arX

iv:1

002.

1885

v2 [

hep-

th]

11

Jun

2010

SHEP-10-01

Holographic Description of the Phase Diagram of

a Chiral Symmetry Breaking Gauge Theory

Nick Evans,∗ Astrid Gebauer,† Keun-Young Kim,‡ and Maria Magou§

School of Physics and Astronomy,

University of Southampton,

Southampton, SO17 1BJ, UK

The large N N=4 gauge theory with quenched N=2 quark matter in the presence of a magnetic fielddisplays chiral symmetry breaking. We study the temperature and chemical potential dependenceof this theory using its gravity dual (based on the D3/D7 brane system). With massless quarks, atzero chemical potential, the theory displays a first order thermal transition where chiral symmetry isrestored and simultaneously the mesons of the theory melt. At zero temperature, these transitionswith chemical potential are second order and occur at different chemical potential values. Betweenthe three there are two tri-critical points, the positions of which we identify. At finite quark massthe second order transition for chiral symmetry becomes a cross over and there is a critical pointat the end of the first order transition, while the meson melting transition remains similar to themassless quark case. We track the movement of the critical points as the mass is raised relative tothe magnetic field.

I. INTRODUCTION

The phase diagram in the temperature chemical po-tential (or density) plane is a matter of great interest inboth QCD and more widely in gauge theory [1–3]. InQCD there is believed to be a transition from a confiningphase with chiral symmetry breaking at low temperatureand density to a phase with deconfinement and no chiralsymmetry breaking at high temperature. In the stan-dard theoretical picture for QCD with massless quarks,the transition is first order for low temperature but grow-ing density, whilst second order at low density and grow-ing temperature. The second order transition becomes across over at finite quark mass. There is a (tri-)criticalpoint where the first order transition mutates into the(second order) cross over transition. In fact though therecould still be room in QCD for a more exotic phase dia-gram [3] as we will discuss in the context of our resultsin our final section.

In this paper we will present a precise holographic [4–6]determination of the phase diagram in the temperaturechemical potential plane for a gauge theory that displaysmany of the features of the QCD diagram, although theprecise details differ. A pictorial comparison of our the-ory to QCD can be made by comparing Fig 5 to Fig 10.

The theory we will consider is the large N N=4 gaugetheory with quenched N=2 quark matter [7–10] whichhas been widely studied [11]. An immediate differencebetween the N=4 glue theory and QCD is that the ther-mal phase transition to a deconfined phase occurs for in-finitesimal temperature since the massless theory is con-

∗Electronic address: evans@soton.ac.uk†Electronic address: ag806@soton.ac.uk‡Electronic address: k.kim@soton.ac.uk§Electronic address: mm21g08@soton.ac.uk

formal [5]. Essentially the entire temperature chemicalpotential phase diagram of our theory is therefore char-acterized by strongly coupled deconfined glue.The quark physics is more subtle though - the phase

diagram in the temperature chemical potential (density)plane for the N = 2 quark matter has been studied in [12–17]. When the quark mass is zero the theory is conformaland the origin of the phase diagram is a special point withconfined matter. Immediately away from that point, ineither temperature or chemical potential, a first ordertransition moves the theory to a deconfined theory (themesons melt [18–20]).When a quark mass is present in the N=2 theory the

meson melting transition occurs away from the origin.This transition has been reported as first order with a sec-ond order transition point where the first order transitionline touches the T = 0 chemical potential axis [15, 16](in the grand canonical ensemble). Interestingly thereis a phase transition line in the temperature versus den-sity plane (in the canonical ensemble) in which the quarkcondensate jumps [12, 13]. This area of the phase dia-gram is intrinsically unstable though and not realizableby imposing any chemical potential [16].The crucial ingredient we will add to the theory is chi-

ral symmetry breaking which will also bring the theorycloser in spirit to QCD. As shown in [21–24] the N = 2theory in the presence of a magnetic field displays chiralsymmetry breaking through the generation of a quarkanti-quark condensate. At zero density the finite tem-perature behaviour has been studied [21, 22] and thereis a first order transition from a chiral symmetry brokenphase at low temperature to a chiral symmetry restoredphase at high temperature. In this paper we will includechemical potential as well to map out the full phase di-agram in the temperature chemical potential plane. Wewill find a chiral symmetry restoration phase transition,which is first order for low density and second order forlow temperature - there is a critical point where these

http://arxiv.org/abs/1002.1885v2mailto:evans@soton.ac.ukmailto:ag806@soton.ac.ukmailto:k.kim@soton.ac.ukmailto:mm21g08@soton.ac.uk

2

transitions meet. This physics is in addition to a mesonmelting transition which is first order at large temper-ature but apparently second order at low temperature.This latter region of transition is interesting because itis associated with a discontinuous jump from an embed-ding off the black hole to one that ends on it and it looksnaively first order. However, when we plot any avail-able order parameter in the boundary theory it appearssecond order.We will also track the movement of these transition

lines and critical points as the quark mass rises relativeto the magnetic field. The infinite mass limit correspondsto the pure N=2 theory without magnetic field [14, 16].The second order chiral symmetry restoration transitionbecomes a cross over the moment a mass is introduced.The first order transition structure though remains, evento the infinite mass limit, with two critical points: one isthe end point of the first order transition and the otheris the the end point of the second order meson meltingtransition. This structure was not reported in the resultsin [14, 16]1 but this is not surprising since the structure,in that limit, is on a very fine scale. We have only found itby following the evolution of the larger structure presentat low quark mass with a magnetic field. In addition wepresent evidence to suggest the parameter space with asecond order meson melting transition extends away fromjust the T = 0 axis, again, even in the infinite mass limit.We have confirmed these results in the strict B = 0 limitalso.The theory we study may appear to be a rather vague

relative of QCD with magnetic field induced chiral sym-metry breaking. On the other hand it is a theory ofstrongly coupled glue with the magnetic field inducingconformal symmetry breaking in the same fashion asΛQCD in QCD. In fact the magnetic field case in thebasic N=4 dual is the cleanest known example of chiralsymmetry breaking in a holographic environment. Otherdeformations of the N=4 gauge theory typically lead toan ill-understood IR singular hard wall - see for exam-ple [25, 26]. The magnetic field case provides a smoothIR wall where we have more control but the results arelikely to be the same in those more complex cases. Wecan hope to learn some lessons for a wider class of gaugetheories.

II. THE HOLOGRAPHIC DESCRIPTION

The N=4 gauge theory at finite temperature has aholographic description in terms of an AdS5 black hole

1 The existence of two critical points is related with the existenceof the black hole to black hole transition. It is actually justvisible in Fig 2c of [14] but the authors had not probed it indetail previously. After discussion of our results with the authorsof [14], they have refined their computations and confirmed ourresults.

geometry (with N D3 branes at its core)[4–6]. The ge-ometry is

ds2 =r2

R2(−fdt2 + d~x2) + R

2

r2fdr2 +R2dΩ25 , (1)

where R4 = 4πgsNα′2 and

f := 1− r4H

r4, rH := πR

2T . (2)

Here rH is the position of the black hole horizon whichis related to the temperature T .

We will find it useful to make the coordinate transfor-mation

dr2

r2f≡ dw

2

w2=⇒ w :=

√r2 +

√r4 − r4H , (3)

with wH = rH . This change makes the presence of aflat 6-plane perpendicular to the horizon manifest. Wewill then write the coordinates in that plane as ρ and Laccording to

w =√ρ2 + L2 , ρ := w sin θ . L := w cos θ , (4)

The metric is then

ds2 =w2

R2(−gtdt2 + gxd~x2)

+R2

w2(dρ2 + ρ2dΩ23 + dL

2 + L2dΩ21) , (5)

where

gt :=(w4 − w4H)2

2w4(w4 + w4H), gx :=

w4 + w4H2w4

. (6)

A. Quarks/D7 brane probes

Quenched (Nf ≪ N) N=2 quark superfields can be in-cluded in the N=4 gauge theory through probe D7 branesin the geometry[7–10]. The D3-D7 strings are the quarks.D7-D7 strings holographically describe mesonic operatorsand their sources. The D7 probe can be described by itsDBI action

SDBI = −TD7∫

d8ξ√−det(P [G]ab + 2πα′Fab) , (7)

where P [G]ab is the pullback of the metric and Fab isthe gauge field living on the D7 world volume. We willuse Fab to introduce a constant magnetic field (eg F12 =−F21 = B) [21] and a chemical potential associated withbaryon number At(ρ) 6= 0 [13, 27].We embed the D7 brane in the ρ and Ω3 directions

of the metric but to allow all possible embeddings mustinclude a profile L(ρ) at constant Ω1. The full DBI action

3

we will consider is then

S =

∫dξ8L(ρ) =

(∫

S3ǫ3

∫dtd~x

)∫dρ L(ρ) , (8)

where ǫ3 is a volume element on the 3-sphere and

L := −NfTD7ρ3

4

(1− w

4H

w4

)

×√(

1 + (∂ρL)2 −2w4(w4 + w4H)

(w4 − w4H)2(2πα′∂ρAt)2

)

×

√√√√((

1 +w4Hw4

)2+

4R4

w4B2

). (9)

Since the action is independent of At, there is a conserved

quantity d(:= δS

δFρt

)and we can use the Legendre trans-

formed action

S̃ = S −∫

dξ8FρtδS

δFρt=

(∫

S3ǫ3

∫dtd~x

)∫dρ L̃(ρ) ,

(10)

where

L̃ := −NfTD7(w4 − w4H)

4w4

√K(1 + (∂ρL)2) (11)

K :=

(w4 + w4H

w4

)2ρ6 +

4R4B2

w4ρ6

+8w4

(w4 + w4H)

d2

(NfTD72πα′)2. (12)

To simplify the analysis we note that we can use themagnetic field value as the intrinsic scale of conformalsymmetry breaking in the theory - that is we can rescaleall quantities in (11) by B to give

L̃ = −NfTD7(R√B)4

w̃4 − w̃4Hw̃4

√K̃(1 + L̃′2) , (13)

K̃ =

(w̃4 + w̃4H

w̃4

)2ρ̃6 +

1

w̃4ρ̃6 +

w̃4

(w̃4 + w̃4H)d̃2 ,(14)

where the dimensionless variables are defined as

(w̃, L̃, ρ̃, d̃)

:=

(w

R√2B

,L

R√2B

,ρ

R√2B

,d

(R√B)3NfTD72πα′

).

(15)

In all cases the embeddings become flat at large ρ tak-ing the form

L̃(ρ̃) ∼ m̃+ c̃ρ̃2

, m̃ =2πα′mq

R√2B

, c̃ = 〈q̄q〉 (2πα′)3

(R√2B)3

.(16)

In the absence of temperature, magnetic field and densitythe regular embeddings are simply L(ρ̃) = m̃, which isthe minimum length of a D3-D7 string, allowing us toidentify it with the quark mass as shown. c̃ should thenbe identified with the quark condensate with the relationshown.

We will classify the D7 brane embeddings by theirsmall ρ̃ behavior. If the D7 brane touches the blackhole horizon, we call it a black hole embedding, other-wise, we call it a Minkowski embedding. We have usedMathematica to solve the equations of motion for the D7embeddings resulting from (13). Typically in what fol-lows, we numerically shoot out from the black hole hori-zon (for black hole embeddings) or the ρ̃ = 0 axis (forMinkowski embeddings) with Neumann boundary condi-

tion for a given d̃. Then by fitting the embedding functionwith (16) at large ρ̃ we can read off m̃ and c̃.

B. Thermodynamic potentials

The Hamilton’s equations from (10) are ∂ρd =δS̃δAt

and ∂ρAt = − δS̃δd . The first simply means that d is theconserved quantity. The second reads as

∂ρ̃Ãt = d̃w̃4 − w̃4Hw̃4 + w̃4H

√1 + (L̃′)2

K̃, (17)

where Ãt :=√22πα′AtR√2B

.

There is a trivial solution of (17) with d̃ = 0 and con-

stant Ãt [16]. The embeddings are then the same as

those at zero chemical potential. For a finite d̃, Ã′t is sin-gular at ρ̃ = 0 and requires a source. In other words theelectric displacement must end on a charge source. Thesource is the end point of strings stretching between theD7 brane and the black hole horizon. The string tensionpulls the D7 branes to the horizon resulting in black holeembeddings [13]. For such an embedding the chemicalpotential(µ̃) is defined as [13, 27]

µ̃ := limρ̃→∞

Ãt(ρ̃)

=

∫ ∞

ρ̃H

dρ̃ d̃w̃4 − w̃4Hw̃4 + w̃4H

√1 + (L̃′)2

K̃, (18)

where we fixed Ãt(ρ̃H) = 0 for a well defined At at theblack hole horizon.

The Euclideanized on shell bulk action can be inter-preted as the thermodynamic potential of the boundary

field theory. The Grand potential (Ω̃) is associated with

the action (9) while the Helmholtz free energy (F̃ ) is as-

4

sociated with the Legendre transformed action (10):

F̃ (w̃H , d̃) :=−S̃

NfTD7(R√B)4Vol

=

∫ ∞

ρ̃H

dρ̃w̃4 − w̃4H

w̃4

√K̃(1 + (L̃′)2) (19)

Ω̃(w̃H , µ̃) :=−S

NfTD7(R√B)4Vol

=

∫ ∞

ρ̃H

dρ̃w̃4 − w̃4H

w̃4

√(1 + (L̃′)2)

K̃×

((w̃4 + w̃4H

w̃4

)2ρ̃6 +

1

w̃4ρ̃6

)(20)

where Vol denote the trivial 7-dimensional volume in-tegral except ρ̃ space, so the thermodynamic potentialsdefined above are densities, strictly speaking. Since

K̃ ∼ ρ̃6, both integrals diverge as ρ̃3 at infinity andneed to be renormalized. Thermodynamic potentials,(18),(19) and (20) are reduced to B = 0 case if we sim-

plify omit all ρ̃6

w̃4and then tildes. See for example (21).

III. CHIRAL SYMMETRY BREAKING ANDTHE THERMAL PHASE TRANSITION

We begin by reviewing the results of [21, 22] on mag-netic field induced chiral symmetry breaking and thethermal phase transition to a phase in which the conden-sate vanishes. While they show the embeddings for fixedT and different values of B, we will show the embeddingsfor fixed B and different values of T . By fixing B we areusing it as the intrinsic scale of symmetry breaking in thesame fashion as ΛQCD plays that role in QCD.Let us digress here to explain how to understand the

figures we will present in this paper. For example, inFig 1 we have three columns. The left is the D7 braneembedding configuration. The middle shows a plot ofthe allowed values of the condensate c̃ as a function ofthe quark mass m̃ - these are thermodynamical conjugatevariables. The right is the corresponding thermodynamicpotential. Each row is for a fixed parameter we are vary-ing - here it’s temperature. The left and middle plotsare plotted by solving the equation of motion (13) withthe black hole boundary condition that the embedding isorthogonal to the horizon.The right hand plot is calculated using (19) or (20).

Both are the same at zero density. We subtractlimρ̃→∞

14ρ̃4 to remove the common infinite component.

Every point in the middle and right plots correspondsto one embedding curve in the left plot. These points arecolor coded with the colors common across each of thethree plots. The order of colors follows the rainbow fromthe bottom embedding as a mnemonic.

In the middle plot we can find any transition point bya Maxwell construction (an equal area law), which is alsoconfirmed by the minimum of the grand potential on theright. The vertical dashed line in the middle and righthand plots corresponds to the transition point.

In the left plots the gray region contains embeddingsthat are excluded since they are unstable, as shown inthe middle and on the right.

The results for the case of a constant magnetic fieldand varying temperature are displayed in Fig 1a-d. TheFig 1a (Left) shows the D7 embeddings when T ≪ Band the black hole is small. The embeddings are driven

away from the origin of the L̃− ρ̃ plane - this behaviouris a result of the inverse powers of w̃, when w̃H ≪ 1,in the Lagrangian (13) which lead the action to grow ifthe D7 approaches the origin (note that the factor of ρ̃3

multiplying the action means the action will never actu-ally diverge). There is also a embedding that end on theblack hole (shown in red) but they are thermodynami-cally disfavoured as shown in Fig 1a (Right).

At large ρ̃ the stable embedding with m̃ = 0 has anon-zero derivative so c̃ is non-zero and there is a chi-ral condensate i.e. chiral symmetry breaking. The U(1)symmetry in the Ω1 direction is clearly broken by anyparticular embedding too. We can numerically read offthe values of m̃ and c̃ from the embeddings and theirvalues are shown in Fig 1a (Middle), where the dottedblue curves are for Minkowski embeddings, whilst thered curves are for black hole embeddings.

If the temperature is allowed to rise sufficiently thenthe black hole horizon grows to mask the area of theplane in which the inverse w̃ terms in the Lagrangian arelarge. At a critical value of T the benefit to the m̃ = 0embedding of curving off the axis becomes disfavouredand it instead lies along the ρ̃ axis - chiral symmetrybreaking switches off. This first order transition occursat w̃H = 0.2516 as shown in Fig 1b by Maxwell’s con-struction (Middle) and by lower grand potential (Right).Our value for the critical temeprature agrees with the

value B̃ = 16 in [22] since our w̃ is the same as√

1

B̃in

[22].

We show an example of the embeddings above the crit-ical temperature, their grand potential and the evolutionof the curves in the m̃− c̃ plane in Fig 1c.The Fig 1d shows a case when T ≫ B when the area of

the plane in which B is important is totally masked by theblack hole and the results match those of the usual finiteT version of the N=2 theory. For m̃ > ũH the embed-dings are Minkowski like whilst for small m̃ they fall intothe black hole. There is a first order phase transition be-tween these two phases which is the meson melting phasetransition discussed in detail in [28–32]. Minkowski em-beddings have a stable mesonic spectrum [10] whilst inthe case of black hole embeddings the black holes’ quasi-normal modes induce an imaginary component to themeson masses [18, 19]. We can see that the previouslyreported “meson melting” transition at large quark mass

5

0.5 1.0 1.5 2.0 2.5Ρ

-0.6

-0.4

-0.2

0.2

0.4

0.6L

w H = 0.15-0.15 -0.10 -0.05 0.05 0.10 0.15

m

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

-c

-0.15 -0.10 -0.05 0.05 0.10 0.15m

2.48

2.50

2.52

2.54

2.56

2.58

2.60W

(a)Low temperature - w̃H = 0.15. Here we see chiral symmetry breaking with the blue embedding thermodynamically preferredover the red at m̃ = 0.

0.5 1.0 1.5 2.0 2.5Ρ

-0.6

-0.4

-0.2

0.2

0.4

0.6L

w H = 0.2516

-0.04 -0.02 0.02 0.04 0.06m

-0.25

-0.20

-0.15

-0.10

-0.05

-c

-0.04 -0.02 0.02 0.04 0.06m

2.53

2.54

2.55

W

(b)Transition temperature - w̃H = 0.2516. This shows the point where the first order chiral symmetry phase transition occursfrom the blue to the red embedding. The transition can be identified by considering Maxwell’s construction(Middle) or the

lowest free energy(Right).

0.5 1.0 1.5 2.0 2.5Ρ

-0.6

-0.4

-0.2

0.2

0.4

0.6L

w H = 0.3 0.05 0.10 0.15 0.20m

-0.25

-0.20

-0.15

-0.10

-0.05

-c

0.00 0.02 0.04 0.06 0.08 0.10m2.490

2.495

2.500

2.505

2.510

2.515

2.520W

(c)Above the transition - w̃H = 0.3. This is the chiral restored phase with the m̃ = 0 curve lying along the ρ̃ axis (red).

500 1000 1500 2000 2500Ρ

-600

-400

-200

200

400

600L

w H = 300

100 200 300 400 500 600m

1´ 106

2´ 106

3´ 106

4´ 106

5´ 106

6´ 106

7´ 106-c

100 200 300 400 500 600m

-4´ 109

-3´ 109

-2´ 109

-1´ 109

0

W

(d)High temperature - w̃H = 300. Here the magnetic field is negligible and the embeddings show the usual finite temperaturemeson melting transition.

FIG. 1: The D7 brane embeddings(Left), their corresponding m̃ − c̃ diagrams(Middle), and the Free energies(Right) in thepresence of a magnetic field at finite temperature. (Parameters are scaled or B = 1/2R2 in terms of parameters without tilde.)

becomes also the chiral symmetry restoring transition atzero quark mass.

IV. FINITE DENSITY OR CHEMICALPOTENTIAL AT ZERO TEMPERATURE

We can now turn to the inclusion of finite density orchemical potential in the theory with magnetic field. Inthis section we consider the zero temperature (ũH = 0)theory only, and will continue to finite temperature inthe next section.

A finite density (chemical potential) at zero tempera-ture has been studied in the N=2 theory without a mag-netic field in [15], where analytic solutions for both ablack hole like embedding and a Minkowski embeddinghave been found. When a magnetic field is turned on, an-alytic solutions are not available any more, but we havefound numerical solutions that continuously deform fromthe known analytic solutions at zero magnetic field.

Minkowski embedding solutions correspond to zerodensity and finite chemical potential. The black hole likeembedding is the embedding deformed by the density - a

spike forms from the D7 down to the origin of the L̃− ρ̃

6

1 2 3 4 5 6Ρ

0.1

0.2

0.3

0.4

L

d= 0.01

-0.2 -0.1 0.1 0.2m

-0.25

-0.20

-0.15

-0.10

-0.05

0.05-c

-0.10 -0.05 0.05m

2.54

2.56

2.58

2.60

F

(a)Low density - d̃ = 0.01. Here we see chiral symmetry breaking (the blue embedding is preferred over the red embedding) anda spiral structure in the m̃ vs c̃ plane.

1 2 3 4 5 6Ρ

0.1

0.2

0.3

0.4

L

d= 0.1

-0.2 -0.1 0.1 0.2m

-0.25

-0.20

-0.15

-0.10

-0.05

0.05-c

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20m

2.52

2.54

2.56

2.58

2.60

2.62

2.64F

(b)Increasing density below the transition - d̃ = 0.1. There is still chiral symmetry breaking here with the orange embeddingpreferred to the red. Note the spiral structure in the m̃− c̃ plane has disappeared.

1 2 3 4 5 6Ρ

0.05

0.10

0.15

0.20

0.25

L

d= 0.3197

-0.2 -0.1 0.1 0.2m

-0.25

-0.20

-0.15

-0.10

-0.05

0.05-c

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20m

2.65

2.70

2.75

2.80F

(c)Transition point - d̃ = 0.3197. This shows the point where the second order chiral symmetry phase transition occurs.

1 2 3 4 5 6Ρ

0.05

0.10

0.15

L

d= 1

-0.2 -0.1 0.1 0.2m

-0.25

-0.20

-0.15

-0.10

-0.05

0.05-c

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20m

3.425

3.430

3.435

3.440

3.445F

(d)High density d̃ = 1. This is the chiral restored phase with the m̃ = 0 curve lying along the ρ̃ axis. For larger m̃ the usualspike like embedding can be seen.

FIG. 2: The D7 brane embeddings(Left), their corresponding m̃ − c̃ diagrams(Middle), and the Free energies(Right) in thepresence of a magnetic field at finite density. (Parameters are scaled or B = 1/2R2 in terms of parameters without tilde.)

plane (Fig 2d (Left)) which has been interpreted as aneven distribution of strings (i.e. quarks) forming in thevacuum of the gauge theory.

First of all it will be interesting to see how the repul-sion from the origin induced by a magnetic field and theattraction to the origin by the density compete. Thuswe start with the canonical ensemble (that is solutions

with non-zero d̃) and consider black hole like embeddingsexclusively. The plot in Fig 2a (Left) shows the embed-

dings for a small value of density. The solutions showthe chiral symmetry breaking behaviour induced by themagnetic field but then spike to the origin by the den-sity at small ρ̃. For m̃ = 0 one should compare the blueand red embeddings - the blue one is thermodynamicallypreferred as shown in Fig 2a (Right) The theory showssimilar behavior to that seen at zero density: there is aspiral structure in the m̃ vs c̃ plane (Fig 2a (Middle)) [21].That will disappear as the density increases.

7

0.5 1.0 1.5 2.0 2.5Ρ

-0.6

-0.4

-0.2

0.2

0.4

0.6L

w H = 0

0.2 0.4 0.6 0.8Μ

0.05

0.10

0.15

0.20

0.25

0.30

0.35d

0.0 0.2 0.4 0.6 0.8Μ

2.52

2.54

2.56

2.58

2.60

W

(a)Zero temperature - w̃H = 0. The second order meson melting transition and then the second order chiral restorationtransition are apparent.

0.5 1.0 1.5 2.0 2.5Ρ

-0.6

-0.4

-0.2

0.2

0.4

0.6L

w H = 0.15

0.1 0.2 0.3 0.4 0.5Μ

0.05

0.10

0.15

0.20

0.25

0.30

d

0.1 0.2 0.3 0.4Μ

2.52

2.54

2.56

2.58W

(b)Low temperature - w̃H = 0.15. The zero temperature structure remains.

0.5 1.0 1.5 2.0 2.5Ρ

-0.6

-0.4

-0.2

0.2

0.4

0.6L

w H = 0.23

0.05 0.10 0.15Μ

0.02

0.04

0.06

0.08

0.10

0.12

d

0.00 0.05 0.10 0.15Μ2.540

2.542

2.544

2.546

2.548

2.550W

(c)Above the first tri-critical point - w̃H = 0.23. The meson melting transitions remains second order but the chiral symmetryrestoration transition is first order.

0.5 1.0 1.5 2.0 2.5Ρ

-0.6

-0.4

-0.2

0.2

0.4

0.6L

w H = 0.24

0.05 0.10 0.15Μ

0.02

0.04

0.06

0.08

0.10

0.12d

0.05 0.10 0.15Μ2.538

2.540

2.542

2.544

2.546W

(d)Above the second tri-critical point - w̃H = 0.24. There is now only a single first order transition for meson melting andchiral symmetry restoration.

0.5 1.0 1.5 2.0 2.5Ρ

-0.6

-0.4

-0.2

0.2

0.4

0.6L

w H = 0.2516

0.02 0.04 0.06 0.08 0.10 0.12 0.14Μ

0.02

0.04

0.06

0.08

0.10

d

0.02 0.04 0.06 0.08 0.10 0.12 0.14Μ2.532

2.534

2.536

2.538

2.540

W

(e)High temperature - w̃H = 0.2516. The ground state preserves chiral symmetry for all values of µ̃.

FIG. 3: The D7 brane embeddings (Left), their corresponding d̃− µ̃ diagrams (Middle), and the grand potentials (Right) formassless quarks in the presence of a magnetic field at a variety of temperatures that represent slices through the phase diagramFig 5. (Parameters are scaled or B = 1/2R2 in terms of parameters without tilde.)

8

As the density increases the value of the condensate forthe m̃ = 0 embeddings falls - we show a sequence of plotsfor growing d̃ in Fig 2b-d (Middle). There is a critical

value of d̃ = 0.3197 where c̃ becomes zero for the masslessembeddings - above this value of d̃, D7 embedding is flatand lies along the ρ̃ axis (2c-d (Left)) . One can see fromthe plots that there is a second order phase transition to aphase with no chiral condensate. In Fig 2c (Left) and 2d

(Left) we show embeddings at the critical value of d̃ andabove it respectively. At very large density the solutionsbecome the usual spike embeddings of the N = 2 theoryat zero magnetic field.

We are not yet done though since there are alsoMinkowski embedding with zero density but constantchemical potential. These can have lower energy and bethe preferred vacuum at a given value of chemical poten-tial - that is, they are important in the Grand CanonicalEnsemble. The relevant analysis is in Fig 3a (Fig 3b-e will be explained in the next section). On the left itshows the three possible types of embedding of the D7for a given chemical potential at zero temperature. The

black curve is the Minkowski embedding (with d̃ = 0),the blue the chiral symmetry breaking spike embedding

(with d̃ 6= 0) and the red the chiral symmetry preservingblack hole embedding (with d̃ 6= 0). Strictly speakingthere is a fourth embedding which lies along the ρ̃ axisand has constant At = µ - its energy is equal for all µ̃ tothat of the red embedding at µ̃ = 0 and is never preferredover the red embedding with density, so we will ignore ithence forth. The trajectory of the three key embeddings

in the d̃ − µ̃ space is shown in the middle plot (notethat again these two variables are thermodynamical con-jugate variables). Finally on the right the grand poten-tial is computed. Clearly at low chemical potential the

Minkowski embedding is preferred and d̃ = 0. There isa critical value of µ̃ = 0.470 at which a transition occursto the spike embedding. This transition looks naivelyfirst order since it is a transition between a Minkowskiembedding and a black hole embedding. However, wecan see that the Grand Potential appears smooth andthe quark density is continuous, which is shown again inFig.4a. The solid lines in Fig 4a are calculated from (18),which is based on the holographic dictionary. The dot-ted lines are obtained by numerically differentiating the

grand potential (d̃ = −∂Ω̃∂µ̃

), which comes from a thermo-

dynamic relation. This is a nontrivial consistency checkof the holographic thermodynamics as well as our calcu-lation [27, 33].

Further in Fig 4b we plot the behaviour of the quarkcondensate through this transition. The density andquark condensate are both smooth and the transitionlooks clearly second order. Here we have tested thesmoothness numerically at better than the 1% level.Whether there is some other order parameter that dis-plays a discontinuity is unclear but it would be surprisingthat any order parameter were smooth, were the transi-tion first order. We conclude the transition is second

(a) Density: the solid lines are calculatedfrom (18) and the dotted lines are

obtained by numerically differentiating

the grand potential (d̃ = − ∂Ω̃∂µ̃

).

(b) The quark condensate.

FIG. 4: Plots of the order parameters vs chemical potentialat zero temperature and finite B. Both are continuous acrossthe Minkowski to spiky embedding transition(µ̃ ∼ 0.47). Thegreen arrows indicate the changes of phase.

order (or so weakly first order that it can be treated assecond order). This second order nature of the transi-tion from a Minkowski to a spiky embedding has beenshown also in the B = 0, m̃ 6= 0 case at zero temperatureanalytically [15] and numerically [16].Finally, above the chemical potential corresponding to

the meson melting transition (µ̃ = 0.470), non-zero den-sity is present and the physics already described in theCanonical Ensemble occurs, which turns out to be equiv-alent to the results from the current Grand CanonicalEnsemble. Both Ensemble predict the second order tran-sition to the flat embedding at the same point, µ̃ = 0.708

or d̃ = 0.3197, which is the chiral symmetry restorationpoint. Notice that for the Canonical Ensemble we used

(m̃,c̃) conjugate variables on constant d̃ slices, while for

the Grand Canonical Ensemble we used (µ̃,d̃) conjugatevariables on constant m̃ = 0 slices. This agreement fromdifferent approaches is another consistency check of ourcalculation.On the gauge theory side of the dual, the description

is as follows. At zero density there is a theory with chiralsymmetry breaking and bound mesons. As the chemi-

cal potential is increased d̃ remains zero and the quarkcondensate remains unchanged. Then there is a secondorder transition to finite density (to a spike like embed-ding) which is presumably associated with meson melt-ing induced by the medium. At a higher density there

9

is then a further second order transition to a phase withzero quark condensate.Finally we note a recent paper [34] that proposed an

alternative ground state for a chiral symmetry breakingtheory at finite density. They proposed that the stringspike might end on a wrapped D5 brane baryon vertex inthe centre of the geometry. We have not considered thatpossibility here but it might be interesting to investigatethis in the future. The magnetic field induced chiral sym-metry breaking provides a system in which this could becleanly computed without the worries of the hard wallpresent in that geometry.

V. THE PHASE DIAGRAM IN THE GRANDCANONICAL ENSEMBLE

We have identified a first order phase transition from achiral symmetry breaking phase with meson bound statesto a chirally symmetric phase with melted mesons in ourmassless theory in the presence of a magnetic field withincreasing pure temperature. On the finite density axisthe meson melting transition is second order and sepa-rate from another second order chiral symmetry restoringphase transition. Clearly there must be at least one crit-ical point in the temperature chemical potential phasediagram. We display the phase diagram of the masslesstheory, which we will discuss the computation of, in Fig5.To construct the phase diagram we have plotted slices

at fixed temperature and varying chemical potential. Wedisplay the results in Fig 3a-e where we show the em-beddings (Left) relevant at different temperatures, their

trajectories in the d̃ − µ̃ plane (Middle) and the grandpotential (Right).The phase diagram agrees with our previous results:

At zero chemical potential we have the transition pointw̃H = 0.2516. At zero temperature we have the transition

point at µ̃ = 0.708, which corresponds to d̃ = 0.3197. Wealso identify µ̃ = 0.470 as the position of the second order

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Μ

0.05

0.10

0.15

0.20

0.25

0.30w H

ΧSB, mesonΧS, meson melted

ΧSB, meson melted

1st order

2nd order

2nd order

H0.129,0.236L

H0.267,0.201LTri-Critical Point

zero density

FIG. 5: The phase diagram of the N = 2 gauge theory with amagnetic field. The temperature is controlled by the param-eter w̃H and chemical potential by µ̃. (Parameters are scaledor B = 1/2R2 in terms of parameters without tilde.)

(a) w̃H = 0.15 (b) w̃H = 0.23

FIG. 6: Quark condensate vs chemical potential at finite B.Both are continuous across the Minkowski (black) to blackhole (orange) embedding transition. At w̃H = 0.23 the blackhole (orange) to black hole(red) transition is discontinous.

transition to a meson melted phase with non-zero d̃ andchiral condensate c̃.

The dotted green line is the line along which d̃ = 0 andcorresponds to the second order meson melting transitionfrom a Minkowski embedding to a black hole embedding.The transition generates density continuously from zero.The quark condensate also smoothly decreases from itsconstant value on the Minkowski embedding. We dis-play the continuous behaviour of the quark condensateacross the transition in Fig 6. Note this means that theslope of the embedding at the UV boundary is continu-ous through the transition even though the embeddingin the IR is discontinuous and topology changing. Againwe have checked the smoothness of these parameters nu-merically to better than the 1% level.

The blue line corresponds to a first order transitionand the red dotted line is a second order transition indensity, chiral condensate etc. The red dotted line israther special in that this is a phase boundary only atm̃ = 0. This is because this phase boundary is relatedto the spontaneous breaking of chiral symmetry whichonly exists at m̃ = 0. At finite m̃ it must be a cross overregion as we will discuss further in section VII

The diagram then displays two tri-critical points. It isstraightforward to identify where the points lie numer-ically. The chiral symmetry tri-critical point where the

0.00 0.05 0.10 0.15 0.20 0.25 0.30 d

0.05

0.10

0.15

0.20

0.25

0.30w H

ΧSB

ΧS

1st order

2nd order

Tri-Critical Point H0.176,0.201L

H0,0.236L

FIG. 7: The phase diagram of the N = 2 gauge theory witha magnetic field. The temperature is controlled by the pa-

rameter w̃H and the density by d̃. (Parameters are scaled orB = 1/2R2 in terms of parameters without tilde.)

10

first and second order chiral symmetry restoration transi-tions join lies at the point (µ̃, w̃H) = (0.267, 0.201). Thesecond tri-critial point where the meson melting transi-tions join is at (µ̃, w̃H) = (0.129, 0.236).

VI. THE PHASE DIAGRAM IN THECANONICAL ENSEMBLE

We can study the phase diagram also in the canonicalensemble. It is shown in Fig 7 and has the same informa-tion as Fig 5. The pale green region in Fig 5 lies in thegreen dotted line along the w̃H axis of Fig 7. The chiralsymmetry breaking region enclosed by the red, green andblue lines in each figure map onto each other. Similarlythe high temperature and density region to the upperright of all the lines in both plots map onto each other.The two double blue lines and the area between them inFig 7 correspond to the single blue line in Fig 5, whichis natural since the blue line in Fig 5 is a first ordertransition line and the density change is discontinuous.Thus the gray region in Fig 7 is an unstable density re-gion which hides in the phase boundary in Fig 5. Thatregion may only be reached by super-cooling or super-heating since it is unstable. The true ground state atthose densities and temperatures should be a mixture ofthe black hole and Minkowski embedding in analogy withthe liquid-gas mixture between the phase transition’s ofwater [16]. It’s not clear how to realize that mixture ina holographic set-up.

VII. FINITE MASS

We next describe the evolution of the phase diagramwith quark mass. If we move away from zero quark massthen the second order chiral symmetry restoration phasetransition at T=0 but growing chemical potential be-comes a cross over transition. This can be seen in Fig2 where for m̃ 6= 0 the non-zero value of the condensatecan be seen to change smoothly with changing µ̃ andthere is no jump in any order parameter. The (chiral)tri-critical point becomes a critical point. However, theother transition lines survive the introduction of a quarkmass.

In Fig 8. we plot the phase diagram for various quarkmass, m̃, at constant B. The colors represent differentquark masses - m̃ = 0, 1, 1.5, 2, 3 from bottom to top areblack, red, orange, green, and blue. The solid lines are forfinite, fixed B. To show the influence of the magnetic fieldwe also display the B = 0 solution as the dotted lines.The gray line shows the motion of the critical points.

In general the magnetic field shifts the transition lineup and right, meaning that the magnetic field makes themeson more stable against the temperature/density me-son dissociation effect. This is important at small m̃ butnegligible at large m̃ as expected.

2nd order

1st order

0.5 1.0 1.5 2.0 2.5 3.0Μ

0.5

1.0

1.5

2.0

2.5

w H

(a)The curves correspond tom̃ = 0(black), 1(red), 1.5(orange), 2(green), 3(blue) frombottom to top. The gray line is the path of the critical

points.

0.042 0.044 0.046 0.048 0.050 0.052 0.054

0.816

0.818

0.820

0.822

0.824

0.031 0.032 0.033 0.034 0.035 0.036 0.0370.756

0.757

0.758

0.759

0.760

(b)A zoom into figure (a) to show the critical point structure.at small chemical potential. Here we show the detail ofm̃ = 1 case, but a similar structure exists for every case.

FIG. 8: The phase diagram at finite current quark mass withfinite B (solid lines) and zero B (dotted lines).

Both critical points survive the introduction of a finitem̃, even though it looks like there is no critical point inFig 8a. Zooming in on the appropriate region at smallchemical potential reveals the two critical point structureas shown in Fig 8b. Their positions, as m̃ changes, aremarked by the gray line in Fig 8a. The one line representsthe two critical points which are indistinguishable closeon the scale of Fig 8a. The chiral symmetry critical pointmoves very close to the other critical point even for a verysmall mass (m̃ ∼ 0.01). The interpretation of the criticalpoints and the phase boundaries are the same as in them̃ = 0 case in the previous section.

Notice that the black hole to black hole transition ex-ists even in the B=0 case as shown in Fig 8b(Right), so itis not purely due to the magnetic field. Nevertheless thistransition seems not to have been reported in the previ-ous works [14, 16]. We believe that this is because thetransition line between the two critical points is too smallto be resolved on the scale of Fig 8a, which agrees qual-itatively with the figures in [16]. In order to find thosetransitions we had to slice the temperature down to or-der 10−3 as shown on the vertical axis in Fig 8b(Right).Any coarser graining would miss it.

The final surprise relative to the previous work is thatthe meson melting transition below the critical point ap-pears second order in our work even in the infinite masslimit. To emphasize this we show a number of plots in

11

0.445 0.446 0.447 0.448 0.449 0.450 0.451Μ0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

-cwH = 0.4

0.445 0.446 0.447 0.448 0.449 0.450 0.451Μ0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030d

0.445 0.446 0.447 0.448 0.449 0.450 0.451Μ-0.00006

-0.000059

-0.000058

-0.000057

-0.000056

-0.000055

-0.000054W

(a) wH = 0.4. At µ̄ ∼ 0.4467 there is a Minkowski to black hole embedding transition, which is second order in both chiralcondensation and density.

0.0338 0.0340 0.0342 0.0344 0.0346 0.0348 0.0350Μ0.022

0.023

0.024

0.025

0.026

0.027

0.028

0.029-c

wH = 0.7575

0.0338 0.0340 0.0342 0.0344 0.0346 0.0348 0.0350Μ0.000

0.002

0.004

0.006

0.008

0.010

d

0.0338 0.0340 0.0342 0.0344 0.0346 0.0348 0.0350Μ-0.0095970

-0.0095965

-0.0095960

-0.0095955

-0.0095950

-0.0095945

-0.0095940

-0.0095935W

(b) wH = 0.7575. There are two transitions. The first (µ̄ ∼ 0.0341) is a Minkowski to black hole transition and second order incondensation and density. The second (µ̄ ∼ 0.03455) is a black hole to black hole transition and first order.

0.0310 0.0315 0.0320 0.0325 0.0330Μ0.022

0.024

0.026

0.028

0.030

0.032-c

wH = 0.7587

0.0310 0.0315 0.0320 0.0325 0.0330Μ0.000

0.002

0.004

0.006

0.008

0.010

d

0.0310 0.0315 0.0320 0.0325 0.0330Μ-0.009734

-0.009732

-0.009730

-0.009728

-0.009726

-0.009724

-0.009722

-0.009720W

(c) wH = 0.7587. At µ̄ ∼ 0.0321 there is a Minkowski to black hole embedding transition, which is first order in both chiralcondensation and density.

0.020 0.022 0.024 0.026 0.028Μ

0.024

0.026

0.028

0.030

0.032

0.034

0.036

-cwH = 0.762

0.020 0.022 0.024 0.026 0.028Μ0.000

0.002

0.004

0.006

0.008

0.010

0.012

d

0.020 0.022 0.024 0.026 0.028Μ-0.01014

-0.01012

-0.01010

-0.01008

-0.01006W

(d) wH = 0.762. At µ̄ ∼ 0.0235 there is a Minkowski to black hole embedding transition, which is first order in both chiralcondensation and density.

0.005 0.010 0.015 0.020Μ

0.025

0.030

0.035

0.040-c

wH = 0.7658

0.000 0.005 0.010 0.015 0.020Μ0.000

0.002

0.004

0.006

0.008

0.010d

0.005 0.010 0.015 0.020Μ-0.01066

-0.01064

-0.01062

-0.01060

-0.01058

-0.01056

-0.01054

-0.01052W

(e)Above wH = 0.7658 only a black hole embedding (Red) is stable configuration.

FIG. 9: Chiral condensation (Left), density (Middle), and the grand potentials (Right) for massive quarks (m = 1) at B = 0at a variety of temperatures that represent slices through the phase diagram Fig 8a.

12

the B = 0 theory in Fig 9.Since the scaled variables (16) cannot be used at B =

0, (20) and (18) read in terms of the original coordinates:

Ω̄(wH , µ̄) :=−S

NfTD7Vol

=

∫ ∞

ρH

dρw4 − w4H

w4

√(1 + (L′)2)

K

(w4 + w4H

w4

)2ρ6 ,

(21)

where

µ̄ =

∫ ∞

ρH

dρ d̄w4 − w4Hw4 + w4H

√1 + (L′)2

K, (22)

K =

(w4 + w4H

w4

)2ρ6 +

w4

(w4 + w4H)d̄2 , (23)

µ̄ :=√23πα′At(∞) , d̄ :=

√23

NfTD72πα′d (24)

By the same procedures as in the previous sections weget Fig 9. Compared to Fig 3, the left column of Fig 9is the chiral condensate instead of the embedding con-figurations. In Fig 3 there is always a red black holeembedding, which corresponds to the flat embedding atzero quark mass. It is not present at finite quark mass.At very low temperature the transition is Minkowski to

black hole and second order in the condensate and den-sity(Fig 9a). As the temperature goes up a new black holeto black hole transition pops up by developing a ‘swal-low tail’ in the grand potential - this transition is firstorder in the condensate and density(Fig 9b). As temper-ature rises the ‘swallow tail’ grows continually and even-tually “swallows” the second order Minkowski to blackhole transition (Fig 9 c,d). i.e. At higher temperature thesecond order Minkowski to black hole transition entersan unstable regime and plays no role any more. Insteadonly the first order Minkowski to black hole transition ismanifest. Finally the Minkowski embedding becomes un-stable compared to the black hole embedding(Fig 9e). Atan even higher temperature the Minkowski embedding isnot allowed and only a black hole embedding is available(Not shown in Fig 9).These results all match with our work at finite B and

increasing mass, confirming those results and our phasediagrams already presented.

VIII. COMPARISON TO QCD

We have computed the phase diagram for a particulargauge theory using holographic techniques. There aremany differences between our theory and QCD: the the-ory has super partners of the quarks and glue present; itis at large N and small Nf , so quenched (and we haveonly computed for degenerate quarks to avoid complica-tions involving the non-abelian DBI action); the theory

ΧSB

ΧS

Μ

T

(a)Standard scenario

ΧSB

ΧS

Μ

T

(b)Exotic Scenario

FIG. 10: Two possible phase diagrams for QCD with theobserved quark masses. (a) is the standard scenario foundin most of the literature but a diagram as different as (b)remains potentially possible according to the work in [3]. Wehave not included any color superconducting phase here atlarge chemical potential.

has deconfined glue for all non-zero temperature; the the-ory has a distinct meson melting transition. In spite ofthese differences the phase diagram for the chiral con-densate shows many of the aspects of the QCD phasediagram so we will briefly make a comparison here.The QCD phase diagram is in fact not perfectly

mapped out since there have only recently been latticecomputations attempting to address finite density [3].The phase structure also depends on the relative massesof the up, down and strange quarks. The standard the-oretical picture [1–3] for physical QCD is shown in Fig10a. At zero chemical potential the transition with tem-perature is second order (or a cross over with massivequarks). At zero temperature there is a first order tran-sition with increasing chemical potential (ignoring anysuperconducting phase). These transitions are joined bya critical point. Comparing to our theory in Fig 5 we seethat the transitions’ orders are reversed and the pictureslook rather different.In fact though as argued in [3] the picture could be very

different in QCD. At zero quark mass the finite temper-ature transition is first order and whether it has changedto second order depends crucially on the precise physicalquark masses. Similarly whether the finite density tran-sition is truly first order or second order depends on theexact physical point in the mu,d, ms, µ, T volume. Ar-guments can even be made for a phase diagram matchingthat in Fig 10b which then matches the structure of thechiral symmetry restoring phase diagram of the theorywe have studied. For the true answer in QCD we mustwait on lattice developments. Clearly our model will notmatch QCD’s phase diagram point by point in mu,d, ms,µ, T volume but it provides an environment in whichclear computation is possible for structures that matchsome points in that phase space.Finally, we note a more general point that seems to

emerge from the analysis. The introduction of a chemi-cal potential weakens the first order nature of the tran-sitions in our analysis. This matches with results foundin QCD on the lattice. The weakening of the first orderphase transition is demonstrated for the chiral transition

13

in the light quark mass regime [35, 36], and is shown forthe deconfinement transition in the heavy quark massregime [37, 38].Acknowledgements: NE and KK are grateful for

the support of an STFC rolling grant. KK would like

to thank Owe Philipsen, Sang-Jin Sin, Yunseok Seo,Johanna Erdmenger, and Ingo Kirsch for discussions.AG and MM are grateful for University of SouthamptonMayflower Scholarships.

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