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Journal of Physics A Mathematical and General

Current correlations in the nonlinear Schrodinger model from conformal field theory

Alexander Bérkovich, Ganpathy N. Murthy

Published Apr 7, 1991• 2 citations
Publisher

Abstract

The authors analyse the nonlinear Schrodinger model (the one-dimensional Bose gas) by means of conformal field theory in conjunction with a newly developed cutoff procedure. This makes it possible to obtain exact expressions for current correlators in the model. A number of new results are presented, including the expression for the two-point current correlator on a finite-size strip and the general n-point current correlator on the plane. All these results have the full time dependence and are presented to order 1/g, where g is the strength of the repulsive delta -function interaction.

DOI
10.1088/0305-4470/24/7/024

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The authors analyse the nonlinear Schrodinger model (the one-dimensional Bose gas) by means of conformal field theory in conjunction with a newly developed cutoff procedure. This makes it possible to obtain exact expressions for current correlators in the model. A number of new results are presented, including the expression for the two-point current correlator on a finite-size strip and the general n-point current correlator on the plane. All these results have the full time dependence and are presented to order 1/g, where g is the strength of the repulsive delta -function interaction.
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.
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The two-point correlation function for the quantum nonlinear Schr\"odinger (one-dimensional $\ensuremath{\delta}$-function gas) model is studied. An infinite-series representation for this function is derived using the quantum inverse-scattering formalism. For the case of zero temperature, the infinitecoupling ($c\ensuremath{\rightarrow}\ensuremath{\infty}$) result of Jimbo, Miwa, M\^ori, and Sato is extended to give an exact expression for the order-1c\frac{1}{c}c1​ correction to the two-point function in terms of a Painlev\'e transcendent of the fifth kind.
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Finite-size scaling analysis is performed for one-dimensional Bose and Fermi systems with long-range interaction g/r2{\mathit{r}}^{2}r2, based on the exact solution of Sutherland. The low-energy behavior of this model is shown to be described by the c=1 conformal field theory. Exact formulas for various correlation exponents are obtained. At a special point g=4, the model belongs to the same universality class as the Haldane-Shastry model, the effective theory of which is the level-1 SU(2) Kac-Moody theory.