Magnetic field driven instability in planar NJL model in real-time formalism

Abstract

It is known that the symmetric (massless) state of the Nambu--Jona-Lasinio model in 2+1 dimensions in a magnetic field B is not the ground state of the system at zero temperature due to the presence of a negative, linear in &|\sigma+i\pi|$,termintheeffectivepotentialforthecompositefields$\sigma\sim\bar{\psi}\psi$and$\pi\sim\bar{\psi}i\gamma^5\psi$,whilethequadratictermisalwayspositive(atachyonisabsent).Wefindthatfinitetemperatureisanecessaryingredientforthetachyonicinstabilityofthesymmetricstatetooccur.UtilizingtheSchwinger--Keldyshreal-timeformalismwecalculatethedispersionrelationsforthefluctuationmodesofthecompositefields$\sigma$and$\pi$. We demonstrate the presence of the tachyonic instability of the symmetric state for coupling constant that exceeds a certain critical value which vanishes as temperature tends to zero in accordance with the phenomenon of magnetic catalysis.

Comments: 13 pages, 2 figures