HIERARCHICAL STRUCTURE OF AZBEL-HOFSTADTER PROBLEM - STRINGS AND LOOSE ENDS OF BETHE-ANSATZ

We present numerical evidence that solutions of the Bethe anstaz equat ions for a Bloch particle in an incommensurate magnetic field (Azbel-H ofstadter or AH model), consist of complexes-''strings'', String solut ions are well known from integrable field theories. They become asympt otically exact in the thermodynamic limit. The string solutions for th e AH model are exact in the incommensurate limit, where the Aux throug h the unit cell is an irrational number in units of the elementary flu x quantum. We introduce the notion of the integral spectral flow and c onjecture a hierarchical tree for the problem. The hierarchical tree d escribes the topology of the singular continuous spectrum of the probl em. We show that the string content of a state is determined uniquely by the rate of the spectral flow (Hall conductance) along the tree. We identify the Hall conductances with the set of Takahashi-Suzuki numbe rs (the set of dimensions of the irreducible representations of U-q(sl (2)) with definite parity). In this paper we consider the approximatio n of non-interacting strings, It provides the gap distribution functio n, the mean scaling dimension for the bandwidths and gives a very good approximation for some wave functions which even captures their multi fractal properties, However, it misses the multifractal character of t he spectrum. (C) 1998 Elsevier Science B.V.