Refined Holographic Entanglement Entropy for the AdS Solitons and
AdS black Holes
Masafumi Ishihara NCTU
Collaborators: Feng-Li Lin and Bo Ning (NTNU)
AdS Black Hole
AdS-soliton and AdS Black Hole with Gauss- Bonnet term.
We divide the total system into two parts; region a and region b.
Entanglememt Entropy Sa is defined as von Neumann entropy with the reduced density matrix ρa which is obtained by tracing out area “b” from the total system density matrix ρtot.
Sa=-Tra (ρa Log ρa ) ρa=Trbρtot
EE counts the number of correlations between region a and regoin b
Holographic Entanglement Entropy
Holographic Entanglement Entropy (Ryu and Takayanagi '06)
A is the area of the minimal surface in the bulk gravity background whose boundary is a.
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
SUV-ind is considered to count the degrees of freedom at the scale R. By using holographic method, we calculated the RG-flow (SUV-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).
First, we consider the AdS5
-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. a4(R) will be determined by solving full equation of motion numerically.
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 dSUV-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: dSUV-ind/dR.
There are two discontinuous jumps at R=0.682 and 0.725.
Since dSUV-ind/dR is negative, SUV-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
Extremal AdS 5 soliton
Next we cosider the metric of extremal AdS5 soliton which is obtained by double wick rotating the extremal charged AdS5
In this case, there is only disk topology solution in all R
region. dSUV-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 dSuv-ind/dR by the same method.
In this case, dSUV-ind
/dR becomes positive for small R
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
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
We also calculate the finite term of entanglement entropy Sfinite by subtracting the divergent term. Blue(Red) line is Sfinite
obtained by the Disk (Cylinder ) topology solutions. When there are blue and red Sfinite with same R, we choose the larger one.
For small R region, Sfinite 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 dSUV-ind/dR. There is also the crossover.
AdS-soliton with Gauss Bonnet correction
We consider the effect of the Gauss-Bonnet term to SUV-ind. The metric is （Ogawa and Takayanagi '11)
This metric is obtained from the following action.
where λGB 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 hb is the induced metric on the boundary of γA and Κ is the trace of its extrinsic curvature.
Then, by using the metric, the functional we need to minimize is
The equation of motion is
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.
By using the same method, we can get
where K and c1 and c2 are defined as
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
By calculating Ra4(R) and H(zm)dzm/dR numerically and subtracting 1/R3 term we can get RG flow.
Blue and red line denote dSUV-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)
We show Sfinite and dSUV-ind/dR. When λGB=0, there is no phase transition but smooth cross over between disk and cylinder
solutions. On the other hand, when λGB becomes finite, there is phase transitions and crossover is violating. This means that
crossover appears in the very strongly coupled field theory.
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.
● We calculate the RG-flow (dSUV-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, SUV-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.