Hierarchical Structure of Azbel-Hofstader Problem: Strings and loose ends of Bethe Ansatz

TL;DR

This paper provides numerical evidence and a hierarchical framework for understanding the string solutions of the Bethe Ansatz equations in the Azbel-Hofstadter model, revealing a connection between spectral flow, topology, and fractal properties.

Contribution

It introduces a hierarchical tree structure for the Azbel-Hofstadter problem, linking spectral flow to string solutions and Hall conductance, and approximates the spectrum's properties.

Findings

String solutions are asymptotically exact in the incommensurate limit.

The hierarchical tree describes the topology of the spectrum.

Approximate methods capture gap distribution and some wave function properties.

Abstract

We present numerical evidence that solutions of the Bethe Ansatz equations for a Bloch particle in an incommensurate magnetic field (Azbel-Hofstadter or AH model), consist of complexes-"strings". String solutions are well-known from integrable field theories. They become asymptotically exact in the thermodynamic limit. The string solutions for the AH model are exact in the incommensurate limit, where the flux through the unit cell is an irrational number in units of the elementary flux quantum. We introduce the notion of the integral spectral flow and conjecture a hierarchical tree for the problem. The hierarchical tree describes the topology of the singular continuous spectrum of the problem. 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 numbers (the set of dimensions of the irreducible representations of $U_q(sl_2)$ with definite parity). In this paper we consider the approximation of noninteracting strings. It provides the gap distribution function, the mean scaling dimension for the bandwidths and gives a very good approximation for some wave functions which even captures their multifractal properties. However, it misses the multifractal character of the spectrum.