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We study the ground state instability of a strongly coupled QFT with the$z=2$ Schr\"odinger symmetry in a constant electric field using probe branesholography. The system is $N_f$ $\mathcal{N}=2$ hypermultiplet fermions at zerocharge density in the supergravity Schr\"odinger background. We show that theinstability occurs due to Schwinger-like effect and an insulator state willundergo a transition to a conductor state. We calculate the decay rate ofinstability and pair production probability by using the $gauge/gravity$duality. At zero temperature for massive fermions, we suggest that theinstability occurs if the critical electric field is larger than the confiningforce between fermions, which is proportional to an effective mass. Wedemonstrate that, at zero temperature, the Schr\"odinger background simulatesthe role of a crystal lattice for massive particles. We also show that atfinite 't Hooft coupling for particles with a mass higher than$\frac{\sqrt{\lambda}}{\pi \b}$, in this background, instability does notoccur, no matter how large the external electric field is, meaning that we havea \textit{perfect insulator}. Moreover, we derive Euler-Heisenberg effectiveLagrangian for the non-relativistic strongly correlated quantum theory fromprobe branes holography in Schr\"odinger spacetime.

Ground State Instability in Non-relativistic QFT and Euler-Heisenberg Lagrangian via Holography