A. G. Izergin

Quantum inverse scattering method and correlation functions

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS

The quantum nonlinear Schrodinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest weight vector permits us to represent correlation function as a determinant of a Fredholm integral operator. This integral operator can be treated as the Gelfand-Levitan operator for some new differential equation. These differential equations are written down in the paper. They generalize the fifth Painleve transcendent, which describe equal time, zero temperature correlation function of an impenetrable Bose gas. These differential equations drive the quantum correlation functions of the Bose gas. The Riemann problem, associated with these differential equations permits us to calculate asymp-totics of quantum correlation functions. Quantum correlation function (Fredholm determinant) plays the role of τ functions of these new differential equations. For the impenetrable Bose gas space and time dependent correlation function is equal to τ function of the nonlinear Schrodinger equation itself, For a penetrable Bose gas (finite coupling constant c) the correlator is τ-function of an integro-differentiation equation.

The quantum inverse scattering method approach to correlation functions

The inverse scattering method approach is developed for calculation of correlation functions in completely integrable quantum models with theR-matrix of XXX-type. These models include the one-dimensional Bose-gas and the Heisenberg XXX-model. The algebraic questions of the problem are considered.

The Inverse Scattering Method Approach to the Quantum Shabat-Mikhailov Model

The Shabat-Mikhailov model is treated in the framework of the quantum inverse scattering method. The BaxtersR-matrix for the model is calculated.

LATTICE VERSIONS OF QUANTUM FIELD THEORY MODELS IN TWO DIMENSIONS

Abstract The quantum inverse scattering method allows one to put quantum field theory models on a lattice in a way which preserves the dynamical structure. The trace identifies are discussed for these models.

Determinant formula for the six-vertex model

The partition function of a six-vertex model with domain wall boundary conditions is considered on the finite lattice. The authors show that the partition function satisfies a recursive relation. They solve the recursion relation by a determinant formula. This gives a determinant representation for the partition function. They use the Quantum Inverse Scattering Method (QISM).

Correlation functions for a strongly correlated boson system

Abstract The correlation functions for a strongly correlated exactly solvable one-dimensional boson system on a finite chain as well as in the thermodynamic limit are calculated explicitly. This system, which we call the phase model, is the strong coupling limit of the integrable q -boson hopping model. The results are presented as determinants.

Critical exponents for integrable models

Abstract Phase transition in quantum systems in two space-time dimensions takes place at zero temperature. A general formula is obtained for the critical exponent describing the power decrease of zero-temperature correlation functions as long distances. This formula is valid for a large class of Bethe ansatz solvable models including the Heisenberg magnet and the one-dimensional Bose gas. The critical exponent is connected with the fractional charge; it is also expressed in terms of macroscopic characteristics of the models.

Correlation functions for the Heisenberg

The general method for calculation of correlation functions in integrable quantum models has been given in papers [1, 2]. The correlation function of the third components of local spins for the Heisenberg one-dimensionalXXZ-antiferromagnet is calculated in this paper. The answer is a series which gives, in particular, an improved version of the usual perturbative expansion in the anisotropy parameter. The remarkable property of the series obtained is that the long-distance asymptotics of the correlator is given already by the first term. The arguments are given in favour of the convergence of the series.

XXZ

The integrable statistical physics model on the rectangular two-dimensional lattice which we call ‘the L-model’ is constructed. This model generates the integrable quantum sine-Gordon model on the one-dimensional lattice in the same way as the ice model generates the XXZ model.