  We develop a Hamiltonian formalism for simulating interacting chiral fermions
on the lattice while preserving unitarity and locality and without breaking the
chiral symmetry. The fermion doubling problem is circumvented by constructing a
Fock space endowed with a semi-definite norm. When projecting our theory on the
the single-particle sector, we recover the framework of Stacey fermions, and we
demonstrate that the scaling limit of the free model recovers the chiral
fermion field. Technically, we make use of a matrix product operator norm to
mimick the boundary of a higher dimensional topological theory. As a proof of
principle, we consider a single Weyl fermion on a periodic ring with
Hubbard-type nearest-neighbor interactions and construct a variational
generalized DMRG code to demonstrate that the ground state for large system
sizes can be determined efficiently. As our tensor network approach does not
exhibit any sign problem, we can add a chemical potential and study real-time
evolution.
