Exact Parent Hamiltonians for All Landau Level States in a Half-flux Lattice

TL;DR

This paper develops exact parent Hamiltonians for higher Landau level states on a half-flux lattice, enabling exploration of topological phases and non-Abelian states in lattice systems.

Contribution

It generalizes the Poisson summation rule to higher Landau levels, deriving exact flat-band Hamiltonians with potential for realizing non-Abelian fractionalized states.

Findings

Derived families of parent Hamiltonians for higher Landau levels.

Identified conditions for gapped and gapless phases based on lattice symmetries.

Proposed models with realistic decay hopping amplitudes for experimental realization.

Abstract

Realizing topological flat bands with tailored single-particle Hilbert spaces is a critical step toward exploring many-body phases, such as those featuring anyonic excitations. One prominent example is the Kapit-Mueller model, a variant of the Harper-Hofstadter model that stabilizes lattice analogs of the lowest Landau level states. The Kapit-Mueller model is constructed based on the Poisson summation rule, an exact lattice sum rule for coherent states. In this work, we consider higher Landau-level generalizations of the Poisson summation rule, from which we derive families of parent Hamiltonians on a half-flux lattice which have exact flat bands whose flatband wavefunctions are lattice version of higher Landau level states. Focusing on generic Bravais lattices with only translation and inversion symmetries, we discuss how these symmetries enforced gaplessness and singular points for odd Landau level series, and how to achieve fully gapped parent Hamiltonians by mixing even and odd series. Our model points to a large class of tight-binding models with suitable energetic and quantum geometries that are potentially useful for realizing non-Abelian fractionalized states when interactions are included. The model exhibits fast decay hopping amplitudes, making it potentially realizable with neutral atoms in optical lattices.