Density driven symmetry breaking and Butterfly effect in holographic superconductors

We study the density driven symmetry breaking in holographic superconductors by considering the positive mass squared case. We show that even for the positive $m^2$, a scalar condensation still forms, provided the chemical potential is high enough. As $m^2$ increases, the phase space folds due to the non-linearity of the equations of motion, and two nearby points in the phase space can represent symmetry breaking and preserving configurations respectively. The phase space defined by the set of initial conditions of field variables at the horizon undergoes a non-linear radial evolution to result in the phase space folding, a characteristic phenomenon in a non-linear system. We then calculate the specific heat, which characterizes superconductors and has been measured in experiments.We observe a discontinuity in the specific heat at the transition point and compare our results with experimentally observed numbers. The electrical conductivity for various $m^2$ is also calculated.