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Journal of Physics A Mathematical and General

Conformal dimensions in Bethe ansatz solvable models

Anatoli G. Izergin, V. E. Korepin, Nicolai Reshetikhin

Published Jul 7, 1989• 116 citations
Publisher

Abstract

Models solvable by the hierarchy of Bethe ansatze (i.e. by the multicomponent Bethe ansatz) are considered. The spectrum of conformal dimensions which determines the long-distance asymptotics of correlations is calculated. This asymptotics in a general case is described by the direct sum of conformal theories, each possessing a central charge equal to one.

DOI
10.1088/0305-4470/22/13/052

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Models solvable by the hierarchy of Bethe ansatze (i.e. by the multicomponent Bethe ansatz) are considered. The spectrum of conformal dimensions which determines the long-distance asymptotics of correlations is calculated. This asymptotics in a general case is described by the direct sum of conformal theories, each possessing a central charge equal to one.
Asymptotic Bethe-ansatz solutions are obtained for one-dimensional quantum systems with a 1r2\frac{1}{{r}^{2}}r21​ long-range interaction by generalizing Sutherland's method to multicomponent quantum systems. We obtain the solutions to the supersymmetric $t\ensuremath{-}J$ model of Kuramoto and Yokoyama, the $\mathrm{SU}(\ensuremath{\nu})$ Haldane-Shastry model, and the multicomponent $t\ensuremath{-}J$ model. The excitation spectrum as well as bulk quantities are computed analytically. Conformal properties of low-energy excitations are discussed in connection with Luttinger liquid theory.
We present a new and efficient method for deriving finite-size effects in statistical physics models solvable by Bethe Ansatz. It is based on the study of the functional that maps a function to the sum of its evaluations over the Bethe roots. A simple and powerful constraint is derived when applying this functional to infinitely derivable test functions with compact support, that generalizes then to more general test functions. The method is presented in the context of the simple spin-1/2 XXZ chain for which we derive the finite-size corrections to leading eigenvalues of the Hamiltonian for any configuration of Bethe numbers with real Bethe roots. The expected results for the central charge and conformal dimensions are recovered.
Introduction The long distance asymptotics of correlation functions in integrable models have attracted long-standing interest. For the gapless one-dimensional systems considered in this book, zero temperature is a critical point. At zero temperature, ( T = 0), correlation functions decay as a power of the distance, but for T > 0, correlation functions decay exponentially. The powers of the distance by which correlation functions decay at zero temperature are called the critical exponents and they are the subject of this chapter. A recent development in the understanding of critical phenomena in (1+1)-dimensional systems is connected with conformal field theory which provides a powerful method for calculating critical exponents. However, many important results in (1+1)-dimensional systems have been obtained from perturbation calculations and renormalization group treatments, and. It should also be noted that the quantum inverse scattering method (QISM) approach to correlation functions confirms the predictions of conformal field theory (CFT), but more importantly QISM techniques show that the critical exponents of integrable models depend only on the underlying R-matrix. The Luttinger liquid approach is very powerful; it is also close to CFT. This approach is essentially based on the representation of a critical one-dimensional model in terms of a gaussian model. In the framework of this approach, the critical exponent was calculated first for Bethe Ansatz solvable models with a single degree of freedom. One should also mention the bosonization technique.
A new family of exactly solvable models is introduced. These models are generalizations of the XXZ chain where the distance among spins up (σzσ^zσz-basis) cannot be smaller or equal to t (t=0,1,2,...). The case t=0 recovers the standard XXZ chain. The coordinate Bethe ansatz is applied and the phase diagram is calculated. Exploring the finite-size consequences of conformal invariance the critical exponents are evaluated exactly at the critical regions of the phase diagram
We describe a method to derive, from first principles, the long-distance asymptotic behavior of correlation functions of integrable models in the framework of the algebraic Bethe ansatz. We apply this approach to the longitudinal spin–spin correlation function of the XXZ Heisenberg spin- 1/2 chain (with magnetic field) in the disordered regime as well as to the density–density correlation function of the interacting one-dimensional Bose gas. At leading order, the results confirm the Luttinger liquid and conformal field theory predictions.