The Thirring Model describes relativistic fermions moving in a two-dimensional plane and interacting via a contact term between covariantly conserved currents. The physical system it most resembles is that of low-energy electronic excitations in graphene. For free electrons at half-filling on a honeycomb lattice, conduction and valance bands form cones just touching at their vertices at two “Dirac points” lying within the first Brillouin zone. Since the density of states vanishes, and the effective fine structure constant is boosted by a factor v_F/c ≈ 1/300, where the pitch of the cones v_F is the Fermi velocity, the resulting physics is described by a strongly-interacting relativistic quantum field theory, with equal likelihood of exciting electrons or holes.

Besides possible applications in layered condensed matter systems, the Thirring model is an interesting theory in its own right, and possibly the simplest theory of fermions requiring a computational solution. A major question is whether or not for sufficiently strong self-interaction a bilinear condensate <ψψ> forms, restructuring the ground state and causing a gap to form between the conduction and valence bands resulting in a phase transition from semimetal to insulator – this process is precisely analogous to spontaneous chiral symmetry breaking in QCD. It is anticipated that this can occur if the number of fermion species N lies below some critical N_c: determination of N_c requires non-perturbative quantum field theory.

In lattice field theory, the problem seems to require a precise rendering of the correct U(2N) fermion global symmetries. We have been using the Domain Wall Fermion formulation, in which U(2N) Ginsparg-Wilson symmetries are recovered in the limit that the domain wall separation L_s to ∞. Simulations have been performed using the RHMC algorithm on 12³ and 16³ systems with L_s ranging from 8 to 48. It turns out that at strong couplings g² and light fermion masses m recovery of GW symmetry is slow – nonetheless at weak-to-moderate couplings our results for <ψψ> as a function of g², m obtained at L_s = 48 are compatible with an extrapolation to the large L_s limit based on the entire dataset. They are well-fitted by an equation of state which assumes a continuous symmetry-breaking transition with non-mean field theory critical exponents at a critical 1/g² ≈ 0.28 (see figure). The fit is less convincing at the strongest couplings (lying within the broken phase), which we attribute to the L_s to ∞ limit not yet being reached.

A major outcome of the project to date is the confirmation that the critical number of flavors for symmetry breaking in the Thirring model N_c>1. This is a significant step towards understanding the formulation of fermion quantum fields in strongly-interacting theories.