Hofstadter spectrum in electric and magnetic fields

TL;DR

This paper analyzes the energy spectrum of two-dimensional Bloch electrons under combined magnetic and electric fields, revealing phenomena like miniband formation, Landau level broadening, and the magnetic Stark ladder with discrete energy levels.

Contribution

It introduces an exact finite difference equation framework for the system, incorporating arbitrary periodic potentials and electric fields in rational flux conditions, advancing understanding of electron dynamics in such fields.

Findings

Weak periodic potential creates minibands and minigaps.

Electric field causes quasienergy crossings and avoided crossings.

Strong electric field leads to a magnetic Stark ladder with discrete energy levels.

Abstract

The problem of Bloch electrons in two dimensions subject to magnetic and intense electric fields is investigated. Magnetic translations, electric evolution and energy translation operators are used to specify the solutions of the Schr\"odinger equation. For rational values of the magnetic flux quanta per unit cell and commensurate orientations of the electric field relative to the original lattice, an extended superlattice can be defined and a complete set of mutually commuting space-time symmetry operators is obtained. Dynamics of the system is governed by a finite difference equation that exactly includes the effects of: an arbitrary periodic potential, an electric field orientated in a commensurable direction of the lattice, and coupling between Landau levels. A weak periodic potential broadens each Landau level in a series of minibands, separated by the corresponding minigaps. The addition of the electric field induces a series of avoided and exact crossing of the quasienergies, for sufficiently strong electric field the spectrum evolves into equally spaced discreet levels, in this "magnetic Stark ladder" the energy separation is an integer multiple of $ h E / a B $, with $a$ the lattice parameter.