Hofstadter-Toda spectral duality and quantum groups
arXiv:2312.14242 · doi:10.1063/5.0202635
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.
arXiv commentAccepted on Journal of Mathematical Physics, 15 pages, no figures, reference added
References in corpus (25)
- Hofstadter's butterfly in moire superlattices: A fractal quantum Hall effect
- Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure
- Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
- Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices
- Cloning of Dirac fermions in graphene superlattices
- Topological States and Adiabatic Pumping in Quasicrystals
- Geometric Engineering of Quantum Field Theories
- Spectral signatures of many-body localization with interacting photons
- Discrete Heisenberg-Weyl Group and Modular Group
- Four-Dimensional Quantum Hall Effect in a Two-Dimensional Quasicrystal
- Disordered Topological Insulators via -Algebras
- Five Dimensional Gauge Theories and Relativistic Integrable Systems
- Quantum Geometry of Refined Topological Strings
- Unitary representations of $U_{q}(\mathfrak{sl}(2,\RR))$, the modular double, and the multiparticle q-deformed Toda chains
- Hofstadter butterfly as Quantum phase diagram
- Fractional quantization of the topological charge pumping in a one-dimensional superlattice
- The absolutely continuous spectrum of the almost Mathieu operator
- Exact quantization conditions for the relativistic Toda lattice
- Hofstadter's Butterfly in Quantum Geometry
- Topologically quantized current in quasiperiodic Thouless pumps
- Quantum Group, Bethe Ansatz and Bloch Electrons in a Magnetic Field
- Calabi-Yau geometry and electrons on 2d lattices
- Hofstadter's Butterfly and Langlands Duality
- Groups with Spanier-Whitehead duality
- Non-Hermitian Hamiltonians and similarity transformations