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, 1411.6213 , with S. He, D. Li and H.-Q. Zhang, work in progress

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

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/0405152 Amico, Fazio, Osterloh and Vedral, quant-ph/0703044 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

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/9404039

A holographic proposal—the HEE Ryu and Takayanagi PRL96, 181602 (06) is the d dimensional static minimal surface in AdS with boundary . B z A boundary

The HEE formulae has been proofed by Fursaev , hep-th/0606184; Method been improved by Lewkowycz and Maldacena, 1304.4926; Another derivation of the HEE for spherical entangling surfaces by Casini, Huerta and Myers, 1102.0440; Other evidences to confirm the HEE proposal within the AdS_3/CFT_2 correspondence are Headrick, 1006.0047, Hartman, 1303.6955, Faulkner, 1303.7221. Many applications: Applications of HEE to black holes Emparan, hep-th/0603081; Probing the confinement/deconfinement phase transition in the large N gauge theories, Klebanov, Kutasov and Murugan, arXiv:0709.2140; 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:1202.2068…

Many generalizations: Higher derivative terms corrections higher curvature effects from bulk (super)gravity \alpha' correction) Gauss-Bonnet term correction, Fursaev , hep-th/0606184; Gravitational Chern-Simons term correction, JRS, 0810.0967; Castro, Detournay, Iqbal and Perlmutter, 1405.2792; Lovelock gravity correction, de Boer, Kulaxizi and Parnachev, 1101.5781; Hung, Myers and Smolkin, 1101.5813; …

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, 1101.2881; Another limit is to consider entanglement entropy of local excited states in which the subsystem is of large size. Nozaki, Numasawa and Takayanagi, 1302.5703; He, Numasawa, Takayanagi and Watanabe, 1403.0702

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, 1212.1164; Strip: Sphere:

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, 1305.2682; 1st order variation of entropy=1st order entanglement Hamiltonian Blanco, Casini, Hung and Myers, 1305.3182 Relation between the ground state entanglement Hamiltonian and the stress tensor derived in path integral formalism Wong, Klich, Pando Zayas and Vaman, 1305.3291; 1st law like relation was shown to be equivalent to the 1st order perturbative Einstein equations in pure AdS Nozaki, Numasawa, Prudenziati and Takayanagi, 1304.7100; Lashkari, McDermott and Van Raamsdonk, 1308.3716 Faulkner, Guica, Hartman, Myers and Van Raamsdonk, 1312.7856 What about the HEE with second order low E excitations?

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, 1305.3182 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, 1401.5089

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

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

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

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

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

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

In the same sense, then the area is

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

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

The energy associated with the subsystem A is

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.

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.

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

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

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

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

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

AAdS_3 spacetime Charged black hole as perturbation away from pure AdS_3

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

From the CFT side Rosenhaus and Smolkin, 1410.653 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.

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 …

Thank You!