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

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 &|σ+iπ|$, term in the effective potential for the composite fields $σ\sim\barψψ$ and $π\sim\barψiγ^5ψ$, while the quadratic term is always positive (a tachyon is absent). We find that finite temperature is a necessary ingredient for the tachyonic instability of the symmetric state to occur. Utilizing the Schwinger--Keldysh real-time formalism we calculate the dispersion relations for the fluctuation modes of the composite fields $σ$ and $π$. 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.