. Remarkably, the formulae for the amplitudes are in both cases of the same form. We also explain how to adapt our formulae to the description of ground-state correlation functions of the finite chain. The usefulness of our novel formulae is demonstrated by working out explicit results in the high- and low-temperature limits. We obtain, in particular, the large-distance asymptotics of the longitudinal two-point functions for small temperatures by summing up the asymptotically most relevant terms in the form factor expansion of a generating function of the longitudinal correlation functions. As expected, the leading term in the expansion of the corresponding two-point functions is in accordance with conformal field theory predictions. Here it is obtained for the first time by a direct calculation.
Using the XX- spin chain as an example we show that in general the asymptotic expansion of the correlator at large distances is not given by the sum over the low-lying particle-hole intermediate states. Only the first two terms in the expansion agree with the predictions of the Luttinger liquid theory. The other terms are in general given by the intermediate states with the particles and holes with the quantum numbers far away from the Fermi- points (at the distances of order of the length of the chain). We argue that the whole expansion cannot be described by the Luttinger liquid theory.
Starting from the massless form factor expansion for the two-point dynamical correlation functions obtained recently, I extract the long-distance and large-time asymptotics of these correlators. The analysis yields the critical exponents and associated amplitudes characterizing the asymptotics. The results are obtained on the basis of exact and first principle based considerations: they do not rely, at any stage, on some hypothetical correspondence with a field theory or the use of any other phenomenological approach. Being based on form factor expansion, the method allows one to clearly identify which contributions to the asymptotics issue from which class of excited states. All this permits to settle the long-standing question of the contribution of bound states to the asymptotics of two-point functions. For instance, when considering the long-distance m behavior of equal-time correlators, the analysis shows that while, in fine, the bound states only produce contributions that are exponentially small in m, they also play a key role in canceling out certain power-law contributions which, should they be present, would break explicitly the universality structure of the long-distance behavior.