An exact formula for the energy spectra of the Hofstadter butterfly

What is it about?

The Hofstadter butterfly is the fractal energy spectra of the Hofstadter model, so called for the striking resemblance to the wings of a butterfly. The Hofstadter model allows to describe and understand several phenomena in a 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. 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.

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Photo by Joshua J. Cotten on Unsplash

Photo by Joshua J. Cotten on Unsplash

Why is it important?

We have proven the spectral duality formula for the Hofstadter butterfly, that is, the fractal energy spectra of the Hofstadret model, using the apparatus of quantum groups. The Hofstadter model describes several phenomena in modern condensed matter systems, such as quasiperiodicity, flat-band superconductivity, fractality, topological states of matter, Landau levels, and Thouless pumps.

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