  Lately there are many SLAC fermion investigations on the (2+1)D Gross-Neveu
criticality of a single Dirac cone [1,2]. While the SLAC fermion construction
indeed gives rise to the linear energy-momentum relation for all lattice
momenta at the non-interacting limit, the long-range hopping and its consequent
violation of locality on the Gross-Neveu quantum critical point (GN-QCP) --
which a priori requires short-range interaction -- has not been verified. Here
we show, by means of large-scale quantum Monte Carlo simulations, that the
interaction-driven antiferromagnetic insulator in this case is fundamentally
different from that on a purely local $\pi$-flux Hubbard model on the square
lattice. In particular, we find the antiferromagnetic long-range order in the
SLAC fermion model has a finite temperature continuous phase transition, which
violates the Mermin-Wagner theorem, and smoothly connects to the previously
determined GN-QCP. The magnetic excitations inside the antiferromagnetic
insulator are gapped without Goldstone mode, even though the state
spontaneously breaks continuous $SU(2)$ symmetry. These unusual results
proclaim caution on the interpretation of the quantum phase transition in SLAC
fermion model as that of GN-QCP with short-range interaction.
