  We explore a large $N$ generalization of the Kitaev model on the honeycomb
lattice with a simple nearest-neighbor interacting Hamiltonian. In particular,
we focus on the $\mathbb{Z}_4$ case with isotropic couplings, which is
characterized by an exact $\mathbb{Z}_4$ one-form symmetry. Guided by symmetry
considerations and an analytical study in the single chain limit, on the
infinitely long cylinders, we find the model is gapped with an extremely short
correlation length. Combined with the $\mathbb{Z}_4$ one-form symmetry, this
suggests the model is topologically ordered. To pin down the nature of this
phase, we further study the model on both finite and infinitely long strips,
where we consistently find a $c=1$ conformal field theory (CFT) description,
suggesting the existence of chiral edge modes described by a free boson CFT.
Further evidence is found by studying the dimer correlators on infinitely long
strips. We find the dimer correlation functions show a power-law decay with the
exponent close to 2 on the boundary of the strip, while decay much faster in
the bulk. Combined with the topological entanglement entropy extracted from
cylinder geometry, we identify the spin liquid is chiral and supports a
$\mathrm{U}(1)_{-8}$ chiral topological order. A unified perspective for all
$\mathbb{Z}_N$ type Kitaev models is also discussed.
