# Anyons are not energy eigenspaces of quantum double Hamiltonians

## TL;DR

This paper introduces a new family of commuting four-body projector Hamiltonians that differentiate among non-trivial charges and fluxes, revealing that anyons are not solely associated with energy eigenspaces in quantum double models.

## Contribution

It generalizes Kitaev's quantum double models by constructing Hamiltonians that distinguish non-trivial charges and fluxes, showing the non-uniqueness of anyons and energy eigenspaces.

## Key findings

- Hamiltonians made of commuting four-body projectors discriminate non-trivial charges and fluxes.
- Anyons are not in one-to-one correspondence with energy eigenspaces.
- Presence of local degrees of freedom affects the topological classification.

## Abstract

Kitaev's quantum double models, including the toric code, are canonical examples of quantum topological models on a 2D spin lattice. Their Hamiltonian defines the groundspace by imposing an energy penalty to any nontrivial flux or charge, but does not distinguish among those. We generalize this construction by introducing a novel family of Hamiltonians made of commuting four-body projectors that provide an intricate splitting of the Hilbert space by discriminating among non-trivial charges and fluxes. Our construction highlights that anyons are not in one-to-one correspondence with energy eigenspaces, a feature already present in Kitaev's construction. This discrepancy is due to the presence of local degrees of freedom in addition to topological ones on a lattice.