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 nonlocality and concomitant singularity at the zone edge have important implications. Large momentum transfers yield spurious features already in the noninteracting case. Upon switching on interactions, nonlocality 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 at 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, the SLAC Hamiltonian does not flow to the 1D helical Luttinger liquid fixed point. Aside from DMRG simulations, we have used auxiliary field quantum Monte Carlo simulations to arrive at the above conclusions.