Abstract
Interacting theories of N relativistic fermion flavors in reducible spinor representations in 2+1 spacetime dimensions are formulated on a lattice using domain wall fermions (DWF), for which a U(2N) global symmetry is recovered in the limit that the wall separation $L_s$ is made large. The Gross-Neveu (GN) model is studied in the large-N limit and an exponential acceleration of convergence to the large-$L_s$ limit is demonstrated if the usual parity-invariant mass $m\bar\psi\psi$ is replaced by the U(2N)-equivalent $im_3\bar\psi\gamma_3\psi$. The GN model and two lattice variants of the Thirring model are simulated for N = 2 using a hybrid Monte Carlo algorithm, and studies made of the symmetry-breaking bilinear condensate and its associated susceptibility, the axial Ward identity, and the mass spectrum of both fermion and meson excitations. Comparisons are made with existing results obtained using staggered fermions. For the GN model a symmetry-breaking phase transition is observed, the Ward identity is recovered, and the spectrum found to be consistent with large-N expectations. There appears to be no obstruction to the study of critical UV fixed-point physics using DWF. For the Thirring model the Ward identity is not recovered, the spectroscopy measurements are inconclusive, and no symmetry breaking is observed all the way up to the effective strong coupling limit. This is consistent with a critical Thirring flavor number $N_c<2$, contradicting earlier staggered fermion results.
Highlights
Interacting theories of N relativistic fermion flavors in reducible spinor representations in 2+1 spacetime dimensions are formulated on a lattice using domain wall fermions (DWF), for which a U(2N ) global symmetry is recovered in the limit that the wall separation Ls is made large
The GN model and two lattice variants of the Thirring model are simulated for N = 2 using a hybrid Monte Carlo algorithm, and studies made of the symmetry-breaking bilinear condensate and its associated susceptibility, the axial Ward identity, and the mass spectrum of both fermion and meson excitations
The Thirring model has been studied using lattice simulations for 2 ≤ N ≤ 18 [23]–[25] which confirm that a symmetry-broken phase is present, that Nc ≈ 7 [25], and that critical exponents extracted from the equation of state close to the fixed point depend sensitively on N, quite distinct from the behaviour of the GN model
Summary
Let’s define the lattice action to be studied. The fermion kinetic term uses the 2 + 1d domain wall operator defined in [31, 32]: Skin = Ψ DΨ ≡. Formulation of the GN model on a lattice with DWF proceeds from the observation that the interaction with the auxiliary in (2.9) formally resembles a mass term [34]. The DWF formulation follows from (2.2) with Uμ ≡ 1, (2.3), (2.4) with the interaction term defined solely in terms of fields on the domain walls, SGNint = σ(x)[Ψ (x, Ls)P−Ψ(x, 1) + Ψ (x, 1)P+Ψ(x, Ls)], x (2.10). In the so-called non-compact approach the interaction between fermion bilinears and the vector auxiliary defined on the lattice links is linear; this has the virtue that only four-fermion terms are generated on integration over Aμ, making the connection with the continuum form (1.2) as transparent as possible. By analogy with (2.10) we study a surface formulation with Uμ ≡ 1 in (2.2) and link fields Aμ(x) defined solely on the walls interacting with point-split bilinears:.
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