Asymptotic critical behavior of holographic phase transition at finite topological charge -- a potentially new quantum phase transition at finite chemical potential

We investigate analytically the asymptotic critical behavior at large chemical potential of the conformal field living at the AdS boundary of a four-dimensional spacetime Einstein gravity. The threshold values of the chemical potential for the appearance of condensate states are discrete, equal spacing, with the gap approaches zero logarithmically in the limit $Trightarrow 0$. Numerical results surprisingly show that, the result apply even for states with low quantum number, as low as for the first or second excited states of the condensate, especially on the liquid side of the black hole van der Waals - like phase transition. We postulate that, at the exact limit $T = 0$ where the gap is zero, all excite states of the condensate are activated above a finite chemical potential, suggesting a new quantum phase transition as a function of the chemical potential.