  The Nielson-Ninomiya theorem states that a chirally invariant free fermion
lattice action, which is local, translation invariant, and real necessarily has
fermion doubling. The SLAC approach gives up on locality and long range hopping
leads to a linear dispersion with singularity at the zone boundary. We
introduce a SLAC Hamiltonian formulation that is expected to realize a U(1)
helical Luttinger liquid in a naive continuum limit. We argue that non-locality
and concomitant singularity at the zone edge has important implications. Large
momentum transfers yield spurious features already in the non-interacting case.
Upon switching on interactions non-locality invalidates the Mermin-Wagner
theorem and allows for long ranged magnetic ordering. In fact, in the strong
coupling limit the model maps onto an XXZ-spin chain with $1/r^2$ exchange.
Here, both spin-wave and DMRG calculations support long ranged order. While the
long-ranged order opens a single particle gap the Dirac point, the singularity
at the zone-boundary persists for any finite value of the interaction strength
such that the ground state remains metallic. Hence, SLAC Hamiltonian does not
flow to the $1$d helical Luttinger liquid fixed point. Aside from DMRG
simulations, we have used auxiliary field quantum Monte Carlo simulations to
arrive to the above conclusions.
