## 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*^{b}*ρ*^{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}*= A/(4G*

^{N}*)*

*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

^{UV-ind.}

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 AdS^{5}

### -

soliton which is dual to the gapped (2+1)-dimensional field theory.*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. a*^{4}*(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/R^{3} term is absorbed in the UV-cutoff as follows.

There is also 1/R^{3} term in a^{4}(R). So, we have to subtract

1/R^{3} term from a^{4}[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 AdS^{5} soliton is

where is a^{4}(R) with the term 1/R^{3} being

*subtracted off. For z*^{0}*=1, a*^{4}*(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/R^{3} from
a^{4}(R)for R<0.682 and -0.0174/R^{3} for R>0.725. Around critical
point, we subtract -0.0435/R^{3} which is just an interpolating

value of the 1/R^{3} 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, r^{1} and r^{2} 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 r^{1}(0)=0 ( ∵
r(0)=R) and , we can get r^{1}(z)=0

From the large R expansion of A and introducing r^{1}(z)=0,
, we can find that topological entanglemen entropy

becomes zero.

### Extremal AdS ^{5} soliton

Next we cosider the metric of extremal AdS^{5} 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 AdS^{4}-soliton metric whose
boundary is (1+1)-dimansional field theroy.

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

### In this case, dS

^{UV-ind}

### /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 a^{3}(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 c*^{1} and c^{2}* 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=z^{0}

f

*By calculating Ra*^{4}*(R) and H(z*^{m}*)dz*^{m}*/dR numerically and *
subtracting 1/R^{3} 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 AdS^{5}-soliton
case.

## AdS black hole with GB term

We also consider the AdS^{5} black hole with GB term. The
metric is obtained as

*where f(z) and f** ^{0}* are the same as previous case.

By using the same method, we can obtain

where, a^{4}(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 AdS^{5}-
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.