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

Recent studies of interacting particles on tight-binding lattices with broken time-reversal symmetry reveal ``zero-field'' fractional quantum Hall (FQH) phases (fractional Chern insulators, FCI). In a partially-filled Landau level, the non-commutative guiding-centers are the residual degrees of freedom, requiring a ``quantum geometry'' Hilbert-space description (a real-space Schr\"odinger description can only apply in the ``classical geometry'' of unprojected coordinates). The continuum description does not apply on a lattice, where we describe emergence of the FCI from a non-commutative 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 renormalize 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. We study quantum geometry at transitions between topologically-distinct instances of a fuzzy lattice, as well as $N$-body states with local Hubbard interactions.