Bethe Ansatz and Classical Hirota Equation

We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be obtained from entirely classical set up. Namely, the eigenvalues of the quantum transfer matrix and the scattering $S$-matrix itself are identified with a certain $\tau$-functions of the discrete Liouville equation. The Bethe ansatz equations are obtained as dynamics of zeros. For comparison we also present the Bethe ansatz equations for elliptic solutions of the classical discrete Sine-Gordon equation. The paper is based on the recent study of classical integrable structures in quantum integrable systems, hep-th/9604080.Comment: 15 pages, Latex, special World Scientific macros include

Hidden Integrability of a Kondo Impurity in an Unconventional Host

We study a spin-1/2 Kondo impurity coupled to an unconventional host in which the density of band states vanishes either precisely at (``gapless'' systems) or on some interval around the Fermi level (``gapped''systems). Despite an essentially nonlinear band dispersion, the system is proven to exhibit hidden integrability and is diagonalized exactly by the Bethe ansatz.Comment: 4 pages, RevTe

On the singular spectrum of the Almost Mathieu operator. Arithmetics and Cantor spectra of integrable models

I review a recent progress towards solution of the Almost Mathieu equation (A.G. Abanov, J.C. Talstra, P.B. Wiegmann, Nucl. Phys. B 525, 571, 1998), known also as Harper's equation or Azbel-Hofstadter problem. The spectrum of this equation is known to be a pure singular continuum with a rich hierarchical structure. Few years ago it has been found that the almost Mathieu operator is integrable. An asymptotic solution of this operator became possible due analysis the Bethe Ansatz equations.Comment: Based on the lecture given at 13th Nishinomiya-Yukawa Memorial Symposium on Dynamics of Fields and Strings, Nishinomiya, Japan, 12-13 Nov 1998, and talk given at YITP Workshop on New Aspects of Strings and Fields, Kyoto, Japan, 16-18 Nov 199

Geometric adiabatic transport in quantum Hall states

We argue that in addition to the Hall conductance and the nondissipative component of the viscous tensor, there exists a third independent transport coefficient, which is precisely quantized. It takes constant values along quantum Hall plateaus. We show that the new coefficient is the Chern number of a vector bundle over moduli space of surfaces of genus 2 or higher and therefore cannot change continuously along the plateau. As such, it does not transpire on a sphere or a torus. In the linear response theory, this coefficient determines intensive forces exerted on electronic fluid by adiabatic deformations of geometry and represents the effect of the gravitational anomaly. We also present the method of computing the transport coefficients for quantum Hall states.Comment: 6 pages, discussion of angular momentum formulas in sec. 7 is amende