An Exact QFT/Lattice Map for Fermions

We construct an exact map between continuum fermion fields and lattice fields in 1+1 dimensions. This map rewrites a continuum Hamiltonian as a lattice Hamiltonian with nearest-neighbor couplings and an infinite-dimensional local Hilbert space. The resulting lattice model reproduces exactly the energy spectrum, global operator algebra, chiral symmetries, and anomalies of the continuum QFT on any finite lattice. Unlike domain wall fermions, truncating the local Hilbert space introduces no extra massless modes, and the entanglement entropy of an interval converges as the truncation is removed. The lattice local algebras are of type I and lie between the type III continuum algebras of nested spatial regions, in accordance with the split property. We demonstrate this construction explicitly for $N$ Majorana fermions, realizing the $\mathrm{O}(N)_\mathrm{L}\times\mathrm{O}(N)_\mathrm{R}$ lattice chiral symmetry. We also construct an exact lattice model for the Schwinger model and numerically reproduce the mass gap $e/\sqrtπ$.