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Asymptotic critical behavior of holographic phase transition at finite topological charge -- the spectrum of excited states becomes continuous at $T=0$

Within the framework of AdS/CFT duality, excited states of the conformal field living at the global AdS boundary of a four-dimensional spacetime Einstein gravity are investigated analytically in the probe limit where the field equations are linearized. At asymptotically large values, the threshold chemical potential for the appearance of excited condensate states are discrete, equal spacing, with the gap approaches zero logarithmically in the limit $T\rightarrow 0$. Remarkably, numerical results show that, this behavior applies even for states as low as for the first or the second excited state of the condensate. This is especially significant on the liquid side of the black hole van der Waals - like phase transition (small or zero topological charge) where there seems to be no gap between the ground state and the first excited state at zero temperature. We postulate that, at the exact limit $T = 0$ where the gap is zero, the spectrum of threshold chemical potentials becomes continuous, all excited states of the condensate are activated above a finite chemical potential, suggesting a new quantum phase transition as a function of the chemical potential. Previous studies have largely missed this continuous spectrum of excited states in the $T\rightarrow 0$ limit. This fact should be taken into account carefully in AdS/CFT duality studies.