NITheP WITS node seminar: Prof Jacob Sonnenschein (Tel Aviv University) TITLE: "Hologrqaphic entanglement of confining theories"

Capetown and Johhnesburg August 2014

O. Ben-‐Ami, D.Carmi. U. Kol, C. Nunez D. Schofield M. Warschavski

Entanglement entropy

! The entanglement entropy is a measure of the amount of entanglement of a system, it measures the quantum correlation between two parts of a system

! The entanglement entropy (EE) of A is given by

! Where is the reduced density matrix traced over B

Entanglement Entropy in QFT

! In QFT the EE measures the entanglement of A a region of space and it complementary region B.

! SA is UV divergent and in d space dim can be expanded

where l is the size of A, a is a UV cutoff and S0 is finite


! The leading divergence term is proportional to the area ( called the “ Area law”)

! Computing in QFT the EE is hard but can be done for simple models like 2d CFTs

! For line segment

! For line segment at T


! A line segment part of a circle of circomference L

! For massive field on a semi-‐infinite length

Holographic Entenglement entropy (HEE)

! Holography relates gravity theory in AdSd+2 to CFTd+1 on the boundary of the AdS bulk.

! The metric of the AdS space in Poincare coordinates

! The boundary CFT resides on R^(1,d) located at z=0 ! The metric diverges at z=0 so we take a cutoff z=a


! Ryu and Takanayagi conjectured that minimal surface whose . boundary coincides with the Newton constant boundary of A ! d+2 dim. Newton constant


! One can expand the HEE

! In agreement with the area law and the general result in QFT.

! The ci=0 for i even/odd for d odd/even ! The result assumes smooth boundary ( no cusps) ! For d odd the logarithmic coefficient c0 is proportional to the central charge of the boundary CFT

Line segment in CFT2 –example of HEE

! Example of HEE-‐ 2d CFT ! Consider a line segment with boundaries

! In AdS3 with the metric ! The minimal surface is the Semi-‐circle

! So the minimal area is

! Using we finally find

CFT2 at finite temperature – example of HEE

! For CFT2 at finite temperature we use the BTZ metric

! The minimal length is

! Which coincides with the CFT2 result

An infinite strip in d-‐dimensions-‐ an example of HEE

! For infinite strip in d dimensions

! The result for HEE is

! Only the leading divergent and a finite term.

Confining Wilson Loop �  In SU(N) gauge theories one defines the following gauge

invariant operator

where C is some contour

�  The quark – antiquark potential can be extracted from a strip Wilson line

�  The signal for confinement is E ~ Tst L

Stringy Wilson loop �  The natural stringy dual of the Wilson line ( which obeys the loop equation) is

where is the renormalized Nambu Goto action, namely the renormalized world sheet area

Review – Wilson loop of confninig backgrounds

! For a wide class of backgrounds we can parameterize the metric that depends only on the radial coordinate as follows


! Define

! Then the renormalized Wilson loop is given by

bare energy of WL Subtraction of the “quark masses minimal value of r


! The length of the WL as a function of ρ0

! The WL admits linear ( confining) behavior if either of the two conditions is obeyed E. Schreiber,Y. Kinar J.S

! α(ρ) has a minimum at ρΛ α(ρΛ)>0 ! g(ρ) diverges at ρΛ


! We proved that L(ρ0) is monotonically decreasing

! The Energy is linear at long distances

Outline

! Review-‐Holographic entanglement Entropy (HEE) ! Review-‐ Wilson loops of confining backgrounds ! HEE of non-‐conformal backgrounds ! Similarities and differences between HEE and WL ! Phase transition in HEE of confining theories ! A necessary condition for the phase transition ! Examples of the phase transition ! Relations between HEE and WL for systems of confinement and finite temperature

! Mutual information: multi-‐strips ! 1/N correction ! Summary

HEE for non-‐conformal backgrounds

! How can one generalize the RT HEE prescription? ! For a AdSd+2 we minimized a d-‐dim surface so in general for 10d background Klebanov, Kutasov, Murugan (KKM) suggested to minimize an 8 dim surface .

! Applying the KKM rule on AdSxS5 we get back the RT HEE formula.

! The minimization is of a surface in the Einstein frame. Upon transforming to the string frame we get

HEE of non-‐conformal backgrounds

! Thus the HEE for non-‐conformal systems is given by

10-‐d Newton constant Induced string frame metric

! The HEE is given by minimizing this action over the surfaces that approach γ-‐ the boundary of the entangled region, on the boundary of the bulk.

HEE of non-‐conformal backgrounds

! The entangling surface was taken to be a strip of width l

! The metric in the string frame is taken


Holographic radial coordinate ! The volume of coordinates ! We also define


! H(ρ) and β(ρ) are the key players ! H(ρ) is typically monotonically increasing function ! H(ρ), due to the volume, shrinks to 0 at ρΛ ! β(ρ) is typically monotonically decreasing function ! Upon substituting HEE is given by

ρ0 Minimal value of ρ

The length is


! For a disconnected solution the HEE does not depend on ρ0 and it is

! The HEE is UV divergent but the difference between the connected and disconnected is finite

! If S is positive the true solution is the disconnected, and if it is negative then it is the connected


! L(ρ0) is non-‐monotonic for confining backgrounds


! HEE as a function of L. The butterfly configuration

! connected solution disconnected one


! The connected solution exists only for 0 < L < Lmax. ! In this range there are two possible values for the connected solution.

! The upper branch is an unstable solution. ! This doublevaluedness corresponds to L(ρ0). ! As a result there is a first order phase transition at L = Lc between the connected and the disconnected solutions.

! For this reason KKM have argued that a signal for a phase transition is the non-‐monotonicity of the function L(ρ0).

! Indeed, every peak in L(ρ0) corresponds to a possible phase transition in the entanglement entropy S(L).

Similarities and differences between WL and HEE

! In spite of the fact that in QFT there is no apparent relation between WL and EE their functional form in holography looks similar.

! The length as a function of ρ0 for both cases is

! Where M(ρ) can be written as

Similarities and differences between WL and HEE

! The energy of the WL and the HEE take the form

! This can be written also as

On the relation between WL and HEE

! There is a striking similarity between WL and HEE of theories compactified on time and space S^1


! what are the differences between WL and HEE? ! A major difference between these two observables is the behavior of the function M(ρ) close to

ρ= ρΛ . ! For both cases M(ρ) is a monotonically increasing function, but the behavior close to ρ=ρΛ is different.

! For the entanglement entropy M(ρ) = H(ρ) shrinks there to zero , since the volume shrinks to zero.

! On the other hand the Wilson loop in confining backgrounds M( ρΛ) = α^2(ρΛ) >0 since this quantity is related to the confining-‐string tension.

! Therefore M(ρ) behaves very differently for the two observables, when calculated in a generic confining background.


! To be concrete, let us focus on Dp branes compactified on a circle.

! These backgrounds are dual to confining field theories in p space-‐time dimension.

! The background metric and dilaton are

! Which implies that


! Rc is the radius of the compact circle

! The surface of the n sphere

! Therefore

! Whereas ! This difference is responsible for the butterfly shape of S(L) versus the linearity of E(L) for the WL.

Sufficient conditions for phase transitions

! What is the condition so that a background admits a phase transition?

! This translates to what is the condition for L(ρ0) to have a maximum and hence to be double-‐valuedness?

! Let us assume that around ρ0 =ρΛ

! For r,t>0 the integrand of L diverges at ρ0 =ρΛ and we can approximate

where

Sufficient conditions for phase transitions

! When t<2 L(ρ0) is monotonically increasing which means that β(ρ) should not diverge faster than

! Close to the boundary we can expand

! The asymptotic behavior of L(ρ0)

! For j>2 the L à 0 when hence there is maximum and double-‐valuedness.

! For there will not be any phase transition.

Examples of the criteria for phase transition

! Let’s examine the criteria on ! The relevant functions are

! Now t=4 so the condition for p.t is not obeyed

! the concavity is obeyed.

Dp brane compactified on a circle

! For the Dp branes background compactified on S^1

! Close to the horizon

! So that t=1 and L à0 near the horizon ! Close to the boundary so only for yjtrrtr there is a phase transition but not for p>4.


! For p=3,4


! For p=5,6 the WL confines with no p.t of HEE. Why?

The story of D5 and D6 branes

! We have realized that in spite of the fact that the WL of the compactified D5 D6 branes confines it seems that there is no phase transition in HEE.

! This is caused by the UV non-‐locality of the QFT. ! We found a way to fix this situation, by introducing a hard UV cutoff and observing that new configurations appear that would not only recover the phase transition but also solve an stability problem of the configurations that miss the phase transition.

The story of D5 and D6 branes

Can we understand the phase transition in QCD? ! It is argued the large Nc QCD has a Hagedorn density of states.

! The number of states of mass M, N(M) behaves as

! The corresponding partition function

! Z diverges at large enough temperature ! The system below the Hagedorn temperature is believed to be with Z ~N^0 whereas above it Z~N^2

Can we understand the phase transition in QCD?

! In analogy to the compactified Euclidean time direction KKM argued that for the HEE there is a critical value of L the entangeling segment.

! The EE of non-‐interacting massive glueballs ! So that EE~N^0 for L<Lc and EE~N^2 for L>Lc

Does the phase transition exists at finite N

! It is interesting to ask what the effect of 1/N corrections ! Does the phase transition occur only at

! In other words, will the jump in the derivative of S (L) persist beyond leading order in N. ! In order to answer this question one need to precisely calculate the higher order corrections to the connected and disconnected surfaces and check if the derivative jumps

! This is beyond our current ability. ! Perhaps a easier question is what approximately is the 1/N correction to the disconnected surface.


! Maldacena et al have a suggestion for computing the 1/N correction to the HEE, by computing the EE through the Ryu-‐Takayathey surface .

! As an example the disconnected surface in the Klebanov-‐Strassler model, there will be a correction in the form of a log divergence proportional to the number of goldstones in the KS model.

! Now in our case we consider a strip entangling region therefore the log term should be absent.

! The "Bulk entanglement" should depend on whether the lightest bulk field is massless or massive.

! For massless the correction should be and for a massive


! For Wilson line at finite temperature the phase transition is smoothed out. In the holographic picture it is due of an exchange of a massive mode

! So probably both the phase transition for the WL at finite temperature and the HEE of confining theories will not survive at finite N

HEE of multi stips

! One can use as an order parameter HEE for multiple strips geometry

HEE of multi stips

! The phase diagram for Ads5 compactified on S^1

Summary and open questions

! We investigated the HEE of confining theories ! The similarities and differences between WL and the HEE were discussed

! Sufficient conditions for the phase transition were stated.

! We addressed the puzzle of the Maldacena Nunez background and Dp>4 branes.

! The phase diagram of multiple strips was derived ! We discussed the phase transition at finite N

Introduction

! Solitons-‐ classical static configurations of finite energy show up in a wide range of physical systems

! Solitons are known for instance in hydrodynamics and non linear optics .

! In field theory we have encountered sine-‐Gordon solitons, ‘t Hooft Polyakov monopoles , Skyrmions and Instantons ( solitons of 5d YM theory)

! In recent years solitons take the form of Wilson-‐lines, Dbranes etc.

Introduction

! Determining soliton solutions typically means solving non linear differntial equations.

! One would like to find tools to handle such configurations without solving for them explicitly.

! Two important issues are: (i)  Existence proofs (ii)  Stability of the solutions.

Derrick theorem

! Consider a scalar field in d+1 dimensions with

non negative, ; vanishes for φ=0 ! The energy associated with a static configuration

! Consider a scaling deformation ( not a symmetry)

Derrick theorem

! The energy of the rescaled configuration

! The minimum of the energy is for the un-‐rescaled soliton with λ=1

Derrick theorem

! We now change the integration variable

! The re-‐scaled energy is

! The variation of the energy has to obey

! For d>2 each term has to vanish separately and for d=2 the potential has to vanish. Both cases occur only for the vacuum.

! Solitons can exist only for d=1

Manton’s integral constraints

! For a static configuration the conservation of the energy-‐momentum tensor implies a spatial conservation of the stress tensor

! Define the vector

! Then

Manton’s integral constraints

! Let’s take

! For this choice we get

! In particular when the surface term vanishes we get Manton integral constraint

Introduction-‐ questions

! The questions that we have explored are

! Can Derrick’s theorem and Manton’s integral constraints be unified?

! Can one generalize these constraints to other types of deformations?

! What are their implications on Solitons, Wilson lines, static solutions of gravity , D branes and spatially modulated configurations.

Outline

! Part I-‐ General formalism ! A. Geometrical deformations of solitons ! B. Deformations by global transformations ! C. Deformation, and stress forces of periodic solutions

! D. “ Elasticity requirements” ( or minimizing and not only extreemizing)

Outline

Part II-‐ Applications-‐ ! (i) Higher derivative actions and sigma models ! (ii) Current constraints on known solitons

! (iii) Solitons of non linear(DBI)Electromagnetism

Outline

! (v) Constraints on D brane and string actions ! (vii) Probe branes in brane backgrounds ! (vii) D3 brane with electric and magnetic fields ! (viii) Adding Wess –Zumino terms ! (ix)Flavor branes in M-‐ theory MQCD ! (x) Application to the Ooguri Park spatial modulation models

General formalism

Part I-‐

General formalism

(i) Geometrical deformations

! Consider a theory of several scalar fields ! Take to be a soliton with ( finite) energy

! We now deform the soliton

! We expand the geometrical deformation

! We take it to be linear

Geometrical deformations

! -‐ rigid translations

! antisymmetic -‐rotations

Geometrical deformations

! diagonal -‐dilatation (not necessarily isotropic)

! Symmetric no diagonal components -‐ shear

Geometrical deformation

! The energy of the deformed soliton is

! The Stress tensor is


! Thus the variation of the energy relates to the stress tensor as

! For theories with scalars and no gauge fields ! Hence the stress and energy momentum tensors are related via


! Since are arbitrary we get Manton’s integral conditions

! More precisely we get

! So that for vanishing surface term we get the constraint of vanishing integral of the stress tensor


! As is well known for Maxwell theory, the canonical energy momentum tensor is not gauge invariant and one has to add to it an improvement term

! Such that which guarantees the conservation of the improved tensor

! For these cases we get that the variation of the energy


! For the modified case the integral constraint reads

! Again when the surface term vanishes we get that the integral of the stress tensor vanishes

BPS configurations and the vanishing of the stress tensor

! With right fall off we have

! What about the vanishing of the stress tensor itself?

! For 1+1 dim. solitions the virial theorem reads

So the stress tensor 0

BPS configurations and the vanishing of the stress tensor ! This result can be related to a 1+1 supersymmetic model

for which Supersymmetry relates the stress tensor Tij to the supercurrent Via the susy Ward Identity From the fact that the BPS solutions are invariant under half of the supersymmetris vanishing of Tij [Moreno Schaposnik]

(ii) Deformations by global symmetry

! Suppose that our system is invariant under a global symmetry.

! The corresponding current conservation for static configurations reads

! Deforming the soliton yields a variation of the energy

! For constant it is obviously a symmetry but again we take the transformation parameter

Deformations by global symmetry

! Thus we get the integral equation

! For vanishing surface term the integral of the space components of the global currents vanishes

Deformations by global symmetry

! In order to have a finite surface integral the current should go as

! At leading order for large radii the current reads

! So there must be a massless mode ! This happens generically when the symmetry is spontaneously broken and the mode is the NG mode

(iii)Geometric deformation of periodic solutions

! Apart from solitons there are also static solutions that break translational invariance but have divergent energy ( but finite energy density).

! The analysis of above does not apply but one can do a local analysis on some restricted region.

! For periodic configurations will take the unit cell ! The total force on the surface surrounding the unit cell of such a solid should be zero.

! The force on a face of the cell is

Geometric deformation of periodic solutions

! We can increase the size of a unit cell and at the same time deform the neighboring cells so the periodic solution remains unchanged farther away.

! The forces on the faces of the unit cell no longer cancel:

! The net force on the surface after the transformation could be pointing

! (i) Out of the unit cell -‐ unstable since the deformed cell will now continue increasing its size.

! (ii) Into the unit cell – restoring stability



(iv) Elastic properties of inhomogeneous solutions

! We need to minimize and not only extreemize the energy.

! We vary the energy to second order. ! We use an analogy with elasticity theory and map the minimization to a positivity condition on the stiffness tensor.

Elastic properties of inhomogeneous solutions

! In general in thermodynamics we have

Stress tensor Strain

! For ideal isotropic fluid displacement vector


! Hook’s law for small deformations

! The energy is minimized if for any two unit vectors a and b the stiffness tensor obeys

``Elastic properties of inhomogeneous solutions

! Consider fluctuations of the coordinates

! The variation of the energy of a scalar field theory to second order is

! The stiffness tensor is


! For a gauge theory the stiffness tensor is

! Where

Part II-‐ Aplications

1. ( warm-‐up) Sigma models

! One can easily generalize Derrick’s theorem to a case of a sigma model

! Repeating the procedure of above yields

! When the signature of the metric is positive then the conclusions for the generalized case are the same .

! If the signature is not positive there is no constraints in any dimension.

2. Higher derivative actions

! Consider the higher derivative lagrangian density

! The corresponding equations of motion

! The conserved energy momentum tensor

Higher derivative actions

! The Hamiltonian of the static system

! Under isotropic re-‐scaling of the coordinates

! Requiring extreemality for

Higher derivative actions

! The higher derivative terms thus ease the restriction on solitonic solutions for pure scalar field theories: we can get solitons for d < 4.

! That’s the mechanism in the Skyrme model ! Generalizing this result to any higher order derivative Lagrangian density, where the derivative terms are quadratic in the fields of the form

! Now the constraint in principle allows solitons for any dimension d < 2N.

3. integral of the current constraint

! Let’s examine this constraints on familiar solitons for: Topological currents, global and local currents.

! Topological currents are conserved without the use of equations of motion.

! The general structure of these currents in d space dimensions is

is a tensor of degree d composite of the underlying fields and their derivatives ! If the current is composed of only scalar fields, abelian or non-‐abelian, the spatial components have to include a time derivative

! So we conclude that

The current constraint in the ‘t Hooft Polyakov monople

! The system is based on SO(3) gauge fields and iso-‐vector scalars described by

! The SO(3) current is

! The equations of motion

The current constraint in the ‘t Hooft Polyakov monople

! The relevant ansatz for the classical configurations

! Asymptotically they behave as

! Substituting the ansatz to the current

! It is obvious that ! The current constraint is obeyed

The constraint on the axial current of the skyrme model

! The two flavor Skyrme model is invariant under both the SU(2) vector and axial flavor global transformations.

! The currents read ! plus higher derivative corrections that follow from the Skyrme term.

! The space integral of the non-‐abelian (axial)current

is In accordance with the fact that there is an SSB

4. Soliton of non linear electromagnetism

! Let us analyze the constraints on EM expressed in terms of a DBI action.

! We check first the ordinary Maxwell theory ! The energy is

! The scaling of E and B are ! The scaled energy

! Derrick’s condition

! No solitons apart from d=3 for self duals

Soliton of DBI non-‐ linear electromagnetism

! The DBI action of EM in d+1 =4 is

! The associated energy density

! Derrick’s constraint

! Electric and magnetic solitons are not excluded

5. Constraints on string and D-‐brane actions

! If the generalized constraints are fulfilled by some string or D brane configuration it may indicate about possible states apart from the trivial ones.

! The constraints are based on comparing configurations with the same boundary conditions

! For finite volume ones, the variation may change the boundary conditions.

! Satisfying the constraints is not a proof of existence ! The constraints my exclude classes of solitons

Constraints on string and D-‐brane actions

! The action of the low energy dynamics of D-‐branes

Dp brane dilaton pull back tension induced metric of the NS form ! D-‐branes can also carry charge that couples to a RR flux. This corresponds to a WZ (CS) action

pullback of the RR k-‐form

Constraints on strings

! Similarly the NG action describes the fundamental string

! The string is charged under the NS two form

! The induced metric is

the embedding coordinates

Fixing diffeomorphism

! The brane ( string) action is invariant under diffeomorphism hence the constraints are trivially satisfied.

! For instance for the NG string the energy

! To get non-‐trivial constraints we must gauge fixed the diffeomorphis invariance

Fixing diffeomorphism

! We use the usual static gauge. ! We split the coordinates -‐ -‐worldvolume -‐-‐transverse We impose (i)  space-‐time translation invariance on the worlvolume (ii)  truly static

Constraints on D brane action without gauge fields

! In the static gauge the pull-‐back metric reads

! The energy is

. -‐Dbrane subtraction for . -‐string . E=0


! Derrick’s condition is now

! Where we have used

! After some algebra we find that the condition is

! This can be obeyed so we can not exclude Dbrane solitons


! However for depending only on a single x

! Since for non-‐trivial the integrand is positive the constraints cannot be satisfied.

! There are no solitons D-‐branes ( even for p=1) that depend on only one coordinate

Probe brane in Dp brane background

! The near horizon background has the metric

! The dilaton

! A RR form

! The DBI+ CS actions read

Probe brane in Dp brane background

! Derrick’s condition is now

! The second derivative condition is

! There are no soliton solutions for any p that obey the stronger condition of vanishing of the integrand.

Generalized conditions for Branes with gauge fields

! When electric field is turned on the energy is not just –LDBI but rather the Legendre transform

! It is convenient to define M such that

! The energy can be written as

Deformation constraints on D branes with gauge fields

! Rather than deriving Derrick’s condition let’s look this time on Manton’s constraints

! The explicit form of the stress tensor reads

Adding the WZ terms

! Again like the DBI action we have first to gauge fix ! The pullback of the RR fields is

! For instance for D1 brane the WZ action reads

! The contribution to the stress tensor is

! In the absence of gauge fields

! Hence we see again that there is no D1 soliton solution

The D3 brane case

! For the D3 brane case the WZ term is

! The contribution of the WZ term to the stress tensor

Application to gravitational backgrounds

! Upon gauge fixing the diffeomorphism and parameterizing the metric the dilaon and fluxes we get an action of a bunch of scalar fields with a potential.

! In case that there is a dependence only on the radial direction it is a 1+1 dimensional action.

! Generically the ``kintic terms” are not positive definite.

! It turns out that the integrand of Derrick’s condition translates to the ``null energy condition”.

! Let’s demonstrate this


! Consider the DC on d brane solutions of gravity ! The bosonic part of the SUGRA action in D dimensions

! We take the metric in the string frame


! In terms of the metric fields and the dilaton

! The ``null energy” condition which is a Gauss law associated with fixing

! It is identical to the integrand of Derrick’s condition


! Consider the following 1+1 dim model with N degrees of freedom

! The extremum condition reads

! The integrand is just the energy of a 0+1 dim. where x is taken to be the time. Thus the vanishing of the integrand is identical to the ``null energy condition”

Flavor branes in MQCD

! The type IIA brane configuration [Aharony,Kutasov,Lunin,Yankielowicz]

! Can be uplifted to M theory background


! The shape of the curved five brane

! The induced metric is

! The Lagrangian density


! The Neother charges associated with the shifts of x6 and α are

! Applying Derrick’s condition yields

! The integrand is identical to the Noether charge E thus the condition translates to `` null energy condition”

Application to spatially modulated models

! Spatial modulation (S.M) was identified in YM+CS theory on an AdS5 black-‐hole

! The background metric is given by

! With the warp factor


! The background electric field is given by

! The spatially modulated solution

! The equations of motion


! Integrating the first equation we end up with

! This equation admits solution with amplitude

! The relation between h0 and k


! The energy density of the boundary field theory is

! It is minimized at

The stiffness tensor

! The energy density is given by

! The expression for the stiffness tensor is complicated ! For the unit vectors

It is given by

The stiffness tensor

positive

negative Thus there are regions which indeed correspond to minima but other ( blue ones) correspond to maxima

Stress foreces

! We check now for the stability against deformation in the x2 direction

! The pressure is negative for all k and has a maximum for

! In the region the system is not restored ! The minimum of the free energy at is in the instability region

Summary and open questions

! We unified and generalized Derrick’s and Manton constraints on solitons.

! We have applied the condtions to sytems of soltions with global currents

! Sigma model and higher derivative actions ! DBI electromagnetism ! Dbranes including the DBI and WZ terms ! The method can be applied to many more `` modern solitons”

! In particular we are investigating the stability of the spatially modulated brane and bulk solutions.

NITheP WITS node seminar: Prof Jacob Sonnenschein (Tel Aviv University) TITLE: "Hologrqaphic entanglement of confining theories"

Education

nite entanglement entropy

entanglement entropy

renormalized wilson

strip wilson line

b entanglement entropy

t entanglement entropy

d dimensions

cft2 example of hee