PI: Simon Hands, University of Liverpool
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 vF/c≈1/300, where the pitch of the cones vF 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 of interest in its own right, as possibly the simplest relativistic theory of fermions requiring a computational solution. A major question is whether or not for sufficient interaction strength 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 chiral symmetry breaking in QCD. It is anticipated that this can occur if the number of fermion species N lies below some critical Nc: accurate determination of Nc requires control over non-perturbative quantum field theory.
In lattice field theory, the problem requires a precise rendering of the correct U(2N) fermion global symmetries. We use Domain Wall Fermion formulation, in which U(2N) GInsparg-Wilson symmetries are recovered in the limit that the wall separation Ls→∞. Simulations have been performed using the RHMC algorithm on 163 systems with Ls ranging from 8 to 80. At strong couplings g2 and light fermion masses m recovery of GW symmetry is slow – nonetheless our data for <ψψ> as a function of g2, m 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 βc≡agc-2≈0.28 (see figure), characteristic of an interacting conformal field theory.
A major outcome of the project is the confirmation that the critical number of flavors for symmetry breaking in the Thirring model Nc >1. This is a significant step towards understanding strongly-interacting fermions from first principles. We are now calculating propagators to understand the quasiparticle and bound-state excitation spectra.
