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

TL;DR

This paper investigates how magnetic fields and finite temperature induce instabilities in the symmetric state of the 2+1 dimensional NJL model, revealing a tachyonic instability linked to magnetic catalysis.

Contribution

It demonstrates the necessity of finite temperature for tachyonic instability and calculates dispersion relations using the Schwinger--Keldysh formalism.

Findings

Tachyonic instability occurs at finite temperature for strong coupling.

Critical coupling for instability vanishes as temperature approaches zero.

Magnetic catalysis influences the stability of the symmetric state.

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|$, term in the effective potential for the composite fields $\sigma\sim\bar{\psi}\psi$ and $\pi\sim\bar{\psi}i\gamma^5\psi$, 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 $\sigma$ 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.