  Quarks in strong magnetic fields (|eB|>>Lambda_QCD^2 ~ 0.04 GeV^2) acquire
enhanced infrared phase space proportional to |eB|. Accordingly they provide
larger chiral condensates and stronger backreactions to the gluon dynamics.
Confronting theories with lattice data at various values of |eB|, one can test
theoretical ideas as well as validity of various approximations, domain of
applicability of the effective models, and so on. The particularly interesting
findings on the lattice are inverse magnetic catalysis and linear growth of the
chiral condensate as a function of |eB|, which pose theoretical challenges. In
this talk we propose a scenario to explain both phenomena, claiming that the
quark mass gap should stay at around ~ Lambda_QCD, instead of ~|eB|^{1/2} which
has been supposed from dimensional arguments and/or effective model
calculations. The contrast between infrared and ultraviolet behaviors of the
interaction is a key ingredient to obtain the mass gap of ~Lambda_QCD.
