Validity of SLAC fermions for the (1 + 1)-dimensional helical Luttinger liquid

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 $1\mathrm{/}{r}^2$ 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.