Fermion Masses without Spontaneous Symmetry Breaking

Masses of free fermions arise from local fermion bilinear terms in the action. If symmetries of the theory prevent such terms, fermions remain massless perturbatively. However, these symmetries can break spontaneously and generate non-zero fermion bilinear condensates that can make fermions massive. This traditional mechanism of fermion mass generation is well known and used in the standard model of particle physics to give quarks and leptons their masses. In this talk we argue that fermion masses can also be generated without spontaneous symmetry through four-fermion condensates when symmetries prevent fermion bilinear condensates from forming.  We show that this less explored mechanism of fermion mass generation is responsible for making four reduced staggered lattice fermions massive at strong couplings in a lattice model with a local four-fermion coupling. The model has a massless fermion phase at weak couplings and massive fermion phase at strong couplings. In particular there is no spontaneous symmetry breaking of any lattice symmetries in both these phases. In three space-time dimensions there is a direct second order phase transition between the two phases, which allows one to take the continuum limit of the exotic massive phase. The same model in four space-time dimensions contains a narrow intermediate phase with fermion bilinear condensates. Our work is closely related to recent developments in the field of topological insulators.