QED-Bloch Theory with Homogeneous Magnetic Fields: Modifications of the Landau Levels and the Hofstadter Butterfly

Probing electronic properties of periodic systems by arbitrary homogeneous magnetic fields has
unravelled fundamental new phenomena in condensed matter physics. Much theoretical work has
been devoted to describe those systems in different regimes, still a general first principles modeling
of such fundamental effects is lacking. Here we propose a solution to the problem of Bloch electrons
in a homogeneous magnetic field by including the quantum fluctuations of the photon field. A
generalized quantum electrodynamical (QED) Bloch theory from first principles is presented. As an
application we show how the well known Landau physics shows up in this framework and we derive
quantum corrections to the Landau levels. These quantum corrections have direct implications for the integer quantum Hall effect. Moreover, in the case of a 2D solid in a perpendicular magnetic field, in the limit where the field fluctuations go to zero, we recover the fractal pattern of the Hofstadter butterfly. Further generalizations and modifications of the Hofstadter butterfly will be presented.