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Correlation functions, entanglement and chaos in the TT¯ / JT¯ -deformed CFTs

Song He, Hongfei Shu

Published Jan 1, 2020• 75 citations
OpenAlex

Abstract

In this paper, we regard the $T\\bar{T}$/$J\\bar{T}$-deformed CFTs as perturbation theories and compute the first order correction of the correlation functions due to the $T\\bar{T}$/$J\\bar{T}$-deformation. As applications, we study the R\\'enyi entanglement entropy of excited state in the $T\\bar{T}$/$J\\bar{T}$-deformed two-dimensional CFTs. We find, up to the perturbation first order of the deformation, the R\\'enyi entanglement entropy of locally excited state will obtain a nontrivial time dependence. The excess of the R\\'enyi entanglement entropy of locally excited state will also be dramatically changed up to order ${\\cal O}(c)$. Furthermore, we investigate the out of time ordered correlation function to confirm that the $T\\bar{T}$/$J\\bar{T}$-deformations do not change the maximal chaotic behavior of holographic CFTs up to the first order of the deformations.

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In this paper, we regard the TbarTT\\bar{T}TbarT/JbarTJ\\bar{T}JbarT-deformed CFTs as perturbation theories and compute the first order correction of the correlation functions due to the TbarTT\\bar{T}TbarT/JbarTJ\\bar{T}JbarT-deformation. As applications, we study the R\'enyi entanglement entropy of excited state in the TbarTT\\bar{T}TbarT/JbarTJ\\bar{T}JbarT-deformed two-dimensional CFTs. We find, up to the perturbation first order of the deformation, the R\'enyi entanglement entropy of locally excited state will obtain a nontrivial time dependence. The excess of the R\'enyi entanglement entropy of locally excited state will also be dramatically changed up to order calO(c){\\cal O}(c)calO(c). Furthermore, we investigate the out of time ordered correlation function to confirm that the TbarTT\\bar{T}TbarT/JbarTJ\\bar{T}JbarT-deformations do not change the maximal chaotic behavior of holographic CFTs up to the first order of the deformations.
Correlation functions, entanglement and chaos in the TT‾/JT‾ T\overline{T}/J\overline{T} TT/JT-deformed CFTs
Abstract In this paper, we regard the TT‾/JT‾ T\overline{T}/J\overline{T} TT/JT T T ¯ / J T ¯ -deformed CFTs as perturbation theories and calculate the first order correction of the correlation functions due to the TT‾/JT‾ T\overline{T}/J\overline{T} TT/JT T T ¯ / J T ¯ ­deformation. As applications, we study the Rényi entanglement entropy of excited state in the TT‾/JT‾ T\overline{T}/J\overline{T} TT/JT T T ¯ / J T ¯ -deformed two-dimensional CFTs. We find, up to the first order perturbation of the deformation, the Rényi entanglement entropy of locally excited states will acquire a non­trivial time dependence. The excess of the Rényi entanglement entropy of locally excited state is changed up to order O(c). Furthermore, the out of time ordered correlation function is investigated to confirm that the TT‾/JT‾ T\overline{T}/J\overline{T} TT/JT T T ¯ / J T ¯ -deformations do not change the maximal chaotic behavior of holographic CFTs up to the first order of the deformations.
In this paper, we study the entanglement entropy of a single interval on a cylinder in two-dimensional TT‾T\overline{T}TT-deformed conformal field theory (CFT). For such case, the (R\'enyi) entanglement entropy takes a universal form in a CFT. We compute the correction due to the deformation up to the leading order of the deformation parameter in the framework of the conformal perturbation theory. We find that the correction to the entanglement entropy is nonvanishing in the finite temperature case, while it is vanishing in the finite size case. For the deformed holographic large ccc CFT, which is proposed to be dual to a AdS3{\mathrm{AdS}}_{3}AdS3​ gravity in a finite region, we find the agreement with the holographic entanglement entropy via the Ryu-Takayanagi formula. Moreover, we compute the leading order correction to the R\'enyi entropy, and discuss its holographic picture as well.
We study the entanglement entropy (EE) and the R\'enyi entropy (RE) of multiple intervals in two-dimensional TT‾T\overline{T}TT- deformed conformal field theory (CFT) at a finite temperature by field theoretic and holographic methods. First, by the replica method with the twist operators, we construct the general formula of the RE and EE up to the first order of a deformation parameter. By using our general formula, we show that the EE of multiple intervals for a holographic CFT is just a summation of the single interval case even with the small deformation. This is a nontrivial consequence from the field theory perspective, though it may be expected by the Ryu-Takayanagi formula in holography. However, the deformed RE of the two intervals is a summation of the single interval case only if the separations between the intervals are big enough. It can be understood by the tension of the cosmic branes dual to the RE. We also study the holographic EE for single and two intervals with an arbitrary cutoff radius (dual to the TT‾T\overline{T}TT deformation) at any temperature. We confirm our holographic results agree with the field theory results with a small deformation and high temperature limit, as expected. For two intervals, there are two configurations for EE: disconnected (sss-channel) and connected (ttt-channel) ones. We investigate the phase transition between them as we change parameters: as the deformation or temperature increases the phase transition is suppressed and the disconnected phase is more favored.
Various aspects of warped conformal field theories (WCFTs) are studied including entanglement entropy on excited states, the Renyi entropy after a local quench, and out-of-time-order four-point functions. Assuming a large central charge and dominance of the vacuum block in the conformal block expansion, (i) we calculate the single-interval entanglement entropy on an excited state, matching previous finite temperature results by changing the ensemble; and (ii) we show that WCFTs are maximally chaotic, a result that is compatible with the existence of black holes in the holographic duals. Finally, we relax the aforementioned assumptions and study the time evolution of the Renyi entropy after a local quench. We find that the change in the Renyi entropy is topological, vanishing at early and late times, and nonvanishing in between only for charged states in spectrally-flowed WCFTs.
We study the holographic entanglement entropy of an interval in a quantum field theory obtained by deforming a holographic two–dimensional conformal field theory via a general linear combination of irrelevant operators that are closely related to, but nonetheless distinct from, TT‾, JT‾ and TJ‾, and compute the Casin–Huerta entropic c–function. In the ultraviolet, for a particular combination of the deformation parameters, we find that the leading order dependence of the entanglement entropy on the length of the interval is given by a square root but not logarithmic term. Such power law dependence of the entanglement entropy on the interval length is quite peculiar and interesting. We also find that the entropic c–function is ultraviolet regulator independent, and along the renormalization group upflow towards the ultraviolet, it is non–decreasing. We show that in the ultraviolet the entropic c–function exhibits a power law divergence as the interval length approaches a minimum finite value determined in terms of the deformation parameters. This value sets the non–locality scale of the theory.
Abstract Various aspects of warped conformal field theories (WCFTs) are studied including entanglement entropy on excited states, the Rényi entropy after a local quench, and out-of-time-order four-point functions. Assuming a large central charge and dominance of the vacuum block in the conformal block expansion, (i) we calculate the single-interval entanglement entropy on an excited state, matching previous finite temperature results by changing the ensemble; and (ii) we show that WCFTs are maximally chaotic, a result that is compatible with the existence of black holes in the holographic duals. Finally, we relax the aforementioned assumptions and study the time evolution of the Rényi entropy after a local quench. We find that the change in the Rényi entropy is topological, vanishing at early and late times, and nonvanishing in between only for charged states in spectrally-flowed WCFTs.