#### Non-equilibrium physics at a holographic chiral phase transition

Nick Evans
0
2
Tigran Kalaydzhyan
1
Keun-young Kim
0
2
Ingo Kirsch
1
Open Access
0
Kavli Institute for Theoretical Physics China
, CAS,
Beijing 100190, China
1
DESY Hamburg, Theory Group, Notkestrasse 85, D-22607 Hamburg,
Germany
2
School of Physics & Astronomy, University of Southampton
, Southampton,
U.K
The D3/D7 system holographically describes an N =2 gauge theory which spontaneously breaks a chiral symmetry by the formation of a quark condensate in the presence of a magnetic field. At finite temperature it displays a first order phase transition. We study out of equilibrium dynamics associated with this transition by placing probe D7 branes in a geometry describing a boost-invariant expanding or contracting plasma. We use an adiabatic approximation to track the evolution of the quark condensate in a heated system and reproduce the phase structure expected from equilibrium dynamics. We then study solutions of the full partial differential equation that describes the evolution of out of equilibrium configurations to provide a complete description of the phase transition including describing aspects of bubble formation.
Contents
1 Introduction
2 Holographic descriptions
2.1 Chiral symmetry transition
2.2 A boost-invariant expanding plasma
3 Out of equilibrium description of the chiral phase transition
3.1 D7 flavour brane action
3.2 Naive equilibrium based approximation
3.3 Adiabatic dynamic D7 brane embeddings 3.4 Full PDE solutions
4 Dependence of the condensate on B
5 Discussion A Partial differential equation
1 Introduction
Thermal phase transitions are a crucial aspect of the evolution of the Universe after the Big
Bang and also in the physics of heavy ion collisions. We have traditionally lacked tools to
study these transitions though in strongly coupled systems such as QCD. The AdS/CFT
correspondence [13], which gives a weakly coupled string/gravity description of a class of
strongly coupled gauge theories, offers the chance to study similar transitions in detail.
In this paper we study a simple dual of a theory with gauge fields and quarks which
has (in the presence of a magnetic field) a first order chiral symmetry restoring phase
transition as it is heated. Previous analysis of this gauge theory (using probe branes
in an AdS-Schwarzschild black hole geometry) has explored the first order transition for
equilibrium (time independent) configurations. The heating or cooling of the system can
be studied thanks to the boost-invariant expanding or contracting plasma geometry of
Janik and Peschanski [4]. That geometry, which we will review below, has a moving black
hole horizon describing the changing temperature in the gauge theory. We will study
probe branes in this geometry to learn more about the first order phase transition out
of equilibrium.
The particular duality we concentrate on is the simplest example of holography with
fundamental quark fields [510]. We do not consider the specific degrees of freedom of the
theory too crucial - it is some strongly coupled gauge theory that displays a chiral phase
transition. We hope, in the spirit of AdS/QCD models [11, 12], that it reflects broad aspects
of many strongly coupled systems. The specific gauge theory is constructed from the D3/D7
system in type IIB string theory which we will describe further below. The theory is the
large Nc N = 4 U(Nc) gauge theory with a small number of quark hypermultiplets. We
will work in the quenched approximation [7] (appropriate when Nf Nc) which on the
gravity dual side corresponds to treating the D7 branes as probes in the metric generated
by the D3 branes. There is a U(1) chiral symmetry (a remnant of the SU(4) R-symmetry
of the N = 4 theory) which is broken when a quark condensate forms [9, 13, 14]. Several
mechanisms for triggering this condensation have been explored. The cleanest is when a
background magnetic field is introduced [1523] so we will use that mechanism here. The
reader might want to loosely view the B field as simply the introduction of a conformal
symmetry and supersymmetry breaking parameter that triggers the strong dynamics to
cause the symmetry breaking. Physically, magnetic fields may be strong during structure
formation in the early universe, in particular during the epoch of the QCD primordial phase
transition, and in non-central high-energy heavy ion collisions studied at RHIC. Running of
the coupling in the holographic dual also causes quark condensation as has been shown in
back-reacted dilaton flow geometries [9, 24] and models with a phenomenologically imposed
dilaton profile [25]. The quark condensate can be determined in these models and an
effective IR quark mass is generated. The theories display a massless pion-like Goldstone
field and a massive sigma field (since we are at large Nc it is stable) that is the effective
Higgs particle.
The equilibrium finite temperature behaviour of the theory with a magnetic field has
been studied in [1517, 22, 23]. Finite temperature can be included through the presence
of an AdS Schwarzschild black hole. At a critical temperature the D7 embedding flips from
a chiral symmetry breaking embedding away from the horizon to a symmetry preserving
embedding that enters the horizon. The transition is therefore also associated with
meson melting [2628] - for embeddings away from the horizon there are regular linearized
fluctuations describing the meson spectrum of the theory. For the embedding that enters
the horizon there are only in-falling quasi-normal modes describing unstable plasma
fluctuations. In terms of the quarks of the theory the high temperature phase is analogous
to the quark gluon plasma phase in QCD whilst the low energy phase is more akin to the
hadronic phase of QCD. It should be noted though that the gluonic degrees of freedom
deconfine at any finite temperature so the analogy is imperfect.
The crucial extra ingredient we shall add to this story in this paper is provided by
the boost-invariant expanding or contracting plasma geometry of Janik and Peschanski [4].
This geometry has a black hole whose horizon moves away from the boundary in time as
the N = 4 plasma it describes expands and cools. The time reversed solution at zero
viscosity describes a heating plasma and we will find it useful to discuss that scenario too
below. The geometry is a late time expansion (when the black hole is small) in powers of
inverse time. However, by controlling the strength of the magnetic field on the D7 probes,
felt by the quarks, we can arrange to place the chiral phase transition at any point in the
evolution so the expansion is sufficient to fully study the transition.
We will place a D7 brane1 into the expanding plasma geometry and determine the
partial differential equation (PDE) that describes the time dependence of its embedding.
A good first approximation to the transition behaviour is provided by the equilibrium
results with the temperature replaced by the temperature as a function of time from the
moving background. In fact we will see that that is an extremely good approximation
when talking about the slow or adiabatic heating or cooling of vacuum configurations at
temperatures even close to the phase transition. If the chiral symmetry breaking embedding
is heated though, the local minima in the effective potential associated with that embedding
is eventually lost and the configuration becomes an out of equilibrium configuration. The
equilibrium results can not describe the subsequent evolution. Similarly excited vacuum
states must be studied through the full PDE system.
We first turn to an approximation to the PDE. The solution can be power expanded in
inverse powers of time. This reduces the PDE to a system of ordinary differential equations
(ODEs) that are much easier to solve. This analysis was first done in [31] where the time
evolution of the high temperature phase of the pure N = 2 theory was studied. Using this
technique we solve the ODE system and find the moving D7 solution. This method assumes
that the initial vacuum state is exactly a maximum or minimum of the zero temperature
effective potential. Again the heating is essentially adiabatic in nature. The expansion
breaks down if the super-heated state ceases to be an extrema of the effective potential.
This method allows us to confirm the success of the equilibrium derived results although
in fact the expansion breaks down before the equilibrium results deviate from the full
PDE solutions.
The most interesting out of equilibrium questions lie beyond the adiabatic
approximation though. In a physical first order transition quantities such as the condensate do not
jump but the vacuum state instead performs a fast roll from one configuration to another.
The timing of that transition can and most likely will be spatially dependent i.e. bubbles
of the true vacuum will form and grow. We turn to solving the full PDEs to study these
phenomena (with care this can be done using in built PDE solvers in for example
Mathematica). For example we are able to watch the super-heated chiral symmetry breaking
phase roll to the symmetric phase. We can also simulate initial conditions with some extra
energy (which might for example come from thermal fluctuations) and see such
configurations transition away from the super-heated vacuum before the meta-stable vacuum has
disappeared in the effective potential. This allows us to confirm some elements of the
transition such as the length of time in which there is a mixed phase. We also are effectively
watching a very large bubble form.
Our main result then is to have developed numerical techniques that let us reproduce
the phase structure using the PDE solutions and to describe non-equilibrium physics that
is necessarily present in the first order transition.
In our final section we also analyze the ODE expansion approximation to the PDE
solutions for the D7 embedding to make clear the full dependence of the solutions on the
1By placing a fundamental string, which corresponds to a heavy quark, into the expanding plasma
geometry, the diffusion constant [29] and drag force [30] was computed.
magnetic field value. In that case the dependence is available analytically. We also show
the effects of the viscosity.
Holographic descriptions
In this section we review the N = 2 gauge theory with a magnetic field and its holographic
description. We discuss the theorys finite temperature chiral phase transition in an
equilibrium description. We then review how to study flavour physics in a nonequilibrium
set-up using holography.
Chiral symmetry transition
We will study the D3/D7 brane model at finite temperature and with a magnetic field [15
17, 22, 23]. The magnetic field, which causes chiral symmetry breaking, competes with the
temperature that prefers to restore chiral symmetry. Consequently there is a first order
phase transition.
The N = 4 SU(N ) gauge theory at finite temperature has a holographic description
in terms of an AdS5 black hole geometry which can be written as
ds2 = Rw22 (gtdt2 + gxd~x2) + Rw22 (d2 + 2d23 + dL2 + L2d2) ,
gx :=
SDBI = TD7
d8pdet(P [G]ab + 2Fab) ,
where P [G]ab is the pullback of the metric and Fab is the gauge field living on the D7 world
volume. We will use Fab to introduce a constant magnetic field [1517],
We embed the D7 brane assuming only dependence: L() at constant . The full DBI
action we will consider is then
Z Z Z Z
SDBI =
where 3 is a volume element on the 3-sphere and
L e Nf TD7(RB)4e3 1 wweH44
e
wH = 0.15
Le(e) me + ece2 , me = 2R2mBq , ce = hqqi (R(2 2B)3)3 , (2.8)
where me and ce are identified with the quark mass and the quark condensate, respectively.
Since we are interested in spontaneous symmetry breaking we impose me = 0 in the UV
(e ). Then the value of ce is determined by requiring regularity in the IR (e 1). If
there is more than one solution then we choose the one giving the minimum free energy.
The results are displayed in figure 1. At low temperatures T B, the black hole
is small and the embeddings are repelled from the origin of the Le e plane (figure 1
(left)). This behaviour is a result of the inverse powers of we4, when weH 1, in the last
term of the Lagrangian (2.6), which causes the action to grow if the D7 approaches the
origin. Consequently ce is non-zero and chiral symmetry is broken. If the temperature
is allowed to rise sufficiently then the black hole horizon grows to mask the area of the
plane in which the inverse we4 terms in the Lagrangian are large. At a critical value of T
(wH = 0.2516, figure 1 (middle)) the benefit to the me = 0 embedding of curving off the
axis becomes disfavoured and it instead lies along the axis - chiral symmetry breaking
switches off. This transition is first order since the condensate vanishes discontinuously,
which corresponds to the embedding change from the blue to the red one. At higher
temperatures (figure 1 (right)), the embedding stays flat, ec = 0, as expected in the chirally
symmetric phase. The corresponding condensate vs. temperature (c-wH ) plot is shown
in figure 1 (bottom).
A boost-invariant expanding plasma
The geometry (2.1) is dual to a system in thermodynamical equilibrium and therefore not
suitable for the description of the chiral phase transition in a rapidly expanding plasma,
in which the transition is basically a non-equilibrium process. Boost-invariant expanding
N = 4 SU(N ) plasmas out of equilibrium can however be described by the time-dependent
background found in [4] (see [32, 33] for a review). In the following we review the basic
features of this background and discuss the embedding of probe D7 branes dual to quenched
flavours (quarks) in the plasma.
The boost-invariant geometry is a 5D spacetime with coordinates {, y, x(= x1, x2), z},
which, apart from the holographic direction z, parameterise the 4D spacetime on the
boundary. The longitudinal position plane is parameterised by the proper time and rapidity
y (related to x0,3 as x0 = cosh y, x3 = sinh y), the transverse coordinates are collected
in x. We also add a five-sphere to obtain a full type IIB supergravity background. The
metric is then of the form
dRs22 = z12 ea(,z)d 2 + eb(,z) 2dy2 + ec(,z)dx2 + dzz22 + d52 , (2.9)
where R is the radius of the AdS5 space. At late times, the coefficients can be expanded
to first order as [4, 34, 35]
20 3 +v4v4 + 20 ln 33 + vv44
20 3 +v4v4 0 ln 33 + vv44
0 is a free parameter of mass dimension 8/3 and is related to the energy density, while
0 is related to the shear viscosity. v is a scaling parameter valid at large . Note that
a(, z), b(, z) and c(, z) are expanded around = in powers of 1/ 2/3 as [4, 34, 35]
and similarly for b(, z) and c(, z). The coefficients are functions of a scaling parameter
v only. Due to this scaling behaviour the complicated PDE Einstein equations can be
reduced to an ODE system, which allows analytic solutions. At early time 1 there is
no scaling behaviour and we should solve the full PDEs.
To find which gauge theory state corresponds to this bulk metric according to the
AdS/CFT dictionary let us expand the metric around z = 0,
The leading terms (of order O(z0)) of the metric elements (2.13) is simply the Minkowski
metric in the y coordinate system, and the subleading terms (of order O(z4)) correspond
to the expectation value of the energy-momentum tensor, i.e.
1
If we assume that the time-dependent entropy density s( ) has the same form as in the
static case, s( ) = (2/2)Nc2T ( )3, then the ratio /s can be computed as [35]
which at large agrees with the known static bound. Note that the numerical value of 0
in (2.11) is crucial to get 1/4.
hTi = 2Nc22 zlim0 z14 (g ) .
Finally we note the uniqueness of the gravity solution. There is a potential singularity
at v = 31/4 in (2.10). The curvature invariant, RR = R0(v) + R1(v) 2/3 +
R2(v) 3/4 + is only regular at each order if one makes the specific choices above.
We must choose 1/3 for the power of in (2.11) to make R0 regular [4]. We must also
choose (2.10) with the specific choice of the numerical value 0 as in (2.11) to make R2
regular [35]. (R1 is always regular and does not give any constraint.)
Flavours in this background were first studied in [31]. Knowing the explicit background
geometry (2.9) we can study flavour physics using the D7 brane DBI action. The action
schematically reads
S
1 + (L)2 B ((2+LL)22)2 ,
This expansion inherently assumes that the late time configuration is precisely the
equilibrium vacuum and when we heat it there is no excess energy. This leads to an adiabatic
approximation. The fraction 1/3 is chosen because all exponents of in A and B are integer
multiples of 1/3 and the constant m reflects the fact that the embeddings get flat at large
. Here we only consider a large bare quark mass (m 1) and Minkowski embeddings,
i.e. D7 configurations which end well above the black hole horizon.
This reduces the partial differential equation to an infinite set of ordinary differential
equations given by
3 3fi() = 9(m82m+202)5 Ii ,
I8 = 1 ,
otherwise Ii13 = 0 ,
We set mi = 0 so the bare quark mass is zero and not time-dependent. We also impose the
condition fi(0) = 0 for regularity. With these boundary conditions, we find fi = 0 except
for f8, f10. As a result, one gets
c( ) = 54m205 8/3 4001/4 10/3 + .
The condensate approaches zero as 8/3 and the viscosity has a dragging effect
+ 10/3.
We close this section with some technical remarks on black hole embeddings. In the
Fefferman-Graham (FG) coordinate system (2.9) one cannot approach the horizon for fixed
as in the Schwarzschild black hole metric. There is also an extended background written in
terms of Eddington-Finkelstein (EF) coordinates, where the spacetime is well defined across
the horizon [36, 37] and black hole embeddings may be described more consistently there.
However we have found that the embedding configuration is not easy to handle in those
coordinates. Below, to enable us to study black hole embeddings, we use FG coordinates
but with a cut-off slightly above the horizon. Since the FG coordinate system is a good
patch near the horizon at large , by restricting ourselves to large , we may capture the
essential physics of the embedding. This will allow us to go beyond the results in [31].
Out of equilibrium description of the chiral phase transition
In the previous section we reviewed the holographic description of dynamical flavours in
an expanding plasma [31]. A boost-invariant background with embedded D7 branes is
however not sufficient to study the chiral transition. In that case the D7 embeddings are
always flat in the chiral limit (m 0) and the system is always in the chiral symmetric
phase with vanishing quark condensate. In order to describe the transition to the chiral
broken phase, we need a repulsive effect to compete against the attractive force of the black
hole. As in the static case reviewed in section 2.1, this can be achieved by turning on a
magnetic field. In this section, we will therefore consider D7 branes with a world-volume
magnetic field in the dual geometry of an expanding plasma. In this way we will find the
holographic dual of a chiral transition and deduce the dynamic effective potential for the
time-dependent quark condensate.
D7 flavour brane action
The background metric for a boost-invariant expanding plasma can be written as
SDBI = TD7
d8pdet(P [G]ab + 2Fab) ,
where we turn on a constant magnetic field [1517],
in order to induce the chiral symmetry breaking.
More explicitly, the D7-brane action reads
1 + L2 B
A 1 v98 exp 4001/4 9 v8v8 2/3 ,
B 1 + vv434 2 exp 2001/4v4 9 + v4 2/3 ,
1 3 9 v8
9 v4
C (3 + v4)2 exp 4001/4 3 + v4 Coth1
N Nf TD7
dyd2x ,
Naive equilibrium based approximation
rH = 31/e401/e41/3 1 201/402/3 . (3.11)
If this heating were very slow (as it is at large ) we would expect to be able to plot the
e
quark condensate against e by simply substituting for the temperature T in the equilibrium
results. We show that plot in figure 2a (black solid curve) which follows directly from the
c-T plot in figure 1.
We can recast the c vs. e plot schematically as a time dependent effective potential by
e
fitting it at each value of e to a potential of the form
V = m2ec2 + ec4 + ce6 (3.12)
using the values of ec at the extrema to fix the parameters (the overall scale is not set but
the figure is intended to only be schematic). We plot this in figure 2b. This is the standard
picture of a first order transition.
It is important to interpret these plots correctly. Firstly note that there is a c = 0
e
solution for all e. If we begin at high temperature with the symmetric flat embedding and
cool slowly or adiabatically (i.e. move to large e) then we can remain in that embedding for
all e. Above the time a = 20.6 two extra solutions for the condensate develop and these,
e
as we will discuss, trigger the first order transition. At large the c = 0 flat embedding
e e
becomes a local maximum of the effective potential. To stay in the flat embedding for
all e is the extreme limit of super-cooling the high temperature phase into an unstable
vacuum state.
The top trajectory in the ec vs. e plot is best thought about in the time reversed
solution that is heating up. At early times (large e in the plot) the solution is the T = 0
symmetry breaking D7 embedding vacuum. Now as we heat adiabatically the condensate
tracks along the trajectory to smaller . The solution ceases to exist at a, where the cusp
e e
in the (black) curve is, indicating that the minimum of the effective potential corresponding
to this solution has ceased to exist at a. At this point the brane will move quickly, as
e
an out of equilibrium configuration, ending as the flat embedding or oscillating about
it. The real-time evolution of the configuration is represented by the path of red dots in
figure 2b. Starting from the global minimum with c 6= 0, the minimum is lifted up and
the configuration moves adiabatically. At ea the minimum disappears and the system rolls
down the potential to the true vacuum with c = 0. The equilibrium results can not tell us
about this motion.
Now we also see the role of the middle arc in the plot - these solutions are
symmetry breaking embeddings that end on the horizon and they are a local maximum of the
potential lying between the two minima (the embeddings of the top trajectory and the
flat embedding).
In fact the first order transition point between the symmetry preserving and the
symmetry breaking embeddings can be computed from the free energy in the equilibrium
computation. This transition occurs at ec = 27.5, where the local and the global minimum
in the effective potential interchange their roles, i.e. for c 27.5 the symmetry breaking
e
vacuum (with c 6= 0) becomes the global minimum. Another important time determined
by the equilibrium computation is eb = 34.9 where the mixed phase ends at large .
e
Our first task in solving the PDEs for the D7 brane motion that result from (3.4) is
to demonstrate that this description is essentially correct - we will see that it is. We will
try to find confirmation of the times eac of these events. The more interesting task is
that we will be able to follow the evolution of a particular initial condition through the
phase transition. Of course in the first order transition the brane configuration does not
discontinuously leap between the symmetry breaking embedding to the flat embedding but
evolves continuously. We will provide solutions for this evolution.
Another interesting phenomena associated with a first order transition is bubble
formation. In real systems thermal energy will lead to volumes of space which have more
energy than an equilibrium like state in a local minimum of the potential at any particular
time. These volumes may climb over the potential hill to the other local minimum during
the mixed phase period shown in figure 2. These bubbles then grow or contract triggering
the phase transition around ec ending any super-heated or cooled phase. We will not look
at (x3) spatial dependent brane embeddings. However, we can inject kinetic energy in the
holographic directions of our description into the brane configuration before we heat or
cool it to see the configurations moving more quickly between the two local minima than
the lowest energy configuration. This will allow us to test the time period in which the
mixed phase exists in the out of equilibrium problem.
Adiabatic dynamic D7 brane embeddings
Our first study of the PDEs describing the D7 embedding in the expanding plasma
geometry will be to study adiabatic expansion. The non-linear partial differential equations
resulting from (3.10) can again be transformed into a set of second-order ordinary
differential equations as we reviewed for the case with no symmetry breaking in section 2.2. For
that, we use the late-time expansion
Thus we will be following the evolution of the symmetry breaking embedding as decreases.
There is also the solution fi = 0 which corresponds to the symmetric embedding. Note
10
fi HL
20
10
0
-10
-20
-30
-40
-50
that, contrary to (2.20), a nontrivial asymptotic embedding f0(e) is assumed because of the
repelling effect of the magnetic field. The equations for every fi(e) can be obtained order
by order in 13 , which will be solved recursively. Unlike (2.21) these equations are quite
lengthy and will not be presented here. They do not allow for an analytic solution such
as (2.24) but can be numerically solved without any difficulty. As in the static case [9], there
are expected to be two types of brane solutions, Minkowski and black hole embeddings,
depending on whether the brane ends on or above the horizon.
For Minkowski embeddings, the boundary conditions are
fi(0) = 0
fi() = 0 ,
as before. With these boundary conditions, we find non-trivial profiles fi = fi(e) for
i = 0, 4, 8, 10, and fi = 0 otherwise. A general dynamical embedding function is therefore
of the form
Le(e, e) = f0(e) + f4(e4;/3e0) + f8(e8;/3e0) + f10(e1;0e/03, 0) , (3.15)
e e e
where we neglect terms with i > 11 since they are beyond the validity regime of our
approximation of the boost-invariant metric.3 The numerical plots of the non-trivial profiles
fi (i = 0, 4, 8, 10) are shown in figure 3.
These profiles have the following qualitative properties. As compared to the
solution (2.24) for B = 0, there are additional non-trivial profiles, f0 and f4. The profile f0
agrees with that in the static case (e ) [1517] (dashed curve in figure 3a) We found
that the equation for f0 is independent of e0 and 0, which is natural since at e the
system achieves equilibrium at low temperature and will not depend on the non-equilibrium
dynamics (0) or specific initial conditions (0) anymore. The profiles f4 and f8 depend on
e
e0 and have a negative sign, which reflects the attraction of the D7 brane to the black hole.
f10 is the first profile which depends on 0. In contrast to f4 and f8, f10 has a positive sign
showing the dragging effect, as in the zero B case around (2.25). The big amplitudes of
3In principle there may be a finite f12, but we will ignore it since it is a higher order term.
L@D
0.5
0.4
0.3
0.2
0.1
L@D
0.5
0.4
0.3
0.2
0.1
L@D
0.5
0.4
0.3
0.2
0.1
L@D
0.5
0.4
0.3
0.2
0.1
f4, f8, and f10 will be suppressed at large by negative powers of . For self-consistency
e e
we will only consider the region where all the sub-leading terms are well dominated by
e
the leading terms, i.e. f0 f4e3/4 . Some of the final embedding profiles Le(e, )
e
are shown in figure 4, where the green lines are plotted by plugging the numerical data of
figure 3 into (3.15).
We would not expect this expansion approach to work beyond the point where the
symmetry breaking embedding ceases to be even a local minimum because higher order
terms will grow. One can nevertheless plot solutions using f0 f10 that reach down all
the way to the horizon. We will use the full PDE solutions in the next section to test the
point where the expansion has broken down.
In fact early in our studies we tried to use the expansion even for black hole embeddings.
The ODEs for the fi do not contain the horizon however. We attempted to put the horizon
in by hand by imposing boundary conditions relevant to a black hole. For each point in
the L plane if we assume a horizon is present we can deduce the value of from (3.11)
- one can impose some boundary condition such as orthogonality to the posited horizon
and seek solutions for each fi. This correctly gives one massless solution at each e and
they look very similar to the equilibrium black hole embeddings. In fact though on solving
the full PDEs we realize that this is far beyond the point where the expansion method has
collapsed! We will include some of the resulting curves below though as evidence of the
break down of the ODE approximation.
Figure 4 shows the embedding profiles at various stages of the evolution of the
expanding plasma. The black hole horizon (3.11) is indicated by a black quarter circle in the figure.
Its size decreases with time corresponding to a cooling and expanding plasma. Figure 4
reflects the division of the quark-gluon plasma into the three phases described above.
The quark condensate as a function of time can be read form the asymptotic form of
Le(e, e) ec(e2) . (3.16)
e
We plot these results in figure 2a (the dashed lines) for comparison to the equilibrium
inspired results. The main result here is that the late time behaviour is indeed just the
equilibrium expectations. Where the curved line deviates from the equilibrium results is
in fact a sign that the expansion used in this section has broken down, the configuration
is out of equilibrium and adiabatic behaviour is no longer possible. Full solutions of the
PDEs will show this in the next section.
Full PDE solutions
In the previous section we interpreted our solution obtained from the ordinary differential
system as the evolution of extremum states in the potential. This evolution also reflects
the real time dynamics of the plasma whenever there is a unique deep global minimum for
the vacuum state, because the embedding time dynamics is expected to be well localized
around the minimum. This is the case in the chiral symmetric phase at early times and
in the broken phase at late times. In the mixed phase at intermediate times there appears
also a local minimum in addition to the global minimum. The small potential barrier
separating these minima is potentially easily overcome by a fluctuation. The path of the
local maximum seen in the adiabatic approach will not be realized as a dynamical solution
since it is unstable. Further below the critical time ea where the symmetry breaking
minimum disappears a heating vacuum will be left in a very non-equilibrium state that
again can not be followed using the expansion technique of section 3.3.
To find the real time evolution of the chiral transition out of equilibrium, we need to
solve the PDE directly and compare the solution with our previous approximate solutions.
In practice it is more convenient to consider a heating process than a cooling one4 because
we can use a well defined starting configuration at large as an initial condition of our
e
partial differential equation, i.e.
The conditions (3.17) and (3.18) completely determine the dynamics with the partial
differential equation derived by varying the action (3.10). Again, the actual expression of the
equation of motion is lengthy and will be shown in the appendix.
4In principle, time reversed heating is justified only at zero viscosity since a finite viscosity results in
decreasing entropy. However, we keep a finite viscosity in our numerics since it has a negligible effect and
the results are relevant to the cooling case.
Figure 5. The embedding profiles L(e, e) from the full PDE solution for a solution starting in the
low temperature symmetry breaking vacuum ( = 1).
Numerically we solve the equation using Mathematicas inbuilt PDE solvers. These
are somewhat temperamental and one needs to spend considerable time adjusting precision
tolerances in order to find smooth solutions in sensible periods of computer time. When
we have such solutions we test their stability to changes in precision settings to ensure they
are reliable.
With the boundary conditions above we can run our simulations until the embeddings
touch the black hole. Beyond that one needs dynamic boundary conditions along the black
hole surface. At least in the coordinates we use here this is a hard problem. We have found
a simple trick that seems to produce sensible black hole embeddings though. After the
D7 has touched the horizon at = 0 we artificially hold the embedding at the top of the
e
horizon. The large evolution of the D7 is local and relatively unaffected by this incorrect
e
embedding at small . Further, as has been observed before, solutions shooting from the
e
black hole horizon experience a numerical attraction onto the unique regular black hole
embedding ending at a given point on the horizon. The result is that we get numerical
solutions like those shown in figure 5 (e.g. the red, blue, or orange curve) where the D7
follows the horizon before shooting out to large e. We believe that these represent very
good approximations to the large embeddings solutions. Since we extract the condensate
e
ce at large we will live with the improper near horizon behaviour. It would of course be
e
interesting to try to improve on this with dynamic boundary conditions in the future.
5 In practice, we fix some emax and modify the initial conditions (3.17) as L(e emax, e) =
LODE(emax, e) and L (e emax, e) = L ODE(emax, e), where LODE is given by (3.15).
breaking one and with extra energy the configuration oscillates about that minimum. The
motion is simple harmonic as can be seen from the independence of the period on the
amplitude of the oscillations. As the solutions approach ec the different solutions begin
to diverge. The solution with = 10 lies close to the equilibrium curve - the oscillations
about the minimum are small and the state stays super-heated. When = 30, 40 Le(0)
falls more quickly suggesting that at least some of the branes length in has escaped
the local potential minimum. The upwards wiggles suggest that some of the length is still
repelled back into the well though. Finally though by = 50 the brane has ridden over
the potential barrier and escaped the local minimum. The difference in arrival times at
the horizon for these configurations (about 6 units of e) is a rough estimate of the period
of the mixed phase. It seems to broadly match the equilibrium inspired picture again.
In figure 8 we plot the motion of the IR = 0 end point of the brane with time for
a low energy configuration ( 1). The brane certainly seems localized in a minimum
down to a time of order = 27 (compare to the equilibrium estimate that the mixed phase
e
begins at eb = 34.9 and becomes meta-stable at ec = 27.5). Further the steepest period of
acceleration is below e = 22 (to be compared with the equilibrium estimate for when the
metastable vacuum ceases to exist a = 20.6). We conclude that this system is indeed a
e
super-heated state that survives in the local minimum until very close to the equilibrium
estimate for a. Note the other obvious feature in the plot is the deceleration just below
e
e = 20. This corresponds to where in our simulation the D7 first impacts on the black
hole - at smaller e we hold the D7 at the horizon as discussed above so this behaviour is
an artefact. A full solution would continue to accelerate along the horizon.
These configurations with excess L are also very much linked to bubble formation.
A bubble forms in the mixed phase when a volume of space has excess energy due to a
thermal fluctuation and escapes the local minima early. Here by treating the whole space
as one we are essentially describing the formation of a large homogeneous bubble. It would
be interesting in the future to try to study x dependent initial conditions to understand
how quickly or slowly bubbles grow.
Dependence of the condensate on B
In the previous section we focused on the chiral transition induced by a magnetic field,
which simply played the role of an intrinsic symmetry breaking scale. In this section we
turn to the physics depending on B more quantitatively.
To make analytic progress we will concentrate on the adiabatic (ODE) approach in
which we rescaled all variables by some power of B, see (3.9). In order to study the
dependence of the quark condensate on B, we use the original parameters, in terms of
which the condensate can be expanded as
Note that c0 is independent of 0, 0 and 0 enters only in c10. The B dependence of the
leading term agrees with that in the static (zero temperature) case [1517], i.e. it scales
as B3/2. The first subleading term ( 4/3) may be compared to the finite temperature
case [1517, 22, 23]. In the adiabatic approximation, where T 1/3, this term scales
approximately like T 4B5/6. However, due to the B dependence of c4, this scaling is
not exact. In general, the effect of the subleading terms is to lower the exponent in
c(, B) B to a value < 3/2. The scaling of the total condensate c(, B) with B will
again be determined numerically.
Figure 9a shows our results for the condensate for various values of the magnetic
field. We have only plotted the condensates associated with the Minkowski embeddings,
which exist for some time > (B). Here is defined as the time, when the Minkowski
solution meets the black hole solution (This corresponds to the cusp in the dashed curve
in figure 2a). We take to mark the time of the phase transition. As can be seen from
figure 2a the early time behaviour is an imperfect approximation to the full PDE solutions
but the later time solutions are consistent with the solutions. We find that for fixed , or
equivalently for fixed temperature, the condensate grows with increasing magnetic field. In
the limit (i.e. at zero temperature), we find the following dependence on B:
which is in agreement with the zero temperature result [1517]. For earlier times, the
dependence on B is shown in figure 9b. We find numerically that c(, B) B where the
power decreases from 1.5 at large to approximately 1.0 at small . In other words,
the dependence on B tends to become linear at high temperatures. Our results hold for
sufficiently strong magnetic fields. At small B the system is in the symmetric phase (c = 0).
The tendency for the condensate to increase with B is in qualitative agreement with
observations in chiral perturbation theory [38, 39] ( B3/2 for strong fields, B for weak
fields), in the Nambu-Jona-Lasinio model [40] ( B2), in a confining deformation of the
holographic Karch-Katz model [19] ( B2), and in SU(2) [41] ( B) and SU(3) [42] lattice
calculations ( B1.60.2). The dependence on B typically ranges from linear to quadratic
behaviour, i.e. the powers of B are in the range 1 2.
We may also study the effect of B on the time of the chiral transition which is marked
by . From the D-brane picture we expect that decreases with B. For large B, the
repelling force caused by the B-field is much stronger than the attractive force of the
black hole. Even at early times, Minkowski embeddings, associated to stable mesons,
are therefore favoured over black hole embeddings, which implies that the meson melting
process sets in at some earlier time, i.e. at higher temperatures.
Figure 10 shows as a function of B. We find that indeed decreases with
increasing B as (B) B1.55. The numerically found exponent 1.55 for the scaling of B is
close to 1.5 for the case of vanishing shear viscosity, which can be explained as follows.
c
0.22
0.21
0.20
0.19
0.18
At = e and = 0, the horizon is located at
e
where for the numerical analysis we chose 0 = 1. The deviation from 1.5 is due to
the effect of the shear viscosity. We also numerically confirmed that the exponent is 1.5
without viscosity.
This scaling of may be compared to results for the critical temperature Tc of the
phase transition in the static case. In the adiabatic approximation (when 0 = 0), T
1/3, which implies T B1/2. This square root behaviour is in agreement with the
result for the critical temperature Tc in the static approach [22, 23] and with [39] for
strong magnetic fields. It is also in qualitative agreement with studies in QCD [43, 44].
Finally we consider the effect of changing the viscosity. The viscosity effect is very
small, since it is doubly suppressed by both large and small 0. This can also be seen
e
in figure 11, where the condensate is plotted for four cases: the green is the leading term,
the blue includes the subleading term, the red includes up to the third term and the black
is the full expression. The viscosity effect is then the difference between the red and black
curves. There is a small visible difference only at small e.
We have analyzed the first order chiral and meson melting phase transition in a warming
or cooling strongly coupled gauge theory with quarks using the AdS/CFT Correspondence.
We have developed numerical techniques to study the PDE that describes the motion of
a D7 brane in a time dependent geometry. In particular this allows us to explicitly find
smooth solutions of the non-equilibrium configurations that are necessarily part of the
transition. These results confirm the equilibrium analysis of the transition but also go
beyond them. For example we have described the formation of large homogeneous bubbles
in the mixed phase of the transition. In the future it would be interesting to study spatially
inhomogeneous bubbles to understand their growth although this would require the solution
of a 2+1 dimensional PDE which is potentially more numerically tricky.
To keep our problem a simpler 1+1 dimensional PDE we also restricted motion of the
D7 brane to the holographic coordinates L and . There could also of course be fluctuations
in the holographic angular direction which we have not described. Such configurations
might be useful in the study, for example, of disordered chiral condensates [45]. We may
also consider the finite density case by turning on the time component of the U(1) gauge
field on the probe brane [22, 23, 46, 47].
We hope that the system we have studied can shed some light on first order transitions
in a range of strongly coupled gauge theories including perhaps QCD. The N = 4
background, however, does deconfine in the presence of an infinitesimal temperature, which is
not expected in simple QCD. We note though that there are several arguments in favour of
a possible splitting of the deconfinement and chiral transitions in QCD in the presence of a
strong magnetic field [43]. While the temperature of the chiral transition increases [43, 48],
the temperature of the deconfining transition either decreases with increasing magnetic
field [49] or increases but much slower than the chiral transition [43]. Both transitions
become first-order transitions and a new phase with broken chiral symmetry and
deconfinement appears for sufficiently strong magnetic fields. So, it is the region of the T -B
phase diagram with strong magnetic fields and above the deconfinement temperature in
which our model might be qualitatively compared with QCD.
Acknowledgments
The authors thank Romuald Janik, Johanna Erdmenger, Jonathan Shock, Michal Heller,
Volker Schomerus and Piotr Surowka for discussions. NE and KK are grateful for support
from STFC.
Partial differential equation
In this appendix we give an explicit expression for the partial differential equation which was
numerically solved in section 3.4. The equation of motion corresponding to the Lagrangian
L defined by (3.10) is given by
Here we denote the derivatives as L(, ) = L(, ), L (, ) = L(, ). After a
straightforward calculation, the functions Ki can be written as
K1 =
3w8 (3w4 1)
K2 =
9w8 (3w4 +1)
3w4 1
9w8 1 2/3 27w4 w4 1 4/3
20 15w4 + 218w8 coth1 3w4 1 !
4 3w4 +1 X+
20 15w4 + 218w8 coth1 3w4 1 !
9w8 1 2/3 27w12 +54w8 Y 4/3 +1 +27w4 Y 4/3 +1 +4
18w40 27w12 +9w8 3Y 4/3 +1 +3w4 Y 4/3 1 1
(13w4)2 X+
where we defined the functions w2, Y and X as
w2 = w2(, ) 2 +L2(, ) 2/3 ,
X exp
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