Caution on Gross-Neveu criticality with a single Dirac cone: Violation of locality and its consequence of unexpected finite-temperature transition

Lately there are many SLAC fermion investigations on the $(2+1)\mathrm{D}$ Gross-Neveu criticality of a single Dirac cone. While the SLAC fermion construction indeed gives rise to the linear energy-momentum relation for all lattice momenta at the noninteracting 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 $\ensuremath{\pi}$-flux Hubbard model on the square lattice. In particular, the antiferromagnetic long-range order has a finite temperature continuous phase transition, which appears to violate 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 point out the fundamental difference between the QCP in SLAC fermion and that of GN-QCP with short-range interaction.