  We propose and study a wave function describing an interacting
three-dimensional fractional chiral hinge insulator (FCHI) constructed by
Gutzwiller projection of two non-interacting second order topological
insulators with chiral hinge modes at half filling. We use large-scale
variational Monte Carlo computations to characterize the model states via the
entanglement entropy and charge-spin-fluctuations. We show that the FCHI
possesses fractional chiral hinge modes characterized by a central charge $c=1$
and Luttinger parameter $K=1/2$, like the edge modes of a Laughlin $1/2$ state.
By changing the boundary conditions for the underlying fermions, we investigate
the topological degeneracy of the FCHI. Within the range of the numerically
accessible system sizes, we observe a non-trivial topological degeneracy. A
more numerically pristine characterization of the bulk topology is provided by
the topological entanglement entropy (TEE) correction to the area law. While
our computations indicate a vanishing bulk TEE, we show that the gapped
surfaces host a two-dimensional topological order with a TEE per surface
compatible with half that of a Laughlin $1/2$ state, a value that cannot be
obtained from topological quantum field theory.
