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In previous works, we have developed an approach to fix the leading behaviorsof the pure AdS$_3$ and BTZ black hole from the entanglement entropies of thefree CFT$_2$ and finite temperature CFT$_2$, respectively. We exclusively useholographic principle only and make no restriction about the bulk geometry, notonly the kinematics but also the dynamics. In order to verify the universalityand correctness of our method, in this paper, we apply it to the $T\bar{T}$deformed CFT$_2$, which breaks the conformal symmetry. In terms of the physicalarguments of the $T\bar{T}$ deformed CFT$_2$, the derived metric is a deformedBTZ black hole. The requirement that the CFT$_2$ lives on a conformally flatboundary leads to $r_{c}^{2}=\ 6R_{AdS}^{4}/(\pi c\mu)$ naturally, in perfectagreement with previous conjectures in literature. The energy spectum andpropagation speed calculated with this deformed BTZ metric are the same asthese derived from $T\bar{T}$ deformed CFT$_2$. We furthermore fix the dualgeometry of highly excited states with our approach. The result contains thedescriptions for the conical defect and BTZ black hole.

Fix the dual geometries of $T\bar{T}$ deformed CFT$_2$ and highly excited states of CFT$_2$