Naively discretizing fermionic field theory on a lattice is known to produce unwanted fermion modes. This phenomenon is often referred to as 'fermion doubling,' which has been a significant issue when simulating chiral fermionic theory. We develop a formalism for simulating one-dimensional interacting chiral fermions on the lattice without breaking any local symmetries by defining a Fock space endowed with a semi-definite norm defined in terms of matrix product operators. This formalism can be understood as a second-quantized form of Stacey fermions [1], hence providing a possible solution for the fermion doubling problem and circumventing the Nielsen-Ninomiya theorem. The chiral states, in the analogue of edge physics, has non-orthogonal structures, and thus calls for more generic frame work called the generalized eigenvalue problem. Using this form, we find that the chiral Hamiltonian has local and hermitian features, allowing us to be simulated using tensor network methods similar to the ones used for simulating local quantum Hamiltonians. 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 demonstrating that the ground states of the system for large system sizes can be determined efficiently [2].
[1] R. Stacey, Phys. Rev. D. 26, 468 (1982)
[2] J. Haegeman, L. Lootens, Q. Mortier, A. Stottmeister, A. Ueda, and F. Verstraete, arXiv:2405.10285 (2024)
