Refined Holographic Entanglement Entropy for the AdS Solitons and AdS black Holes

Refined Holographic Entanglement Entropy for the AdS Solitons and

AdS black Holes

Masafumi Ishihara NCTU

Collaborators: Feng-Li Lin and Bo Ning (NTNU)

arXiv:1203.6153

Contents

●

Entanglement Entropy

●

AdS-soliton.

●

AdS Black Hole

●

AdS-soliton and AdS Black Hole with Gauss- Bonnet term.

●

AdS/MERA

●

Summary

Entanglement Entropy

We divide the total system into two parts; region a and region b.

Entanglememt Entropy S^a is defined as von Neumann entropy with the reduced density matrix ρ^a which is obtained by tracing out area “b” from the total system density matrix ρ^tot.

S^a=-Tr^a (ρ^a Log ρ^a ) ρ^a=Tr^bρ^tot

EE counts the number of correlations between region a and regoin b

Holographic Entanglement Entropy

Holographic Entanglement Entropy (Ryu and Takayanagi '06)

S

= A/(4G

)

A is the area of the minimal surface in the bulk gravity background whose boundary is a.

c

Entanglement Entropy S has UV-divergence since QFT has infinite number of degrees of freedom.

Thus we have to subtract the UV divergent piece

appropriately to get the term which is independent of UV- cutoff ε. We call this UV-independent entanglement

entropy S

S^UV-ind is considered to count the degrees of freedom at the scale R. By using holographic method, we calculated the RG-flow (S^UV-ind/dR ) for the disk region on (2+1)-

dimensional gapped filed theory and finite temperature field theory which are dual to AdS-soliton and AdS Black Hole respectively.

We also consider AdS-soliton and AdS-Black Hole with higher derivative correction ( Gauss-Bonnet term).

AdS-Soliton

First, we consider the AdS⁵

-

r is a radial direction on the boundary.

The area of minimal surface A and the equation of motion for r(z) are given by

There are two types of solutions; Disk topology solutions (blue line) and Cylinder topology solutions (red line)

Solution r(z) is expanded around UV (z~0) region as

where μ　is arbitrary number.　a⁴(R) will be determined by solving full equation of motion numerically.

f

By Hamilton-Jacobi method, dA/dR with UV-cutoff (z=ε) fixed becomes (Liu and Mezei '12 )

The first term of the above equality is dropped because of the IR boundary condition;

for disk topology

for cylinder topology

By using the formula of the previous page and UV expansion of the solution r(z) , we can obtain

By redefining　μ　as

1/R³ term is absorbed in the UV-cutoff as follows.

There is also 1/R³ term in a⁴(R). So, we have to subtract

1/R³ term from a⁴[R] to obtain independent term of UV-cutoff ε in dA/dR. This can be done by solving the full equation of motion numerically.

Overall, the RG flow of the UV-indepedent part of the holographic entanglement entropy for AdS⁵ soliton is

where is a⁴(R) with the term 1/R³ being

subtracted off. For z⁰=1, a⁴(R) becomes as follows for small R (Disk topology) and large R (cylinder topology) region respectively.

Around the transition point between disk and cylinder

solutoin, we chose a solution which has the larger on-shell action.

We find that for R<0.682 and 0.702<R<0.725 disk topology dominates and for 0.725<R cylinder topology dominates

In order to obtain dS^UV-ind/dR, we subtract -0.0687/R³ from a⁴(R)for R<0.682 and -0.0174/R³ for R>0.725. Around critical point, we subtract -0.0435/R³ which is just an interpolating

value of the 1/R³ terms of the above two values.

Then, We can get UV-independent term: dS^UV-ind/dR.

There are two discontinuous jumps at R=0.682 and 0.725.

Since dS^UV-ind/dR is negative, S^UV-ind is monotonically decreasing.

Topological Entanglement Entropy

Topological entanglement entropy (TEE) γ is the constant piece of entanglement entropy which is independent of R. (e.g. S=αR-γ) To obtain γ, we expand the action in terms of R as

where, r¹ and r² is given as the coefficient functions of large R expansion of r(z).

The equation of motion in the large R expansion becomes

By solving it with the boundary condition r¹(0)=0 ( ∵ r(0)=R) and , we can get r¹(z)=0

From the large R expansion of A and introducing r¹(z)=0, , we can find that topological entanglemen entropy

becomes zero.

Extremal AdS ⁵ soliton

Next we cosider the metric of extremal AdS⁵ soliton which is obtained by double wick rotating the extremal charged AdS5

Black Hole.

In this case, there is only disk topology solution in all R

region. dS^UV-ind/dR is calculated by the same method and it becomes negative.

We also find that TEE is zero.

AdS ⁴ soliton

We also consider the AdS⁴-soliton metric whose boundary is (1+1)-dimansional field theroy.

and calculated dS^uv-ind/dR by the same method.

In this case, dS

/dR becomes positive for small R

region.

AdS ⁴ Black Hole

We consider the AdS4 black hole which is dual to the (2+1)- dimensional field theory with finite temperature T and

chemical potential μ

(hartnoll '11).

Thermal entropy density is given by

The area of minimal surface is given by

dA/dR is obtanined in the same way as

where　a³(R) is given by UV-expansion of r(z)

Then, by using the same method, we can get

For large R, it is consistent with the linear running of the f

thermal entropy.

f

We also calculate the finite term of entanglement entropy S^finite by subtracting the divergent term. Blue(Red) line is S^finite

obtained by the Disk (Cylinder ) topology solutions. When there are blue and red S^finite with same R, we choose the larger one.

For small R region, S^finite obeys area law. For large R, it obeys volume law which is same as thermal entropy.

There is no phase transition but a smooth crossover from small R region to large R region (B. Swingle and T. Senthil '10).

Extremal AdS Black Hole

We also consider the extremal AdS Black Hole metric which is given by

In this case, there is only disk topology solutions. We show the dS^UV-ind/dR. There is also the crossover.

AdS-soliton with Gauss Bonnet correction

We consider the effect of the Gauss-Bonnet term to S^UV-ind. The metric is ^（Ogawa and Takayanagi '11)

This metric is obtained from the following action.

where λ^GB 　is the coupling constant and Gauss Bonnet term is given as

The holographic entanglement entropy in the GB gravity is given by minimizing the following functional

where R is the intrinsic curvature of the induced metric h h^b is the induced metric on the boundary of γ^A and Κ is the trace of its extrinsic curvature.

Then, by using the metric, the functional we need to minimize is

d

The equation of motion is

d

The UV expansion of solutoin r(z) becomes where

There are three types of solutions; Disk topology solutions (small R, blue), Cylinder topology solution(large R, red) and solutions with a cusp shape(middle R region green). About the solutions with a cusp shape, there are infinite number of solutions with the same R. Thus we cannot chose the unique solution for this region. We skip the discussion about the cusp shapes this time.

d

By using the same method, we can get

where K and c¹ and c² are defined as

d

The first term is calculated for disk topology solution as

and for cylinder topology solution, it becomes zero because of the boundary condition on z=z⁰

f

By calculating Ra⁴(R) and H(z^m)dz^m/dR numerically and subtracting 1/R³ term we can get RG flow^.

Blue and red line denote dS^UV-ind/dR for Disk topology solutions and cylinder topology solution respectively.

We can also get that topological entanglement entropy is zero by using the same method on the AdS⁵-soliton case.

AdS black hole with GB term

We also consider the AdS⁵ black hole with GB term. The metric is obtained as

where f(z) and f⁰ are the same as previous case.

By using the same method, we can obtain

where, a⁴(R) is definied as follows from the UV-expansion of solution r(z)

d

We show S^finite and dS^UV-ind/dR. When λ^GB=0, there is no phase transition but smooth cross over between disk and cylinder

solutions. On the other hand, when λ^GB becomes finite, there is phase transitions and crossover is violating. This means that

crossover appears in the very strongly coupled field theory.

AdS/MERA

Recently, the relation between AdS and MERA (multi-scale entanglement renomalization anstaz) is proposed (B. Swingle '09).

MERA is one of the Tensor network states which describe ground states of quantum Many-body Hamiltonians on a lattice.

The vertical line of MERA represents RG scale.

AdS soliton corresponds to MERA with top layer.

non-extremal AdS soliton : Cylinder topology in large R region which corres ponds to product states

Extremal AdS Soliton: Disk topology in large R region which corresponds to

entangled state even though its topolocial entanglement entropy is zero.

Summary

● We calculate the RG-flow (dS^UV-ind/dR) on a disk region for the holographic dual theories to the AdS-solitons and AdS black holes, including the corrected ones by the

Gauss-Bonnet term. RG-flow is negative for most of the cases.

● There is no topological entanglement entropy for AdS⁵- soliton even with Gauss-Bonnet correction

● For the AdS black holes, S^UV-ind obeys the area law at small R and volume law at large R. The transition

between them is a crossover. However, the crossover will turn into the phase transition by the Gauss-Bonnet term.

● We propose that non-extremal (extremal) AdS-soliton

state has the product (entangled) state for large R region by using AdS/MERA.