Dynamical Flavor Symmetry Breaking by a Magnetic Field in $2+1$ Dimensions

It is shown that in $2+1$ dimensions, a constant magnetic field is a strong catalyst of dynamical flavor symmetry breaking, leading to generating a fermion dynamical mass even at the weakest attractive interaction between fermions. The essence of this effect is that in a magnetic field, in $2+1$ dimensions, the dynamics of fermion pairing is essentially one-dimensional. The effect is illustrated in the Nambu-Jona-Lasinio model in a magnetic field. The low-energy effective action in this model is derived and the thermodynamic properties of the model are considered. The relevance of this effect for planar condensed matter systems and for $3+1$ dimensional theories at high temperature is pointed out.