zero for odd dimensions, a reminiscent of vacuum correlator. Turning to the singularities of the correlator, we see that the first term has singularities at t = t0 = 1. The hypergeometric function in the second term can be expressed in terms of elementary functions, but this is not necessary for our purposes. Singularities appear when the argument (tt00−1)t−t becomes0,1or∞, corresponding tot=t0, t= 1ort0 = 1respectively. The singularity att=t0 is just the usual lightcone one. The singularities att = 1andt0 = 1are closely related to the starting time of the mirror. No other singularity is found in the correlator, consistent with the geometric optics expectation. This example provides a further nontrivial check for our approach.
We note that the relatively simple procedure leading to the two-point corre-lator is related to theSO(1,1)symmetry preserved by the ‘mirror’ trajectory t2 −z2 = 1. We expect that a method for solving the wave equation exists which is similar to that of Section 2, introducing variables which make the symmetry manifest. It will be interesting to see whether the two-point correla-tor following from such a method agrees with the result given by the general prescription, away from the singularities.
for large masses, or alternatively for large conformal dimension, the correlator is dominated by the contribution of the space-like geodesic joining the two points. Our result is that the pole structure of the correlator is determined by the geometric optics limit, i.e. by the trajectory of a light ray, is an analogous statement for the light-like case valid also for the massless case corresponding to small conformal dimension. We expect that our proposed study of correlators in thermalization geometries along the lines developed in this thesis will also allow for a further study of the role of geodesics for correlators in these geometries.
Moreover, our analysis provides essential technical tools for the study of two-point function in time-dependent geometries. These tools may now serve as a starting point for studying the behaviour of two-point functions during thermalization in strongly coupled field theories. A possible generalization of our results is to apply our prescription to the boost-invariant background of [166]. The idea is again to match the bulk WKB solution to a near-boundary solution, which is expressed as a superposition of the eigenmodes in pure AdSbackground. Due to the additional proper timeτ appearing in the boost invariant metric, both the WKB solution and the eigenmode set consistent with scaling symmetry will take a different form from the one discussed in this thesis. This will give rise to a proper time dependent correlator with rapidity and transverse space coordinates integrated out, instead of a time dependent spatially integrated correlator. A direct application to [166] with infalling boundary conditions of the wave imposed on the horizon will allow us to obtain the quasi-normal modes beyond the adiabatic approximation used in [181].
Conclusions and outlook
We have presented some applications of the gauge/gravity duality towards studying field theory systems at strong coupling. The systems studied in this thesis might be used as models for describing condensed matter physics near the quantum critical point and the physics of quark-gluon plasma which has been recently created at RHIC. Moreover, we follow the line of considering different gravity setups whose dual descriptions show interesting phenomena in thermal equilibrium, slightly out-of-equilibrium and far-from-equilibrium.
In particular, starting with looking at phenomena of strongly coupled sys-tems in thermal equilibrium, in chapter 3 we study quantum phase transitions in holographic superfluidity at finite baryon density and isospin density in an imbalanced mixture. We apply two approaches, namely a bottom-up approach in section 3.3 using anU(2)Einstein-Yang-Mills theory with back-reaction and a top-down approach in section 3.4 using a D3/D7model setup. The results obtained in chapter 3 are discussed in detail in section 3.4.4. Here, we summa-rize some of the main results. Studying the phase diagrams, in both approaches we observe quantum phase transitions from a normal to a superfluid phase and indications for the existence of quantum critical points. The quantum phase transition, however, is different in the two systems. In the D3/D7brane setup we always find a second order phase transition. In the Einstein-Yang-Mills theory, depending on the strength of the back-reaction, we obtain a continuous or first order transition. While in the D3/D7model setup, the phase diagram and the position of the QCP can be determined only via a numerical method, in the Einstein-Yang-Mills setup we can relate the position of the QCP with the instability due to the violation of the Breitenlohner-Freedman bound and found a critical ratio of the baryon and isospin chemical
µB
µI
c
=
p1−3α2YM
√3αMW
. (6.1)
at the QCP. In the semi-probe limit, where only the back-reaction of theU(1)
Maxwell field is taken into account, i. e.αYM = 0andαMW6= 0, we are able to solve the equations of motion analytically at small baryon chemical potential and reproduce the phase diagram in this regime, see fig. 3.7. Comparing our results with those obtained from QCD in [49, 50], we find many interesting similarities, but also differences concerning the order of phase transitions whose origin has not been resolved completely.
Taking a first step away from studying dual field theories in thermal equi-librium, we use another D3/D7model setup to describe properties of flavor transport in aN = 4SYM plasma. Motivated by the work in [51] where an uni-versal non-linear behavior near the quantum critical point has been found, such as in conductivity. In chapter 4 we use a holographic method introduced [52]
to compute the conductivity tensor of flavor fields beyond linear response.
Thereby, we generalize the results in [53, 54] by including an additional compo-nent of the magnetic field, just extending the case ofperpendicularorientation between the magnetic and electric field studied in [53, 54] to the case of arbi-traryorientations. We calculate the energy and momentum loss rates of the flavor fields to the SYM plasma. The results obtained in this part of the thesis have been summarized in section 4.4. As the main results in this chapter, we compute the full conductivity tensor with the new transport coefficient σxz which could not be obtained with the setups in [52, 53]. Computing the energy and momentum loss rates as in [54], we find a covariant form of a current with anE~ ·B~ anomaly
I2µ=hTµνivν2 ∝ hTµνiναβγFαβhJγi, (6.2) whose corresponding kinetic coefficient is of relevance to a new transport coefficient in hydrodynamics which has been only recently discovered in the context of gauge/gravity duality [156].
Following the line of using gauge/gravity duality for studying phenom-ena of strongly coupled systems in a thermal equilibrium towards an out-of-equilibrium state, in chapter 5, we derive a prescription for computing two-point correlation functions of scalar fields in a time-dependent background geometry which is modeled by a mirror moving in the bulk ofAdSspace. By studying this model we want to make a further step towards a holographic description of thermalization. The reason for that is that thermalization corresponds to time-dependent backgrounds which evolve to form black hole horizon giving notion of temperature. For mirror trajectories preserving the scaling symmetry of the AdSspace we obtain time-dependent two-point correlators whose singularity structure is related to the physics of bouncing light ray between the moving mirror and the AdSboundary. Such a relation is known so far only to the time-independent case, i. e. static mirror [69]. For arbitrary trajectories of the mirror along the radial direction of theAdSspace, we use a WKB analysis and obtain a general prescription for calculating the two-point correlator (5.52).
Outlook Modeling the process of thermalization of strongly coupled medium like quark-gluon plasma is extremely complicated, even using gauge/gravity methods, because early stages of the thermalization require the understanding far-from-equilibrium physics. Another reason is the time-dependence of the process. Our toy model of mirror moving in the bulk ofAdSis very far away from a realistic model for holographic thermalization. One big disadvantage of our model is the absence of any notation for the temperature. This is because we work inAdSspace and not, for example, inAdSblack hole geometries.
Our analysis, however, provides essential technical tools for the study of two-point functions in time-dependent geometries. These tools may serve as a starting point for studying the behavior of two-point functions in a more complex time-dependent backgrounds which are more suitable to describe the process of thermalization in strongly coupled field theories. For instance, the two-point function obtained for a moving mirror in AdS space can be generalized to the case of a gravitational collapsing geometry [40, 41]. The time-dependent geometry is then described by a collapsing spherical shell where inside the shell the geometry isAdSand outside the shell the geometry is AdSblack hole. Using the Israel junction conditions [182], the matching of the coordinates inside and outside the shell gives the equation of motion of the collapsing shell. A direct generalization of our work in chapter 5 would be replacement of the mirror trajectory by a collapsing shell trajectory. This will involve solving the wave equation in a thermalAdSbackground outside the collapsing shell, and the Dirichlet boundary condition used here has to be replaced by a matching condition of a scalar outside and inside the shell.
Many other possible direct extensions of our work have been mentioned earlier in the concluding sections called ‘Summary and outlook’ at the end of each chapter, e. g. in sections 3.4.4, 4.4 and 5.5. Here, we would like to mention one further possible extension of the work which is relevant to both subjects presented in chapter 3 and 4.
One of the motivation for studying the conductivity of flavor fields in a SYM plasma as presented in chapter 4 comes from the results of the work in [51], where it was shown that scaling arguments lead to universal non-linearities in transport such as in conductivity if the system is near the quantum critical point.
In chapter 4 we study the transport of flavor fields moving in aN = 4SYM plasma, which is dual a D3/D7brane setup described in section 4.2.2. This setup does not possess a quantum critical point in the phase diagram. In chapter 3, however, we find two gravity setups which show continuous quantum phase transitions which indicate the existence of a quantum critical point in these systems. Thus the next logical step is studying conductivity with the U(2) Einstein-Yang-Mills theory in section 3.3 or with the D3/D7model setup in section 3.4. In both setups, there are already notations for charge densities.
For studying conductivity we need to turn on additional spatial components of the Maxwell fields which give rise to the electromagnetic fields. Analytical
results for the conductivity in theU(2)Einstein-Yang-Mills setup seem quite promising, since the position of the quantum critical point can be determined analytically.
First of all, I want to express my gratitude to Johanna Erdmenger for her tremendous efforts in supervising my work during the last three years. I am very thankful for her patience, constant willingness to help, intensive care, great support and encouragement. Also I have benefited a lot from her ability for creating a very stimulating and solidary atmosphere in our work group.
Furthermore, I would like to thank Dieter L¨ust for providing excellent working conditions in his groups at the Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut) and at the Ludwig- Maximilians- Universit¨at in Munich. I also thank him for working through this thesis as an interested second referee.
Moreover, I would like to thank my external supervisor Michael Haack for his constant interest on the progress of my work. I would like to thank Rosita Jurgeleit and Monika Goldammer for their administrative assistance and I want to thank Thomas Hahn for his great computer support.
I thank Martin Ammon, Johanna Erdmenger, Viviane Grass, Patrick Kerner, Shu Lin and Andy O’Bannon for the fruitful collaboration.
I would like to thank the International Max Planck Research School for providing various seminars and promoting interchanges among the Ph. D.
students. In particular, I would like to thank Frank Steffen und Otmar Biebel for their great efforts in organizing many interesting colloquia at many pleasant places.
I also thank my fellows Saeid Aminian, Martin Ammon, Veselin Filev, Jan Germer, Viviane Grass, Constantin Greubel, Sebastian Halter, Daniel H¨artl, Johannes Held, Stephan H¨ohne, Matthias Kaminski, Patrick Kerner, Phillip Kostka, Shu Lin, Ren´e Meyer, Steffen M¨uller, Dao Thi Nhung, Le Duc Ninh, Andy O’Bannon, Felix Rust, Jonathan Schock, Katja Seidel, Martin Spinrath, Migael Strydom and Hansj¨org Zeller for sharing many pleasant moments during the last few years.
I am indebted to Johanna Erdmenger, Jan Germer, Constantin Greubel, Patrick Kerner, Migael Strydom and Hansj¨org Zeller for many comments on the manuscript.
Finally, I want to thank Na, Sophie, my parents and family for their constant care, support and encouragement at any time.
Flavor transport
A.1 Derivatives of the on-shell action
In this appendix we write explicit expressions for derivatives of the on-shell action with respect to various fields, as mentioned in section 4.3.
For notational simplicity, we first define a function
d(u) = guu|gtt|gxx3 −gxxA2−A4, (A.1) where A2 and A4 were defined in eqs. (4.17), which we repeat here for completeness:
A2 =guugxxE˜2+gttguu( ˜Bx2+ ˜Bz2) +gxx2 A˜0t2 +gttgxx
A˜0x2+ ˜A0y2+ ˜A0z2 , A4 =gxxE˜2
A˜0y2+ ˜A0z2
+gxxA˜0t2
B˜2x+ ˜Bz2
+guuE˜2B˜x2+gttB˜z2A˜0z2 +gttB˜x2A˜0x2+ 2gttB˜xB˜zA˜0xA˜0z−2gxxE˜B˜zA˜0tA˜0y.
(A.2) Recall from section 4.2 that in our notationguurepresents theuucomponent of the induced D7-brane metric: guu = u12 +θ0(u)2.
The derivatives ∂F∂L
µν, evaluated on our solution, are
∂L
∂Ex =N cos3θ pd(u)
hgxxB˜zA˜0tA˜0y−E˜ guu
g2xx+ ˜Bx2 +gxx
A˜0y2+ ˜A0z2i ,
∂L
∂Ey
=N cos3θ pd(u)
hE˜A˜0xA˜0y + ˜BxA˜0tA˜0z−B˜zA˜0tA˜0x gxx
i,
∂L
∂Ez
=N cos3θ pd(u)
hgxxE˜A˜0xA˜0z−gxxB˜xA˜0tA˜0y −guuE˜B˜xB˜zi ,
∂L
∂Bx
=N cos3θ pd(u)
hB˜x
guu|gtt|gxx+|gtt|A˜0x2−gxxA˜0t2−guuE˜2 +|gtt|B˜zA˜0xA˜0zi
,
∂L
∂By
=N cos3θ pd(u)
h|gtt|B˜xA˜0xA˜0y +|gtt|B˜zA˜0yA˜0z−gxxE˜A˜0tA˜0zi ,
∂L
∂Bz
=N cos3θ pd(u)
hB˜z
guu|gtt|gxx+|gtt|A˜0z2−gxxA˜0t2 +|gtt|B˜xA˜0xA˜0z+gxxE˜A˜0tA˜0yi
.
(A.3)
The variations with respect to the∂µθ(withµ=t, x, y, z), evaluated on our solution, are
δL
δ∂tθ = −Ncos3θ pd(u)
hB˜xB˜zA˜0z + ˜A0x
gxx2 + ˜Bx2i E θ˜ 0, δL
δ∂xθ = −Ncos3θ pd(u)
h|gtt|gxxB˜zA˜0y−E˜A˜0t
gxx2 + ˜Bx2i θ0, δL
δ∂yθ = −Ncos3θ pd(u)
hB˜xA˜0z
|gtt|gxx −E˜2
− |gtt|gxxB˜zA˜0xi θ0, δL
δ∂zθ = +N cos3θ pd(u)
hE˜B˜zA˜0t+ ˜A0y
|gtt|gxx−E˜2i B˜xθ0.
Toy model for holographic thermalization
B.1 The UV limit of (5.25)
The core of the correlator (5.25) is the following integral 1
2πi Z
dλ Γ(λ+ 1)A
Γ(λ+1−d2 )+ Γ(1−λ)B Γ(1−d2 −λ)
! Γ(λ+ 1)A
Γ(λ+1+d2 ) + Γ(1−λ)B Γ(1+d2 −λ)
!−1
t t0
λ
. (B.1) For definiteness, from (5.12) we take
A = (u0+ 1)λF(1−d
2 ,1 +d
2 ;−λ+ 1;1−u0
2 ), (B.2)
B =−(u0−1)λF(1−d
2 ,1 +d
2 ;λ+ 1;1−u0 2 ).
First of all, we note that the integrand (denoted asF(λ, t, t0)) has the property:
F(λ∗, t, t0) =F∗(λ, t, t0). Combined with the fact that the integration path is the imaginary axis, we see the integral is manifestly real, which is consistent with the reality condition of the correlator. We will use the residue theorem to evaluate (B.1). The UV part of the correlator is given by the contribution from poles with large|λ|. For this, we do an asymptotic expansion of the integrand.
The following properties of Γ-functions and Hypergeometric functions are useful [183]:
Γ(λ+α)
Γ(λ+β) ∼λα−β; Γ(−λ+α)
Γ(−λ+β) ∼ sinπ(λ−β) sinπ(λ−α)λα−β;
γ→∞lim F(α, β;γ;z) = 1.
(B.3)
A branch cut at the negative real axis is needed, with the argument ofλfixed by|arg z|< π. Using the above asymptotic behavior, we obtain the integrand
∼λd(tt0)λasλ→ ∞. Ift > t0, we close the contour counter-clockwise and the integral receives contribution from poles in the left half complex plane, while ift < t0, we close the contour clockwise then the integral receives contribution from poles in the right half complex plane.
The possible poles in the whole complex plane are poles of the Gamma function, Hypergeometric function and roots of
Γ(λ+ 1)
Γ(λ+ 1+d2 )A+ Γ(−λ+ 1)
Γ(−λ+1+d2 )B = 0. (B.4)
Note thatF(α, β;γ, z)as a function ofγ has the same singularities asΓ(γ) [183], we can show that all the poles of the Gamma function and Hypergeo-metric function are removable. Thus we are only left with roots of (B.4). Due to the non-algebraic nature of (B.4), finding analytic expression of all the roots is not possible. However, we can deduce a general property of the roots: (B.4) can be equivalently written as
Γ(λ+ 1) Γ(λ+1+d2 )
u0+ 1 u0−1
λ2
F(1−d
2 ,1 +d
2 ;−λ+ 1;1−u0
2 ) = Γ(−λ+ 1)
Γ(−λ+1+d2 )
u0+ 1 u0−1
−λ2
F(1−d
2 ,1 +d
2 ;λ+ 1;1−u0
2 ), orR(λ) =R(−λ)with
R(λ) = Γ(λ+ 1) Γ(λ+1+d2 )
u0+ 1 u0−1
λ2
F(1−d
2 ,1 +d
2 ;−λ+ 1;1−u0
2 ). (B.5) We note thatR(λ∗) = R∗(λ). It is easy to show that if λis a root of (B.4),
−λ, λ∗, −λ∗ are also roots. We plot the left hand side of (B.4) in the complex λplane, and find the zeros lie nearly equally spaced on the imaginary axis.
Therefore, we conclude that the roots must be purely imaginary. Now let us determine the asymptotic form of the roots. In the limitλ → ∞(Λ → ∞), (B.4) has the following asymptotic expression
λ1−d2
(u0+ 1)λ−(u0 −1)λe±iπd−12
, (B.6)
and the root is given by λ=±iπ
d−1 2 + 2k
lnuu00+1−1 , (B.7)
with integerk ≥0. Our approximate roots indeed are consistent with numer-ical plots in the sense that they are symmetric with respect to the real axis and equally spaced. Furthermore we expect (B.7) to be more accurate when lnuu00+1−1 → 0, i.e. u0 → ∞. As the mirror moves more and more slowly, essentially all modes are effectively UV.
The poles lie precisely along the integration contour of λ. In order to obtain a well defined result, we have to deform the contour to circumvent the poles. The ambiguity associated with the detour corresponds to the different causal natures of the resulting correlator1. In practice, it is easy to calculate the advanced correlator, for which we shift the integration ofλslightly to the left.
Then all the UV poles lie to the right of the contour. The integration contour has to be closed counter-clockwise, which requires t0 > t. In this way, we can avoid the branch cut on the negative real axis. The residue at each root is obtained with asymptotic expressions as
Γ(λ+1)A
Γ(λ+1−2d)+Γ(1−λ)B Γ(1−2d−λ)
! d dλ
Γ(λ+1)A
Γ(λ+1+2d)+Γ(1−λ)B Γ(1+2d−λ)
t t0
λ
→λd t
t0 λ
1−e∓iπd lnuu0+1
0−1
. (B.8) We are happy to see the emergence of the factor(1−e∓iπd), which will precisely cancel the pole from Γ(−d)
Γ(1−d2 ). Denotea= lnuu00+1−1 andb = lntt0. The correlator is given by the sum of residues
h Z
dd−1xO(t, x)O(t0,0)iA=−dθ(t0−t)
∞
X
k=0
2dλdebλ1−e∓iπd a(tt0)d+12
Γ(−d)Γ(1+d2 ) Γ(1−d2 )Γ(d)
(B.9) evaluated at λ = ±iπd−12 a+2k. The subscript ‘A’ stands for the advanced correlator. We find that the sum overkcan be expressed in terms of the Lerch transcendent functionΦ(z, s, α)
h Z
dd−1xO(t, x)O(t0,0)iA=−dX
+,−
θ(t0 −t)
(tt0)d+12 e±i(d−1)πb2a (±2iπ
a )d1−e∓iπd a
× Γ(−d)Γ(1+d2 )
Γ(1−d2 )Γ(d) 2dΦ(e±2iπba ,−d,d−1 4 ).
1The ambiguity is familiar in the standard calculation of the vacuum correlator. It can be fixed by a prescription of the integration contour of the frequency
Asd→integer, this reduces to h
Z
dd−1xO(t, x)O(t0,0)iA=−dX
+,−
θ(t0−t)(tt0)−d−12 2
a d
1−e∓iπd a
× Γ(−d)Γ(1+d2 )
Γ(1−d2 )Γ(d) e±iπ(d−1)2 (ba−c)
×
d!
(−c)d+1(±i2π)−
∞
X
r=0
Bd+r+1(d−41)cr(±i2π)d+r r!(d+r+ 1)
, (B.10) where theBn(x)are the Bernoulli polynomials [184]. The constantcis defined as ei2πc = ei2πba with|c| < 12. To obtain the retarded correlator, we note a useful property of the integrand: F(λ, t, t0) =F(−λ, t0, t). This leads to the following relation between retarded and advanced correlators,
h Z
dd−1xO(t, x)O(t0,0)iA=h Z
dd−1xO(t0, x)O(t,0)iR. (B.11)