Schrieffer-Wolff Transformation for Periodically-Driven Systems: Strongly-Correlated Systems with Artificial Gauge Fields

I will discuss the celebrated Schrieffer-Wolff transformation for the Fermi-Hubbard model and show how to generalise it to periodically-driven systems using Floquet theory and ideas from atomic physics. I will demonstrate how the method works using the periodically-driven, strongly-interacting Fermi-Hubbard model. One can identify two regimes resulting in different effective low-energy Hamiltonians. In the non-resonant regime, an interacting spin model can be realised coupled to a static gauge field with a non-zero flux per plaquette. In the resonant regime, where the Hubbard interaction is a multiple of the driving frequency, I will show how to derive an effective Hamiltonian featuring doublon association and dissociation processes. The ground state of this Hamiltonian undergoes a phase transition between an ordered phase and a gapless Luttinger liquid phase. One can tune the system between different phases by changing the strength of the driving potential.