Non-Abelian anyons — quasiparticles whose exchange generates non-commuting unitary transformations — are the foundation of topological quantum computation. They appear naturally in continuous Landau-level physics, but realizing them on a lattice, where experiments actually happen, requires constructing lattice models whose flat bands mimic the analytic structure of Landau levels.
Cruise and Seidel build exactly solvable lattice Hamiltonians with degenerate flat bands that replicate the first p Landau levels, restricted to the lattice. With local bosonic interactions, these become parent Hamiltonians for fractional Chern insulators with Chern number greater than one — the regime where non-Abelian physics lives.
The model stabilizes the bosonic Jain-2/1 state (Abelian), the 2/2-state (supporting Ising anyons), and the 3/3-state (supporting Fibonacci anyons). The Fibonacci anyon is the prize — its braiding alone is sufficient for universal quantum computation. The exact diagonalization confirms the predicted zero-mode counts for each state, validating the construction.
The key technical achievement: the lattice Hamiltonian has exponentially localized hopping. The bands are exactly flat, the interactions are two-body and local, and the resulting ground states are the exact lattice versions of the parton quantum Hall states. No approximations.
An exactly solvable route to Fibonacci anyons on a lattice. The model is a proof of principle — not a material design, but a mathematical demonstration that non-Abelian topological order can live on a lattice through exact two-body interactions.