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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.
Gusynin et al. (Sun,) studied this question.