The Hofstadter Butterfly in a Dynamic Cavity-Induced Synthetic Magnetic Field

TL;DR

This paper explores how a cavity-induced synthetic magnetic field affects the energy band structure of a Fermi gas in a 2D optical lattice, revealing a dynamic Hofstadter butterfly pattern and phase transition behaviors.

Contribution

It introduces a method to realize a dynamical synthetic magnetic field in a cavity system, demonstrating its impact on the Hofstadter butterfly and phase transition order.

Findings

Superradiant phase transition is first order at low flux and second order at higher flux.

The cavity-induced magnetic field causes a non-trivial deformation of the Hofstadter butterfly.

At high pump strength, the Hofstadter butterfly resembles that of a static magnetic field.

Abstract

Energy bands of electrons in a square lattice potential threaded by a uniform magnetic field exhibit a fractal structure known as the Hofstadter butterfly. Here we study a Fermi gas in a 2D optical lattice within a linear cavity with a tilt along the cavity axis. The hopping along the cavity axis is only induced by resonant Raman scattering of transverse pump light into a standing wave cavity mode. Choosing a suitable pump geometry allows to realize the Hofstadter-Harper model with a cavity-induced dynamical synthetic magnetic field, which appears at the onset of the superradiant phase transition. The dynamical nature of this cavity-induced synthetic magnetic field arises from the delicate interplay between collective superradiant scattering and the underlying fractal band structure. Using a sixth-order expansion of the free energy as function of the order parameter and by numerical simulations we show that at low magnetic fluxes the superradiant ordering phase transition is first order, while it becomes second order for higher flux. The dynamic nature of the magnetic field induces a non-trivial deformation of the Hofstadter butterfly in the superradiant phase. At strong pump far above the self-ordering threshold we recover the Hofstadter butterfly one would obtain in a static magnetic field.