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Modern Physics Letters B

FREDHOLM DETERMINANT REPRESENTATION OF QUANTUM CORRELATION FUNCTION FOR SINE-GORDON AT SPECIAL VALUE OF COUPLING CONSTANT

H. Itoyama, V. E. Korepin, H. B. Thacker

Published Sep 20, 1992• 3 citations
Publisher

Abstract

Correlation functions of the Sine-Gordon model (which is equivalent to the Massive-Thirring model) are considered at the free fermion point. We derive a determinant formula for local correlation functions of the Sine-Gordon model, starting from Bethe ansatz wave function. Kernel of integral operator is trigonometric version of the one for Impenetrable Bosons.

DOI
10.1142/s0217984992001113

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Correlation functions of the Sine-Gordon model (which is equivalent to the Massive-Thirring model) are considered at the free fermion point. We derive a determinant formula for local correlation functions of the Sine-Gordon model, starting from Bethe ansatz wave function. Kernel of integral operator is trigonometric version of the one for Impenetrable Bosons.
Introduction Impenetrable bosons in one space dimension are a special case (at coupling constant c → ∞) of the one-dimensional Bose gas (NS model) considered in detail in Chapter I. The results obtained there for the NS model can be specialized to the case of impenetrable bosons. It is to be said that all the formulæe are considerably simplified in this case. From the point of view of physics this is due to the fact that the δ-function potentials in the N -particle Hamiltonian (I.1.11) are now infinitely strong and the bosons cannot penetrate one another, so that the wave function should be equal to zero if the bosons' coordinates coincide. In this respect impenetrable bosons are rather similar to free fermions (see Appendix I.I). In this chapter correlation functions of impenetrable bosons are considered and their representations as Fredholm determinants of linear integral operators are given. The (very essential) simplification with respect to the general case is due to the fact that now all the auxiliary quantum fields can be set equal to zero (see IX.6), so that the kernels of these operators do not contain quantum fields. The determinant representation for the time-dependent correlator is easy to obtain in this case. Temperature correlation functions are considered as well as zero temperature ones.
We derive the leading asymptotic behavior and build a new series representation for the Fredholm determinant of integrable integral operators appearing in the representation of the time and distance-dependent correlation functions of integrable models described by a six-vertex R-matrix.This series representation opens a systematic way for the computation of the long-time, long-distance asymptotic expansion for the correlation functions of the aforementioned integrable models away from their free fermion point.Our method builds on a Riemann-Hilbert based analysis.