Confinement, phase transitions and non-locality

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Published for SISSA by Springer

Received: May 16, 2014 Accepted: May 17, 2014 Published: June 3, 2014

Uri Kol,a Carlos Nez,b,c Daniel Schoeld,b Jacob Sonnenscheina and Michael Warschawskib

aSchool of Physics and Astronomy, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University,Ramat Aviv 69978, Israel

bDepartment of Physics, Swansea University,

Singleton Park, Swansea SA2 8PP, U.K.

cCP3-Origins and DIAS, University of Southern Denmark, Odense, Denmark

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: In this paper we study the conjectural relation between connement in a quantum eld theory and the presence of a phase transition in its corresponding entanglement entropy. We determine the su cient conditions for the latter and compare to the conditions for having a conning Wilson line. We demonstrate the relation in several examples. Supercially, it may seem that certain conning eld theories with a non-local high energy behavior, like the dual of D5 branes wrapping a two-cycle, do not admit the corresponding phase transition. However, upon closer inspection we nd that, through the introduction of a regulating UV-cuto , new eight-surface congurations appear, that satisfy the correct concavity condition and recover the phase transition in the entanglement entropy. We show that a local-UV-completion to the conning non-local theories has a similar e ect to that of the aforementioned cuto .

Keywords: Connement, Gauge-gravity correspondence, Wilson, t Hooft and Polyakov loops, AdS-CFT Correspondence

ArXiv ePrint: 1403.2721

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP06(2014)005

Web End =10.1007/JHEP06(2014)005

Connement, phase transitions and non-locality in the entanglement entropy

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Contents

1 Introduction 11.1 General idea of this paper 2

2 Entanglement entropy and Wilson loops as probes of connement 32.1 Review entanglement entropy in conning backgrounds 32.2 Review Wilson loops in conning backgrounds 52.3 Similarities and di erences 62.4 Su cient conditions for phase transitions 82.5 Examples of the criteria for phase transitions 102.5.1 AdS5 [notdef] S5 102.5.2 Dp brane on a circle 102.5.3 Hard and soft walls 112.5.4 Klebanov-Strassler 132.6 Connement and phase transitions 14

3 Volume and area laws, UV-cuto s and connement 163.1 A useful quantity 173.2 Study of the D5 branes on S1 system 183.3 Finding the short congurations 20

4 The absence of phase transitions in (some) conning models 224.1 More on non-locality 25

5 Recovering the phase transition: the baryonic branch 275.1 Losing our phase transition: adding sources 30

6 Getting back the phase transition(s): sources with a decaying prole 31

7 Conclusions and future directions 34

A Wilson loop-entanglement entropy relation an exercise 36A.1 Small radius expansion of the Wilson loop 37A.2 Large radius expansion of the Wilson loop 37

B A taxonomy of behaviours for systems with sources 37B.1 Linear P sourced systems with sigmoid proles 39B.2 Linear P sourced systems with bump-like proles 39B.3 Rotated sourced systems with sigmoid proles 41B.4 Rotated sourced systems with bump-like proles 41

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C Other models without phase transitions 43C.1 D5 on wrapped on a three-cycle 44C.2 Deformed D4 on a S1, D3 on a S1 and AdS5 [notdef] S5 45

C.3 D6 wrapped on a three-cycle 47 C.3.1 Illustrating the dependence on the UV cuto 49

D Hints at invariances of the entanglement entropy 50D.1 The D4-D8 system 50D.2 A background with a cyclic RG ow 51

1 Introduction

The holographic gauge-strings duality [13] provides an e ective and controllable toolkit to study the dynamics of non-perturbative phenomena. An important such tool is the Entanglement Entropy (EE). This quantity dened originally in quantum mechanical systems has found a wide-range of applications in di erent branches of Physics. We refer the reader to the papers [48] for a review of these applications and formalism.

The holographic prescription to calculate the EE was proposed by Ryu and Takayanagi [9, 10]. For a d dimensional conformal eld theory dual to an AdSd+2 background, the holographic EE is given by minimizing the d dimensional area in AdSd+1 whose boundary coincides with the boundary of the region that denes the EE.1

Klebanov, Kutasov and Murugan (KKM) [11] generalized the prescription of [9] to non-conformal eld theories. In particular they found that certain backgrounds, which are holographically dual to conning systems, admit a rst order phase transition upon varying the width of a strip that sets the entangled regions. Following the work of KKM, the literature on holographic EE in non-conformal theories (including phase-transitions and connement) has grown (see for example [12][16]) In this paper we further explore various aspects of the holographic EE of conning systems.

Another well known observable that in holography involves minimizing an area is the Wilson line (WL). We show that the functional forms of the length of strip associated with EE and of a Wilson line are similar and further so are the EE and the energy of the WL as a function of the length. We discuss the similarities and the di erences of these forms for systems that admit connement. In an analogous manner to the determination of su cient conditions for an area law WL [17], we nd the su cient condition for a rst order phase transition of the holographic EE. We apply these conditions to several examples including the AdS5 [notdef] S5, Dp branes compactied on S1, the hard and soft wall models and the

Klebanov-Strassler model [18].

A special feature of the holographic EE as a diagnostic tool for connement is the fact that it relates not only the IR behavior of the geometry, but also has implications on the

1For a review and for a partial list of references to follow up papers see [48].

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UV behavior of the background.2 We will make clear what sort of connection we expect between connement and the UV-behavior of the QFT.

Making these points and connections explicit will bring us to study the calculation of the EE in non-local QFTs. We deal with this complicated problem by using holographic duals based on D5 or higher Dp branes (p > 5). Then, performing the usual calculation we nd that in spite of the background in question having the IR-geometry suitable to be dual to a conning QFT [17], the phase transition in the EE is absent. We will observe that this is an e ect of the UV non-locality of the QFT. We will propose a way to x this situation, by introducing a hard UV cuto and observing that new congurations appear that would not only recover the phase transition argued in [11], but also solve an stability problem of the congurations that miss the phase transition. Finally, since the UV-cuto may look like a bad x for the problem, we will show (with two examples) how a suitable UV-completion to give a local theory, plays a similar role to the UV-cuto , at least for the purposes of the transition.

This suggest that the EE is not only a quantity useful to diagnose connement, but also to determine if the QFT in question is local in the far UV. Let us explain in a bit more detail some ideas that motivated this paper.

1.1 General idea of this paper

Consider a theory like QCD: it was argued that this theory has a Hagedorn density of states. For a mass M, the number of states N(M) is

N(M)

[parenleftbigg]

Here TH is some energy scale and b a number. The Partition Function will roughly be,

[integraldisplay]

We see that for a high enough temperature, this Partition Function is divergent and not well-dened.

In the case of QCD, this is not an e ect one would actually measure, because hadrons have a natural width and at energies high enough, they would start decaying into each other, there would also be pair creation, etc. But in a truly conning QFT, like for example Yang-Mills or minimally SUSY Yang-Mills, we would have if we take the large Nc limit innite hadrons (glueballs) that will be very narrow and they will present the Hagedorn behaviour above (although generically, the deconnement occurs before the Hagedorn transition). Just like it happens in a typical theory of strings.

The authors of [11], have argued that something similar happens to the EE of a conning theory. An intuitive reasoning lead them to write the EE as

[integraldisplay]

DM M2be+ HM2MLEE,

2Here and below, we follow the common parlance according to which the large and small radial position in the string background are associated with the UV and IR of the dual QFT.

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2be+MTH .

DM M2be+

TH M .

where they used that for scalar non-interacting degrees of freedom our large Nc glue-balls for example the EE goes like e2MLEE, where LEE is the separation between the entangled regions. So it was argued in [11], that truly conning QFTs should present a phase transition in the EE, when 2L > H.3 This phase transition is phenomenologically similar to the connement-deconnement one. So, it is expected that for a given LEE,crit

the EE behaves in a constant manner that is N0 for L > LEE,crit, or grows very

rapidly, like N2, for for L < Lcrit respectively. We will see this in various examples below.

One may argue that in some QFTs the reasoning above might fail, if for example the density of states grows di erently or if the behavior of the EE for many scalars changes and hence the argument breaks down. We will see that a way of having some analytic control over the problem is to rst introduce a cuto at high energy, nd the phase transition exploring regions close to the cuto and then take the cuto to innity. These limits need not commute. A similar situation but in a di erent context was encountered in [19] and we will nd this to be the case in theories with non-local UV-behavior. Similarly we will observe that a local and honest UV-completion of non-local theories plays the same role as the regularisation with a cuto .

The material of this paper is organised as follows: in section 2, we will provide a criteria (valid beyond the examples presented in this paper) to decide if the EE can present a phase transition. The criteria will be carefully illustrated using conning models based on Dp branes compactied on circles. Specically, the understanding of the case of D5 branes on S1, where the phase transition is curiously absent, will be the subject of section 3. We will argue that a naive holographic calculation of the EE misses some key-congurations. We also motivate these congurations by introducing the UV-cuto already mentioned. In section 4, we make explicit the analysis above, on a four dimensional model of conning QFT, but with a non-local UV the need for a cuto appears here also. In section 5, we will see that a suitable UV-completed and local QFT version of the conning model above, has an EE that behaves like the one in the cuto -QFT does. This emphasises the point that the cuto is capturing real Physics.

The paper is of high technical content, as reected in various appendixes, where interesting material was relegated to ease the reading of the main body of this work.

2 Entanglement entropy and Wilson loops as probes of connement

2.1 Review entanglement entropy in conning backgrounds

We start with a brief summary of the results of Klebanov, Kutasov and Murugan (KKM) [11], who studied the Entanglement Entropy (EE) in gravity duals of conning large Nc gauge theories. KKM have generalized the Ryu-Takayanagi conjecture to non-conformal theories, and suggested that the EE in these cases is given by

S = 1

G(10)N

Z d8e2[radicalBig]

G(8)ind (2.1)

3A support for this argument can bee found in [20], where the entanglement entropy of a lattice gauge theory was computed numerically, and a signature of phase transition was observed.

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where G(10)N is the 10-dimensional Newton constant and G(8)ind is the induced string frame metric on . The EE is obtained by minimizing the action (2.1) over all surfaces that approach the boundary of the entangling surface. KKM have considered, as the entangling surface, a strip of length L. In this case, they found that there are two local minima of the action (2.1) for a given L. The rst is a disconnected surface, which consists of two cigars which are separated by a distance L. The second is a connected surface, in which the two cigars are connected by a tube whose width depends on L.

The gravitational background in the string frame is of the form

ds2 = ()

()d2 + dxdx

[bracketrightbig]+ gijd id j (2.2)

where x ([notdef] = 0, 1, . . . , d) parameterize Rd+1, is the holographic radial coordinate

< < 1 (2.3)

( can be zero in some cases) and i (i = d + 2, . . . , 9) are the 8 d internal directions.

There is also a dilaton eld that we denote with . Some RR and NS uxes complete the background, but they will not be relevant to our analysis. The volume of the internal manifold (described by the [vector]

coordinates) is Vint =

[integraltext]

pdet[gij]. We also dene the quantity

H() = e4V 2int d (2.4)

The functions H() and () will play an essential role in the following and all along the rest of this paper. We will mention few important properties of these functions. KKM have argued that in conning backgrounds H() is typically a monotonically increasing function while () is typically a monotonically decreasing function. Since H() includes a factor of the volume of the internal manifold, it typically shrinks to zero size at = ,

in agreement with the vanishing of the central charge at zero energies. On the other hand, () is less restricted and it can either diverge or approach a nite value at = .

Denoting the minimal value of along the connected surface in the bulk by 0, its EE is given by

SC(0) = Vd1 2G(10)N

[integraldisplay]

d[vector]

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0 d[radicaltp]

[radicalvertex]

[radicalbt]

()H() 1 H(0)H()

(2.5)

The length of the line segment for the connected solution as a function of 0 is

L(0) = 2

[integraldisplay]

0 d[radicaltp]

[radicalvertex]

[radicalbt]

()

. (2.6)

On the other hand, the EE of the disconnected solution does not depend on 0 and is given by

SD(0) = Vd1

2G(10)N

[integraldisplay]

H() H(0)

1 d[radicalbig] ()H()

(2.7)

Lmax

Lc Lmax

Figure 1. The phase diagram for the entanglement entropy in conning theories. On the left, the length of the connected solution as a function of the minimal radial position in the bulk L(0),

which is a non-monotonic function in conning theories. On the right, the entanglement entropy of the strip as a function of its length. The solid blue line represent the connected solution while the dashed red line is the disconnected solution. At the point L = Lc there is a rst order phase transition between the two solutions. This type of rst-order phase transition behavior is called the buttery shape in the bibliography.

The EE is in general UV divergent, but the di erence between the EE of the connected and disconnected phases is nite

2G(10)N Vd1

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S(0)

2G(10)N Vd1

(SC SD) =

[integraldisplay]

0 d[radicaltp]

[radicalvertex]

[radicalbt]

()H() 1 H(0)H()

[integraldisplay]

1 d[radicalbig] ()H()

(2.8)

Depending on the value of L (or alternatively of 0) S would either be positive or negative. In the rst case the true solution will be a disconnected surface, while in the later case the connected solution will be the true one. The phase transition between the two solutions is a characteristic of conning theories, and is described in gure 1.

The connected solution exists only in a nite range of lengths 0 < L < Lmax. In

this range there are two possible values for the connected solution, corresponding to the two branches in gure 1. The upper branch is an unstable solution. This double valuedness, which is called the buttery shape, corresponds to the double valuedness in the graph of L(0). As a result of the double valuedness, there is a rst order phase transition at the point L = Lc between the connected and the disconnected solutions. For this reason KKM have argued that a signal for a phase transition, and therefore also for connement, is the non-monotonicity of the function L(0). Indeed, as we will show later in di erent examples, every peak in L(0) corresponds to a possible phase transition in the entanglement entropy S(L).

2.2 Review Wilson loops in conning backgrounds

In this subsection we will review the results of [17] for the rectangular Wilson loop in conning backgrounds. We dene the following function

g()

p . (2.9)

The authors of [17] then found that the regularized energy of the Wilson loop is given by4

EWL(0) = 2

[integraldisplay]

g() ()

[radicaltp]

[radicalvertex]

[radicalbt]

[integraldisplay]

2()

dg(), (2.10)

where 0 is the minimal radial position of the dual string in the bulk. The rst term in the equation above is the bare energy of the Wilson loop while the second term is the energy of two straight string segments (stretched from the horizon to the boundary) which is subtracted from the bare expression in order to regularize its divergence. The length of the Wilson loop as a function of 0 is given by

LWL(0) = 2

[integraldisplay]

2(0) 2()

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g() ()

(2.11)

The authors of [17] found that the background admits linear connement if one of the following conditions is satised

1. () has a minimum

2. g () diverges (2.12)

and the corresponding string tension is ( ) [negationslash]= 0, where is either the point where

minimizes or where g diverges. In other words, if one of the two conditions above is satised, the potential at long distances will behave linearly in L

EWL (LWL) = ( )LWL + . . . (2.13)

where . . . stand for subleading corrections in 1

LWL .

The authors of [17] proved that in conning backgrounds LWL(0) is a monotonically decreasing function. This behavior corresponds to the monotonicity of EWL (LWL), which

always increase with the length. This features, as well as the linearity of EWL (LWL) at

long distances, is described in gure 2.

2.3 Similarities and di erences

So far we have reviewed well-known results for the entanglement entropy and the Wilson loop in conning backgrounds. A priori, these two quantities are not related to each other, and indeed, they show very di erent behaviors - the rst presents a phase transition in the form of a buttery shape, while the later is monotonic. However, the fact that both are probes of connement suggests that there maybe some deeper relation between them. Moreover, as we will discuss below, the functional form of both quantities is very similar.

The length of the entangling strip (2.6) and the length of the Wilson loop (2.11) can both be written in the form

L(0) = 2

[integraldisplay]

[radicalBig]

2()

2(0) 1

0 d[radicaltp]

[radicalvertex]

[radicalbt]

()

(2.14)

M() M(0)

4In the notations of [17] = f.

Figure 2. On the left, the length of the Wilson loop as a function of the minimal radial position in the bulk, which is a monotonically decreasing function. On the right, the energy of the Wilson loop as a function of its length. At long distances the energy is linear in length, which correspond to linear connement.

where M() is di erent for the two cases and is given by

MEE() = H() (2.15)

MWL() = 2() (2.16)

The entropy of the strip (2.8) and the energy of the Wilson loop (2.10) can also be written in a similar way as

4G(10)N

Vd1 S

EWL

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[parenrightBigg]

= 2

[integraldisplay]

0 d[radicaltp]

[radicalvertex]

[radicalbt]

()M() 1 M(0)M()

[integraldisplay]

1 d[radicalbig] ()M

() (2.17)

The expression (2.17) can also be written in the following form

4G(10)N

Vd1 S

EWL

[parenrightBigg]

= pM(0)L(0) 2K(0) (2.18)

K(0)

[integraldisplay]

1 d[radicalbig] ()M

()

[integraldisplay]

1 p () (M() M(0)) (2.19)

The form (2.18) emphasizes the linear nature of EWL at long distances. In both cases, M() is a monotonically increasing function. Therefore, the functional form of the EE and the Wilson loop is very similar. Of course, these mathematical similarities in equations (2.14) and (2.17) are based on the fact that both observables are solutions to a minimization problem.

The question arises as to what is the di erence between them. We claim that the qualitative di erence between these two observables (at least in the case of conning QFTs) is due to the behaviour of the function M() close to = . For both cases M() is a monotonically increasing function, but the behavior close to = is di erent. For the entanglement entropy M() = H() shrinks to zero close to , since H() includes a factor

of the internal volume see (2.4) which always goes to zero at the end of the geometry.5 On the other hand, for the Wilson loop in conning backgrounds M( ) = 2( ) [negationslash]= 0 since this quantity is related to the conning-string tension. Therefore M( ) behaves very di erently for the two observables, when calculated in a generic conning background. This is the source for the qualitative di erence between these two quantities.

To be concrete, let us focus on Dp branes compactied on a circle. These backgrounds are dual to conning eld theories in p space-time dimension. The background metric and dilaton are a generalisation of those written by Witten, as a dual to a Yang-Mills-like four dimensional QFT [2124] (with [prime] = gs = 1),

ds2 =

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7p 2

[bracketleftBigg][parenleftbigg]

7p d2h() +dx21,p1[bracketrightBigg] +h()

7p2d'2c+

p32R2d 28p, (2.20)

h() = 1

7p , e4 =

(3p)(7p) , 2()=

7p , g2()= 1 1

7p .

which implies that,

MEE() = (S7p [notdef] 2Rc)2R7p9ph(), MWL() = [parenleftBig]

7p . (2.21)

Rc is the radius of the compact cycle (see for example [25]),

Rc = 2 7 p

[parenleftbigg]

7p2 (2.22)

and Sn1 =

(n2) is the surface area of the n-sphere. Therefore MWL( ) 7p [negationslash]= 0,

while MEE( ) = 0. This di erence of behavior in the function M(), creates a buttery shape (and a phase transition) in S(L), whereas it gives place to a monotonic behavior and a long-separation linear law in EWL(LWL).

2.4 Su cient conditions for phase transitions

The conditions a background must satisfy so that the Wilson loop shows a conning behaviour (2.12), were derived in [17]. The di erence in behavior, of the Wilson loop and the Entanglement Entropy as probes of the phenomenon of connement, leads to the following question:

Question: what are the conditions on the background to present a phase transition

in the EE?

Using a similar logic to that of [17], in the following we will derive su cient conditions on the background to present a phase transition in the Entanglement Entropy. More explicitly, we will derive the conditions on the background such that L(0) (the length of the entangled strip as a function of the minimal radial position) will increase for 0 close to (the IR

5The vanishing of the internal volume is in agreement with the vanishing of the central charge at zero energies. This is characteristic of conning eld theories.

of the dual QFT) and decrease for asymptotically large value of 0 (the UV of the dual QFT). Hence, the quantity L will present (at least) a maximum and the required double valuedness needed for a phase transition.

We start by deriving the conditions on the background such that L(0) is an increasing function close to 0 = . Let us assume that the functions H() and () have the following expansions around 0 =

H() = hr( )r + O( )r+1, () = t( )t + O( )t+1 (2.23)

with r, t > 0. Then, the integrand corresponding to L in (2.6) is divergent close to = ,

and therefore the integral gets most of its contribution from this region. That allows us to approximate the integrand using eq. (2.23),

lim

L(0) = 2

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p t

[integraldisplay]

0 d( )

t 2

[bracketleftbigg][parenleftbigg]

r 1

12 (2.24)

= (0 )1

t2 2p t

[integraldisplay]

1 dz(z1)

2 z

t+2r2

2r =(0 )1

t2 2p t

t+r2

r t+2r2

We have changed variables to z

[parenleftBig]

r. We nd that L(0) is monotonically increasing

when

t < 2 (2.25)

which means that () should not diverge faster than 1

( )

2 close to .

Next, we derive the conditions on the background such that L(0) is a decreasing function at asymptotically large value of . Close to the boundary = 1 we can expand

H() = hkk + O(k1) () = jj + O(j+1). (2.26)

Plugging these expansions in (2.6) we nd,

L(0) = 2

p j

[integraldisplay]

0 d

j 2

[bracketleftBigg][parenleftbigg]

k 1

# 12. (2.27)

Changing variables to z

0 we nd the asymptotic behavior of L(0) near the boundary

p j01j 2

[integraldisplay]

j 2

p j

k+j2 2k[parenrightBig]

L(0) = 2

j2 . (2.28)

1 pzk 1

2k+j22k[parenrightBig]

We see that for

j > 2 (2.29)

the length L will go to zero as 0 ! 1. There is a maximum somewhere in the middle,

hence a double valuedness for 0(L) and the possibility of a phase transition in the quantity S(0[L]). On the other hand, for j 2, the quantity L will either diverge, or saturate at

a constant value, in the same limit and in this case we do not expect to have a phase transition.

Let us present some examples of absence or presence of phase transitions in the EE, in agreement with the criteria of this section.

3.0

[Minus]10

2.5

2.0

[Minus]20

1.5

[Minus]30

1.0

0.5

[Minus]40

0.0

0 2 4 6 8

[Minus][notdef]0.0 0.5 1.0 1.5 2.0 2.5 3.0

Figure 3. The case of AdS5 [notdef] S5 Here we plot L(0) and S(L).

2.5 Examples of the criteria for phase transitions

As anticipated, we will study here di erent non-conning and conning models to illustrate our criteria above. AdS5 [notdef] S5, Dp branes compactied on a circle, hard and soft walls and

the Klebanov-Strassler model [18] will serve as conrmation of our treatment.

2.5.1 AdS5 S5As a rst example we discuss the EE of N = 4 Super-Yang-Mills, to demonstrate that it does not present a phase transition. The metric and other relevant functions for the case of AdS5 [notdef] S5 are given by

ds2AdS = R2

2 d2 +

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2R2 dx21,3 + R2d 25, () =

4 , H() =

82 3

2R46. (2.30)

Notice that in this case t = 4 and the condition (2.25) is not attained. The length of the strip L(0) as well as the S can be exactly computed and are plotted in gure 3. We see that

L(0) is a monotonically decreasing function which diverges at the origin and goes to zero at the boundary. The EE shows two possible phases: in the rst, it monotonically grows with L while in the second (the disconnected phase) it is constant. The second phase is not favoured, since the EE in this phase is always larger than the EE in the connected phase. Therefore there is no phase transition, as appropriate for a conformal eld theory. Incidentally, also notice that the concavity of the S(L) is such that the correct condition [26, 27],

d2S

dL2 < 0, (2.31)

is achieved.

2.5.2 Dp brane on a circle

Next we consider the background generated by Dp brane compactied on a circle described in eq. (2.20). The functions () and () are,

() =

7p2, () = 1 1

7p (2.32)

Close to the horizon = , we can expand

() =

7 p

[parenleftbigg]

7p 1 + . . . (2.33)

where . . . stands for subleading (nite) terms.

Comparing with (2.23) we nd that in this case t = 1. This means that for this background, and for any value of p, the condition (2.25) will be satised and L(0) will always go to zero at the horizon. Close to the boundary (0) 0(7p) and therefore in this

case, comparing with eq. (2.26), we have j = 7 p. Hence, for p 4, the condition (2.29)

will be satised and L(0) will decrease to zero close to the boundary. However, for p = 5, L(0) will saturate to a nite value and for p 6 it will increase towards the boundary.

We conclude that for p 4 there will be a phase transition since L(0) is a non-

monotonic function, but for p > 4 L(0) is monotonic and there will be no phase transition. In gure 4 we draw the functions L(0) and S(L) for the cases p = 3, 4, 5, 6. We observe a crossing (and hence a phase transition) between the connected and disconnected solutions for the EE, in the case p < 5. We also see that the concavity of S is the correct one for p 4, but does not satisfy eq. (2.31) for p > 4.

2.5.3 Hard and soft walls

The Hard Wall model was proposed in [28] as a holographic description of low-energy properties of QCD. It is described by the AdS metric with the radial coordinate cut at some value = . The results for the entanglement entropy are shown in gure 5.

The authors of [29] have improved the Hard Wall model by cutting of the AdS space smoothly, instead of a hard-wall cuto in the IR. The metric of the Soft Wall model is the same as the AdS metric but there is a non-trivial dilaton

e = e

12 (2.34)

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Then we have

() = R4

4 (2.35)

H() =

82 3

2R46e42 (2.36)

The results are shown in gure 5. We see that the soft wall model admits a similar behavior to the D3 and D4 branes with a phase transition.

We cannot check the conditions for connement in these examples. The Hard Wall background is just a cut AdS. The Soft Wall background does not admit the expansion (2.23) for the function H() in the IR since it includes an exponential factor. Therefore, even though () diverges strongly, we still have a phase transition since the exponential decay of H() takes-over the divergence of ().

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Figure 4. The function L(0) and S(L) in the near extremal Dp brane backgrounds for p = 3, 4, 5, 6 moving down the page. The location of the horizon was set to = 1 in the gures. The dashed red line is the disconnected solution. The D3 and D4 branes shows a phase transition behavior while in the D5 and D6 branes there is no phase transition.

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Figure 5. The function L(0) and S(L) in the Hard (top row) and Soft Wall (bottom row) models. The location of the hard wall was set to = 1 in the gures. The dashed red line is the disconnected solution and the dashed blue line represents the continuation of the AdS solution beyond the hard wall.

2.5.4 Klebanov-Strassler

The entanglement entropy of the background dual to a cascading supersymmetric gauge theory, the deformed conifold [18], was analysed in details in [11]. The authors of [11] have shown that in this case there is a phase transition of the same form as in the D3 and D4 branes, as expected in a conning theory. We would like to demonstrate how this background follows the conditions for phase transition, which we derived above.

The supergravity solution of the deformed conifold is of the following form [18, 30]

ds2 = h

2 () dx21,3 + h

2 () ds26 (2.37)

where ds26 is the metric of the deformed conifold

ds26 = [epsilon1]4/3

2 K ()

[parenleftbigg]

1 3K3 ()

hd2 + g5

[bracketrightBig]

2 + cosh2

2[parenrightBig] [bracketleftBig] g3

2 +

[bracketrightBig]

+ sinh2

2[parenrightBig] [bracketleftBig] g1

2 +

(2.38)

[bracketrightBig] [parenrightbigg]

[epsilon1] is the energy scale and the functions h () and K () are given by

h () = gsM [prime]

2 22/3[epsilon1]8/3I () , (2.39)

I ()

[integraldisplay]

1 x coth x 1

sinh2 x (sinh 2x 2x)

13 , (2.40)

0.25

0.20

0.15

0.10

0.05

0.00

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0 2 4 6 8

Figure 6. The function () in the Klebanov-Strassler background. () saturates to a nite value at the origin = 0 and therefore meets the condition for a phase transition.

K () = (sinh (2) 2)

13 sinh

2 (), () = h()[epsilon1]4/3

6K2() , (2.42)

Vint = 43p6 h5/4[epsilon1]10/3K() sinh2(), (2.43)

H() = e4V 2int 3 = 86

3 [epsilon1]20/3h()K2() sinh4(). (2.44)

The function () approaches a nite value in the far IR = 0, and therefore we have in this case t = 0, corresponding to a monotonically increasing L(0) in the IR, in accordance with (2.25). Near the boundary, i.e. at large , the H () and () functions take the form

H( 1) = [epsilon1]4(gsM [prime])26e2 (2.45) ( 1) = [epsilon1]

43 (gsM [prime])22

In this region the functions does not admit the power expansion we have assumed in (2.26) and therefore we cannot directly check the conditions we have found, but a direct computation [11] shows that L(0) indeed goes to zero close to the boundary and therefore there is a phase transition. The intuition is that () decays exponentially fast close to the boundary, and therefore meets the requirement for su ciently strong decay (j > 2) of eq. (2.29).

2.6 Connement and phase transitions

We are now in a position to compare the conditions for connement on the Wilson loop, see eq. (2.12) derived in [17] and the conditions for a phase transition in the entanglement

1 3

, (2.41)

where is a dimensionless radial coordinate running from zero to innity on the boundary. Then we have,

= h

43 e

3 (2.46)

entropy as were suggested in [11] and further developed in the previous subsection. On physical grounds, since both observables are probes of connement, we expect both conditions to coincide. We will not be able to prove the last statement, but we will give a avour of why it should be true in some examples. On the other hand, we will emphasise a puzzle which arises in other cases. The solution for this puzzle will be the aim of rest of the paper.

Let us start with the conditions in the IR. The conditions on the Wilson loop, eq. (2.12) are really a statement about the IR and therefore we will compare them to the condition we derived for the entanglement entropy in the IR, see eq. (2.25). The condition of eq. (2.25) means that () should diverge slower than 1

( )

2 in order to observe a phase transition in the EE. Using eq. (2.9) we relate the divergence of to the divergence of g, which is one of the conditions for linear connement in the Wilson loop, see eq. (2.12) (remember that since ( ) is the string tension it must be nite, and therefore does not play a role in the discussion about divergences). If we take the case of the Dp brane on S1 see eq. (2.20) as an example, we see that close to =

g2() =

[parenleftbigg]

[parenrightbigg]

1 2 , see eq. (2.9).

While we have presented an intuition (but not a proof) of the equivalence between the conditions on the EE and the Wilson loop in the IR, there is a puzzle concerning the UV. First, we note that there are no UV conditions on the Wilson loop to obey connement. As far as the Wilson loop is concerned, the only condition we demand is linearity at long distances (IR). On the other hand, in order to observe a phase transition in the EE, the background also has to satisfy the condition in the UV, eq. (2.29). This last condition is not satised in certain conning backgrounds Dp branes on S1 with p > 4 as discussed in the previous subsection. The question that arises is

Question: why does certain cases show linear connement in the Wilson loop but do

not show a phase transition in the entanglement entropy?

Answering this question will take us to a nice detour that will include: area and Volume law behaviors for Entanglement Entropy, local and non-local QFTs, theories with a UV cuto and theories that are UV-completed approaching a near conformal point. This material will be carefully presented in section 3. Then, we will apply all that information

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7 p

+ . . . (2.47)

where . . . stand for sub-leading nite corrections. This divergence is in agreement with the condition on in eq. (2.25), for any value of p. A possible violation of (2.25) would correspond to a stronger divergence of g. We are not aware of such examples. Based on the intuition of backgrounds with compact circles, such a case would correspond to a situation with two compact circles with a topology of a cigar, where the tip of both cigars located in the same radial position. It would be interesting to try to rule out that case (or alternatively nd one such example), but we leave this for future work. When approaches a nite value at = , the corresponding connement-condition on the Wilson loop is the rst among the two in eq. (2.12), which is that has a minimum at = . In this case it seems that the maximum of () at = (as in gure 6) corresponds to the minimum of (), via the relation

to answer what happens in a trademark model of Connement in 4-d QFT, namely the twisted compactication of D5 branes on a two-cycle of the resolved conifold ( with or without fundamental-matter elds). We will introduce a cuto that will recover the phase transition in the EE and solve the concavity problem mentioned above. Understanding how to UV-complete these non-local QFTs, nicely recovering the phase transitions without appealing to cuto -e ects, will be the goal of section 4.

3 Volume and area laws, UV-cuto s and connement

The goal of this section is to set-up the elements that will allow us to answer the last question posed above. We will briey analyse the case of AdS5 [notdef] S5, then quickly move

into NS5 and D5 branes, where an important role will be played by the non-locality of the associated eld theory. The analog to the conning Wittens model, but with D5 branes that wrap S1 will close the analysis of this section.

Let us discuss the well understood case of AdS5[notdef]S5, as it will be a basis for comparison

for more complicated cases. In AdS5 [notdef] S5 we found that the connected solution is always

the minimal solution for the EE and is always preferred to the disconnected solution for all values of L which has a higher EE see gure 3. The connected solutions asymptote the disconnected ones from below for large L. For a local eld theory, the EE follows what is called a Heisenberg-like relation, such that L(0) 10 for some region (typically

for large 0) of the minimal solution. This type of behaviour can be seen in gure 7 by considering only the navy blue lines (both solid and dashed). We should further note that they have the correct concavity for stability, see eq. (2.31). The AdS5 [notdef] S5 has the usual

Area Law for the EE when we analyse the divergent parts.

Introducing connement in this theory can be thought of as an e ect on the IR region of the corresponding AdS5 [notdef] S5 plot. This e ect can be modelled by a soft-wall solution

as we have already seen. We replace the IR region with the usual L 10 behaviour for

the connected solution, with an unstable branch (cf. the green lines in gure 7).6 This has the e ect of moving the disconnected branch down such that it now meets the stable connected branch at a nite value of L = Lc (see the right panel of gure 7). Thus within backgrounds of this type we nd that there exists a critical value Lc, such that for L > Lc, the minimal solution is now the disconnected branch, and for L < Lc we still have the original AdS-like connected behaviour in the UV (stable branch). The presence of the unstable branch will occur for all theories we will study that exhibit connement (for zero temperature at least). Again these solutions follow the usual Area Law for the divergent parts of the EE. Something peculiar occurs if we now consider the case of D5/NS5 branes, as discussed in [31, 32]. In this case, we nd that the separation of the connected branch is constant, given by L(0) = R/2 where R2 = [prime]gsNc. Thus there is an innite number of connected solutions which are parameterised by the depth to which they probe but all have the same xed value of L (as in the left panel of gure 8).

The authors of [31, 32] argued that the solutions for smaller values of L < Lc (using the approximation of a capped cylinder similar to the one we will discuss shortly), are exactly

6Notice that this solution does not satisfy the criteria for the concavity given in eq. (2.31).

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Figure 7. Here are cartoons of the change in L and S when we have a theory with connement. The navy blue lines (solid and dashed) in both plots represent the behaviour of the connected part of conformal solutions like that of AdS5[notdef]S5 (i.e. L(0) 10), the green line is the unstable branch intro

duced by connement (like in the soft-wall model). The dotted red and navy lines represent the disconnected solution. We can see that in the conning case there is a phase transition at the point Lc.

those that must live near the UV-cuto , with the contribution to the EE coming from the cap, which present a Volume-Law, once the divergent part of the EE is considered.

Another interesting case occurs in the context of D5 branes wrapped on S1, which as explained, is a model of a conning 4 + 1-d QFT. Here we nd that the connected branch of the EE is similar to that of the IR unstable branch of the soft-wall model, but that there is no stable branch as we move into the UV. This would be akin to only keeping the green line in gure 7 (in this specic case the connected branch asymptotically approaches that of the NS5/D5, i.e. L = R2) see the third row in gure 4. The example of D5 wrapped on S1 then presents the IR features of a conning model (like a soft-wall model), but the

UV behavior of a non-local QFT (like at NS or D5 branes). It presents only an unstable branch and a disconnected one and the absence of a phase transition in spite of displaying a conning Wilson loop. We will appeal to cuto -e ects to solve this issue. Below, we will clarify the details of this example. Before that, we present a useful approximation to some of the quantities involved in the calculations.

3.1 A useful quantity

As an aside, we would like to introduce a combination of background functions that (we have checked) approximates very well the function L(0) in all the cases studied in this paper. This is useful, because in the examples dealt with in the following sections, the functions dening the background are only known numerically (or in semi-analytic expansions). Hence the integrals dening L and S are very time-consuming. Instead, the quantity

Y(0) = 2 H()

p () H[prime]()

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]=

(3.1)

can be seen to approximate very precisely the complicated integral in (2.6) that denes L(0). The analog of this function also appeared in studies of Wilson loops and other probes [33, 34].

3.2 Study of the D5 branes on S1 system

In this section, we will emphasize that the presence of a phase transition for the entanglement entropy in conning theories is sensitive to the UV behaviour of the eld theory. We will make this point by considering the simplest conning eld theory in 4 + 1 dimensions that one can construct by wrapping Nc D5 branes on a circle and imposing periodic (anti-periodic) boundary conditions for the bosons (fermions) of this eld theory. This is in analogy with the example introduced by Witten in [24] by double-Wick rotating a black-brane solution.

Indeed, specialising the results of eq. (2.20), we will have that the string frame metric reads,ds2 [prime] = [parenleftBig]

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u R

[parenrightBig]

[dx21,4 + h(u)d'2c] + R

uh(u)du2 + Rud 23, R2 = gs [prime]Nc.

where the functions h(u) and the dilaton are,

h(u) = 1

2, e = gs [prime]

[parenleftBig]

u R

[parenrightBig]

It is more convenient to change from the energy variable u to the radial variable r = [prime]u.

The background and relevant functions for our calculations are,

ds2 =

[parenleftBig]

r R

[parenrightBig]

[dx21,4 + [prime]h(r)d'2] + R

rh(r)dr2 + Rrd 23, R2 = gs [prime]Nc.

h(r) = 1

R r

2, e = gs r

R, R = [prime] . (3.2)

(r) = r

R2r2h(r), V 2int = (4)4l2'R2h(r)r4,

H(r) = (4)4R2l2'

g4s h(r)r4. l' =

R, (r) =

p [prime]

[contintegraldisplay]

d'.

Using the approximation discussed in eq. (3.1), one can compute that the function L(r0)

asymptotes (from below) to a constant value

L(r0 ! 1) Y(r ! 1) = lim

r0!1

qr2(r2 R2 ) (2r2 R2 )

pgs [prime]Nc (3.3)

hence preventing any form of double-valuedness and phase transition. See the third row in gure 4 for a plot of the calculation done with the background in eq. (3.2). Notice again, that the connected solution has the wrong concavity, hence it is unstable. This would lead us to believe that the disconnected solution would always be the minimal EE solution for all values of L. With this case in mind one can instead ask the question

Question: are there other solutions which have smaller EE that we should consider? As we anticipated above, the answer goes like this: in [31, 32], the authors discuss how non-locality a ects EE calculations. They argue that one should add a UV-cuto and also

Figure 8. Here are cartoons of the types of solutions we shall be considering in the left panel and details of the approximation in the right panel. In both, U represents the UV boundary and is the IR end of the space. In the left panel, the red lines (including the dashed line at ) represent the disconnected solution (D), the green line represents a generic connected solution(C) which probes down to a depth 0 and nally in blue are the solutions which live close to the boundary (B) and behave under the Volume Law for the EE. In the right panel, we outline the various sections of the approximation. The purple solid lines map out the approximation to the connected dashed green solution, which we split into three parts: two vertical contributions labelled as A1 and a horizontal contribution labelled L. The surface mapped out by the dashed purple lines, which is useful when we regularise our approximation, consists again of three parts: the two vertical contributions labelled A2 and the horizontal contribution labelled L0.

consider solutions which live close to it (represented by B in gure 8). These solutions can minimise the EE in cases which exhibit non-locality. There is a di erence between these solutions and the ones we have discussed so far, in that these solutions no longer follow the standard Area Law but instead follow a Volume Law for the divergent part of the EE. This observation was also made in other contexts by [3537]. From this insight we can try to understand in which cases these new Volume Law solutions may be relevant to our question.

Furthermore, it is possible to have a phase transition between these two types of behaviour (Volume Law $ Area Law). We will nd that the Volume Law behaviour is

always linked with the non-local UV behaviour of our theories. So, we may wonder what this implies for the case of the D5 branes wrapped on S1? In this case, we are introducing a connement scale and thus as discussed above (with the example of the soft-wall) we will introduce an unstable branch, joining our disconnected solutions, which have a degenerate point at one end of the connement branch (point X in the right panel of gure 9), to our near UV solutions, which have a degenerate point at the other end of the connement branch (point Y in the right panel of gure 9). The practicalities of realising this UV branch of solutions will be discussed shortly.

Note that in the cartoon in gure 9, the introduction of solutions like B (cf. short solutions in gure 8) mean that for the EE, there is now a phase transition between the disconnected solutions and the extensive solutions at the point ~Lc. In the cases we shall consider, we can argue that a transition to a region in the UV where an extensive behaviour of the EE is the minimal solution, is a sign of non-locality [31, 32, 35, 36], but

A2 A2

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Figure 9. Here are cartoons of the behaviour in a background like that of the D5 branes wrapped on an S1. The dashed red line represents the disconnected solution, and the green that of the connement branch, which join at the point X. With nite UV cuto U, we would nd something similar to the dashed navy blue branch in both plots for solutions near the cuto scale. If we increase the UV cuto (meaning the point Y moves to larger 0), we nd that the gradient of the

UV branch becomes steeper, such that in the limit that we remove the cuto completely, it becomes the vertical solid navy blue line and we reproduce exactly the extensive solutions.

further to that, a sign that one may want to look to UV complete these theories in a non-trivial way, if they are to be correct duals to nice eld theories. Let us now be more precise about the short solutions.

3.3 Finding the short congurations

Now we would like to motivate the existence of these new short congurations explicitly and then use them as a completion for some specic EE diagrams. To this end we will use a particular approximation. We let our surface be rectangular in shape, such that the sides follow the same path as the disconnected surface from = U to = 0 (we can take U ! 1

later). Then we connect the two vertical surfaces A1 with the horizontal surface L at constant 0 (depicted as the solid purple lines in the right panel of gure 8). Initially, we shall rewrite the EE of the disconnected solution SD, by splitting it into two parts joined at 0,

SD = 2(A1 + A2) + L0 (3.4)

where A1 is the contribution from U down to 0, while A2 is the contribution from 0 down to the end of space . Note that L0 would be a contribution from the horizontal piece at the end of space which is vanishing in the cases we consider. From this, we can now write the surface area of our approximating surface as

Sapp = 2A1 + L. (3.5)

It should be noted here that the approximate surfaces Sapp are not extremal surfaces as they are not proper solutions of the equations of motion. Nevertheless, we note that for a xed value of 0, A1 and A2 are strictly constant, while L / L, thus when L ! 0 then

L ! 0. This means that, no matter how small A2 is, there exists small enough values of

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~Lc

L such that we have L < 2A2, and thus Sapp < SD. The existence of these congurations would indicate, that for small L there will exist solutions that have lower EE than the disconnected case, and that there exists actual extremal solutions also with lower EE than the disconnected one.

Now let us derive the precise formula for the area of Sapp. Starting with L, we notice that since the surface is volume lling in all but the [notdef]x1, [notdef] directions, we have

L = [integraldisplay]

Yi=1

Yj=1 d idxjpgind

[integraldisplay]

pg x x . (3.6)

Using the parametrisation x = [notdef]

, u[notdef], with u 2 [notdef]L2, L2[notdef], we can easily deduce that L / Vint

2 e2[notdef]=0L =

pH(0)L (3.7)

The two sides A1 are given by

2A1 = 2

[integraldisplay]

0 d

p ()H() (3.8)

Note that A1 is divergent for U ! 1. Thus we renormalise using the same ap

proach as for the extremal solutions; we subtract the disconnected surface area SD =

[integraltext]

U d

p ()H(). Thus overall we have7

Sapp(L) = 2

Vint

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2 e2[notdef]0L 2 [integraldisplay]

d[radicalbig] ()H().

(3.9)

Notice that this last formula is an approximation of the expression in (2.17). We now study what happens in the examples of the last section.

If we take 0 ! we will have that Vint d/2e2 ! 0 as we observed, it happens in all our models that HEE( ) ! 0 see eq. (2.21). We then recover the disconnected solutions and due to our renormalisation scheme, it is easy to see that the solution will always sit on top of the L-axis of our Sapp(L) plots. This is a feature of all backgrounds studied here. In the other limit 0 ! U our surface becomes the Sapp-axis of the Sapp(L)

plot, and we have a smooth interpolation in between see the top left of gure 10. Note that these lines map out the actual connected solution and thus approximate this case very well. A quantity that proves to be very useful is the point at which the surfaces cross the L-axis. We know that Sapp = 0 whenever

L =

[integraltext]

0 d

p ()H()

Vint d2 e2[notdef]0 T (

) (3.10)

Thus in cases like that of AdS5 [notdef] S5, T (

) will be monotonically decreasing function with T varying between (1, 0] as 0 varies between [0, 1).

7Note that we have added the extra multiplicative factor to take care of the sharp edges of our surfaces that otherwise make the surface a worse approximation to the actual minimal solutions.

If we now study the soft-wall case (see the top-right panel of gure 10), we can see that the solutions again near the UV look similar to that of the AdS5 [notdef] S5 case above,

as expected. The di erence lies in the solutions near the IR, where the surfaces initially begin to move into the positive plane of the Sapp plot, meaning they have higher EE than the disconnected solution. A phase transition appears as one would expect. This change can also be seen in the behaviour of T (0). Indeed, now T ! [notdef]0, 0[notdef] as 0 ! [notdef]0, 1[notdef].

Additionally T (0) is an increasing function for small 0 (below the connement scale) and a decreasing function above it.

Now let us discuss backgrounds which exhibit non-locality. First we look at the case of D5 branes as studied in [31, 32] and depicted in the bottom-left panel of gure 10. We see that the surfaces all intercept the axis at L = /2 (as we have set R = 1) with increasing gradient as we move towards the UV. Thus in the limit we would expect to nd a vertical line at L = /2. This agrees with the expectations of our discussion above.

Finally, we move to the D5 wrapped on S1 that mostly occupied us in this section. The associated plot can be seen in the bottom-right panel of gure 10. Here we nd that the surfaces initially move up into the positive plane of the Sapp plot and they then asymptote the same value as the at D5 solution leading again to a vertical line at L = /2 (choosing R = 1) in the UV. The two D5 cases discussed di er in a subtle point. While the phase transition in the two cases is always between Area $ Volume Law behaviour, in the at

D5 case, all the connected solutions sit at the transition point.

These short congurations living at the cuto appear and play an important role, every time we have a non-local QFT. They are needed in order to avoid having only a connected-unstable and the disconnected solutions. The short congurations imply the existence of a phase transition between connected-stable and disconnected branches of the EE.8

In the next section, we will study another example of this cuto e ect when studying the behavior of the EE in one of the trademark models of conning eld theory (but with a non-local UV). We will also learn that a similar job as the one done by the cuto can be done by a UV-completion of the QFT. This gives a well-behaved EE-phase transition, with an area law for the divergent part of the EE, etc.

4 The absence of phase transitions in (some) conning models

The reader might object that the example of D5 wrapped on S1 discussed above is not such a good model for a conning eld theory. In principle theories in 4 + 1 dimensions have strongly coupled UV behavior, their IR tends to be weakly coupled and what we observed is just an e ect of these features, in contradiction with us imposing the model to be conning. One may imagine that for duals to eld theories in 3 + 1 dimensions the phase transition should reappear.

Below, we will analyse this claim, by rst studying the case of a dual QFT obtained by wrapping Nc D5 branes on a two-cycle of the resolved conifold [38, 39]. We will discover

8Notice that we are taking limits in a given order; rst 0 ! U followed by U ! 1. This is very

reminiscent of the treatment in [19].

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Sapp

[Minus]20

[Minus]40

[Minus]60

[Minus]80

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[Minus]100

0.0 0.5 1.0 1.5

Sapp

Figure 10. In the above we have plotted a number of the surfaces Sapp(L). The colour scheme

is such that purple lines are surfaces with 0 approaching 0, and the red solutions which have 0 approaching U. The top-left panel is that of AdS5 [notdef] S5, then in the top-right the Soft-Wall, the

bottom-left is at D5 branes and the bottom-right is D5 branes wrapped on S1.

that the behaviour in the UV is not much di erent from the one of the at-D5 brane just analysed above. The reader may argue that this is due to the fact that at energies high enough, the dual QFT becomes higher dimensional, an innite set of KK-modes coming from the compactication of the D5 branes on the two-cycle indicate the higher dimensional character of the QFT. This is of course correct, but the point is subtle. Indeed, as Andrews and Dorey [40] have shown (in the perturbative regime), the eld theory is completely equivalent to four-dimensional N = 1 Yang-Mills, expanded at a particular point of its Higgs branch, which is a well-dened 4-d QFT. Also, the same sort of KK-modes appear if we compactify a stack of D4 branes on S1 and in that conning model the phase transition is present, see [11] and section 2. We will then carefully calculate the entanglement entropy for this QFT based on wrapped D5 branes.

Let us start by writing explicitly the metric describing D5 branes wrapping a two-cycle inside the resolved conifold. In string frame, we have a (dimensionless) vielbein,

exi = e

[prime]gs dxi , e = e

2 +kd , e = e

2 +hd , e' = e

2 +h sin d' ,

e1 = 12e

2 +g(~

!1 + a d ) , e2 = 12e

2 +g(~

!2 a sin d') , e3 =

2 +k(~

!3 + cos d') .

The quantities ~

!i, i = 1, 2, 3 are the left-invariant forms of SU(2). In the string frame we have a metric,

ds2str = [prime]gs

Xi=1(ei)2 . (4.1)

The background is completed with a dilaton () and a RR three-form that we will not need to write here see, for example section II in the paper [41] for a complete description of the system.

It is sometimes more e cient, to change the basis to describe the background and RR elds from the functions [a, h, g, k, ] to another set of functions [P, Q, , Y, ]. This is useful because in terms of the second set, the BPS equations decouple. It is then possible to solve the non-linear ordinary BPS equations, so that everything is left in terms of a function P (), that satises a non-linear ordinary second order di erential equation. The change of basis is explicitly given in eq. (3) of the paper [41]. We summarise it here for future reference. After having solved for [Y, Q, , ] and choosing integration constants to avoid singularities, we have

4e2h = P 2 Q2P coth(2) Q

, e2g = P coth(2) Q, 2e2k = P [prime], ae2g sinh(2) = P, e4 4 0 = e(2h+2g+2k) sinh(2)2, Q() = Nc(2 coth(2) 1). (4.2)

The function P () satises a second order non-linear di erential equation, sometimes called master equation in the bibliography [42]. Di erent solutions to the master equation have been discussed and classied in [4244].

The functions needed to calculate the entanglement entropy in this string-frame background are,

=e , = [prime]gse2k, V 2int =(2)6( [prime]gs)5e4h+4g+5 +2k, H =(2)6( [prime]gs)5e4 +4g+4h+2k.(4.3)

One simple solution of this master equation for the function P (), leading to a smooth background, is known analytically and given by P () = 2Nc. This is the solution in [38, 39]. One can check by replacing in eq. (4.2) that the dilaton behaves as e4

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. We

obtain a behavior similar to the one around eq. (3.2), considering the change in the radial variable between one description and the other (that for large radius is log r), both are

examples of linear dilaton backgrounds.

Indeed, calculating the entanglement entropy S(0), the separation between regions L(0) and then plotting parametrically S(L), we nd that there is no minimal solution present, only the disconnected and the unstable connected ones. The latter exists over a nite range for 0 < L <

2 pgs [prime]Nc with Sc(0) = 0 and Sc(2 pgs [prime]Nc) = 1. Using the

approximation of eq. (3.1), we nd,

L() Y() =

p [prime]gs ek2 [prime] + 2h[prime] + 2g[prime] + k[prime] = (4.4)

Y() = p [prime]gsP [prime] p2

(P 2 Q2)2P 2 coth(2) + P P [prime] QQ[prime] 2Q2 coth(2)

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Figure 11. The system of D5s on a two-cycle Here we plot L(0) and S(L).

Which for the exact solution

P = 2Nc, Q = Nc(2 coth(2) 1),

gives the approximated asymptotics for the function L(0),

L(0 ! 1)

p [prime]gsNc 2

, L(0 ! 0)

p [prime]gsNc2 0.

notice that the Heisenberg-like relation L 10 is violated here. Also note that the

entropy scales asG10Sc

V2 ( [prime]gs)3N

3 2

c .

1 1 40

Like in the example of the compactied D5 branes on S1 above, we see that the connected conguration is unstable not satisfying the concavity condition of eq. (2.31). We do not see the possibility of a phase transition. But here is where the short congurations and the e ects of the UV cuto enter to cure the problem.9

In gure 12 we show the result of considering these short congurations at the UV-cuto . We see that these are the correct congurations to consider as they avoid the instability issue. The phase transition in S(L) is recovered, as it corresponds to a conning model.

4.1 More on non-locality

To make the point of the non-locality clearer, we will consider another solution describing D5 branes compactied on the two-cycle of the resolved conifold (the eld theory has di erent operators driving the dynamics). This second solution is well described in the papers [42, 43] see section 4 of the paper [44] for a good summary.

In contrast with the simple analytic solution P = 2Nc; this second solution is not know analytically. Indeed, only a large radius/small radius expansion is known. A

9As an aside, it should be noted that the nite size e ect reected in the non-zero value of L(0 ! 1),

was observed also for the Wilson loop when calculated in this sort of backgrounds in [4548].

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Figure 12. The system of D5s on a two-cycle Here we plot L(0) and S(L) again, but this time introducing a UV cuto . The green line is the solution without UV cuto , the dashed blue line is with the cuto at U = 19, the dotted blue is with the cuto at U = 19.5, and nally the dashed red line represents the disconnected solution. Notice that increasing the value of U leads to an increase in the gradient in the visible branch in the S(L) plot in the right panel.

numerical interpolation between both asymptotic behaviours is quite easy to obtain. The large and small radius expansions for the function P () read,

P ( ! 0) h1 +

4(h21 4N2c)15h1 3 +

16 1575h31

(3h41 4h21N2c 32N4c)5 + O(7)

P ( ! 1) ce4/3 +

N2cc e4/3[parenleftbigg]

42 4 +

[parenrightbigg]

+ O

e8/3

[parenrightBig]

(4.5)

where h1, c are two integration constants of the master equation mentioned above.

We see that the small radius expansion is quite similar to the one of the exact solution P = 2Nc. Indeed, for the constant h1 = 2Nc we recover the exact solution. On the other hand, the large radius expansion of the solution for the function P () is quite di erent from the linear behaviour of the exact solution. These di erences and similarities suggest that the dynamics of the QFT dual to the second solution is actually a icted by an irrelevant operator. In the paper [41], this point was made precise (see also the discussion in [44]). The operator can be seen to be of dimension eight. The situation is not so di erent from the case of keeping the constant factor in the warp factor of the D3 branes solution = 1+ L4r4 .

Indeed, it can be shown that the factor 1 makes the background of Nc D3 branes dual to N = 4-SYM with a dimension eight operator inserted. In order to UV-complete this QFT one needs to insert back the whole tower of string modes.

The eld theory dual to the background obtained with the solution P = 2Nc is

a icted by less severe non-localities than the eld theory dual to the second semi-analytical solution of eq. (4.5). We can see how the entanglement entropy reects this. We recalculate the numerics for the entanglement entropy S(0) and separation L(0) (we do this for a sample numerical solution of the form given by eq. (4.5), where the irrelevant operator is inserted with a small coe cient h1 = 2Nc + [epsilon1]). The result is described in gure 13, showing that L(0) deviates even more from the needed double valuedness. The saddle point solution does exist for all values of L and there is not any other connected solution.

Figure 13. The system of D5s on a two-cycle but with exponential behaviour in P (h1 = 203100Nc) Here we plot L(0) and S(L). The grey line is the linear P solution (h1 = 2Nc) for comparison.

Note that whilst L(0) vanishes linearly for small values of 0, it grows exponentially in the UV, L(0 ! 1) e20/3. Of course, this departs even more from the Heisenberg-like

scaling L 10 characteristic of local eld theories. All these results can be obtained from

eq. (4.4) with the solution in eq. (4.5) and are reected by gure 13.

As described above, even when the string dual shows a conning Wilson loop or area law behaviour,10 the system is not showing a phase transition in the entanglement entropy. All this relies on the UV properties of the eld theory. The same phenomena can be displayed in other systems with qualitative similar eld theoretic high energy behaviour, see appendix C.

One may wonder if all hope for observing a phase transition in the entanglement entropy (without appealing to the above discussed regulating cuto and short solutions) is lost. In the next section, we will discuss a possible UV completion of the eld theory on the compactied D5 branes. This completion, in terms of an inverse Higgs mechanism is discussed in the papers [41, 50, 51] and [43]. To this we turn now.

5 Recovering the phase transition: the baryonic branch

As we discussed above, the non-local UV-properties of the eld theory dual to a stack of Nc D5 branes compactied on the two-cycle of the resolved conifold, needs of the introduction of a UV-cuto (and associated e ects) for the existence of a phase transition for the entanglement entropy (even when the Wilson loop displays an area law). A cleaner way of recovering the phase transition that the Klebanov-Strassler system displays [11], will be to appeal to the connection between both systems. This connection is well understood and discussed in the papers [41, 50, 51] and [43]. A simple U-duality (or equivalently, a rotation of the SU(3)-structure, characterising the D5 brane solution) connects both backgrounds and this constitutes the string/geometric version of an inverse Higgs mechanism; see the discussion in the paper [41].

10A subtle point is that the Wilson loop cannot be strictly calculated with the solution of eq. (4.5). This is due to the boundary condition for the strings at innity, that cannot be satised. See [49].

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There is a simple and subtle point we wish to emphasise. In performing this U-duality or rotation of the SU(3)-structure, there is a constant to be chosen. Choosing it to a precise value amounts to UV-completing with the precise matter content such that the UV of the resulting QFT is as healthy as the Klebanov-Strassler QFT. See the discussion in [41] for a careful explanation of this point. In other words, choosing this constant in the appropriate way implies that the warp factor of the background asymptotes to zero and the switching-o of the dimension-eight irrelevant operator discussed in the previous section.

The new background, generated by the U-duality or the rotation on the SU(3)-structure forms is described by a (dimensionless) vielbein, in string frame,

2 +k1/4d, E = e

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= e

21/4 dxi

[prime]gs , E = e

2 +h1/4d , E' = e

2 +h1/4 sin d', (5.1)

E1 = 12e

2 +g1/4(~

!1 + a d ) , E2 = 12e

2 +g1/4(~

!2 a sin d') , E3 =

2 +k1/4(~

!3 + cos d') .

The function is,

= 1 e2 2 (1). (5.2)

The crucial choice of constant mentioned above reects the fact that the warp factor vanishes at large radius. Notice also that this expression for implies that this U-duality or rotation of the SU(3)-structure can only be performed in the case in which the dilaton is stabilised at large radius, namely (1) is a nite value. In other words, we cannot use this

for the analytical solution P = 2Nc, that has linearly growing dilaton. Only the solutions described around eq. (4.5) can be used.

We only quote the background vielbeins after the U-duality, from which the full background can be obtained. In this case the elds H3, F5, F3, are generated and the solution is precisely the dual to the Baryonic Branch of the Klebanov-Strassler QFT [52].

The string frame metric is,

ds2str = [prime]gs

Xi=1(Ei)2 , (5.3)

and the background functions are again determined by the functions P , Q as in eq. (4.2), where P satises the same di erential master equation as before, solved by the function P () in eq. (4.5). In other words, both solutions are the same; this U-duality or rotation of SU(3)-forms is a solution generating technique.

The quantities needed for the calculation of the entanglement entropy are,

= e 1/2, = gs [prime]e2k, V 2int = (2)6( [prime]gs)5e4h+4g+5 +2k5/2,

H = (2)6( [prime]gs)5e4 +4g+4h+2k. (5.4)

We see that the di erence to eq. (4.3) is the presence of the factor dened in eq. (5.2). This warp factor has a large radius asymptotic given by see section 4 of [44],

3N2c

8c2 e8/3(8 1) + . . . . (5.5)

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Figure 14. A typical solution on the Baryonic Branch of Klebanov-Strassler (h1 = 203100Nc) Here we plot L(0) and S(L). The grey line is the linear P solution (h1 = 2Nc) for comparison.

and it is precisely this decaying behaviour that will bring back our phase transition. Indeed, we can calculate using eq. (3.2), that the small and large asymptotics of the function L(0) vanish,

L() Y() = 2

p [prime]gs ek[radicalbig]

h(4 [prime] + 4g[prime] + 4h[prime] + 2k[prime]) +[prime]h, (5.6)

L(0 ! 0) 0, L(0 ! 1) e20/3.

As anticipated, the IR behavior for L(0) is the same as the one for the analytic solution

P = 2Nc or the one of eq. (4.5); this is because the warp factor is constant for small radial coordinate. The UV behaviour instead, is quite di erent and driven by the factors of in eq. (5.4). Notice also, that in a convenient radial variable r = e2/3, we have

L(r0 ! 1)

1r0 , (5.7)

This is a signal of locality according to [31, 32]. The UV-completion provided by the baryonic branch eld theory has recovered locality. What about the phase transition in the EE?

Calculating with the expressions of eq. (5.4), the entanglement entropy associated with the Baryonic Branch of the KS-eld theory gives the result displayed in gure 14.

As one can see, here we get the nice phase transition behaviour, also expected from the calculation in the paper [11]. The intuitive description of what is going in terms of a Van der Waals gas analogy is consistent with this result as well. Figure 15 serves as a good summary of the discussion in the previous and the present sections.

We have settled the problem of recovering the phase transition. We would like now to do a couple of calculations that will lead to an improved understanding of the Klebanov-Strassler system (in nice agreement with the ideas presented in [53]). We will move our eld theory to a mesonic branch. We will do this rst in a way, that as was argued in [43], implies that towards high energies the evolution of the QFT is described in terms of Seiberg dualities, but more importantly successive Higgsings that change the matter content very fast. This quick growth of the matter content is equivalent to the addition of another

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Figure 15. These are various plots of L(0) for comparison. The grey graph is the solution P = 2Nc with linear dilaton. The green and blue lines represent numerical solutions before and after the U-duality respectively (solved for the same value of h1 = 203

100 Nc with asymptotically

constant dilaton.

irrelevant operator (of dimension-six in this case, di erent from the dimension-eight one we have discussed above). This was discussed in detail in [43] and [44].

As a consequence of the insertion of this new irrelevant operator, with the added nonlocalities to the QFT, we will loose the phase transition achieved (in nice agreement with the discussion of the previous section). We will then explain how to get the phase transition back, with a precise way of switching o that irrelevant operator. We turn to that now.

5.1 Losing our phase transition: adding sources

As described above, we will move our Klebanov-Strassler QFT from the baryonic branch to a mesonic branch. In order to do this, we need to de-tune the ranks of the two gauge groups that in the baryonic branch is SU(kNc) [notdef] SU(kNc + Nc). The unbalance is achieved by

adding matter that in the dual string theory is represented by D5 branes with induced D3

brane charge. This will change the gauge group to SU(kNc +nf)[notdef]SU(kNc +Nc +nf + Nf2 ).

The Nf-D5 and nf-D3 branes are added as sources; namely the background is a solution to the equations of motion of Type IIB Supergravity plus the Born-Infeld Wess-Zumino action for the sources. The associated solutions were discussed in detail in [43, 44]. One characteristic is that the sources need to be added with a prole that vanishes close to ! 0, in order to avoid curvature singularities (see the discussion in [48, 54] and [44]). A

prole that preserves the same SUSY as the background and avoids all singularities, can be derived to be [48],

S() = Nf tanh(2)4.

Also, this prole can be translated according to

S() = Nf ( ) tanh(2 2 )4 (5.8)

and still preserve SUSY and avoid singularities everywhere.

The formalism used in deriving these backgrounds runs parallel to the one described above, in the sense that a change of basis and a second order di erential master equation can be written for a single function P () that in this case will contain the e ect of the (nf, Nf) D3-D5 sources. A rotation of the SU(3)-structure forms is used to generate the solutions dual to the mesonic branch of the KS eld theory. See section 3 of the paper [44] for a clear summary of the set-up and its subtleties.

In the papers [43, 44] a solution encoding the e ect of the sources was found. The large radius asymptotics for the warp factor in eq. (5.2) is given in eq. (2.25) of the paper [44], using the radial coordinate r = e2/3 we have,

lim

r!1

Nfr2 + 3N2c log r

r4 , (5.9)

This deviates (by what seems to be the addition of an irrelevant operator of dimension-six) from the cascading behaviour of the Klebanov-Strassler QFT.

The eld theoretic logic is again, that the number of D3 sources, nf, grows very fast as was discussed in [44]

nf S()(sinh(4) 4)1/3 e4/3, (5.10) and this rapid growth of the gauge group ranks going to higher energies (due to Higgsing every time a source D3 is crossed) implies that the QFT looses the 4-d character of the KS system.

Aside from this, the calculation of the Wilson loop in the solution mentioned above will produce an area-law behaviour, indicating connement. Following the logic of the previous sections, we should expect that in spite of the conning behaviour, a phase transition in the entanglement entropy is wiped-o by the irrelevant operator (the presence of this irrelevant can also be associated with nite-size e ects in computing the Wilson loop).

This is indeed the case as one can explicitly calculate. While the small radius asymptotics for L(0 ! 0) 0 is unchanged, we obtain that

L(0 ! 1) = Y( ! 1)

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p [prime]gsNf 8 .

Figure 16 illustrates this point, with = 0, so no phase transition, unless of course a

UV-cuto is introduced and the study of short congurations sorts out the problem as in the examples above.

This brings us to the conclusion that the eld theory may be in a mesonic branch with good IR properties, but that we should look to somewhat localise the sources to avoid this too-fast growth of degrees of freedom expressed in eq. (5.10), if we wish to recover the high energy 4-d behaviour the baryonic branch was displaying (which is UV completing the system). We do this in the next section.

6 Getting back the phase transition(s): sources with a decaying prole

As discussed above, the lost phase transition in the entanglement entropy (without introducing the UV-cuto ), together with the asymptotic form of the warp factor in eq. (5.9),

Figure 16. Plots of L(0) and S(L) for a solution with S ! Nf in the UV and Nc = 4, Nf = 9.

No phase transition is present. Note that we have chosen a value of h1 such that we have hardly any linear behaviour in P .

strongly indicates that the QFT is not behaving as a 4-d QFT, in the sense of locality being lost. It was understood in [44] that this is due to a very rapid growth of degrees of freedom.

In backgrounds where the prole for the sources is S tanh(2)4, this is reected by

the fact that the ow to the UV of the QFT is described by a superposition of Seiberg dualities the logarithmic term in eq. (5.9) and a Higgsing, represented by the term quadratic in the radial variable. Another interplay between Higgsing and Seiberg cascade was previously observed in [55, 56]. In the particular solution with source prole S

tanh(2)4, the rate of Higgsing becomes too fast at high energies, the UV of the eld theory does not behave like a 4-d QFT. This is reected in the entanglement entropy, which does not display the nice phase transition achieved in section 5.

We would like this mesonic branch solution of the KS eld theory to behave like a 4-d QFT. In order to do this, we will slow-down the growth of degrees of freedom, by proposing a phenomenological prole for the sources. This is phenomenological in the sense that is not derived from rst principles (as a kappa symmetric embedding of sources with this prole). Nevertheless, the prole we will propose has the following properties [44]:

The background still satises BPS equations, suggesting SUSY preservation.

The energy density of the sources T00 is positive denite for proles that decay at

most as fast as the one we will propose.

The central charge of the dual QFT when calculated with this prole is a monotonic

and growing function.

The prole we will adopt, following [44] is,

S() = Nf tanh(2)4e4/3. (6.1)

Notice that now, one can nd a new background solution, where the sources are somewhat localised. The dual QFT is in a mesonic branch as explained in [44], where the background solution was explicitly written.

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Figure 17. Plots of L(0) and S(L) for a solutions with S ! 0 in the UV and Nc = 4, Nf = 9. A

single rst order phase transition is present. Note that we have chosen a value of h1 such that we have hardly any linear behaviour in P .

The relevant functions for the calculation of the entanglement entropy are those in eq. (5.4). The explicit expressions for the asymptotics (and numerical solutions) of all the participating functions are described in section 4 of [44]. In particular

( ! 1)

38c2 e8/3

hN2c(8 1) + 2cNf 4NcNfS1[bracketrightBig]+ . . . .

S1 =

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[integraldisplay]

0 d S tanh(2)2.

Comparing this with eq. (5.5) we observe that the functional decay of the warp factor is the same. The cascade of Seiberg dualities is still present (in this radial coordinate is represented by the term N2c). The constant terms represent the e ects of the sources, that even when they are very suppressed at large values of still contribute. Indeed, using the radial coordinate r e2/3, they contribute to the warp factor as

Nfr4 , which was

expected for a localised stack Nf nf D3 branes.

It is not hard to believe that if we follow the full numerical calculation with the expressions of eq. (5.4), we will nd a phase transition in the entanglement entropy. This is indeed the case, the plots of gure 17, make this point concrete.

The function L(0) is well approximated by the function Y(0) and satises a

Heisenberg-like relation, as shown in eq. (5.7), with the numerical di erences of the case induced by the factors of Nf and S1 in the warp factor

, and in the other background functions. Still, we have L(r0 ! 1) 1/r0, in agreement with the locality criteria

proposed in [31, 32].

An interesting observation is that since the sources can be translated at a given point as discussed above

S() = Nf ( ) tanh(2 2 )4e4/3 (6.2)

we can introduce another scale to the system, represented by , the point where the

source become activated. Numerical solutions (that are a bit more time-expensive to

Figure 18. Plots of L(0) and S(L) for a solutions with S ! 0 in the UV and Nc = 4, Nf = 9

but = 2510. Two rst order phase transitions are present. Note that we have chosen a value of h1

such that we have hardly any linear behaviour in P .

nd) can now show a double phase transition, as can be seen in gure 18 (for further discussion see appendix C).

This is a phenomena that probably was observed in other contexts, but we are unaware of them in the bibliography. There might be gas, generalisation of Van der Waals one which includes two di erent interactions of similar strength between the particles composing the gas. This could lead to a double or even multiple rst order transitions. The gure 18, illustrate our results for the entanglement entropy, L(0) and S(L), when calculated in the mesonic branch of the KS eld theory in the presence of a localised bunch of matter represented by a D3-D5 bound state.

We wish to close this section by emphasizing that our ndings are clearly making the point of KKM [11]: a phase transition in the EE is a sign of a conning QFT. But one should be careful about the UV behavior of this eld theory. If non-local, either cuto e ects or a UV completion will recover the phase transition. We can also think our ndings for the EE as a diagnostic to decide if a QFT is showing or not the high energy behaviour expected from a four-dimensional (or lower) QFT, free of non-localities.

7 Conclusions and future directions

Let us rst summarise the contents and ideas explained in this paper.

We started by observing the obvious analogy between the holographic calculation of Wilson loops and Entanglement Entropy. Indeed, being both minimisation problems for two- and eight-surfaces, they display very similar general solutions. In spite of the analogous formulas derived when minimising surfaces, the two observables are such that when evaluated on particular backgrounds the results are quite di erent. For example, in conning backgrounds (the topic that mostly occupied us in this work) the Wilson loop holographic calculation gives a linear dependence between the quark-antiquarks Energy and separation EQQ LQQ (for large LQQ), while for the Entanglement Entropy, we observe a very di er

ent behavior, including a rst order phase transition. Aside from these observations, either obvious or made previously in the bibliography, the rst contribution of this paper was to

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develop a simple and operative criteria to test under which conditions the Entanglement Entropy would display a phase transition. This was done in section 2, where aside from the criteria for phase transitions, some examples were given and an important question was posed: Why the relation between Connement and the existence of a phase transition in the Entanglement Entropy breaks down, for models based on D(p > 4) branes?

Section 3 answered this question, after a detour into non-local QFTs, QFT with a cuto and the realisation that, when calculating with string duals to non-local QFTs, we are potentially missing a set of congurations that are very important in the calculation of the Entanglement Entropy. These congurations that become apparent when considering the non-local QFT with a cuto , solve the Physics problem of having only a disconnected and an unstable solution. This is the material discussed in section 3, together with the explicit solution for a simple conning model D5s on S1 where the problem originally appears.

It may be unpleasant to some physicists that we need a UV-cuto to resolve the problem of stability and to regain the phase transition argued to be present in conning models in [11]. Nevertheless, we want to point out that this UV cuto is actually capturing the behavior that the QFT, once UV-completed, will display. Indeed, this is the point made in section 4. We studied a trademark model of connement in four dimensions, the string background corresponding to D5 branes wrapping a two-cycle of the resolved conifold. We explained how this model would not display a phase transition if taken at face value. But upon the introduction of the cuto at high energies, we observed the phase transition and the whole behavior of a four dimensional conning eld theory with a Hagedorn density of glueball-states. The e ect of this cuto is the same as the one found in the UV-completed QFT on the D5 branes. Indeed, an inverse Higgs mechanism takes place completing the non-local QFT into the Klebanov-Strassler eld theory (in a generic point of its Baryonic Branch). This point is made clear, with calculations and plots of the S(L) phase transition in section 5. Further to this, in section 5, we also pointed out that some backgrounds describing the Mesonic Branch of the Klebanov-Strassler eld theory (these backgrounds include a large number of D5-D3 sources) are also a icted by a non-locality, unless the sources are introduced with a particular prole proposed (using completely di erent arguments) in [44].

This completes a very pleasant picture advocated in section 6, linking conning, nonlocal QFTs and their local UV-completed counterparts. Further a link between the Entanglement Entropy and its phase transition, that act as a measure of both locality and connement. Numerous appendices complement the presentation, and study a wide variety of other examples, to further illustrate our ideas above.

Let us describe a couple of ideas that this work suggests as possible extensions of what we have learnt here.

First of all, since we found that conning models typically imply a phase transition for the EE, it would be interesting to ask what happens to the EE when one considers a conning model that presents also a phase transition for the Wilson loop. Indeed, models with various scales have shown this behavior see for example [41, 57]. It may be, as we argued in section 6, a multi-phase transition is present for the EE in these cases. It is also worth analysing if the conning behavior implies that H( ) = 0, since the argument

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we gave involving the central charge of the eld theory may be evaded. In this same line, the criteria for phase transitions in the EE, discussed in section 2, may lead to interesting extensions. One could try to nd out if connement is actually needed for the EE to present a phase transition. Studying the invariances of the EE under di erent dualities seems like another small and nice project. We already know that S-duality and non-Abelian T-duality [5860], are invariances of the EE.

On more general grounds, an observation that this paper suggests is the following: we know that black holes for Dp branes (with p > 4) turn out to have negative specic heat. We also know that the holographic renormalisation program can be successfully applied to backgrounds based on Dp branes with p < 5 [61]. We found that the connection between non-locality of the QFT and the absence of the phase transition in the EE, is there for solutions based on Dp branes with p > 4, and this signals that probably, as it happens with the EE, one may nd a way to x the density of states of the nite Temperature QFT and also, with a UV-cuto , or better with a suitable UV-completion, one may be able to implement the program of holographic renormalisation. Sorting out which observables turn out to behave similarly with the cuto , or the UV-completion, seems another interesting problem.

Acknowledgments

Discussions with various colleagues helped to improve the contents and presentation of this paper. We wish to thank: Ofer Aharony, Adi Armoni, Carlos Hoyos, Prem Kumar, David Kutasov, Yolanda Lozano, Niall T. Macpherson, Patrick Meessen, Maurizio Piai, Alessandro Pini, Diego Rodriguez-Gomez and Johannes Schmude.

This work started while Carlos Nunez was a Feinberg Foundation Visiting Faculty Program Fellow, he thanks the hospitality extended at Weizmann Institute and The Academic Study Group for the Isaiah Berlin Travel award.

A Wilson loop-entanglement entropy relation an exercise

The similarities between the EE and the Wilson loop, summarised in equations (2.14), (2.17) suggest an interesting small exercise. We can ask what are the conditions on a given background, so that the EE and the Wilson loop have the same L dependence. In order to solve this exercise, we will consider situations in which the EE behaves, for large separations, as S Lp, being p some positive number. For example, in the case

of conformal eld theories in d + 1 dimensions, one nds that p = d 1. This is referred

to as an Area law for the EE. In some other examples characteristically in non-local (d+1)-dimensional theories one nds p = d, in which case the name of Volume law is used. We will rst study, based on the similarities alluded to above and summarised by equations (2.14), (2.17) the characteristics that a background must have such that the Wilson loop behaves like the EE, namely EWL LpWL. Let us start with an IR analysis.

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A.1 Small radius expansion of the Wilson loop

The functions (), g() characterising the Wilson loop computations, can be expanded around = as follows

() = ( ) + akk + O(k+1), g() = bjj + O(j+1).

If ( ) [negationslash]= 0 the Wilson loop exhibits linear connement with ( ) the string tension.

We are not interested in a linear law EWL f( )LWL and therefore assume ( ) = 0.

Further assuming k > 0, j > 1 and few other reasonable assumptions on the functions

(), g(), the authors of [17] found that

E L

for large L and for the case k > j + 1 (the case k j + 1 will not result in a negative

power of [lscript]).

A.2 Large radius expansion of the Wilson loop

On the other hand, the functions (), g() can be expanded around the boundary = 1

as follows

() = cnn + O(n1),

g() = g(1) + dmm + O(m+1),

with n, m > 0. In the region close to the boundary, i.e. small distances, the Wilson loop then behaves as [25]

E L

The UV-behavior of the Wilson loop will take the same functional form as the EE (for the case of a d + 1 QFT with Area law) S L1d, when

n = d

d 1

For the Dp brane we then have n = 7p2. The UV-behavior of the Wilson loop in the p-dimensional QFT, coincides with the functional dependence of the EE, calculated for a strip of lenght L in d-space dimensions and with Area law, when

p = 5d 7

d 1

. (A.2)

B A taxonomy of behaviours for systems with sources

Adding sources to the D5 wrapped on a two-cycle as discussed in sections 5.1 and 6, creates a rich and complex family of backgrounds, with many free parameters which inuence the

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j+1

kj1 ,

1n1 . (A.1)

behaviour of the Entanglement Entropy of the dual QFT. In this section we will systematically categorize the di erent cases the parameter space allows.11 The behaviour of the Entanglement Entropy is most severely inuenced by the following choices:

Rotation - The U-duality/rotation of the SU(3)-structure described in section 5 and 5.1, that we will refer to as rotation for brevity, will only be applicable to cases where the solution to the master equation exhibits exponential behavior in the UV see eq. (4.5). However, even after the addition of sources, solutions that are linear in the UV can still be found. In what follows we will make a clear distinction between the linear (and thus unrotated) and rotated exponential behaviour of P () . Note that here we shall not study unrotated exponentially growing solutions, representing QFTs coupled to gravity and string modes, as they are known to require a non-trivial UV completion and in these cases the Entanglement Entropy S(L) will always diverge as in gure 13.

Prole - The second important choice is the type of prole to be used. In what follows we will focus on two types of prole, those proles in which lim!1 S() = 1, or those that instead have lim!1 S() = 0. The rst type are given by eq. (5.8) and we will refer

to proles of this type as sigmoid proles. The proles we will adopt for the second case are given by eq. (6.1) and will be referred to as bump-like proles.

Thus we will divide this section into 4 subsections, each discussing one possible combination of the above choices. The analysis in each subsection involves the study of the interplay of the three relevant scales in the background:

- The scale at which the source prole S() becomes non-zero

- The scale at which P changes from having linear behavior (when <

exponential asymptotic behavior (when >

x = NfNc - the ratio of source branes to color branes present

In each section, the discussion will usually start with the case where = 0, to see what

e ect the addition of sources has. Once this case is understood, predicting the behaviour of the system with non-zero becomes trivial. For < , the system will behave like

the corresponding solution on the Baryonic Branch, and then at will smoothly switch

to the associated behaviour of the sourced system.

Further, note that the scale

is only nite in the rotated case. Thus, in a similar fashion to the discussion in the last paragraph, we know that for <

the system will behave like its corresponding linear case (with or without the addition of sources). Thus it makes sense to study the linear P solutions with added sources rst.

Finally, although we do not always discuss them here directly, the short congurations play an important role in the examples below that exhibit non-locality in the corresponding QFT, in a similar way to the cases presented in the main body of the paper.

11It should be noted that we will focus here on the connected solution. In all cases, we chose a renormalization scheme such that the disconnected case will be a horizontal line at S = 0 in the S(L) plots.

12Note this only applies in cases with P e4/3 but not in cases with P .

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) to having

)12

B.1 Linear P sourced systems with sigmoid proles

All the solutions of this form are such that L ! const [negationslash]= 0 from below with no phase transi

tion in S(L) as we move toward the UV. That said, there exists an interesting dependence on x = NfNc , with x = 2 playing an important role. For x < 2 and setting [prime] = gs = 1 we have,

L(0 ! 1) =

which is while for x > 2 we get

L(0 ! 1) =

For x = 2 we get precisely

pNc. (B.3)

We see that for x < 2, L approaches 2 pNc (as in the sourceless case), while for x > 2, L approaches 2

pNf Nc. In the case of x = 2, the UV expansion is exact, indicating that L reaches its bound at a nite value of . Taking into account the analysis of this class of solutions found in [62], this behaviour can be explained through a Seiberg Duality picture, which involves taking the function Q ! Q and Nc ! Nf Nc, relating solutions below

and above x = 2. For x = 2, we have an invariance under the aforementioned duality transformation, and this e ectively freezes the entanglement entropy in place (away from the IR), causing the S(L) plot of this solution (see gure 19) to stop at a nite value and not grow without bound as S(L) for all other solutions of this class.

If we now consider proles with non-zero , for 0 < , the solution will follow the

sourceless case, whose limit is L ! 2 pNc, and thus smaller than or equal to the limit of the

equivalent solution with sources. This, and the fact that adding sources leads to the solution approaching its UV limit faster than the sourceless solution, guarantees that adding sources at will always cause an increase in L (as shown in gure 19). Hence, it is not possible

to produce a phase transition, due to the addition of sources, for these kinds of solutions.

B.2 Linear P sourced systems with bump-like proles

The most important di erence between this and the previous subsection is that the source prole is suppressed in the UV. Thus we expect the deviation from the sourceless case to diminish for large . Nevertheless, the prole still introduces an interesting dynamic into the system, especially when we look at non-zero . Although so far we have described this

as the scale at which the sources become active (forcing the system to deviate from the sourceless case), it now further dictates the point at which the sources e ectively become inactive, thus bringing the system back in line with the corresponding sourceless case.

A simple check agrees with the above analysis: in all cases we have L ! 2 pNc in the UV, as expected. As in the previous subsection, the behaviour has an interesting dependence on x. For low x, we see no phase transition, only a small deformation on L around

is noticeable. But for large enough x, this deformation becomes sharp enough such that a local maximum forms in L, and thus a we have a phase transition in S(L). The critical value

L(0 ! 1) =

pNc

1 140[parenrightbigg]+ O

[parenleftbigg]1 20

, (B.1)

pNf Nc

1 140[parenrightbigg]+ O

[parenleftbigg]1 20

. (B.2)

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Figure 19. Various Plots of L(0) and S(L) in linear backgrounds with sigmoid source proles. Continuous lines represent solutions for = 0, while dashed lines represent = 2. The coloring

is such that the solutions with x = [notdef]1, 2, 3[notdef] are given by [notdef]purple, green, red[notdef] respectively. The grey

line represents the sourceless P = 2Nc solution.

1500

1000

500

[Minus]500

[Minus]1000

[Minus]1500

Figure 20. Various Plots of L(0) and S(L) in linear P backgrounds with bump-like source proles. Continuous lines represent solutions for = 0, while dashed lines represent = 2. The coloring

is such that the solutions with x = [notdef]5, 10, 15[notdef] is given by [notdef]purple, green, red[notdef] respectively. The

grey line represents the sourceless P = 2Nc solution.

xc at which a phase transition forms is a monotonically decreasing positive-denite function of . This is logical as a deformation in L means a change in L[prime]. As we can see from g

ure 11, L[prime] is a monotonically decreasing positive-denite function, so at larger any e ect in L is more pronounced. It is possible to estimate the value at which a phase transition is introduced. For = 0, the critical value is xc 5.3, while for = 2, xc 2.3 already.

Further, it should be noted that while we are able to form a phase transition via the addition of sources, the fact that L in the UV approaches a nite limit from below (which we term LUV ), guarantees that for every rst order phase transition, from the disconnected to the connected solution, would be accompanied by a discontinuous jump, from the continuous solution back to the discontinuous solution. Thus, here we again require the introduction of the branch of solutions living close to the boundary (that we termed short congurations), which will remove the problem, as it is the preferred branch for L < LUV .

1400

1200

1000

800

600

400

200

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0.0 0.5 1.0 1.5 2.0

B.3 Rotated sourced systems with sigmoid proles

All solutions of this class share the following UV asymptotics for L:

L(0 ! 1) =

3 8

e40/3[parenrightBig](B.4)

From this we can immediately see that L will always eventually approach 3

pNf from below. This means there are only the two possibilities (depicted in gure 20): either we will have a phase transition (connected - disconnected) coupled with a discontinuity, or there will be no phase transition at all (disconnected always preferred). Again, bear in mind, that we would employ the branch of solutions near the boundary to resolve any issues with the stability of the conguration and the above discontinuity, or lack of phase transition.

Let us study in detail which case occurs for di erent values in the parameter space.13

As expected from the van der Waals gas analogy, we will never nd a phase transition when the two scales and

pNf

1 (Nf 2Nc)24cNf 0 e40/3[parenrightbigg]+ O

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are too close together. Thus we will assume that there always is a large enough separation between the two scales in the following.

First let us assume 0 <

. We know from our previous analysis that we cannot have a phase transition before <

. We also know that lim!1 L = [notdef]

pNc

2 , pN

Nc 2

[notdef]

for [notdef]x 2, x > 2[notdef] respectively and beyond

, L tends to 38

pNf. Thus we can see that L will (for almost all values of x) rst approach a larger value than its UV limit, guaranteeing a phase transition. The exception is the range 16

9 x 167, for which a phase transition

does not occur, no matter the separation between and

, making the region around

x = 2 quite special.

Let us study the case 0

< . We know that before the solution will behave like

its corresponding (rotated) sourceless solution. Thus the Entanglement Entropy will behave as depicted in gure 14 in this region. Thus, as long as we push su ciently far away to

make sure we pick up the maximum in L, we will always produce a phase transition.

B.4 Rotated sourced systems with bump-like proles

For all rotated systems with bump-like prole we have L ! 0 in the UV. This guarantees

that we have at least one phase transition in every case. It turns out, one can construct solutions with two phase transitions, one between the disconnected solution and the connected, and one between two di erent branches of the connected solution.

As we know for <

the solution will behave exactly as in section B.2. Thus we know, that we can pick up a phase transition for large x, that is produced through the interplay of the scales where the sources become relevant and are suppressed. Further, now that we have a nite

, we e ectively introduce a third scale, and it is exactly this scale that gives us the second phase transition. The phase transition due to

is always present.

Thus we know that we will always have two phase transitions for x > xc. If we choose a non-zero , we additionally need the condition [notdef]

[notdef] 0. However, there is a third, more

subtle condition. It follows from the fact, that while the above conditions are su cient to

13Keep in mind that as mentioned, for < the solution will follow its corresponding (rotated) sourceless case, and for <

the solution will behave exactly as the solutions discussed so far in this appendix.

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Figure 21. Various Plots of L(0) in rotated backgrounds with sigmoid source proles. The left-hand panel shows solutions for = 0,

= 4. In the right-hand panel, continuous lines are for

solutions with = 4,

= 1, while dashed lines represent = 1,

= 4. The coloring is such that the solutions with x = [notdef]1, 2, 3[notdef] is given by [notdef]purple, green, red[notdef] respectively. The grey graphs

represent the corresponding rotated sourceless solutions.

produce two phase transitions, it is possible for the system to overshoot, leading to one of the phase transitions being in the unphysical branches of the Entanglement Entropy. A case like this is presented in gure 22 in the middle-left panel. It is easy to see, that the physical branch of the Entanglement Entropy will contain two phase transitions if the second maximum of L is noticeably lower than the rst maximum. Armed with this knowledge we deduce that the third condition is:

0 <

- x must be very large. The critical value xcc above which we have two phase transition is always larger than xc. Thus, here, this condition is always strictly stronger than the rst one.

< - x increases the size of the second bump, so we know that we will have two

physical phase transitions only for a bounded range of values of x, namely xc < x <

xcc.

Last, but not least it should be noted, that we can increase the number of phase transitions by introducing additional scales. For example, we could use source proles such as [47]:

() = 8

tanh4(2( 1))e

4( 1 )

3 + tanh4(2( 2))e

4( 2 )

3 if 2

(B.5)

Each time we let the sources become relevant again, we will introduce additional phase transitions that can be tuned to lie on the physical branch.

Notice that in this paper, we have used the word source branes instead of avor branes. We made this distinction because in many cases studied here, the uctuations on the branes (the mesons) are non-normalisable, and hence non-dynamical. The connection with studies and ideas presented by the authors in [6365] is worth pursuing and is left for future work.

tanh4(2( 1))e

4( 1 )

3 otherwise

Figure 22. In all gures above: continuous graphs have = 0 and x = 10 and dashed ones have

= 5 and

= 2. For the continuous lines, the red line has

C Other models without phase transitions

Here we will summarise the situation in other known conning models that do not produce a phase transition in the EE, unless of course, a cuto is introduced as explained in the main body of this paper.

We will start o by discussing the cases of D5 branes wrapped on a three-cycle and see how it is similar to that of the D5 branes wrapped on two-cycle. We shall then go on

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= 0, while the green line has

= 4.

For the dashed lines, blue indicates x = 10 and purple x = 2. In all of the panels involving S(L), the disconnected solution lies at S = 0, but is not shown. As we can see in the top-right panel, the continuous red solution has a rst order phase transition. The left-hand panel in the middle row is an example of overshooting - the branch that is relevant in the dashed purple case between the transitions, is above the disconnected solution and is thus unphysical. The right-hand gure in the middle row represents a solution that has been tuned to have 2 physical phase transitions. The graph has been enhanced in the two panels in the bottom row to clearly show each of the phase transitions.

to look at the backgrounds discussed in KKM, but modied by changing the warp factor in such a way that can be thought of as adding a relevant operator that changes the UV, but leaves the IR una ected. We then nally go on to discuss D6 branes wrapped on three-cycle, discuss the similarities, and make a comment about hard cut-o s.

We will notice, that making a more natural choice of the radial coordinate implies, that when lim0!1 L(0) = 1, L diverges linearly. This is true for many of the divergent

cases we have studied in this paper. For example, in systems generated by D5 branes, a closer analysis reveals that a coordinate transformation e

3 = r is needed to properly compare those cases to the ones presented below. It is easy to check that this will recover the linear behaviour for L(0).

C.1 D5 on wrapped on a three-cycle

Here we look at the backgrounds presented in [? ], and generalised in [68? ]. We start by dening the ansatz: there are two sets of SU(2) left-invariant 1-forms, i andi (i = 1, 2, 3), which obey

2[epsilon1]ijkj ^k. (C.1)

Each parametrises a three-sphere, and can be represented by three angles, ( , ', ),

1 = cos d + sin sin d', 2 = sin d + cos sin d', 3 = d + cos d' (C.2)

and similarly, three angles (~

, ~

, ~) for, which take a similar explicit form.14 Our spheres will also be bered with a one-form Ai. The Ai take the form

Ai = 12 (1 + w) i (C.3)

where w is a function of the radial coordinate. We can then write down our Type IIB metric ansatz (in Einstein Frame), in terms of the following vielbeins,

Exi = efdxi, E = ef+gd, E = ef+h

2 1, E' =

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di =

2[epsilon1]ijkj ^ k, di =

ef+h

2 2, E =

2 3,

E1 = ef+g

ef+g

2 (3 A3) (C.4)

where xi represents the Minkowski metric in 2 + 1 dimensions, is the radial coordinate, and [notdef]f, g, h[notdef] are only functions of . This means we can write the metric compactly as

ds2E =

Xi(Ei)2 (C.5)

The theory also contain a non-trivial dilaton . There is also a RR three-form F3 but we shall not require its expression here. From the usual SUSY requirements we nd = 4f.

There is a solution generating procedure [? ] (similar to the one discussed in the case of D5 branes wrapped on a two-cycle in the main text) which takes us from this solution

14The range of the angles here is 0 ,

< , 0 ', ~

' < 2 and 0 ,

2 (1 A1), E2 =

2 (2 A2), E3 =

~ < 4.

to one of Type IIA (with extra uxes). Here we write the relevant parts: we can write our metric in the string frame using ds2s = e /2ds2, then use an S-duality. The S-duality takes

ds2s ! e ds2s = ds2str, F3 ! H3, ! , (C.6)

leaving us in the common Type II NS-sector. Then after applying the dualities we generate the following Type IIA solution

d2str =1/2dx2i+1/2ds27, e2 =1/2e2 ,= 1

cosh2 1tanh2 e2( 1)[parenrightBig]

, (C.7)

where hatted quantities denote the new rotated solution and the unhatted are the original Type IIB functions. Again, we shall not require the explicit forms of F4 and H3 for what follows. We can recover the original string frame metric by taking ! 1.

We again read o the relevant quantities to calculate the EE as

Vint = 44 [notdef]

3/2e3g+3h, H() = e4 V 2int 2, () =1/2, () =e2g (C.8)

and making the appropriate substitutions we nd that

pH() = 22 [notdef] 1/2e3g+3h+2 ,

p =1/2eg. (C.9)

We can now discuss the behavior of the EE in each case. For the Maldacena-Nastase (MNa) case, we nd the same as in the D5 wrapped on a two-cycle (with linear P = 2Nc), such

that the separation L grows with 0, and nds a maximum at L1 = /2. The unrotated

case initially follows the MNa behavior in the IR, but then blows up, whereas the rotated result (which again follows the MNa result up to around the same scale as the unrotated), goes to zero for larger 0. This can all be seen in gure 23.

This means that for the EE, in both the MNa and unrotated case, the disconnected branch is always the lower than the connected branch and thus we require the short congurations. This is not true in the rotated case, where we nd a behavior like that of a rst order phase transition (thanks to the presence of the function), akin to what happened with the D5 branes wrapped on a two-cycle, after completing the system into the Baryonic Branch solution.

C.2 Deformed D4 on a S1, D3 on a S1 and AdS5 S5Initially, we start with the deformation of the D4 branes on S1, where we write the metric as

ds2 =()

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dxdx + f()(dx4)2[bracketrightbig]+()1 2

[bracketleftbigg]

d2f() + 2d 24[bracketrightbigg]

, e4 =() (C.10)

We then get as usual

() =()1/2, () =

()

f(), Vint() =

323R

3 2

()

4 4

pf(). (C.11)

It is easy to check that if we instead choose the warp factor as (this choice emphasizes the non-locality discussed in the main body of the paper),

= 1 +

(C.12)

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Figure 23. Here we have plots of the separation L(0) on the left and the entanglement entropy S(L) on the right. The MNa background is in grey (g0 = 1), the unrotated in green and rotated in blue (both have g0 = 1 + 102).

we nd () = 3 + R3

3 3

, H() = 102462R3 3 + R3

[parenrightbig][parenleftBigg][parenleftBigg]3

81 , (C.13)

This leads to the behavior presented in the left-hand panel of gure 24. Note that now, L grows linearly in the UV. This is easy to see if we look at L in the UV as

LdD4(0 ! 1) =

20 30 + R3

3/2

p30 3860 + 530 (R3 3) 2R3 3

0. (C.14)

Similarly for the deformed D3, with

ds2 =()

dxdx + f()(dx3)2[bracketrightbig]+()1 2

[bracketleftbigg]

d2f() + 2d!25[bracketrightbigg]

, e4 = const, (C.15)

and () and () as in (C.11), but with the warp factor modied as

() = 1 +

(C.16)

Vint() = 4R2

()5

pf(), (C.17)

and thus

() = 4 + R4 4 4

, H() = 84R8

pR4 + 4 4 3 2 . (C.18)

For deformed AdS5 [notdef] S5, we simply neglected the standard periodic association of the x3

coordinate (be aware that this leaves a conical singularity at ). As x3 is not compact, it no longer is a part of Vint but of dxdx. Note that the metric is no longer of the form

of (2.2). But this can trivially be dealt with by modifying the equation of H() to

H() = e4V 2int df() . (C.19)

We getVint() = 3()

54 5, H() = 62 4 + R4

[parenrightbig][parenleftBigg][parenleftBigg]4

[parenrightbig]

(C.20)

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Figure 24. In the left panel we have L(0) for the deformed D4 on S1 and in the right panel we have L(0) for the deformed D3 on S1. The blue lines represented the undeformed solutions compared with those with the deformation in green.

The results are qualitatively equal to what we found for the deformed D4 on S1. Choosing the warp factor as we did, causes L to grow linearly in the UV , preventing the phase transition. The UV expansion for L yield the following

LdD3(0 ! 1) =

[radicalBig][parenleftBigg]

40 + R4

3 40 4 580 2R4 4 240

[parenrightbig]

0, (C.21)

340 4

[radicalBig][parenleftBigg]

40 + R4

3 40 4 580 + 340 (R4 4) R4 4

0. (C.22)

The right-hand panel of gure 24 shows the deformed D3 result and the deformed AdS5 [notdef] S5 behaviour is very similar.

C.3 D6 wrapped on a three-cycle

Here we look at the behaviour of the solutions recently discussed in [60]. The full metric and dilaton are given by [70, 71],

ds2IIA,st = Ne2/3

"dx21,3N + dr2 + b2(d 2 + sin2 d'2)+

+ a2(~

!1 + gd )2 + a2(~

!2 + g sin d')2 + h2(~

!3 cos d')2[bracketrightBigg]

LdAdS(0 ! 1) =

(C.23)

h2 = c2f2f2 + c2(1 + g3)2

e4/3 = c2f2 4Nh2

The background functions a, b, c, f, g and g3 are determined through the BPS equations

a =

c2a +

a5f2

8b4c3 ,

b =

c2b

a2(a2 3c2)f2

8b3c3 ,

2500

2000

1500

1000

500

0 500 1000 1500

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Figure 25. Here we have plots of the dilaton in the left panel and the separation L(0) in the

right panel in the backgrounds based on the D6 branes wrapped on a three-cycle. The range of c is given in the text, with c = 32 in purple and larger values c in the range of colours to up to red.

c = 1 +

c2 2a2 +

c2 2b2

3a2f2

8b4 ,

f =

a4f3

4b4c3 , (C.24)

as well as the relations

g(r) = a(r)f(r)

2b(r)c(r) , g3(r) = 1 + 2g(r)2. (C.25)

The forms of the functions required for the calculation of the Entanglement Entropy are given by

Vint =

[integraldisplay]

qdet[gij] = (4)3b2a2h( [prime]gsN)5/2e5/3,

= [notdef]e2/3, = [prime]gsN[notdef] , , d = 3, (C.26)

H = e4V 2int d = (4)6[notdef]3b4a4h2( [prime]gsN)5e4/3,

ds25 = [dx21,3 + dr2], 3 = H

The solutions in which the dilaton stabilises are interesting because the associated backgrounds do not need an M-theory completion, so we will focus on them. We will re-express the expansion parameters used in [60] as follows,

q0 = 2

2 + c

, R0 = 12 + c. (C.27)

So the parameter space is then dened through c thus we choose to explore the following values for c:

d8dy

32 + 105. (C.28)It appears that c = 32 is a limiting case and the dilaton diverges, while for c > 32 it eventually stabilises. The results are presented in gure 25. Note that in all the cases, L grows without bound for large 0. This agrees with the UV expansion of L which is given by

L(0 ! 1) =

c = 32,

32 + 105,

32 + 101,

32 + 103,

5 0

35(q1R1) + O

[parenleftbigg]

1 0

. (C.29)

20 40 60 80 100 120 140

Figure 26. These are various plots of L(0) in the D6 wrapped on a three-cycle solution with r = 10. The linear graph is the solution without cuto , while the other three are numerical solutions with cuto 50, 100, 150 respectively.

In these solutions we nd initially that L will shoot o at di erent gradients depending on c, but eventually curves down to approach a line with the same gradient, but shifted by a constant.

C.3.1 Illustrating the dependence on the UV cuto

The equations (C.24) also have a known analytical solution [72], that reads

a(r) =

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r 3

r3 r3

, b(r) = r 2p3,

c(r) =

r 3

r3 r3

, f(r) = r2p3. (C.30)

From this form, one can work out the relevant functions and nd that

L(0 ! 1) =

2p ( 5

12 )

( 112)

0. (C.31)

Note that here L also grows linearly into the UV. In the main body of the paper we discuss the need for additional solutions, especially in backgrounds that have some form of nonlocality. We found that these solutions are given by short congurations that never go far from the boundary. In section 3.2, we explicitly mention that one can nd these solution by studying the behaviour of the system close to the UV cuto .

In the cases discussed where L diverges, if we look at the short congurations, we nd that they change as we vary the UV cuto . Thus, we may want to think of this as viewing a system with strong IR/UV mixing, which causes the divergence, as discussed in [35, 36] (see to gure 26).

One could easily conclude that this solution has a phase transition, at a particular separation L, for a particular value of the cuto , when in fact it is entirely cuto dependent. Another case where the cuto has caused similar e ects was presented in [73].

D Hints at invariances of the entanglement entropy

When calculating the Holographic Entanglement Entropy Density, we always seem to be able to reduce the problem to a one-dimensional system such that

SA Vd1

[integraldisplay]

= dL(,

) (D.1)

For backgrounds of the form given in eq. ((2.2)) this is a trivial observation. However, as we will show in the two examples examples, this remains true even for more complicated backgrounds, with warp factors depending on the coordinates of the internal space and various brations.

D.1 The D4-D8 system

Let us look at the system presented by Brandhuber and Oz in [74], the D4-D8 system. Here the ten dimensional space is a bration of AdS6 over S4. Let us study if this causes complications with the entanglement entropy calculation. The metric can be written as

ds2str = M(!) U2dx21,4 + 9Q4dU2U2 + Q d 24[bracketrightbigg]

, (D.2)

where

M(!) = [prime]Q1/2

[bracketleftbigg]

4 C(8 Nf) sin !

JHEP06(2014)005

1/3 , d 24 = d!2 + cos2 ! d 23, (D.3)

and the dilaton has a prole given by

e = Q1/4C

5/6 . (D.4)

Choosing the appropriate 8-dimensional surface 8 = [notdef]!, 1, 2, 3, x2, x3, x4, [notdef] and allow

ing the radial coordinate to be U = U(x1) and for xed t we nd that the induced metric takes the form

ds2 8 =M(!)

[bracketleftbigg]

4 C(8 Nf) sin !

U2(dx22+dx23+dx24)+U2dx21

1+ 9Q4(U[prime])2 U4

+Q d!2+cos2 ! d 23

[parenrightbig][bracketrightbigg]

. (D.5)

We can then write

1/2

pdet g8 = sin2 1 sin 2 M(!)4 cos3 ! U4

1 + 9Q4(U[prime])2 U4

(D.6)

From here it is easy to see that we can perform the integrals in the action for the entanglement entropy and the result will be of the form

S = 22( [prime])4

[bracketleftbigg]

4 C(8 Nf)

r1 + 9Q4(U[prime])2U4 (D.7)

where the last part which is now in the usual form (the factor of 9

20 comes from the ! integral) and is the standard result for AdS6. Thus the D4-D8 system can be solved using the method we have employed previously.

1/3 Q1/2C2 [notdef]9 20

[integraldisplay]

d U4

D.2 A background with a cyclic RG ow

We now turn our attention to the setup described by Balasubramanian in [75]. In this paper they construct non-singular solutions of a six dimensional theory which is a warped product of AdS5 and a circle. These solutions have very non-trivial warp factors which break the symmetries of AdS5 to discrete scale invariance and also break the translational symmetry along the circle. Let us study if these causes troubles in our calculation of the entanglement entropy of a strip. The metric takes the form

ds26 = e2C[!, ]

he2!/L(dt2 + dx2i) + d!2[bracketrightBig]+ e2B[!, ](d + A[!, ]d!)2 (D.8)

where the functions B, C and A are non-trivial functions of the Jacobi Elliptic functions

sn, cn and dn. Their exact form will not be important in what follows. We are interested in whether the mixing in the metric due to the bration represented by A causes any issue

in the calculation of the EE. If we calculate the form of the corresponding pullback of the metric onto the now 4-dimensional surface 4 = [notdef]x2, x3, x1, [notdef], and setting the radial

coordinate ! = !(x1) we nd that when we take the determinant it gives

pdet g4 = eB[!, ]+3C[!, ]+ 3! L

q1 + e2!

L (![prime])2. (D.9)

From this we see that there are no terms involving the function A and that this again falls

into the simple form and we can again use the standard procedure.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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JHEP06(2014)005

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SISSA, Trieste, Italy 2014

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In this paper we study the conjectural relation between confinement in a quantum field theory and the presence of a phase transition in its corresponding entanglement entropy. We determine the sufficient conditions for the latter and compare to the conditions for having a confining Wilson line. We demonstrate the relation in several examples. Superficially, it may seem that certain confining field theories with a non-local high energy behavior, like the dual of D5 branes wrapping a two-cycle, do not admit the corresponding phase transition. However, upon closer inspection we find that, through the introduction of a regulating UV-cutoff, new eight-surface configurations appear, that satisfy the correct concavity condition and recover the phase transition in the entanglement entropy. We show that a local-UV-completion to the confining non-local theories has a similar effect to that of the aforementioned cutoff.

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Title

Confinement, phase transitions and non-locality in the entanglement entropy

Author

Kol, Uri; Núñez, Carlos; Schofield, Daniel; Sonnenschein, Jacob; Warschawski, Michael

Pages

1-57

Publication year

2014

Publication date

Jun 2014

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP06(2014)005

ProQuest document ID

1652874739

SISSA, Trieste, Italy 2014

Confinement, phase transitions and non-locality in the entanglement entropy

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