1. On Holographic Entanglement Entropy with Second Order Excitations
Jia-Rui Sun
Institute of Astronomy and Space Science
Sun Yat-Sen University
Holographic duality for condensed matter physics
KITPC, July 2015
with S. He and H.-Q. Zhang, ,
with S. He, D. Li and H.-Q. Zhang, work in progress

2. Outline The entanglement entropy and HEE
HEE with first order excitations
HEE with second order excitations
Remarks

3. The entanglement entropy and HEE
The quantum entanglement is of great importance in characterizing the correlation between non-local physical quantities in quantum many-body systems.
Calabrese and Cardy, hep-th/
Amico, Fazio, Osterloh and Vedral, quant-ph/
For a pure state with density matrix \rho, divide the system into A+B, the entanglement entropy can be calculated by the von Neumann entropy A B

4. For a 2d field theory at critical point (zero T), the EE is
At finite T, the EE is
For higher dimensional QFT, the EE is proportional to the area of the boundary—area law+sub-leading corrections.
Since EE also describes the lack of information for observer in A to B, it has been used to explain the origin of the black hole Bekenstein-Hawking entropy
Bombelli etc, PRD34, 373 (1986); Jacobson, gr-qc/

5. A holographic proposal—the HEE
Ryu and Takayanagi PRL96, (06)
is the d dimensional static minimal surface in AdS
with boundary B z A
boundary

6. The HEE formulae has been proofed by Fursaev , hep-th/0606184;
Method been improved by Lewkowycz and Maldacena, ;
Another derivation of the HEE for spherical entangling surfaces by Casini, Huerta and Myers, ;
Other evidences to confirm the HEE proposal within the AdS_3/CFT_2 correspondence are
Headrick, ,
Hartman, ,
Faulkner,
Many applications:
Applications of HEE to black holes Emparan, hep-th/ ;
Probing the confinement/deconfinement phase transition in the large N gauge theories, Klebanov, Kutasov and Murugan, arXiv: ;
Probing phase structures in condensed matter systems, Takayanagi, Gen.Rel.Grav. 46 (2014) 1693 ;
Using HEE to study the renormalization group in QFTs, Myers and Singh, arXiv: …

7. Many generalizations:
Higher derivative terms corrections
higher curvature effects from bulk (super)gravity \alpha' correction)
Gauss-Bonnet term correction,
Fursaev , hep-th/ ;
Gravitational Chern-Simons term correction,
JRS, ;
Castro, Detournay, Iqbal and Perlmutter, ;
Lovelock gravity correction,
de Boer, Kulaxizi and Parnachev, ;
Hung, Myers and Smolkin, ; …

8. HEE with first order excitations
Quantum corrections
From the QFT side, the universal behavior of entanglement entropy in low-energy excited states is very important in understanding the quantum entanglement nature of the system,
For 2d CFT with small subsystem
Alcaraz, Berganza and Sierra, ;
Another limit is to consider entanglement entropy of local excited states in which the subsystem is of large size.
Nozaki, Numasawa and Takayanagi, ;
He, Numasawa, Takayanagi and Watanabe,

9. How to calculate the low E excitation corrections holographically?
From the HEE proposal, this kind of quantum corrections to the boundary field theory are associated with the perturbations from the bulk gravitational theory.
First law-like relation for the HEE
Bhattacharya, Nozaki, Takayanagi and Ugajin, ;
Strip:
Sphere:

10. 1st law-like relation in higher derivative gravity
Recently, the First law-like relation for the HEE has been studied for many cases
1st law-like relation in higher derivative gravity
Guo, He and Tao, ;
1st order variation of entropy=1st order entanglement Hamiltonian
Blanco, Casini, Hung and Myers,
Relation between the ground state entanglement Hamiltonian and the stress tensor derived in path integral formalism
Wong, Klich, Pando Zayas and Vaman, ;
1st law like relation was shown to be equivalent to the 1st order perturbative Einstein equations in pure AdS
Nozaki, Numasawa, Prudenziati and Takayanagi, ;
Lashkari, McDermott and Van Raamsdonk,
Faulkner, Guica, Hartman, Myers and Van Raamsdonk,
What about the HEE with second order low E excitations?

11. HEE with second order low E excitations
When the bulk minimal surface is the sphere, the quadratic correction to the relative entropy has been discussed when the bulk is an neutral black brane
Blanco, Casini, Hung and Myers,
Also valid with additional operators such as scalar and current.
The relation has been checked for entangling sphere in the presence of Gauss-Bonnet term correction, constraints on the coupling constant also been studied.
Banerjee, Bhattacharyya, Kaviraj, Sen and Aninda Sinha,

12. We will study the HEE with 2nd order quantum corrections when the boundary is a strip
The area of the bulk co-dimensional 2 surface in d+1-dimensional spacetime is
Up to 2nd order perturbation of the bulk metric

13. For the perturbations of the induced metric, there are two parts of contributions, one is from the bulk metric perturbation, another is from the change of the embedding

14. We choose the background to be the d+1 dim RNAdS black brane in EMD theory
The cosmic censorship conjecture requires
In Poincare coordinate

15. We treat the charged black brane as the perturbed geometry deviates away from the pure AdS via the mass and charge, and consider perturbations up to 2nd order of the small mass and charge.
Computing the minimal surface: dividing the boundary CFT into two intervals A+B, A be a strip in x1 ∈ [−l/2, l/2] and xb ∈ [−L0/2,L0/2], the are of bulk co-dimenisonal-2 surface is
The solution is

16. z_* :turning point of the bulk minimal surface, reflecting the embedding.
The area is
where from the UV/IR relation in AdS/CFT
The solution can be written as

17. Which can be solved perturbatively in terms of the dimensionless mass and charge

19. In the same sense, then the area is

20. the 1st order HEE is always positive, the 2nd order HEE is always negative and it’s bounded by the cosmic censorship conjecture
Computing the boundary stress tensor:
The tress tensor of the boundary CFT is obtained from the Brown-York tensor,
up to 2nd order bulk metric perturbation

21. For the spatially flat boundary, the boundary counterterm is
For the d+1 dim RNAdS black brane

22. The energy associated with the subsystem A is

23. The 1st order result gives the 1st law-like relation
Naively, at the 2nd order, the result shows
Similar to the relative entropy for spherical entangling surface.

24. However, recall that the bulk gravity dual we studied is a
stationary RN-AdSd+1 black brane, so it is expectable that the perturbed geometries are also stable, order by order away from the pure AdSd+1 spacetime under small fluctuations.
Supporting evidence:
1. the bulk perturbed Einstein equations are satisfied both at the first and second orders;
2. the equivalent relation between the 1st law-like relation with the 1st order Einstein equation.
it is natural to expect that at higher orders perturbation, the 1st law-like relation should dual to bulk perturbed dynamical equations at the same order.

25. Namely, at 2nd order perturbation, the 1st law-like relation should be

26. AAdS_3 spacetime
Spinning BTZ black hole as perturbation away from pure AdS_3

27. To calculate the HEE for the spinning BTZ black hole, the covariant HEE method is required. Note that the BTZ black hole is locally identical to the pure AdS3
Then the HEE for the subsystem A is
and

28. The bulk minimal surface is a minimal curve
where ML>=J is used and

29. For the same reason, at 2nd order perturbation, the 1st law-like relation should be
recall that the conformal dimension of the boundary stress tensor of the CFT2 dual to the bulk massless graviton is
which is consistent with the estimation from CFT_2

30. AAdS_3 spacetime
Charged black hole as perturbation away from pure AdS_3

31. But as before, at 2nd order, the correct 1st law-like relation should be

32. From the CFT side
Rosenhaus and Smolkin,
This relation can be seen from the variation of the area functional from the field/operator duality in the gauge/gravity duality.
The 2nd order excitation of the HEE corresponds to the 2-point correlation functions of the associated operators in the dual CFT, such the stress tensor-stress tensor, current-current, scalar-scalar, etc.

33. Remarks
We showed that the 1st law-like relation is hold for the strip-shape boundary CFT up to 2nd order perturbations.
The HEE will be naturally bounded at quadratic order perturbations by the requirement from the bulk cosmic censorship conjecture.
Many interesting problems:
The 2nd order dynamics of HEE, gauge invariant form;
Deriving a more general formalism for 2nd order HEE ;
Highly excited quantum states’ contribution to EE …

34. Thank You!