On holographic phase transitions at finite chemical potential
aa r X i v : . [ h e p - t h ] D ec Preprint typeset in JHEP style - PAPER VERSION
On holographic phase transitions at finite chemicalpotential
Shunji Matsuura
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, CanadaDepartment of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyoku, Tokyo113-0033, JapanE-mail: [email protected]
Abstract:
Recent Holographic studies have shown that N=4 super Yang-Mills theory cou-pled to fundamental matter with finite chemical potential undergoes a first order phase tran-sition. In this paper, we study N f D6 probe branes with or without electric field on it in theblack D4 brane background compactified on a circle with supersymmetry breaking bound-ary condition. At energy scales much lower than the compactification scale, the dual gaugetheory is effectively four dimensional non-supersymmetric SU( N c ) Yang-Mills theory coupledto fundamental matter with or without baryon number charge. Within the supergravity ap-proximation, the decoupling of the Kaluza-Klein modes is not fully realized. For chemicalpotential µ b < N c M q there is a line of first order phase transitions from stable meson phaseto unstable meson phase. On the other hand for µ b > N c M q there is no phase transition andmesons are unstable. A peculiar and interesting property of this system is that for a certainrange of chemical potential µ b < N c M q , a new phase transition appears in the unstable mesonphase. This phase transition is characterized by a discontinuous change of unstable mesonlifetime. ontents
1. Introduction 12. Holographic framework 4
3. Thermodynamics 10
4. Conclusion and Discussion 17
1. Introduction
Thermal phase structure of strongly coupled SU( N c ) super Yang-Mills theory at finite chem-ical potential and finite temperature may be studied by using holographic duality [3, 4]. Theholographic method provides a powerful framework to study a broad class of large N c , stronglycoupled gauge theories with a small number N f of fields in the fundamental representations[9, 10, 11, 12, 13, 14]. The gravity dual of these gauge fields appear as N f probe Dq-braneson the near horizon geometries of N c black Dp-branes.Particularly interesting physics appears when the system undergoes a confinement/de-confinement phase transition. Above this phase transition temperature T dec , the gluons andthe adjoint matters are deconfined and the dual geometry contains a black hole [5]. Since ourmain understanding of QCD largely comes from the behaviour of fundamental matter, it isinteresting to ask how they behave around the deconfinement phase transition. The thermalproperties of fundamental matter in the deconfinement phase can be extracted by studyingprobe D-branes in this black hole background.In [11, 12], it was show that at zero baryon density, fundamental matter undergoes a firstorder phase transition at T fun . In the lower temperature phase, the probe D-brane sits entirelyoutside the black hole horizon, which is called a Minkowski embedding (see Figure 1), and themass spectrum of meson in this phase is discrete and has a mass gap[11, 12]. In the highertemperature phase, a part of the probe D-brane fall through the black hole horizon, which iscalled a black hole embedding, and the mass spectrum of meson in this phase is continuousand gapless, characterized by quasinormal modes [12, 15, 16]. We emphasize that this isin the deconfinement phase. That means that after the gluons and the adjoint matters aredeconfined, the fundamental matter is still in bound states at temperature T dec < T < T fun .– 1 –he meson bound states in the deconfinement phase are also illustrated in related models[18]. On the other hand if T fun is smaller than T dec , the confinement/deconfinement phasetransition and the meson melting take place simultaneously. This phenomenon, the stablemeson bound states in a deconfinement phase T dec < T < T fun , is also found in latticeQCD[20], which suggests that bound states of heavy quarks survive after the deconfinementphase transition up to a few T fun .Motivated by these agreements with Figure 1:
Probe D-brane configurations in the pres-ence of the black hole. Gray circle represents the blackhole horizon. For the black hole embeddings, tempera-ture decreases from the bottom to top. The dark greencolour represents Minkowski embedding which is en-tirely outside the horizon. At lower temperature, theprobe D-brane is entirely outside the black hole hori-zon, i.e., the Minkowski embedding.(dark green line).At higher temperature, the probe D-brane partially fallthrough the black hole horizon, i.e., the black hole em-bedding.
QCD, other region of the QCD phase di-agram, i.e., finite temperature with fi-nite baryon density n b or finite chemicalpotential µ q = µ b /N c was investigatedin [2, 21, 22, 23, 24, 25, 26], see also[19]. The introduction of a finite baryondensity corresponds to the presence ofa U(1) gauge field, which is a diagonalpart of the gauge group U(N f ), on theprobe D-brane.In [2], this direction is explicitly in-vestigated in the D3/D7 system. Theboundary gauge theory is SU( N c ) N =4super Yang-Mills theory with N f N =2hypermultiplets in the fundamental rep-resentation. There, physical quantitiessuch as quark condensate in finite n b con-tinuously change from those in n b = 0.One of the most crucial difference be-tween zero and nonzero baryon densitysystems is in the presence finite baryon density, Minkowski embeddings are unphysical anddo not play any role, while black hole embeddings cover the whole range of the temperatureabove deconfinement phase transition. This is different from n b = 0 case where black holeembeddings cover only high temperature region and Minkowski embeddings cover only lowtemperature region[11, 12]. In the overlapping region, there is a phase transitions from aMinkowski embedding to a black hole embedding.The reason why Minkowski embeddings are unphysical in the presence of finite baryondensity is as follows; The electric field on the D-brane represents dissolved fundamentalstrings. Since Wess-Zumino coupling is inactive in this brane configuration, fundamentalstrings never ‘leak’ from the brane and the local baryon number density is still nonzero abovethe horizon. Since the fundamental strings cannot terminate, we have to have fundamentalstrings stretching from the probe D-brane to the horizon. At this junction point, the tensionof the probe D-brane is always smaller than that of the fundamental strings. That means thatthere is no force balanced configurations in the Minkowski embeddings at finite baryon den-– 2 –ity and the brane must fall down through the horizon. See [2, 1] for more detail discussions.Instead at very low temperature a black hole embedding mimics a Minkowski embedding, i.e.,a very long and narrow spike, which corresponds to a bundle of dissolved fundamental stringson the probe D-brane, stretches down to the horizon. For smaller n b , there is a first orderphase transition from a black hole to another black hole embedding, which is very similar toa Minkowski to a black hole embedding in the zero baryon density case. Above the criticaldensity n ∗ b , there is no phase transition anymore.However it was shown in [2] that there is an unstable region near the phase transitionline. It is expected that this unstable region finally decays to other stable state. Howeversince this region minimizes the free energy, there is no state to decay and the final state wasmissing.This puzzle was recently addressed in [1]. In the grand canonical ensemble, we fix nota baryon density but a chemical potential. Since a Minkowski embedding can have a finitechemical potential without a baryon density, this embedding is physical in the grand canonicalensemble and plays an important role. The black hole embeddings cover only larger chemicalpotential or higher temperature region. On the other hand, the Minkowski embeddings coverthe whole value of µ with temperature lower than T fun (see fig.2 in [1]). There are phasetransitions inside the overlapping region from Minkowski to black hole embeddings. One ofthe most remarkable results is that the unstable region is thermodynamically unfavourablein the grand canonical ensemble. Since in the thermodynamical limit, the grand canonicaland the canonical ensemble should give the same answer[28], the unstable region is not a trueground state but it should be replaced by an inhomogeneous phase of a stable black holeembedding and a Minkowski embedding in the canonical ensemble.The aim of this paper is to investigate the generality of the phase structure in Dp/Dqsystem. We analyze the D4/D6 system. Physical properties such as the phase transitions froma black hole to a black hole embedding in the canonical ensemble and from a Minkowski to ablack hole embedding in the grand canonical ensemble and the existence of the unstable regionare the same. However the inhomogeneous phase has a more complicated and interestingstructure. There is a new additional phase transition. Figure 2 shows the phase diagram ofthe D4/D6 system. As in the case of the D3/D7 system, the black hole embeddings coveronly high temperature or high chemical potential region. The boundary of the black holeembeddings is plotted in the green line. In the region surrounded by the blue line, there aremore than one value of n q for each µ q /M q and T / ¯ M . The red line is the line of the phasetransitions. On the scale of (a), the difference between the red and the green line is very subtle.An interesting structure is found in (b) and (c). Near µ q /M q = 0 . T / ¯ M = 0 .
77, the red lineseparates into two branches. The upper branch represents phase transitions from black holeto black hole embeddings. In the field theory side, this corresponds to a discontinuous changeof meson lifetime. Below this phase transition temperature, the spectral function of mesonswould show sharp peaks representing longer lifetime quasiparticles, while above the phasetransition, the peaks would be flatter and the lifetime of the quasiparticles would be shorter.The lower branch represents phase transitions from Minkowski to black hole embeddings. We– 3 –ill discuss the relation between these phases and the unstable region in Section 3. n q = 0 n q = 0(a) (b)(c) Figure 2:
Phase diagram: Quark chemical potential µ q /M q versus temperature T / ¯ M . The red lineseparates the phase of Minkowski embeddings (small temperatures, small µ q /M q ) from black holeembeddings. Figure (b) zooms in on the region near the end of this line and also depicts the boundaryof the region accessed by the black hole embeddings (green) and a small region (enclosed by the bluecurve) where more than one black hole embedding in available for a given value of µ q and T . Figure(c) zooms in on the region near the branches of the red lines. The two branches of the red lines showthe phase transitions, the lower one is from the Minkowski embeddings to the black hole embeddingsand the upper one is from the black hole embeddings to the black hole embeddings.
2. Holographic framework
In this section we explain the D4/D6 system. Closed string modes on the near horizongeometry of D4 brane is dual to maximally supersymmetric five dimensional super Yang-Mills theory with large N c and at strong coupling on the boundary . In order to obtain fourdimensional gauge theory at low energy scale, we compactify one space direction, say x , witha radius M − KK . For energy scale E << M KK , the boundary gauge theory is effectively fourdimensional. Since we want to study the thermal properties of fundamental fields coupling to– 4 –ang-Mills field, we choose the supersymmetry-breaking spin structure, i.e., the anti-periodicboundary condition for fermions in x direction. The fermions in the vector multiplet getmasses of order M KK at tree level and the scalars possibly get masses of order g N c M KK atone loop level. So the gauge theory is effectively non-supersymmetric four dimensional Yang-Mills theory at low energy scale. By fixing the boundary geometry, this system undergoes aphase transition from a thermal AdS phase to an AdS black hole phase at certain temperature T dec . In the dual language, this transition is deconfinement/confinement phase transition[5].This system describes only adjoint matters. To approach QCD in this holographic framework,quarks or fundamental matter must be included. The introduction of fundamental matterin the holographic framework is demonstrated in [8]. According to their work, a probe D-brane plays a role of fundamental matter in the dual picture. In our case, inserting N f D6branes into this black-D4 background corresponds to the coupling of N f hypermultiplets inthe fundamental representation to the Yang-Mills fields with the gauge group SU( N c ). Thequarks arise from the lightest mode of the fundamental strings connecting between D4 andD6 branes. In the decoupling limit, the mesons are dual to the fundamental strings whoseboth ends are attaching on D6 branes. This means that the fluctuations on the D6 branesdescribe the mesons in the dual gauge theory. The supergravity solution corresponding to the decoupling limit of N c coincident D4-branesin the black hole phase is in the string frame ds = 12 (cid:16) ̺L (cid:17) / (cid:20) − f ˜ f dt + ˜ f dx (cid:21) + (cid:18) L̺ (cid:19) / ˜ f / / (cid:2) d̺ + ̺ d Ω (cid:3) (2.1) e Φ = (cid:16) uL (cid:17) / = (cid:18) ̺ L (cid:19) / ˜ f / (2.2)where f = 1 − u ̺ and ˜ f = 1 + u ̺ . u is the location of the horizon and the AdS radius L is given by L = g s N c πl s . (2.3)Hawking radiation appears in this background with the temperature given by the surfacegravity T = κ/ π , or by the regularity of the Euclidean section, T = 34 π r u L . (2.4) However the Kaluza-Klein modes do not decouple within the supergravity approximation[5]. For example,the lightest glueball spectrum is of the same order as the strong coupling scale[7]. This metric (isotropic coordinate) is related to the ordinary coordinate as ( u ρ ) / − u / = p u − u – 5 –his temperature is identified with the temperature of the boundary gauge theory. Thedual boundary gauge theory is maximally supersymmetric Yang-Mills theory on R , withcoordinates { t, ~x } .As we explained above, we compactify on a circle with radius 1 /M KK with supersymmetrybreaking boundary condition. At low energy E << M KK the effective field theory is fourdimensional non-supersymmetric Yang-Mills theory. The string coupling constant in the bulktheory and the gauge coupling constant in the dual boundary theory is related through g = (2 π ) g s l s , g = g M KK (2.5)where g s = e φ ∞ and g , are the Yang-Mills coupling constants in four and five dimensions. We introduce N f D6-branes in this background in the following configuration;0 1 2 3 4 5 6 7 8 9D4 × × × × × D6 × × × × × × × We split the AdS radial direction and S into two parts so that SO (3) × SO (2) symmetryis manifest. We define the radial coordinate r in ( x , x , x ) directions and R in ( x , x )directions where ρ = ̺/u is a dimensionless coordinate. ρ = r + R , r = ρ sin θ, R = ρ cos θ, (2.6)and dρ + ρ d Ω = dρ + ρ ( dθ + sin θd Ω + cos θdφ ) (2.7)= dr + r d Ω + dR + R dφ (2.8)The probe D6 brane embedding is parametrized by a function, χ ( ̺ ) ≡ cos θ ( ̺ ). The inducedD6 metric is ds = 12 (cid:16) ̺L (cid:17) / (cid:20) − f ˜ f dt + ˜ f dx (cid:21) + (cid:18) L̺ (cid:19) / ˜ f / / (cid:20)(cid:18) ̺ ˙ χ − χ (cid:19) d̺ + ̺ (cid:0) − χ (cid:1) d Ω (cid:21) (2.9)Now we introduce U(1) gauge field A ( ̺ ) dt on the probe D6 branes. The D6 brane actionper unit spacetime volume of the gauge theory is I D = − N f T Z e − Φ p − det( P [ G ] ab + 2 πα ′ F ab ) (2.10)= − N f T Ω Z d̺ ˜ f / ̺ (cid:0) − χ (cid:1) s f ˜ f − / (cid:18) ̺ ˙ χ − χ (cid:19) − k ˙ A (2.11)– 6 –here k = (2 / πl s ) , (2.12)and the D6 brane tension is T = 2 πg s (2 πl s ) . (2.13)Let us analyze the behaviour of the electric field A . The electric displacement d is definedby d = N f T Ω f / ̺ (1 − χ ) k ˙ A q f ˜ f − / (1 + ̺ ˙ χ − χ ) − k ˙ A (2.14)The electric displacement represents the number density of dissolved fundamental stringsin the D6 branes. With the similar analysis to [2, 1] the electric displacement identified withthe string number density n q : d = n q . Since the action does not explicitly depend on A , theequation of motion for A is d =constant, or˙ A = df ˜ f − q ̺ ˙ χ − χ r d k + (cid:16) N f T Ω k (cid:17) ˜ f / ̺ (1 − χ ) (2.15)(2.16)Asymptotically χ goes to zero and f and ˜ f go to one. The leading order of the equation isthen ˙ A = d N f T Ω k ̺ + O (cid:18) ρ (cid:19) (2.17)So the asymptotic behaviour of the gauge field is A ∼ µ q − dN f T Ω k ̺ + · · · (2.18)where µ is µ q = d Z ∞ u d̺ f ˜ f − q ̺ ˙ χ − χ r d k + (cid:16) N f T Ω k (cid:17) ˜ f / ̺ (1 − χ ) . (2.19)Here we set A ( u ) = 0. This condition comes from the regularity of the one form at thehorizon. The horizon contains a bifurcation surface where a Killing vector ∂ t vanishes. Inorder for the gauge field, which is a one form A t dt , to be well defined, the component A t mustvanish there. See also [2, 12].According to the holographic dictionary, the dual operator is schematically O q = ψ † ψ + q † D t q (2.20)– 7 –here ψ and q are left and right Weyl fermions and scalar fields (see also [2]). From (2.18)and (2.20), we identify µ q with a quark chemical potential.For convenience, we also use dimensionless quantities and a gauge field,˜ d = 4 N f T Ω √ k du , ˜ µ = √ ku µ, ˜ A = √ ku A. (2.21)The equation of motion for χ is dd̺ f ˜ f ̺ ˙ χ q ̺ ˙ χ − χ − k ˙ A f ˜ f − / (2.22)+ f ˜ f ̺ χ q ̺ ˙ χ − χ − k ˙ A f ˜ f − / ̺ ˙ χ − χ − k ˙ A f ˜ f − / ! = 0 (2.23)The boundary conditions for χ ( ρ ) at the horizon is determined by the regularity and it gives χ (cid:12)(cid:12) ρ =1 = χ and dχ/dρ (cid:12)(cid:12) ρ =1 = 0 for 0 ≤ χ <
1. For Minkowski embeddings it is convenientto use
R, r coordinates instead of χ, ρ coordinates. The boundary conditions for Minkowskiembeddings at r = 0 are R (cid:12)(cid:12) r =0 = R and dR/dr (cid:12)(cid:12) r =0 = 0 for 1 < R . The asymptotic forms( ρ → ∞ ) for χ is χ ∼ m̺ + c̺ + O (cid:16) ̺ (cid:17) (2.24)= ˜ mρ + ˜ cρ + O (cid:16) ρ (cid:17) , (2.25)where we define dimensionless quantities˜ m = mu = 3 m (4 π ) L T , ˜ c = cu = 3 c (4 π ) L T . (2.26)Holography relates these quantities to a quark mass and a condensate[10] by M q = u ˜ m πl s (2.27) < O m > = − / π l s N f T D u ˜ c (2.28)A bare quark mass M q is an asymptotic distance between D4 and D6 brane in flat space.The operator O m is the variation of the mass term in the microscopic Lagrangian, i.e., O m = − ∂ M q L , and the schematic form is O m = ¯ ψψ + q † Φ q + M q q † q, (2.29)where Φ is one of the adjoint scalars. See also [2, 12]. We assume that the vacuum expectationvalue of the fundamental scalar fields q vanishes because whenever the scalar fields acquirethe nonzero expectation value, the energy density increases[10]. In that case, < O m > is equal– 8 –o < ¯ ψψ > and represents a quark condensation. Given the above result, we show variousfigures in terms of T / ¯ M ≡ / √ ˜ m .Let us study more on the chemical potential. Once the equation of motion for χ ( ̺ )(2.23) is solved, the chemical potential is obtained from (2.19). In general, we have to resortto numerical calculation. However, we can extract analytic properties in the limiting casesof small and high temperatures. First we consider low temperature or large bare quark masslimit T / ¯ M →
0. As shown in Figure 1, at very low temperature the probe D-brane goes upstraight from the horizon u to ∼ m . The main contribution to the distance between the D4and D6 brane, or T / ¯ M , comes from this part. Under this approximation, µ q = d Z ∞ u d̺ f ˜ f − q ̺ ˙ χ − χ r d k + (cid:16) N f T Ω k (cid:17) ˜ f / ̺ (1 − χ ) (2.30) ≃ d Z mu d̺ f ˜ f − √ d k (2.31) ≃ m/ √ k = M q (2.32)Regardless of d , µ q goes to M q . This is consistent with Figure 3 where all curves with different˜ d meet on the vertical axis at µ q /M q = 1. On the opposite limit, i.e., high temperature or (cid:144) M (cid:143)(cid:143)(cid:143)€€€€€€€€€€€Μ q M q Figure 3:
Chemical potential µ q /M q versus T / ¯ M for various values of ˜ d , increasing from bottom up:˜ d = 10 − , , , , . small bare quark mass limit T / ¯ M → ∞ , the probe D6 branes are almost flat and intersectthe horizon at the equator. Plugging χ ≃ u >> µ q ≃ d ( N f T Ω k ) 12 / u + O ( d ) (2.33)= 94 N f N c n q T + O ( n q ) (2.34)There are some notable points. The first point is that for any value of n b , µ q goes to zeroas temperature goes to infinity. In this sense, Figure 3 might be misleading. Each line inthe figure shows µ q /M q versus T / ¯ M for fixed ˜ d not d . In each line, higher temperaturecorresponds to a higher baryon number density and lower temperature corresponds to a lowerbaryon number density. The second point is for fixed µ q , n b changes as ∼ T . Since in thegrand canonical ensemble we survey fix µ q planes, this equation suggests that the baryondensity monotonically increases at very high temperature for any fixed µ q /M q .
3. Thermodynamics
We move on to the thermodynamics of the D6 brane, or the fundamental matter. As wewill see the canonical ensemble is not a suitable one for studying the phase diagram since itincludes an unstable region. Our main focus here is the grand canonical ensemble. A similarunstable region can also be found in the D3/D7 system [1]. We will briefly mention on thecanonical ensemble.Euclidean path integral of D6 branes give thermal partition function[27]. Since the clas-sical solution of the equation of motion is the saddle point of the path integral, the on-shellaction gives main contribution to the Gibbs free energy W , i.e., W = T I E . From the grandcanonical point of view, a brane configuration which minimizes the Gibbs free energy isthermodynamically favourable. However, naive calculation of thermal quantities include di-vergences from IR region. For example, we can clearly see that the action diverges as Z dρρ ≃ ρ max . (3.1)In order to obtain physically meaningful, finite quantities, we need to renormalize them.The renormalization procedure of probe D-brane is studied in [29]. According to that, renor-malization is done by introducing boundary terms for probe branes on the cut-off plane sothat the divergences cancel. For the D4/D6 case, this boundary term is concretely studied in[12]. Before starting the detail analysis, we define a normalization constant N = πT N f T D u . (3.2)Inserting the asymptotic expansion of χ and ˙ A into (2.11), the action is I reg N = Z dρρ (cid:18) − ˜ m ρ − m ˜ cρ (cid:19) m ρ · · · − ˜ d ρ ! . (3.3)– 10 –he contribution to the action from the gauge field is the term proportional to ˜ d . This givesno new divergence. So we can use the same boundary term as that of ˜ d =0 case to renormalizethe divergence. We apply the following boundary term per unit spacetime volume of the gaugefield [12], I bound = − Ω L T N f √ γ (cid:18) − χ (cid:19) | ρ = ρ max (3.4)= − u Ω T N f (cid:18) ρ max −
32 ˜ m ρ max − m ˜ c (cid:19) , (3.5)where γ is the induced metric at ρ = ρ max ds γ = 12 (cid:16) u ρ max L (cid:17) / (cid:18) − f ( ρ max ) ˜ f ( ρ max ) dt + ˜ f ( ρ max ) dx (cid:19) . (3.6)The total action is then, I tot N = I reg N + I bound N (3.7)= (cid:20) G ( ˜ m, ˙˜ A ) − (cid:18) ρ min −
32 ˜ m ρ min − m ˜ c (cid:19)(cid:21) , (3.8)where G ( ˜ m, ˙˜ A ) is G ( ˜ m, ˙˜ A ) = Z ∞ ρ min dρ ˜ f / ρ (cid:0) − χ (cid:1) s f ˜ f − / (cid:18) ρ ˙ χ − χ (cid:19) − ˙˜ A − ρ + ˜ m ! (3.9)From the thermodynamical point of view, this action is identified the Gibbs free energy W ( T, µ q ) via W = T I E .In the canonical ensemble with fixed n q , we use the Helmholtz free energy. Similar to[2, 30], the Helmholtz free energy is associated with the Legendre transform of I E .˜ I E N = I D N + R d ˙ A N (3.10)which is function of the temperature and the baryon density. We identify F ( T, n q ) = T I E where F ( T, n q ) is the Helmholtz free energy. Since there is no contribution from the electricdisplacement to the boundary term , the Legendre transformed action is˜ I E N = H ( ˜ m, ˜ d ) − (cid:18) ρ min −
32 ˜ m ρ min − m ˜ c (cid:19) (3.11)where H ( ˜ m, ˜ d ) is H ( ˜ m, ˜ d ) = Z ∞ ρ min dρ " f ˜ f − / s ρ ˙ χ − χ q ˜ f / ρ (1 − χ ) + ˜ d − ρ + ˜ m . (3.12)– 11 –e evaluated the free energy numerically. The qualitative feature is very similar to that ofthe D3/D7 system (see Figure 9 and 10 in [2]). For smaller value of ˜ d/ ˜ m , there is a lineof first order phase transition from black hole to black hole embeddings and above a criticalvalue (cid:16) ˜ d/ ˜ m (cid:17) ∗ there is no phase transition. Note that since ˜ d depends on a temperature n q /T , we use a temperature independent quantity ˜ d/ ˜ m ∼ n q /M q as a parameter. Theblue line in Figure 9 shows the phase transition line in the canonical ensemble. The criticaltemperature and baryon density is T / ¯ M = 0 . d/ ˜ m = 0 . As we mentioned in Section 1, the black hole embeddings cover only above the green line inFigure 2 while the Minkowski embeddings cover whole value of µ q /M q below ∼ T fun . Thissuggests that there is a line of first order phase transitions for µ q /M q < µ q /M q .(a) (b) Figure 4:
Free energy versus temperature for µ q /M q = 0 .
14. The blue dotted (red solid) linerepresents the Minkowski (black hole) branch. The vertical line marks the temperature of the phasetransition.
Figure 4(a) shows the Gibbs free energy W normalized by ¯ N versus temperature T / ¯ M for µ q = 0 .
14 in a broad view and Figure 4(b) shows a zoomed in around the phase transition.Here the normalization is ¯ N = N ¯ M . The red line is the black hole embedding and the dashedblue line is the Minkowski embedding. The starting point of the red line ( T / ¯ M = 0 . W/ ¯ N = − . d = 10 − . As ˜ d becomes larger, the temperature becomes higher and thefree energy becomes smaller, then at ˜ d = 5 × − the temperature starts to decrease and thefree energy starts increase. And at ˜ d = 0 .
17, the temperature turns to increase and the freeenergy turns decrease again. As µ increases, the starting point of the free energy of the blackhole embedding at very small ˜ d goes in the left above direction and the three fold structurestarts to form a swallow tail shape. At a critical value of µ q /M q =0.15, the crossing point of– 12 –a) µ q /M q =0.165 (b) µ q /M q =0.175 Figure 5:
Free energy versus temperature for (a) µ q /M q =0.165 and (b) µ q /M q =0.175. (a) shows thatthere are two phase transitions. One is from the Minkowski embedding to the black hole embeddingat T / ¯ M = 0 .
765 and the other is from the black hole embedding to the black hole embedding at
T / ¯ M = 0 . T / ¯ M = 0 .
76. The vertical lines mark the temperature of the phasetransitions. the swallow tail goes down below the line of the Minkowski free energy. Above this criticalvalue, there are two phase transitions. The first one is from a Minkowski embedding to ablack hole embedding. The second one is from a black hole embedding to another black holeembedding. A representative of this phase is shown in Figure 5(a). At µ q /M q = 0 . T / ¯ M = 0 . T / ¯ M = 0 . µ becomes larger, the swallow tail shrinks smaller and smaller and finallyat µ q /M q = 0 .
175 the black hole to black hole phase transition disappears. The free energyat this value of µ q /M q is shown in Figure 5(b). There is a phase transition at T / ¯ M = 0 . µ q < M q in the D3/D7 system (see also Figure 4 in [1]). The only phase transition is froma Minkowski embedding to a black hole embedding and there is no phase transition from ablack hole to a black hole embedding.This feature can also be seen from other perspective. Figure 7(a) shows ˜ d/ ˜ m versus T / ¯ M diagram near the phase transition for µ q /M q = 0 .
16. As explained above, below a certaintemperature there are only Minkowski embeddings. Around
T / ¯ M = 0 . T / ¯ M = 0 . d/ ˜ m = 0 to ˜ d/ ˜ m = 0 . T / ¯ M = 0 . d/ ˜ m = 0 . d/ ˜ m = 0 . T / ¯ M becomes larger ˜ d/ ˜ m becomes larger.We also show a ˜ d/ ˜ m versus T / ¯ M Figure 6:
Free energy versus temperature for µ q /M q =0 .
005 in the D3/D7 system. There is only one turnoverin the black hole embeddings. This should be comparedwith the D4/D6 system (Figure 4) where there are twoturnovers. diagram for µ q /M q = 0 .
005 in the D3/D7system in Figure 7(b). As above, be-low a certain temperature there are onlyMinkowski embeddings. At
T / ¯ M = 0 . d/ ˜ m = 0 to ˜ d/ ˜ m =0 . µ q / ¯ M q is a two-valuedfunction of T / ¯ M while it is a triple-valuedfunction in the case of the D4/D6 sys-tem, and the region corresponding to (A)-(B) in Figure 7(a) is missing.(a) Baryon density versus temperature in D4/D6 (b) The same in D3/D7 Figure 7:
Baryon density versus temperature for (a) µ q /M q = 0 .
16 in the D4/D6 system and (b) µ q /M q = 0 .
005 in the D3/D7 system. In the D4/D6 system, baryon density is a triple-valued functionof temperature while in the D3/D7 system, baryon density is a double-valued function of temperature.
In the rest of this section, we address another interesting property, i.e., the instabilityin the canonical ensemble, and combine it with the phase transitions discussed above. Theconditions for the stability of the system are given ∂S∂T (cid:12)(cid:12)(cid:12) µ q > , ∂n b ∂µ q (cid:12)(cid:12)(cid:12) T > . (3.13)The unstable configuration can be found by examining µ q /M q versus T / ¯ M and ˜ d/ ˜ m dia-gram(Figure 8). This figure covers only a small range of the surface near the phase transitionin the canonical ensemble. For fixed n b , or ˜ d/ ˜ m , the system jumps from the top to thebottom of the surface at certain temperature. Thermodynamical point of view, this means– 14 – q M q T / ¯ M ˜ d/ ˜ m Figure 8:
Three-dimensional plot of the chemical potential, the temperature and the charge densitydetermined by black hole embeddings. that only the region of the top of this fold with temperature lower the phase transition and ofthe bottom of this fold with temperature higher than the phase transition is favoured. Otherregion are thermodynamically unfavoured do not come into the stability discussion. An in-teresting point is that on the top of the fold and just below the phase transition temperaturewe can actually find a region where thermodynamically favourable but electrically unstable.This region should play a central role in the canonical ensemble. In [1], the similar problemwas addressed. The resolution of the problem in the D3/D7 system was that minimizingthe free energy in the grand canonical ensemble always picks out either a stable black holeembedding or Minkowski embedding. Hence the system never suffer from the instability. Thequestion is whether the same is true for the D4/D6 system. After explaining Figure 9, wecome back to this question.Figure 9 shows the black hole to Minkowski phase transition, ∂µ q /∂n b = 0, and thecanonical phase transition line. The blue line is the canonical phase transition line. Forfixed value of d , there is a black hole to black hole phase transition at this line. The green– 15 – BCD E FG (cid:144) M (cid:143)(cid:143)(cid:143)€€€€€€€€€€ d Ž m Ž Figure 9:
Baryon density at the phase transition in the grand canonical ensemble (red) The blue lineof phase transitions identified in the canonical ensemble at fixed n b . The region enclosed by the greenand blue curves corresponds to the unstable region in which ( ∂µ q /∂n q ) T < line is a part of the ∂µ q ∂n b (cid:12)(cid:12) T = 0 line. The region surrounded by green and blue line is thethermodynamically favourable but electromagnetically unstable region. The red line is themapping of the phase transition line in Figure 2 to ˜ d/ ˜ m − T / ¯ M plane. Let us explain thisline more detail. In Figure 4(b), there is one phase transition from the black hole embeddingto the Minkowski embedding at T = 0 . d/ ˜ m = 0 . d/ ˜ m forfixed T / ¯ M and at this point the phase transition occurs from a Minkowski embedding to ablack hole embedding, i.e., from a point at ˜ d/ ˜ m =0 to a point on the red line with the sametemperature.In Figure 5(a), there are two phase transitions. One is from the Minkowski embeddingto the black hole embedding at T / ¯ M = 0 . d/ ˜ m = 0 . T / ¯ M = 0 . d/ ˜ m = 0 . T / ¯ M = 0 . d/ ˜ m = 0 . d/ ˜ m =0 to a point on the red line between E and G. Then the secondphase transition occurs at a different temperature from a point on the red line between E andF to a point on the red line between F and B.In Figure 5(b), there is only one phase transition from the Minkowski embedding to theblack hole embedding at T / ¯ M = 0 .
76 and ˜ d/ ˜ m =0.003. This corresponds to the left part ofG on the red line.As in the D3/D7 system[1], the whole unstable region is surrounded by the red line. Hence– 16 –he grand canonical ensemble picks out either stable black hole embedding or Minkowskiembedding; the thermodynamically favourable phases discussed above are all stable. Theunstable region appeared in the canonical ensemble is misidentified the ground state sincewe restricted our analysis to the homogeneous configurations. The true ground state is aninhomogeneous phase of stable phases [1].The true ground state of the region surrounded by the red curve A-B and the ˜ d/ ˜ m = 0axis is an inhomogeneous phase of stable black hole (on the red curve A-B) embedding andMinkowski embedding (on the ˜ d/ ˜ m = 0 axis). The true ground state of the region surroundedby B-F-E-B is an inhomogeneous phase of stable black hole embedding (on the red curve B-F)and another stable black hole embedding (on the red curve F-E). The true ground state ofthe region surrounded by the red curve E-G and the ˜ d/ ˜ m = 0 axis, is an inhomogeneousphase of stable black hole embedding (on the red curve E-F) and a Minkowski phase (on the˜ d/ ˜ m = 0 axis). Hence both in the grand canonical ensemble and the canonical ensemble,the unstable region is thermodynamically unfavourable.
4. Conclusion and Discussion
We have seen that there are several common physical properties in the Dp/Dq systems withfinite charge density and chemical potential. In the grand canonical ensemble there is aline of the first order phase transitions from Minkowski to black hole embeddings found in[24, 25, 26, 1]. The black hole embeddings cover high
T / ¯ M or high µ q /M q region while theMinkowski embeddings cover the whole µ q /M q region with T / ¯ M smaller than ∼ T fun . Inthe supersymmetric limit T / ¯ M → µ q /M q always goes to one for any value of n b , i.e., theenergy which is necessary to add one quark in the system is equal to its bare mass. In thislimit black hole embeddings exactly look like Minkowski embeddings. There is a very thinand long spike stretching from Dq branes down to Dp branes. In the dual field theory pointof view[12, 15, 16], the Minkowski embeddings correspond to the stable meson phase and hespectral function consists of a series of delta-function-like peaks, i.e., resonances centred onmass eigenvalues. On the other hand, the black hole embeddings correspond to the unstablemeson phase and the spectral function is continuous. In the canonical ensemble, there isa line of first order phase transitions from black hole to black hole embeddings for chargedensity less than n ∗ b and above this critical density there is no phase transition. Just belowthe phase transition temperature, there is an electrodynamically unstable region. Howeverthis unstable region is not the true ground state and should be replaced by an inhomogeneousphase.We also have seen that there is a difference between the D3/D7 and D4/D6 system. Inthe D3/D7 system for fixed µ q < M q , there is only one phase transition, from a Minkowskito a black hole embedding. On the other hand, in the D4/D6 system for certain range of µ q < M q , there are two phase transitions, one is from a Minkowski to a black hole embeddingand the other one is from a black hole to another black hole embedding. Following the inves-tigation of [15, 17, 23], we can give a physical interpretation of this black hole to black hole– 17 –mbedding phase transition in the dual gauge theory. The feature of the spectral function ischaracterized by the poles of the corresponding retarded correlators in the complex frequencyplane. In the Minkowski embeddings, the poles are on the real axis and the spectral functionis, as mentioned above, a series of delta-function-like peaks. In the black hole embeddings,the poles are located apart from the real axis. At lower temperature, the imaginary part ofthe poles are close to the real axis and the spectral function exhibits distinct peaks. As tem-perature increases, the poles move away from the real axis and the spectral function becomesfeatureless. Hence our black hole to black hole phase transition from lower temperature tohigher temperature would correspond to poles jumping from closer locations to the real axisto farther locations. The peaks of the spectral function would become lower and the life timeof the quasiparticles would become shorter. It would be very interesting to investigate thispoint concretely. Lattice QCD studies[20] suggest that meson bound states survive the de-confinement phase transition and the bound states dissolve at T fun . As explained, Minkowskito black hole phase transition represents this meson dissolving phase transition. A very in-teresting question is then what the phase transition in QCD corresponding to the black holeto black hole phase transition is. It would be an exotic phase transition if it exists in QCD.Before ending the section, we comment one more common property in the D3/D7 andthe D4/D6 systems. As mentioned above, in the T / ¯ M → µ q goes to M q . This result is reasonable in the case of free fermions with zero density. Howeversince our system is strongly interacting system with bosons and fermions, the reason is notvery clear. In fact, the chemical potential is very close to the constituent mass at the phasetransition points not only at T / ¯ M = 0 but also for a broad range of temperature except thecanonical phase transition region. Hence the quarks behave as if they were ‘free’ particles. Wehave put ‘free’ om quotes because the constituent mass takes into account all the correctionsfrom the quarks and the adjoint plasma. The constituent quark mass is defined by the energyof a fundamental string stretching from a probe D-brane to the horizon in a Minkowskiembedding. This constituent quark mass at T / ¯ M = 0 on Minkowski embeddings with n q = 0was calculated in [12, 31, 1] The induced metric of the fundamental string stretching from aprobe D-brane to the horizon in R direction is ds = 12 (cid:18) u RL (cid:19) / (cid:20) − f ˜ f dt (cid:21) + (cid:18) Lu R (cid:19) / ˜ f / / u dR . (4.1)The Nambu-Goto action for the fundamental string is I = − πl s Z R dRdt p − det P [ G ] (4.2)= − πl s Z dt u / (cid:18) R (cid:19) / R − / ! (4.3)Identifying the constituent mass to minus the action per unit time of this static configuration,– 18 –e have M c = 12 πl s u / (cid:18) R (cid:19) / R − / ! . (4.4)(a) (b) Figure 10:
Comparison of the ratio M c /M q (blue) with the ratio µ q /M q (red) at which the phasetransition from a Minkowski to a black hole embedding takes place. The two curves essentially coincideon the scale of the Figure (a). We can see that the constituent mass is almost identical to the chemical potential exceptnear the canonical phase transition region. This result is the same as that of the D3/D7system[1] and the similar result is obtained in [25]. As mentioned in [1], this result is surprisingbecause the quark density is much larger than the size of an individual quark; n q = 2 / N f N c g eff ( M q ) ˜ d ˜ m ! n crit , (4.5)where g eff ( M q ) = λM q = g N c M q and n crit = ¯ M . Structure functions[32] or a quark’sdisturbance of the adjoint fields [33] in the D3/D7 system are well studied. The mass spectrumof mesons in the Dp/Dq system is discussed in [35]. Motivated by their works, we assumethat the individual quark size is ∼ m gap . Combining it with the relation ¯ M /m gap ≃ . n crit = ¯ M at which quarks start to overlap each other. Forthe effective coupling, supergravity approximation is valid only in the region[34]1 << g eff << N / c . (4.6)Hence although (cid:16) ˜ d/ ˜ m (cid:17) may be of order 10 − , n q is much larger than n crit and interactionswould not be negligible. Acknowledgments
The author thanks Robert C. Myers, David Mateos and Rowan F.M. Thomson for usefulconversations. This research was supported by Perimeter Institute for Theoretical Physics.– 19 –esearch at Perimeter Institute is supported by the Government of Canada through IndustryCanada and by the Province of Ontario through the Ministry of Research and Innovation.The author also acknowledge support from a JSPS Research Fellowship for Young Scientists.
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