Symmetries of Thirring Models on 3D Lattices

We review some recent developments about strongly interacting relativistic Fermi theories in three spacetime dimensions. These models realize the asymptotic safety scenario and are used to describe the low-energy properties of Dirac materials in condensed matter physics. We begin with a general discussion of the symmetries of multi-flavor Fermi systems in arbitrary dimensions. Then we review known results about the critical flavor number $N_\mathrm{crit}$ of Thirring models in three dimensions. Only models with flavor number below $N_\mathrm{crit}$ show a phase transition from a symmetry-broken strong-coupling phase to a symmetric weak-coupling phase. Recent simulations with chiral fermions show that $N_\mathrm{crit}$ is smaller than previously extracted with various non-perturbative methods. Our simulations with chiral SLAC fermions reveal that for four-component flavors $N_\mathrm{crit}=0.80(4)$. This means that all reducible Thirring models with $\Nr=1,2,3,\dots$ show no phase transition with order parameter. Instead we discover footprints of phase transitions without order parameter. These new transitions are probably smooth and could be used to relate the lattice Thirring models to Thirring models in the continuum. For a single irreducible flavor, we provide previously unpublished values for the critical couplings and critical exponents.