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arXiv:1412.0761 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Universal relation between thermal entropy and entanglement entropy in conformal field theories

Bin Chen, Jie-qiang Wu

Published Apr 22, 2015• 24 citations
arXivHTMLPDF

Abstract

Inspired by the holographic computation of large interval entanglement entropy of two-dimensional (2D) conformal field theory at high temperature, it was proposed that the thermal entropy is related to the entanglement entropy as ${S}_{th}=\underset{l\ensuremath{\rightarrow}0}{\mathrm{lim}}({S}_{EE}(R\ensuremath{-}l)\ensuremath{-}{S}_{EE}(l))$. In this paper, we prove this relation for 2D CFT with a discrete spectrum in two different ways. Moreover, we discuss this relation for a 2D noncompact free scalar, which is a gapless CFT with continuous spectrum. We show that it could be recovered, after appropriately regularizing the theory.

DOI
10.1103/physrevd.91.086012

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Inspired by the holographic computation of large interval entanglement entropy of two-dimensional (2D) conformal field theory at high temperature, it was proposed that the thermal entropy is related to the entanglement entropy as Sth=liml\ensuremath→0(SEE(R\ensuremath−l)\ensuremath−SEE(l)){S}_{th}=\underset{l\ensuremath{\rightarrow}0}{\mathrm{lim}}({S}_{EE}(R\ensuremath{-}l)\ensuremath{-}{S}_{EE}(l))Sth​=l\ensuremath→0lim​(SEE​(R\ensuremath−l)\ensuremath−SEE​(l)). In this paper, we prove this relation for 2D CFT with a discrete spectrum in two different ways. Moreover, we discuss this relation for a 2D noncompact free scalar, which is a gapless CFT with continuous spectrum. We show that it could be recovered, after appropriately regularizing the theory.
We analyze how excitations affect the entanglement entropy for an arbitrary entangling interval in a 2d conformal field theory (CFT) using the holographic entanglement entropy techniques as well as direct CFT computations. We introduce the excitation entanglement entropy ${\mathrm{\ensuremath{\Delta}}}_{h}S$, the difference between the entanglement entropy generic excitations and their arbitrary conformal descendants denoted through hhh. The excitation entanglement entropy, unlike the entanglement entropy, is a finite quantity (independent of the cutoff), and hence a good physical observable. We show that the excitation entanglement entropy for any given interval is uniquely specified by a local second order differential equation sourced by the one point function of the energy momentum tensor computed in the excited background state, and two boundary and smoothness conditions. We analyze low and high temperature behavior of the excitation entanglement entropy and show that ${\mathrm{\ensuremath{\Delta}}}_{h}S$ grows as a function of temperature. We prove an ``integrated positivity'' for the excitation entanglement entropy, that although ${\mathrm{\ensuremath{\Delta}}}_{h}S$ can be positive or negative, its average value is always positive. We also discuss the mutual and multipartite information and (strong) subadditivity inequality in the presence of generic excitations and their conformal descendants.
In this paper, we use entanglement entropy as a tool to explore the universal properties of conformal field theories (CFTs) dual to extremal BTZ black holes. We demonstrate that the entanglement entropies computed in the CFTs at the boundary of the extremal BTZ and the boundary of the near-horizon limit of the extremal BTZ are in perfect agreement and have the form appropriate to a two-dimensional CFT with only the chiral part excited and the antichiral in the ground state. Furthermore, we analyze the universal limits of the entanglement entropy and recover the correct value of the thermal entropy for large entangling intervals and the first-law-like relation for the small interval.
Time-dependent entanglement entropy (EE) is computed for a single interval in two-dimensional conformal theories from a quenched initial state in the presence of spatial boundaries. The EE is found to be periodic in time with periodicity equal to the system size LLL. For large enough LLL, the EE shows a rise to a thermal value (characterized by a temperature 1/β1/β1/β determined by the initial state), followed by periodic returns to the original value. This works irrespective of whether the conformal field theory (CFT) is rational or irrational. For conformal field theories with a holographic dual, the large ccc limit plays an essential role in ensuring that the EE computed from the CFT is universal (independent of the details of the CFT and of boundary conditions) and is exactly matched by the holographic EE. The dual geometry is computed and it interpolates between a BTZ black hole at large LLL and global AdS at large βββ.
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 whether the relations between the Weyl anomaly, entanglement entropy (EE), and thermal entropy of a two-dimensional (2D) conformal field theory (CFT) extend to 2D boundaries of 3D CFTs, or 2D defects of D≥3 CFTs. The Weyl anomaly of a 2D boundary or defect defines two or three central charges, respectively. One of these, b, obeys a c theorem, as in 2D CFT. For a 2D defect, we show that another, d_{2}, interpreted as the defect's "conformal dimension," must be non-negative if the averaged null energy condition holds in the presence of the defect. We show that the EE of a sphere centered on a planar defect has a logarithmic contribution from the defect fixed by b and d_{2}. Using this and known holographic results, we compute b and d_{2} for 1/2-Bogomol'nyi-Prasad-Sommerfield surface operators in the maximally supersymmetric (SUSY) 4D and 6D CFTs. The results are consistent with b's c theorem. Via free field and holographic examples we show that no universal "Cardy formula" relates the central charges to thermal entropy.
We evaluate one-interval Rényi entropy and entanglement entropy for the excited states of two-dimensional conformal field theory (CFT) on a cylinder, and examine their differences from the ones for the thermal state. We assume the interval to be short so that we can use operator product expansion (OPE) of twist operators to calculate Rényi entropy in terms of sum of one-point functions of OPE blocks. We find that the entanglement entropy for highly excited state and thermal state behave the same way after appropriate identification of the conformal weight of the state with the temperature. However, there exists no such universal identification for the Rényi entropy in the short-interval expansion. Therefore, the highly excited state does not look thermal when comparing its Rényi entropy to the thermal state one. As the Rényi entropy captures the higher moments of the reduced density matrix but the entanglement entropy only the average, our results imply that the emergence of thermality depends on how refined we look into the entanglement structure of the underlying pure excited state.