Split degenerate states and stable p \(+i\) p phases from holography

In this paper, we investigate the p\(+i\)p superfluid phases in the complex vector field holographic p-wave model. We find that in the probe limit, the p\(+i\)p phase and the p-wave phase are equally stable, hence the p and ip orders can be mixed with an arbitrary ratio to form more general p\(+\lambda i\)p phases, which are also equally stable with the p-wave and p\(+i\)p phases. As a result, the system possesses a degenerate thermal state in the superfluid region. We further study the case on considering the back-reaction on the metric, and we find that the degenerate ground states will be separated into p-wave and p\(+i\)p phases, and the p-wave phase is more stable. Finally, due to the different critical temperature of the zeroth order phase transitions from p-wave and p\(+i\)p phases to the normal phase, there is a temperature region where the p\(+i\)p phase exists but the p-wave phase does not. In this region we find the stable holographic p\(+i\)p phase for the first time.