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Within the framework of AdS/CFT duality, excited states of the conformalfield living at the global AdS boundary of a four-dimensional spacetimeEinstein gravity are investigated analytically in the probe limit where thefield equations are linearized. At asymptotically large values, the thresholdchemical potential for the appearance of excited condensate states arediscrete, equal spacing, with the gap approaches zero logarithmically in thelimit $T\rightarrow 0$. Remarkably, numerical results show that, this behaviorapplies even for states as low as for the first or the second excited state ofthe condensate. This is especially significant on the liquid side of the blackhole 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 excitedstate at zero temperature. We postulate that, at the exact limit $T = 0$ wherethe gap is zero, the spectrum of threshold chemical potentials becomescontinuous, all excited states of the condensate are activated above a finitechemical potential, suggesting a new quantum phase transition as a function ofthe chemical potential. Previous studies have largely missed this continuousspectrum of excited states in the $T\rightarrow 0$ limit. This fact should betaken into account carefully in AdS/CFT duality studies.

the journal of high energy physics/the journal of high energy physics

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