Quantum Group, Bethe Ansatz and Bloch Electrons in a Magnetic Field

Abstract

The wave functions for two dimensional Bloch electrons in a uniform magnetic field at the mid-band points are studied with the help of the algebraic structure of the quantum group $U_q(s{l}_2)$. A linear combination of its generators gives the Hamiltonian. We obtain analytical and numerical solutions for the wave functions by solving the Bethe Ansatz equations, proposed by Wiegmann and Zabrodin on the basis of above observation. The semi-classical case with the flux per plaquette $\varphi =1/Q$ is analyzed in detail, by exploring a structure of the Bethe Ansatz equations. We also reveal the multifractal structure of the Bethe Ansatz solutions and corresponding wave functions when $\varphi$ is irrational, such as the golden or silver mean.

30 pages, 11 GIF figures(use xv, or WWW browser)

References in corpus (1)

• Explicit Solutions of the Bethe Ansatz Equations for Bloch Electrons in a Magnetic Field

Cited by in corpus (17)

• On the reduced density matrix for a chain of free electrons
• Ferromagnetism in the Hubbard model with Topological/Non-Topological Flat Bands
• Exact ground state of the sine-square deformed XY spin chain
• Self-similar occurrence of massless Dirac particles in graphene under magnetic field
• Hierarchical Structure of Azbel-Hofstader Problem: Strings and loose ends of Bethe Ansatz
• Asymptotically exact solutions of Harper equation
• Bethe ansatz for the Harper equation: Solution for a small commensurability parameter
• Effects of the field modulation on the Hofstadter's spectrum
• Hofstadter Problem on the Honeycomb and Triangular Lattices: Bethe Ansatz Solution
• Landau Levels from the Bethe Ansatz Equations
• An extended q-deformed su(2) algebra and the Bloch electron problem
• Hofstadter problem in higher dimensions
• Hofstadter-Toda spectral duality and quantum groups
• Golden mean renormalization for the almost Mathieu operator and related skew products
• The quantum group, Harper equation and the structure of Bloch eigenstates on a honeycomb lattice
• Localization and Fractality in Inhomogeneous Quantum Walks with Self-Duality
• Azbel-Hofstadter model on triangular lattice revisited