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In this study, we investigate the Nambu--Jona-Lasinio (NJL) model at finitetemperature and finite density in an external magnetic field using thefunctional renormalization group. We investigate the dependence of the positionof the ultraviolet fixed point (UVFP) of the four-Fermi coupling constant onthe temperature, density, and external magnetic field, and we obtain the chiralphase structure. The UVFP at low temperature and finite chemical potentialoscillates in a small external magnetic field, which can be interpreted as thede Haas--van Alphen effect. We also obtain phase diagrams with complexstructures, where the phase boundary moves back and forth as the externalmagnetic field increases in the low temperature and high density region.

Functional renormalization group study of the Nambu--Jona-Lasinio model at finite temperature and density in an external magnetic field