XVII - The Algebraic Bethe Ansatz and Asymptotics of Correlation Functions

Summary

Introduction

In the previous chapter we evaluated the asymptotics of the field correlation function using differential equations. In this chapter we present a completely different approach. Instead of determinant representations we shall write some special series (which emphasize the role of the R-matrix). This series is especially efficient for the current (j(x) = Ψ^†(x)Ψ(x)) correlation function. It helps us to evaluate the asymptotics at zero temperature. The asymptotics of temperature correlation functions also can be obtained at any value of coupling constant for the Bose gas. The series for the correlation function

is based on the classification of all exactly solvable models (section VII.6) related to the fixed R-matrix. The series explicitly separates the contribution of the R-matrix and of the arbitrary functions a(λ) and d(λ). The Fourier coefficients of the irreducible part depend only on the R-matrix. Let us emphasize once more that in this chapter we shall consider the penetrable Bose gas (0 < c < ∞).

In section 1 the algebraic foundation of the new approach to correlation functions is given. In section 2 the series representation for the current correlator 〈j(x)j(0)〉 at zero temperature (in the thermodynamic limit) is given. In section 3 temperature correlations (for the penetrable Bose gas, 0 < c < ∞) are constructed. In section 4 explicit formulæe for asymptotics are presented. In section 5 the emptiness formation probability (the probability of absence of particles in some space interval due to thermal fluctuations) is evaluated.