P. Wiegmann

Theory of nonabelian goldstone bosons in two dimensions

Abstract We present an exact theory of an O(4)-σ-model based on its relation to a certain fermionic model. The S-matrix and the vacuum energy in a constant external field are computed.

Exact results in the theory of magnetic alloys

Abstract Recently it has been shown that many traditional models used for a description of dilute magnetic alloys are completely integrable and may be solved exactly without any approximation. In this article we summarize the results which have been obtained in this way. The main part of the article is devoted to the consistent and detailed account of the Bethe-Ansatz technique for the s-d exchange (Kondo) model with arbitrary impurity spin, s-d model with anisotropic exchange, degenerate exchange model and for the canonical Anderson model. The thermodynamic properties of a magnetic impurity in a non-magnetic metal host obtained by the Bethe method are considered in detail. Mainly attention is paid to the analysis of singularities associated with the formation of a localized moment, Kondo effect and mixedvalence phenomenon, which can be treated analytically. In the introductory part of the article we discuss the applicability of the models which we study in the paper to the real alloys and consider some m...

Goldstone fields in two dimensions with multivalued actions

Abstract We present an exact solution for the two-dimensional chiral fields with multivalued action, and consider the relation of this theory to the CP -asymmetric Thirring model, free fermions, surrent algebras and Heisenberg antiferromagnets. Also, the parity violating lagrangian for the n -fields based on the Hopf topological invariant is briefly discussed.

Quantum Integrable Models and Discrete Classical Hirota Equations

Functional relation for commuting quantum transfer matrices of quantum integrable models is identified with classical Hirota’s bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. The standard objects of quantum integrable models are identified with elements of classical nonlinear integrable difference equation. In particular, elliptic solutions of Hirota’s equation give complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter’s Q-operator are solutions to the auxiliary linear problems for classical Hirota’s equation. The elliptic solutions relevant to Bethe ansatz are studied. The nested Bethe ansatz equations for Ak−1-type models appear as discrete time equations of motions for zeros of classical τ -functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota’s equation and a new determinant formula for eigenvalues of the quantum transfer matrices are obtained. Department of Mathematics of Columbia University and Landau Institute for Theoretical Physics Kosygina str. 2, 117940 Moscow, Russia James Franck Institute of the University of Chicago, 5640 S.Ellis Avenue, Chicago, IL 60637, USA James Franck Institute and and Enrico Fermi Institute of the University of Chicago, 5640 S.Ellis Avenue, Chicago, IL 60637, USA and Landau Institute for Theoretical Physics Joint Institute of Chemical Physics, Kosygina str. 4, 117334, Moscow, Russia and ITEP, 117259, Moscow, Russia 1

Integrable Structure of Interface Dynamics

We establish the equivalence of 2D contour dynamics to the dispersionless limit of the integrable Toda hierarchy constrained by a string equation. Remarkably, the same hierarchy underlies 2D quantum gravity.

Conformal Maps and Integrable Hierarchies

Abstract: We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy singled out by the conditions known as “string equations”. The same hierarchy locally solves the 2D inverse potential problem, i.e., reconstruction of the domain out of a set of its harmonic moments. This is the same solution which is known to describe 2D gravity coupled to c= matter. We also introduce a concept of the τ-function for analytic curves.

Neutral fermions in paramagnetic insulators

Abstract We discuss sources of neutral fermions in paramagnetic insulators which apparently play a major role in high- T c superconductivity

The principal chiral field in two dimensions on classical lie algebras: The Bethe-ansatz solution and factorized theory of scattering

Abstract The Bethe-ansatz solution, the exact factorized complete S -matrix and the particle spectrum for a two-dimensional chiral field on principal manifolds associated with the classical Lie groups SU( k + 1), SO(2 k ), SO(2 k + 1), Sp(2 k ) are presented. It is shown that the elementary particles are massive and form the basis of the ring of representations of the diagonal of the direct product G × G. The exact results are obtained in the framework of both the Bethe-ansatz approach and the factorized bootstrap program. The bootstrap properties of the S -matrices and relations between the simple roots of the algebras and the solution are discussed.

Factorized S-matrix and the Bethe ansatz for simple lie groups

Abstract The factorized S -matrices as well as the eigenvalues of the transfer matrices for symmetrical degrees of fundamental representations of all Lie groups are presented in the form of the Bethe ansatz in terms of roots systems. As an application the explicit expressions of the S -matrices for the lowest dimension representations of some exceptional Lie groups and the solution of the principal chiral model are calculated.

Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations

In spite of the diversity of solvable models of quantum field theory and the vast variety of methods, the final results display dramatic unification: the spectrum of an integrable theory with a local interaction is given by a sum of elementary energies