Spectroscopy in the 2+1$d$ Thirring Model with $N=1$ Domain Wall Fermions

We employ the domain wall fermion (DWF) formulation of the Thirring model on a lattice in 2+1+1 dimensions and perform $N=1$ flavor Monte Carlo simulations. At a critical interaction strength the model features a spontaneous $\mathrm{U}(2)\rightarrow\mathrm{U}(1)\otimes \mathrm{U}(1)$ symmetry breaking; we analyse the induced spin-0 mesons, both Goldstone and non-Goldstone, as well as the correlator of the fermion quasiparticles, in both resulting phases. Crucially, we determine the anomalous dimension $\eta_\psi\approx 3$ at the critical point, in stark contrast with the Gross-Neveu model in 3$d$ and with results obtained with staggered fermions. Our numerical simulations are complemented by an analytical treatment of the free fermion correlator, which exhibits large early-time artifacts due to branch cuts in the propagator stemming from unbound interactions of the fermion with its heavy doublers. These artifacts are generalisable beyond the Thirring model, being an intrinsic property of DWF, or more generally Ginsparg-Wilson fermions.