Hofstadter-Toda spectral duality and quantum groups

TL;DR

This paper explores a spectral duality between the Hofstadter model and the relativistic Toda lattice, linking it to quantum groups and providing a new formula for the Hofstadter spectrum using symmetric and Chebyshev polynomials.

Contribution

It uncovers a spectral relationship between two important models and connects it to quantum group theory, offering a novel parametrization of the Hofstadter spectrum.

Findings

Established a spectral duality between Hofstadter and Toda models.

Connected the duality to Langlands duality of quantum groups.

Derived a formula for Hofstadter spectrum using symmetric and Chebyshev polynomials.

Abstract

The Hofstadter model allows to describe and understand several phenomena in condensed matter such as the quantum Hall effect, Anderson localization, charge pumping, and flat-bands in quasiperiodic structures, and is a rare example of fractality in the quantum world. An apparently unrelated system, the relativistic Toda lattice, has been extensively studied in the context of complex nonlinear dynamics, and more recently for its connection to supersymmetric Yang-Mills theories and topological string theories on Calabi-Yau manifolds in high-energy physics. Here we discuss a recently discovered spectral relationship between the Hofstadter model and the relativistic Toda lattice which has been later conjectured to be related to the Langlands duality of quantum groups. Moreover, by employing similarity transformations compatible with the quantum group structure, we establish a formula parametrizing the energy spectrum of the Hofstadter model in terms of elementary symmetric polynomials and Chebyshev polynomials. The main tools used are the spectral duality of tridiagonal matrices and the representation theory of the elementary quantum group.