Holographic Schwinger effect with Translational Symmetry Breaking

We investigate the holographic Schwinger effect in a background with translational symmetry breaking (TSB) at finite chemical potential. The gravitational background is characterized by two independent parameters: the TSB parameter $\alpha$, which controls momentum relaxation, and the chemical potential $\mu$, which determines the finite density of the dual field theory. Using the potential analysis method, we derive the total potential governing the pair production process and examine its dependence on $\alpha$, $\mu$, the external magnetic field, and the ratio $\beta=E/E_c$. Our results show that the effects of $\alpha$ and $\mu$ on the Schwinger process strongly depend on the dynamical regime. In the subcritical regime, increasing either $\alpha$ or $\mu$ lowers the potential barrier and facilitates pair production. However, near and above the critical electric field, the roles of these two parameters become qualitatively different. While increasing the chemical potential lowers the total potential and enhances the Schwinger pair production process, increasing the translational symmetry breaking parameter shifts the potential upward and suppresses the production process. We further show that the external magnetic field enhances the Schwinger effect by lowering the effective potential barrier and facilitating pair production. This enhancement persists in both the critical and supercritical regimes. In addition, we qualitatively investigate the corresponding pair production rate through its relation to the total potential and find qualitative consistency between the rate behavior and the potential analysis. Overall, our analysis provides a comprehensive picture of how translational symmetry breaking, finite density, and external magnetic fields influence holographic non-perturbative pair production.