Quantum Geometry of the "Fuzzy-Lattice" Hubbard Model and the Fractional Chern Insulator

Abstract

The fractional quantum Hall effect (FQHE), describing the quantization of the Hall conductance in terms of rational fractional multiples of a fundamental constant, has been experimentally observed in two-dimensional systems of charged fermions confined to Landau orbits in the presence of a strong magnetic field. The effect is particularly interesting from a theoretical standpoint as an example of an interacting topological phase of matter that cannot be described in terms of the Landau symmetry-breaking picture of phase transitions. Properties of FQH states such as quasi-particle excitations obeying fractional statistics and topological ground-state degeneracy have been explained, in part, by proposing trial many-body wavefunctions such as the Laughlin and Read-Rezayi series. Recent studies of interacting particles on tight-binding lattices with broken time-reversal symmetry reveal FQH phases at zero magnetic field (fractional Chern insulators, FCI). In a partially-filled Landau level, the non-commutative Landau-orbit guiding-centers are the residual degrees of freedom, requiring a "quantum geometry" Hilbert-space description, since a real-space Schrödinger description can only apply in the "classical geometry" of unprojected coordinates. The continuum description does not apply on a lattice, where we describe the emergence of the FCI from a noncommutative quantum lattice geometry. We define a "fuzzy lattice" by projecting a one-particle bandstructure with more than one orbital per unit cell into a single band, and then renormalizing the orbital on each site to unit weight. The resulting overcomplete basis of local states is analogous to a basis of more than one coherent state per flux quantum in a Landau level. The overlap matrix characterizes "quantum geometry" on the "fuzzy lattice", defining a "quantum distance" measure and Berry fluxes through elementary lattice triangles. By studying quantum geometry at transitions between topologically-distinct instances of a fuzzy lattice, we numerically observe features of the FCI, including the emergence of topological edge states consistent with the chiral Luttinger liquid theory of the FQH edge, and power-law decay of the overlap between "fuzzy" states, suggesting the presence of strongly-correlated phase. We comment on current work to generalize this "fuzzy-orbital" construction for lattice models with higher Chern number.