ENTANGLEMENT ENTROPY FROM SURFACE TERMS IN GENERAL RELATIVITY
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International Journal of Modern Physics D Vol. 22, No. 12 (2013) 1342020 (6 pages) c World Scientific Publishing Company DOI: 10.1142/S0218271813420200 ENTANGLEMENT ENTROPY FROM SURFACE TERMS IN GENERAL RELATIVITY ∗ ARPAN BHATTACHARYYA † and ANINDA SINHA ‡ Centre for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India † [email protected] ‡ [email protected] Received 16 May 2013 Revised 23 August 2013 Accepted 9 September 2013 Published 1 October 2013 Entanglement entropy in local quantum field theories is typically ultraviolet divergent due to short distance effects in the neighborhood of the entangling region. In the context of gauge/gravity duality, we show that surface terms in general relativity are able to cap- ture this entanglement entropy. In particular, we demonstrate that for 1 + 1-dimensional (1 + 1d) conformal field theories (CFTs) at finite temperature whose gravity dual is Banados–Teitelboim–Zanelli (BTZ) black hole, the Gibbons–Hawking–York term pre- cisely reproduces the entanglement entropy which can be computed independently in the field theory. Keywords : Holography; entanglement entropy; AdS/CFT. PACs Number: 11.25.Tq 1. Introduction The Einstein–Hilbert action for gravity needs to be supplemented by the Gibbons– Hawking–York surface term 1,2 to make the variational (Dirichlet boundary value) problem well-defined. Namely, the total action (in Euclidean signature) is given by I tot = − 1 2 d-1 P d d+1 x √ g(R − 2Λ) + I GHY , I GHY = − 1 d-1 P d d x √ hK, (1) where g is the determinant of the bulk metric, R is the scalar curvature for the bulk spacetime Λ is the cosmological constant, I GHY is the Gibbons–Hawking– York surface term and K is the extrinsic curvature defined on the boundary surface ∗ This essay received an honorable mention in the 2013 Essay Competition of the Gravity Research Foundation. 1342020-1 Int. J. Mod. Phys. D Downloaded from www.worldscientific.com by PENNSYLVANIA STATE UNIVERSITY on 10/05/13. For personal use only.
Transcript of ENTANGLEMENT ENTROPY FROM SURFACE TERMS IN GENERAL RELATIVITY
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International Journal of Modern Physics DVol. 22, No. 12 (2013) 1342020 (6 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218271813420200
ENTANGLEMENT ENTROPY FROM SURFACE TERMSIN GENERAL RELATIVITY∗
ARPAN BHATTACHARYYA† and ANINDA SINHA‡
Centre for High Energy Physics,Indian Institute of Science,Bangalore 560 012, India†[email protected]‡[email protected]
Received 16 May 2013Revised 23 August 2013
Accepted 9 September 2013Published 1 October 2013
Entanglement entropy in local quantum field theories is typically ultraviolet divergentdue to short distance effects in the neighborhood of the entangling region. In the contextof gauge/gravity duality, we show that surface terms in general relativity are able to cap-ture this entanglement entropy. In particular, we demonstrate that for 1+1-dimensional(1 + 1d) conformal field theories (CFTs) at finite temperature whose gravity dual isBanados–Teitelboim–Zanelli (BTZ) black hole, the Gibbons–Hawking–York term pre-cisely reproduces the entanglement entropy which can be computed independently inthe field theory.
Keywords: Holography; entanglement entropy; AdS/CFT.
PACs Number: 11.25.Tq
1. Introduction
The Einstein–Hilbert action for gravity needs to be supplemented by the Gibbons–Hawking–York surface term1,2 to make the variational (Dirichlet boundary value)problem well-defined. Namely, the total action (in Euclidean signature) is given by
Itot = − 12�d−1
P
∫dd+1x
√g(R − 2Λ) + IGHY ,
IGHY = − 1�d−1P
∫ddx
√hK,
(1)
where g is the determinant of the bulk metric, R is the scalar curvature for thebulk spacetime Λ is the cosmological constant, IGHY is the Gibbons–Hawking–York surface term and K is the extrinsic curvature defined on the boundary surface
∗This essay received an honorable mention in the 2013 Essay Competition of the Gravity ResearchFoundation.
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A. Bhattacharya and A. Sinha
with the determinant of the induced metric given by h. Quite remarkably, evenbefore Gibbons and Hawking or York, Einstein had written down a first-orderlagrangian for gravity given by gab(Γn
maΓmnb − Γm
mnΓnab) which differs from Itot by
a surface term.3,4 The surface terms in general relativity are crucial not only toproduce a well-defined variational principle, but also to produce correct black holethermodynamics. These terms are also important to compute the correct Noethercharges arising from diffeomorphism invariance.5 Furthermore, by evaluating theseterms on a black hole horizon, one reproduces black hole entropy.6,7
In this paper, we will argue that the Gibbons–Hawking–York surface term givesthe entanglement entropy in gauge/gravity duality. Entanglement entropy is a use-ful nonlocal probe of how much the degrees of freedom in a region of spacetimeare entangled with the rest. The original motivation for considering entanglemententropy was the hope that such considerations would shed light on the microscopicorigin of black hole entropy.8–10 However, entanglement entropy is a useful tool inother areas of physics also, such as condensed matter systems. Recently, it has beenused to shed light on time-dependent physics as well, where direct computationaltechniques are not available.
It is known that entanglement entropy in conformal field theories (CFTs) ineven d-dimensions take the form
SEE = cdld−2
εd−2+ O
(ld−3
εd−3
)+ ad log
l
ε+ O
((l
ε
)0)
, (2)
where l is a length scale parametrizing the size of the entangling region and ε isa short-distance cutoff. The leading ld−2 term gives the famous area law with anonuniversal proportionality constant — when d = 2 the leading term is the logterm. In even dimensions, the coefficient of the log term is a universal quantitytypically related to a function of the conformal anomalies in the theory while inodd dimensions the log term is replaced by a constant which is considered to bea measure of the degrees of freedom.11 In the context of quantum field theories, adirect computation of entanglement entropy is hard and has been possible only invery specific examples.12,13 The leading term in SEE is proportional to the area andgives the famous area-law. This was the main motivation for trying to relate blackhole entropy with entanglement entropy of quantum fields.
The gauge/gravity correspondence or the AdS/CFT correspondence gives a wayto relate a quantum (typically conformal) field theory in d-dimensions to a theory ofgravity in anti-de Sitter (Ads) backgrounds in d+1-dimensions.14 The prescriptionin gauge/gravity duality to compute entanglement entropy in conformal field theory(CFT) is the following. The CFT is supposed to live on the boundary (CFT) ofthe AdS space. One considers t = 0 slice of this boundary. Then, a spatial region∂N of this boundary is considered and a minimal surface M extending into thebulk is found which ends on ∂N . Ryu and Takayanagi (RT) proposed15 that thearea of this minimal surface is the entanglement entropy. To determine this minimalsurface one considers an “area functional” which is then minimized. This functional,
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Entanglement Entropy from Surface Terms in General Relativity
when evaluated on the black hole horizon, would lead to the black hole entropy.At the onset, we should emphasize that this is a prescription, which passes severalchecks, with no general proof. Unlike the Wald entropy formula which is valid fora general theory of gravity, no such analog exists for the entanglement entropy.This is an unfortunate state of affairs which needs remedy in the near future sinceholographic methods are becoming a popular tool to gain intuition about physicsat strong coupling.a
For computational purposes, one typically uses Einstein gravity in a weaklycurved AdS background which corresponds to a strongly coupled CFT. Accordingto the RT prescription, in order to derive the holographic entanglement entropyfor d-dimensional quantum field theory, one has to minimize the following entropyfunctional on a d − 1-dimensional hypersurface (a bulk co-dimension two surface),
S =2π
�d−1P
∫dd−1x
√h, (3)
where �P is the Planck length and h is the induced metric on the hypersurface.The resulting surface is a minimal surface with vanishing extrinsic curvature. Thegravity dual theory is simply Einstein gravity with a negative cosmological constantΛ = −d(d − 1)/(2L2) where L is the AdS radius. This procedure can be also usedto compute entanglement entropy for the finite temperature field theory. It corre-sponds to the presence of a black hole in the bulk spacetime. We will consider theBanados–Teitelboim–Zanelli (BTZ) black hole as the result for the corresponding1 + 1-dimensional (1 + 1d) CFT is well-known. Let us consider the following metricfor the (nonrotating) BTZ black hole,
ds2 = (r2 − r2H)dt2 +
L2
(r2 − r2H)
dr2 + r2dφ2, (4)
where r = rH is the horizon. Next, we put φ = f(r) into the metric and evaluateS. After finding the Euler–Lagrange equation for f(r) from S, we can determinef(r) i.e. how the entangling surface extends into the bulk spacetime (see Fig. 1):
f(r) =L
rHtanh−1
√r2 − r2
0√r2 − r2
H
rH
r0, (5)
where r0 = rH coth rH l/L2, l being the length of the entangling surface in the fieldtheory. Then using this solution, assuming rH l/L2 � 1 which corresponds to thehigh temperature phase of the field theory, and evaluating S one gets the following
aWhen the entangling surface is spherical, there is a way to compute entanglement entropy inholography given in Ref. 16. A releated earlier observation was given in Ref. 17. After this essaywas submitted for the competition, there was a suggestion for a proof in Ref. 18. Previous attemptsinclude Ref. 19.
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Fig. 1. Entangling surface extending into the bulk AdS. One has to add a Gibbons–Hawking–York term for both the upper and lower branch.
well-known result for the log part of the entanglement entropy:
SEE =2π
�P
∫ 1ε
r0
dr2rL√
(r2 − r20)(r2 − r2
H)=
c
3log(
β
πεsinh
(πl
β
))+ O(ε), (6)
where c = 12πL/�P is the central charge of the two-dimensional CFT, ε is theUV cut-off, l is the length of the entangling surface and β = 1/(LT ) = 2πL/rH isidentified as the periodicity in the time coordinate which is related to the inversetemperature T of the field theory. An independent calculation in 1 + 1d producesexactly this result12 which is taken to be a strong evidence for the validity of theminimal area prescription. Any proposal for the entanglement entropy should agreewith this.
Although this prescription passes certain nontrivial checks such as the strongsub-additivity condition, how does one reproduce the entangling surface in field the-ory? According to the AdS/CFT dictionary, the AdS radius is to be identified withan renormalization group (RG) scale. So naively one would expect that the radiusof the entangling region should become a function of the RG scale. But what arethe rules? Furthermore, what observable do we use in field theory to probe entan-glement entropy? The RT prescription does not appear to provide direct answersto these questions. We can partially remedy this with the following observation.Consider field theory on the bulk co-dimension one slice φ = f(r). We will not sett = 0. Since the time direction is a direct product with the rest, the trace of theextrinsic curvature satisfies (d)Ka
a = (d)Ktt +(d−1)Ki
i . Thus, (d)Ktt − ht
t(d)Ka
a = 0leads to (d−1)Ki
i = 0 which is the same as the minimal surface condition for the d−1slice used in the RT calculation. But the combination (d)Kt
t −htt(d)Ka
a is nothing butthe tt component of the usual Brown–York (holographic20) stress tensor evaluated
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Entanglement Entropy from Surface Terms in General Relativity
on the co-dimension one slice in d-dimensions! Here, htt is simply the tt componentof the pullback metric on the co-dimension one slice — in the Brown–York tensorthe indices are raised and lowered using this metric.
Thus, an alternate way to compute entanglement entropy in gauge/gravity dual-ity presents itself.21 We first compute the time–time component of the Brown–Yorkstress tensor on the co-dimension one entangling surface given by φ = f(r). Set thisto zero and determine f(r). The above argument guarantees that f(r) will workout to be the same as what follows from the RT prescription. Now it is intuitive,that since entanglement entropy is related to the common boundary between thedegrees of freedom living in the two regions, one of which is traced over, it mustbe related to the surface terms in general relativity. Now, recall that the RT areafunctional was such that when it was evaluated on the black hole horizon, we gotthe black hole entropy. The Gibbons–Hawking–York term evaluated on the hori-zon of a black hole is also known to yield black hole entropy.6,7 Let us work outthe Gibbons–Hawking–York surface term in our case explicitly. Unlike the RT areafunctional, there is a time integral in this case. But we know that time has to beperiodic, with period β = 1/T , T being the temperature of the BTZ black hole.After some straightforward algebra, we get
IGHY =1�P
∫ β= 2πLrH
0
dt
∫ 1ε
r0
dr2rr0√
(r2 − r20)(r2 − r2
H),
=c
3log(
β
πεsinh
(πl
β
))+ O(ε),
(7)
wherein going to the second line we have assumed that the field theory is in thehigh temperature phase which makes r0 ≈ rH . Thus, the RT result is identical towhat comes from the Gibbons–Hawking–York surface term. This agreement canbe shown to hold even at zero temperatures if one makes time periodic with theperiodicity related to the inverse Unruh temperature. Also, this connection holds forany dimensions not just 1+1d. For the computations in higher dimensions, readersare referred to Refs. 21 and 22. This method can also be applied for stationarycases such as the rotating BTZ. The explicitly time-dependent situations are leftfor future work.
2. Conclusions
We have shown that entanglement entropy for field theories having holographicduals are related to the surface terms arising in general relativity. This may pointat a more systematic way of computing entanglement entropy by relating it toNoether charges in general relativity. Since the procedure for computing entangle-ment entropy was given in terms of the Brown–York stress tensor, this naturallysuggests a possible way to find out about the entangling surface using field theorymethods. Some evidence for this has been presented in Ref. 21.
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Acknowledgments
We thank Gautam Mandal, Rob Myers and Tadashi Takayanagi for their usefuldiscussions and correspondence.
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