Strongly coupled fermions on the lattice

Since its inception with the pioneering work of Ken Wilson, lattice field theory has come a long way. Lattice formulations have enabled us to probe the non-perturbative structure of theories such as QCD and have also helped in exploring the phase structure and classification of phase transitions in a variety of other strongly coupled theories of interest to both high energy and condensed matter theorists. The lattice approach to QCD has led to an understanding of quark confinement, chiral symmetry breaking and hadronic physics. Correlation functions of hadronic operators and scattering matrix of hadronic states can be calculated in terms of fundamental quark and gluon degrees of freedom. Since lattice QCD is the only well-understood method for studying the low-energy regime of QCD, it can provide a solid foundation for the understanding of nucleonic structure and interaction directly from QCD. Despite these successes problems remain. In particular, the study of chiral gauge theories on the lattice is an outstanding problem of great importance owing to its theoretical implications and for its relevance to the electroweak sector of the Standard Model. However the construction of these theories is plagued by the emergence of massless chiral modes in the lattice theory which have no counterpart in the continuum theory. This is a topological obstruction known as the Nielsen-Nimonoya theorem and can be proven to hold under assumptions of translation invariance, chiral invariance and hermiticity of the lattice Hamiltonian. One strategy that was advocated by Eichten and Preskill (EP) early on in the field was to generate large masses for these additional chiral states by coupling them to additional composite fermions. These composite fermions would arise as bound states via an auxiliary Yukawa interaction. Hence in the continuum limit mirror modes will be forced to decouple from the spectrum without breaking chiral symmetry. This proposal led to many numerical studies of different models. Golterman et.al critiqued this proposal by showing that in specific realizations of the EP model the required phase containing a four fermion condensate was separated from the massless phase needed for a chiral theory by an intermediate phase in which the gauge symmetry was spontaneously broken. This was proven in the large N limit where the model contains N flavors of fermion. It led to the idea that the four-fermion phase was a lattice artifact and further work on these models stopped. However in recent few years this picture has changed. In three dimensions several studies of an SU(4) invariant four-fermion model have provided evidence in favor of a direct transition between massless(PMW) phase and four-fermion(PMS) phase. The generation of a mass without breaking symmetries via a symmetric four fermion condensate has received a lot of interest within the condensed matter community. Indeed this mechanism has been employed to gap out edge modes of topological insulators without breaking any symmetries. Of course the question for high energy physics is whether these new four fermion models exhibit this same structure in four dimensions and, if so, can it be used in the context of the original EP proposal to create a lattice theory whose low energy excitations are chiral. In this thesis I discuss the progress towards this goal