Viscosity Bound, Causality Violation and Instability with Stringy Correction and Charge

Recently, it has been shown that if we consider the higher derivative correction, the viscosity bound conjectured to be $\eta \mathrm{/}s=1\mathrm{/}4\pi$ is violated and so is the causality. In this paper, we consider medium effect and the higher derivative correction simultaneously by adding charge and Gauss-Bonnet terms. We find that the viscosity bound violation is not changed by the charge. However, we find that two effects together create another instability for large momentum regime. We argue the presence of tachyonic modes and show it numerically. The stability of the black brane requires the Gauss-Bonnet coupling constant $\lambda$($=2{\alpha }^{\mathrm{\prime }}\mathrm{/}{l}^2$) to be smaller than 1/24. We draw a phase diagram relevant to the instability in charge-coupling space.