Incorporating Arbitrarily Strong On-Site Correlations into Lattice Models

Abstract

In Chapter 9 I described a protocol to efficiently generate small clusters of fractional quantum Hall states at sites of a deep optical lattice, in the limit where particles were unable to tunnel between sites on experimentally relevant timescales. One motivation for this chapter is to construct a theory of deep lattices that are nevertheless weak enough to allow particle tunneling between sites (analogous to the Bose-Hubbard limit of bosons in an optical lattice introduced in Chapter 2.). I construct a theory—specifically an effective lattice model—that captures the dominant behavior for bosons, and essentially identical techniques lead to a description of fermions. In the bosonic case, I construct an analog of the Gutzwiller mean field theory (introduced in Chapter 2) for this generalized model. I show that the resulting model has a phase diagram topologically the same as the Bose-Hubbard model: there are globally insulating phases with fractional quantum Hall puddles at each site and no coherence between them occupying “Mott lobes” in the chemical potential-lattice depth phase diagram, and a superfluid phase with coherence between the fractional quantum Halls states at each site.