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arXiv:1101.1626

Large-distance and long-time asymptotic behavior of the reduced density matrix in the non-linear Schrödinger model

Karol Kajetan Kozlowski

Published Jan 8, 2011• 6 citations
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Abstract

Starting from the form factor expansion in finite volume, we derive the multidimensional generalization of the so-called Natte series for the zero-temperature, time and distance dependent reduced density matrix in the non-linear Schrödinger model. This representation allows one to \textit{read-off} straightforwardly the long-time/large-distance asymptotic behavior of this correlator. This method of analysis reduces the complexity of the computation of the asymptotic behavior of correlation functions in the so-called interacting integrable models, to the one appearing in free fermion equivalent models. We compute explicitly the first few terms appearing in the asymptotic expansion. Part of these terms stems from excitations lying away from the Fermi boundary, and hence go beyond what can be obtained by using the CFT/Luttinger liquid based predictions.

DOI
10.48550/arxiv.1101.1626

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We present a new method allowing one to derive the long-time and large-distance asymptotic behavior of the correlation functions of quantum integrable models from their exact representations. Starting from the form factor expansion of the correlation functions in finite volume, we explain how to reduce the complexity of the computation in the so-called interacting integrable models to the one appearing in free fermion equivalent models. We apply our method to the time-dependent zero-temperature current–current correlation function in the non-linear Schrödinger model and compute the first few terms in its asymptotic expansion. Our result goes beyond the conformal field theory based predictions: in the time-dependent case, excitations other than the ones on the Fermi surface contribute to the leading orders of the asymptotics.
We propose a form factor approach for the computation of the large distance asymptotic behavior of correlation functions in quantum critical (integrable) models. In the large distance regime we reduce the summation over all excited states to one over the particle/hole excitations lying on the Fermi surface in the thermodynamic limit. We compute these sums, over the so-called critical form factors, exactly. Thus we obtain the leading large distance behavior of each oscillating harmonic of the correlation function asymptotic expansion, including the corresponding amplitudes. Our method is applicable to a wide variety of integrable models and yields precisely the results stemming from the Luttinger liquid approach, the conformal field theory predictions and our previous analysis of the correlation functions from their multiple-integral representations. We argue that our scheme applies to a general class of non-integrable quantum critical models as well.
We describe a Bethe ansatz-based method to derive, starting from a multiple integral representation, the long-distance asymptotic behavior at finite temperature of the density–density correlation function in the interacting one-dimensional Bose gas. We compute the correlation lengths in terms of solutions of non-linear integral equations of the thermodynamic Bethe ansatz type. Finally, we establish a connection between the results obtained in our approach and the correlation lengths stemming from the quantum transfer matrix method.
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We consider the low-temperature limit of the long-distance asymptotic behavior of the finite-temperature density–density correlation function in the one-dimensional Bose gas derived recently in the algebraic Bethe ansatz framework. Our results confirm the predictions based on the Luttinger liquid and conformal field theory approaches. We also demonstrate that the amplitudes arising in this asymptotic expansion at low temperature coincide with the amplitudes associated with the so-called critical form factors.