Hairy Reissner-Nordstrom Black Holes with Asymmetric Vacua

We minimally coupled a scalar potential $V(\phi)$ with asymmetric vacua to the Einstein gravity to numerically construct the hairy Reissner-Nordstrom black hole (RNBH) as a direct generalization of RNBHs to possess scalar hair. By fixing the electric charge to mass ratio $q$, a branch of hairy RNBHs bifurcates from the RNBH when the scalar field $\phi_H$ is non-trivial at the horizon. The values of $q$ are bounded for $0 \leq q \leq 1$, which contrast to a class of hairy black holes with $q>1$ in the Einstein-Maxwell-scalar theory. We find that the profiles of solutions affected by the competition between the strength of $\phi_H$ and $q$, for instance, the gradient of scalar field at the horizon can increase very sharply when $q \rightarrow 1$ and $\phi_H$ is small but its gradient can be very small which independent of $q$ when $\phi_H$ is large. Furthermore, the weak energy condition of hairy RNBHs, particularly at the horizon can be satisfied when $q>0$.