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Journal of Physics A Mathematical and General

Finite-size effects and infrared asymptotics of the correlation functions in two dimensions

N. M. Bogoliubov, Anatoli G. Izergin, Nicolai Reshetikhin

Published Oct 21, 1987• 82 citations
Publisher

Abstract

The finite-size corrections approach to the calculation of the asymptotics of correlation functions containing the oscillating terms in the asymptotics in (1+1)-dimensional theories is presented. Explicit formulae for the critical exponents in integrable models are obtained.
DOI
10.1088/0305-4470/20/15/047

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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.
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We calculate the asymptotics of correlation functions for the one-dimensional model of fermions with correlated hopping recently proposed by the author. The method of calculation is based on predictions of conformal field theory, the Bethe ansatz solutions, and a finite-size analysis of the gapless excitations. The critical exponents are given in terms of dressed charge matrix. Components of this matrix are solutions of a set of integral equations obtained from the Bethe ansatz. The effective transport mass is calculated for an arbitrary band filling.
Using results on the scaling of energies with the size of the system and the principles of conformal quantum field theory, we calculate the asymptotics of correlation functions for the one-dimensional Hubbard model in the repulsive regime in the presence of an external magnetic field. The critical exponents are given in terms of a dressed charge matrix that is defined in terms of a set of integral equations obtained from the Bethe-Ansatz solution for the Hubbard model. An interpretation of this matrix in terms of thermodynamical coefficients is given, and several limiting cases are considered.