Three-dimensional four-fermion theories with exact chiral symmetry on the lattice

Abstract

Four-fermion theories are quantum field theories that describe interactions between fermions via a fourth power of the field in the Lagrangian. Formulated in three spacetime dimensions, their purpose is twofold. On the one hand, they serve as low-energy descriptions of newly discovered materials like graphene. On the other hand, they are interesting as models for spontaneous symmetry breaking. Four-fermion theories allow various different realisations of chiral symmetry and the present work investigates the conditions of their spontaneous breaking. In this work, four-fermion models are formulated on a discrete spacetime lattice, which allows computer simulations. Most previous lattice regularisations did not respect the full chiral symmetry of the corresponding continuum models. Here, we follow a superior approach using the SLAC derivative, allowing an exact implementation of all internal symmetries. We first study the well-known Z_2-symmetric Gross-Neveu model, where a second-order phase transition exists for any number of fermion flavours Nf. Here, new values for the critical exponents of this transition for Nf=1,2,4 and 8 are calculated. For Nf=1 we provide the first values from a lattice field theory setup. The second model studied in this thesis is the Thirring model. Most previous works only found a spontaneously broken phase for a small number of fermion flavours below a critical value Nfc. Various approaches to investigate chiral symmetry breaking for the Thirring model are presented here. In summary, we never observe chiral symmetry breaking in our current simulations. Finally, a new formulation of four-fermion theories is introduced using dual variables acting as occupation numbers for the lattice points.