  We investigate the chiral symmetry and its spontaneous breaking at finite
temperature and in an external magnetic field with four-fermion interactions of
different channels. Quantum and thermal fluctuations are included within the
functional renormalization group approach, and properties of the set of flow
equations for different couplings, such as its fixed points, are discussed. It
is found that external parameters, e.g. the temperature and the external
magnetic field and so on, do not change the structure of the renormalization
group flows for the couplings. The flow strength is found to be significantly
dependent on the route and direction in the plane of couplings of different
channels. Therefore, the critical temperature for the chiral phase transition
shows a pronounced dependence on the direction as well. Given fixed initial
ultraviolet couplings, the critical temperature increases with the increasing
magnetic field, viz., the magnetic catalysis is observed with initial couplings
fixed.
