Paper
0
Kibble-Zurek Scaling in Holographic Quantum Quench : Backreaction
Authors
Sumit R. Das,Takeshi Morita
Keltie McDonald,Julio Torales,daniel dias,Francisco Segovia,Alain Hartmann,Roger Urgeles,Loriano Bonora,Elham Afify,Ferdinando Gliozzi,Chris Quigg,Brett Hambly,Daniela Giuliani,Xiaolei LI,Arcadi Santamaria,Andi Hektor,Vladimir Smirnov,Jorge Guillermo Russo,Ugo Gastaldi,Jacek Kondratowicz,Henryk Okarma,Ana Crespo Blanc,Gregor Fussmann,Jonathan Chase,Renzo Mancuso,Erlend B. Nilsen,Joachim Dengler,David Meeker,michail kuptsov,Joachim Kaiser,Srinivasan Jagannathan,Keisuke Yano,Szu-Yu Lee,Marisol Josefina Sarmiento Alvarado,Mahendra Ramachandran,Peter Wolfsteiner,Garrett Poe,Jesse Thaler,Wei Li,Alain Jaccard,Carolyn Dzierba,Francesco Cilurzo,Erdal Can Alkoclar,Marc Torrens,Andrew Degnan,Andy Van Brocklin,Jing Shu,Joseph Fotsing,David Van Wie,Seok Ki Choi,Christoph Luhn,Vishnu Ram OV,Jose Endrino,Cemile Bardak,Urfat Nuriyev,Gideon Wolfaardt,Christopher Herzog,Martin Köhne,Dan Andersen,Peter Kotanen,Víctor Manuel Palacios Macías,Maciej Majewski,Dmitry Golubkov,Gianluigi Casse,Valerio Bocci,Jens Rolff,Aysel Kayis Topaksu,JESUS CRUZ VILLALON,Jacopo Pinzino,Giovanni Francesco TUZZOLINO,Elisabeth Slooten,Alexander Schenkel,Kenny Erleben,Marius de Leeuw,Enrico Herrmann,Eran Palti,Nathaniel Butlin,Roberto Franceschini,Alison Murray,Mohamed Anber,Aninda Sinha,Irene Valenzuela,Ingo Jordan,Engel Roza,Pablo Gonzalez Camara,Justyna Zwolak,Aleksi Vuorinen,Claudia Hagedorn,Jose J. Fernandez-Melgarejo,Predrag Dominis Prester,Sebastian Sapeta,Nico Callewaert,Iosif Bena,Felisa Smith,Stephane Coulombe,Harri Koivusalo,Antonio Marti,Jorge David Fernández Gómez,Brian Cousens,Matthias Gaberdiel,Christos Hatzis,Carlos Lehn,Brian Palik,Miguel Fiolhais,Vadim Kostyukhin,Timothy Hobbs,Gert Aarts,Jon-Ivar Skullerud,Guy Moore,Arantza Oyanguren,Jeremy Dalseno,Huaiyu Duan,Jui-Yu Chiu,Cheng-Wei Chiang,Walter Tangarife,Rukmani Mohanta,Mihoko Nojiri ,Jose L. F. Barbon,N/A ,Antonio Pich,Luc Ver Donck,Peter Kamp,Lelis Noelia Morales Clemotte,Pedro Pérez,Kristan Jensen,Gabriele Travaglini,Andreas Brandhuber,Yuji Tachikawa,Luca Merlo,Claude Duhr,Berenice Santos Goncalves,Peter Stoffer,Leszek Szewczyk,Paolo Cignoni,Richard Conant,Cristian Ravariu,Daniel Nogradi,Charles Canham,Alicja-Joanna Siegień-Matyjewicz,Mohamed Gouighri,Nathan Berkovits,Thomas Doherty,Paul Bell,Alessandro Mapelli,Brian Walsh,John Blair,Nuno Anjos,Gavin Salam,Oktay Doğangün,Bruno Galhardo,Reina Coromoto Camacho Toro,Ken Hawick,Gerhard Kresbach,Bianca Letizia Cerchiai,Vitaly Velizhanin,Tomasz Lukowski,Jacobo Lopez Pavon,Laura Andrianopoli,Antonia Raya Tena,Juan Herrero Garcia,David G. Cerdeño,Veronica Sanz,S. Prem Kumar,Josser Delgado Almandoz,James Fordyce,Melissa Lucash,Keun-Hyeok Yang,Jes Søe Pedersen,Tim Payn,Amalia Virzo De Santo,Martin Hirsch,Julie M Grossman,Osvaldo Aquines,Hisashi TANAKA,Hyunchul Nha,David Kastor,Antônio Neto,Diego Rodriguez-Gomez,enrico calloni,Milka Brdar-Jokanovic,Sinead Ryan,Syksy Räsänen,Máximo Pló Casasús,Mónica González,Elena Graverini,Arundhati Nag,Tatsuma Nishioka,Priyotosh Bandyoapdhyay,Timm Wrase,Oliver Kutz,Daniele Oriti,Miguel-Angel Monjas,Andrei Tsaregorodtsev,Francesca CASTAGNETO,Jennifer Balmer,Giuseppe Dibitetto,Stewart Martin-Haugh,Antón Faedo,Sophie de Buyl,Daniel Mayerson,Matthew Weel,Marco Serone,Raúl Carballo-Rubio,Florian Staub,Carlos Hoyos Badajoz,Stefano Borgo,Cristina Ribeiro,Norbert Ostermann,David Fenard,Audrey Mayer,Jeffrey Conner,Luis J. Garay,Sonia Front,Amanda Kennedy,Kari Enqvist,Marcin Kucharczyk,Michael Alexander,Simon Lang,Nejc Košnik,Kerry Gibson,ala ,Simon Ross,Maria Fiori,Rossella BRUNETTI,Enrique Fernandez-Martinez,Dhavalkumar Thakker,Wolfgang Dick,Alessandro Feliciello,Stephen Grant,Yusuf Kenan Coban,Horacio Paz,Xavier LE ROUX,Fernando Rodrigues,Celestina Satriano,Cibrán Santamarina Ríos,Abraham Antonio Gallas Torreira,Tamara Štemberga,Rhorry Gauld,Craig Osenberg,Tristan McLoughlin,Maro Cvitan,Martino Borsato,Marek Walesiak,Colin Braithwaite,Fung Lay,Alex Buchel,Pedro Ruiz-Femenia,Igor Samsonov,Jesper Lykke Jacobsen,Dmytro Volin,Masaki Shigemori,Michael Long,Ricardo Graciani Diaz,Ilja Dorsner,Valentin Reys,Vittorio FIORE,Andrei Onishchenko,Matteo Fael,Jeannine Cavender-Bares,Jakub Kvapil,Lillian Smestad,Suk-Yoon Kwon,Andrew O'Bannon,Miguel S Costa,Serguei Golovan,Matti Heikinheimo,Johannes Albrecht,Suzanne Klaver,Vladimir Gligorov,Agnieszka Dziurda,Claire ,Heinrich Schindler,David Dossett,Joseph Cook,Dario Benedetti,Diego Martinez Santos,Jeong-Hyuck Park,Samuel Coquereau,Eric Cogneras,Matthew Vadeboncoeur,Eluned Smith,Robert Driver,Pasquale Di Nezza,Alexandru Catalin Ene,Massimiliano Fiorini,Miriam Calvo Gomez,Abhishek Chowdhury,Andreas Papaefstathiou,Rosen Matev,Emilie Bertholet,Yanxi Zhang,Sameer Murthy,Alexander Vagner,Jeroen van Tilburg,Bartosz Małecki,Giampiero Mancinelli,Irina Nasteva,Paula Alvarez Cartelle,Antonio Augusto Alves Junior,Lorenzo Ubaldi,Conor Fitzpatrick,Andrea Contu,David Chamont,Daniel Craik,Pablo Vázquez Regueiro,Paul Seyfert,Kazuyoshi Carvalho Akiba,KÜBRA ONERGE,Kirovné Rácz Réka,Luca Federici,Andreas Starmans,Sherry Bumpus,James Mead,Eduardo Picatoste,Michał Dziewiecki,Fedor Baryshnikov,Carla Marin Benito,Alejandro Alfonso,Ángeles Solanes-Corella,Méril Reboud,Christophe Martin,Melissa Maria Cruz Torres,Maxime Schubiger,Tomohiro Abe,Lorraine Brennan,Michele Caselle,Emma Finch,Roldao da Rocha,Eva Millesi,Dawn Sumner,Roy Lemmon,Sambit Kumar Dash,Valerio Matozzo,Taushif Ahmed,Taofeek Abiodun Otunla,Juan F. 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arXiv:1409.7361v3 [hep-th] 20 Jan 2015
UK/14-05
Kibble-Zurek Scaling in Holographic Quantum
Quench : Backreaction
Sumit R. Das (a,b)1 and Takeshi Morita (a,c)2
(a)Department of Physics and Astronomy,
University of Kentucky, Lexington, KY 40506, USA
(b) Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, JAPAN
(c) Department of Physics,
Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, JAPAN
Abstract
We study gauge and gravity backreaction in a holographic model of quan-
tum quench across a superfluid critical transition. The model involves a com-
plex scalar field coupled to a gauge and gravity field in the bulk. In earlier work
(arXiv:1211.7076) the scalar field had a strong self-coupling, in which case the back-
reaction on both the metric and the gauge field can be ignored. In this approxi-
mation, it was shown that when a time dependent source for the order parameter
drives the system across the critical point at a rate slow compared to the initial gap,
the dynamics in the critical region is dominated by a zero mode of the bulk scalar,
leading to a Kibble-Zurek type scaling function. We show that this mechanism for
emergence of scaling behavior continues to hold without any self-coupling in the
presence of backreaction of gauge field and gravity. Even though there are no zero
modes for the metric and the gauge field, the scalar dynamics induces adiabaticity
breakdown leading to scaling. This yields scaling behavior for the time dependence
of the charge density and energy momentum tensor.
1e-mail:das@pa.uky.edu
2e-mail:morita.takeshi@shizuoka.ac.jp
UK/14-05
Kibble-Zurek Scaling in Holographic Quantum
Quench : Backreaction
Sumit R. Das (a,b)1 and Takeshi Morita (a,c)2
(a)Department of Physics and Astronomy,
University of Kentucky, Lexington, KY 40506, USA
(b) Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, JAPAN
(c) Department of Physics,
Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka 422-8529, JAPAN
Abstract
We study gauge and gravity backreaction in a holographic model of quan-
tum quench across a superfluid critical transition. The model involves a com-
plex scalar field coupled to a gauge and gravity field in the bulk. In earlier work
(arXiv:1211.7076) the scalar field had a strong self-coupling, in which case the back-
reaction on both the metric and the gauge field can be ignored. In this approxi-
mation, it was shown that when a time dependent source for the order parameter
drives the system across the critical point at a rate slow compared to the initial gap,
the dynamics in the critical region is dominated by a zero mode of the bulk scalar,
leading to a Kibble-Zurek type scaling function. We show that this mechanism for
emergence of scaling behavior continues to hold without any self-coupling in the
presence of backreaction of gauge field and gravity. Even though there are no zero
modes for the metric and the gauge field, the scalar dynamics induces adiabaticity
breakdown leading to scaling. This yields scaling behavior for the time dependence
of the charge density and energy momentum tensor.
1e-mail:das@pa.uky.edu
2e-mail:morita.takeshi@shizuoka.ac.jp
Contents
1 Introduction and summary 2
2 The Basic Setup 4
3 The Probe Approximation 5
3.1 The equilibrium critical point and its exponents . . . . . . . . . . . . . . . 7
3.2 The Adiabatic Expansion and its Breakdown . . . . . . . . . . . . . . . . . 9
3.3 Scaling in the Critical Region . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Gravity Backreaction 14
4.1 Static Solutions and Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Adiabaticity Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Scaling Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Conclusions and Discussions 22
6 Acknowledgements 22
A Complete equations of motion in section 4 23
1
1 Introduction and summary 2
2 The Basic Setup 4
3 The Probe Approximation 5
3.1 The equilibrium critical point and its exponents . . . . . . . . . . . . . . . 7
3.2 The Adiabatic Expansion and its Breakdown . . . . . . . . . . . . . . . . . 9
3.3 Scaling in the Critical Region . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Gravity Backreaction 14
4.1 Static Solutions and Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Adiabaticity Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Scaling Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Conclusions and Discussions 22
6 Acknowledgements 22
A Complete equations of motion in section 4 23
1
1 Introduction and summary
Quantum (or thermal) quench across critical points is an interesting problem in many
areas of physics. Consider starting in the gapped phase of a system and turning on a time
dependent external parameter which drives it to a critical point at a rate slow compared to
the initial gap. While the initial time evolution will be adiabatic, adiabaticity will break
down close to the critical point and the subsequent time evolution is expected to carry
universal signatures of the critical point. Many years ago, Kibble [1], and subsequently
Zurek [2], argued that observables like defect density indeed show scaling behavior. These
arguments - which were first developed for thermal quench and recently generalized to
quantum quench [3] [5] - imply that for a driving involving a single relevant operator, the
time dependence of the one point function of an operator O with conformal dimension x
is of the form [6]
O(t, v) ∼ v xν
zν+1 F (tv zν
zν+1 ) (1.1)
where v is the rate of change of the coupling, ν is the correlation length exponent and
z is the dynamical critical exponent. The arguments which lead to (1.1) involve (i) an
assumption that once adiabaticity breaks the system evolve in a diabatic fashion and (ii)
in the critical region the instantaneous correlation length is the only length scale in the
problem. The first assumption is rather drastic. The second assumption is reasonable,
but unlike equilibrium critical behavior there is really no well understood conceptual
framework like the renormalization group which explains why all other scales decouple
from the problem. This is particularly so for strongly coupled systems. Nevertheless,
Kibble-Zurek scaling has been verified by explicit calculations in many models and is now
being seen experimentally as well [3, 4].
In [7] a study of this problem in strongly coupled field theories which have gravity duals
via AdS/CFT was initiated and continued in [8] and [9]. The idea is to use holographic
techniques to investigate scaling behavior for slow quench without making any of the above
assumptions. In the AdS/CFT correspondence a time dependent coupling of a strongly
coupled boundary field theory corresponds to a time dependent boundary condition for the
bulk dual field, so that the problem reduces to differential equations with time dependent
boundary conditions. A mechanism for emergence of scaling emerges in these studies.
These models involve bulk scalar fields which are dual to order parameters and the critical
point is characterized by a zero mode of the scalar, i.e. a solution of the linearized
equations of motion which satisfy zero source boundary conditions at the AdS boundary
and regularity in the interior. It turns out that in the critical region where adiabaticity is
broken (so that a Taylor expansion in v breaks down) , there is a new small-v expansion
2
Quantum (or thermal) quench across critical points is an interesting problem in many
areas of physics. Consider starting in the gapped phase of a system and turning on a time
dependent external parameter which drives it to a critical point at a rate slow compared to
the initial gap. While the initial time evolution will be adiabatic, adiabaticity will break
down close to the critical point and the subsequent time evolution is expected to carry
universal signatures of the critical point. Many years ago, Kibble [1], and subsequently
Zurek [2], argued that observables like defect density indeed show scaling behavior. These
arguments - which were first developed for thermal quench and recently generalized to
quantum quench [3] [5] - imply that for a driving involving a single relevant operator, the
time dependence of the one point function of an operator O with conformal dimension x
is of the form [6]
O(t, v) ∼ v xν
zν+1 F (tv zν
zν+1 ) (1.1)
where v is the rate of change of the coupling, ν is the correlation length exponent and
z is the dynamical critical exponent. The arguments which lead to (1.1) involve (i) an
assumption that once adiabaticity breaks the system evolve in a diabatic fashion and (ii)
in the critical region the instantaneous correlation length is the only length scale in the
problem. The first assumption is rather drastic. The second assumption is reasonable,
but unlike equilibrium critical behavior there is really no well understood conceptual
framework like the renormalization group which explains why all other scales decouple
from the problem. This is particularly so for strongly coupled systems. Nevertheless,
Kibble-Zurek scaling has been verified by explicit calculations in many models and is now
being seen experimentally as well [3, 4].
In [7] a study of this problem in strongly coupled field theories which have gravity duals
via AdS/CFT was initiated and continued in [8] and [9]. The idea is to use holographic
techniques to investigate scaling behavior for slow quench without making any of the above
assumptions. In the AdS/CFT correspondence a time dependent coupling of a strongly
coupled boundary field theory corresponds to a time dependent boundary condition for the
bulk dual field, so that the problem reduces to differential equations with time dependent
boundary conditions. A mechanism for emergence of scaling emerges in these studies.
These models involve bulk scalar fields which are dual to order parameters and the critical
point is characterized by a zero mode of the scalar, i.e. a solution of the linearized
equations of motion which satisfy zero source boundary conditions at the AdS boundary
and regularity in the interior. It turns out that in the critical region where adiabaticity is
broken (so that a Taylor expansion in v breaks down) , there is a new small-v expansion
2
in fractional powers of v. To leading order in this expansion, the dynamics is dominated
by the zero mode, and the resulting bulk equations of the zero mode lead to a scaling
solution. The analysis can be also used to determine the corrections to scaling.
These studies did not include the effect of fluctuations (i.e. 1/N corrections in the
boundary field theory). More recently [10] have studied the problem by modelling these
fluctuations with a noise in the time evolution in a manner consistent with the fluctuation-
dissipation theorem and found consistency with the Kibble Zurek mechanism. Other
aspects of quantum quench which involve critical points have been investigated in [11], [12].
The models considered in [7] and [8] have scalar fields in the bulk with strong self-
couplings, together with gravity and a Maxwell field. The strong self-coupling allows a
probe approximation in which the backreaction of both the gravity and the bulk gauge
field can be ignored, as in [13]. It is important to examine the effects of backreaction.
First, as we will see below, the zero mode is present only in the scalar sector - not for the
gauge field or the metric perturbations. It is therefore of interest to know whether the
critical dynamics of the gauge field and metric also simplifies and lead to scaling properties
of the charge density and energy-momentum tensor in the boundary theory. Perhaps more
importantly, it is interesting to know whether the system thermalizes in any sense at late
times. This requires a complete treatment of the dynamics of the bulk metric. For a slow
driving far away from any critical point, the evolution is essentially adiabatic. If we start
from the ground state, as in the zero temperature cases of [8] and [9], this means that
there is no collapse into a black hole. If the quench crosses a critical point, the system
gets excited and it would be interesting to know what happens in the bulk.
In this paper, we take the first step in incorporating backreaction by addressing the
first question above. We will find that even though there is no zero mode in the gauge and
gravity sectors, the scalar zero mode feeds in through nonlinearities and leads to a break-
down of adiabatic evolution of the gauge and gravity fields. In the critical region there is
again an expansion in fractional powers of v. The scalar dynamics is dominated by the
zero mode and the nonlinear coupling with the gauge field and the metric leads to scaling
solutions for all the fields. The AdS/CFT dictionary then yields scaling functions for the
expectation value of the order parameter, the charge density and the energy momentum
tensor. We will not address the question of late time behavior and thermalization : this
would require detailed numerical work which we postpone to a later investigation.
For this purpose, we consider the holographic superfluid model of [14], subsequently
studied by [15]. The model considered in [8] is a variation of this model : the scalar
has a self-coupling in addition to minimal coupling to the gauge field and the metric.
One of the boundary space directions is compact with some radius R. Quantum quench
3
by the zero mode, and the resulting bulk equations of the zero mode lead to a scaling
solution. The analysis can be also used to determine the corrections to scaling.
These studies did not include the effect of fluctuations (i.e. 1/N corrections in the
boundary field theory). More recently [10] have studied the problem by modelling these
fluctuations with a noise in the time evolution in a manner consistent with the fluctuation-
dissipation theorem and found consistency with the Kibble Zurek mechanism. Other
aspects of quantum quench which involve critical points have been investigated in [11], [12].
The models considered in [7] and [8] have scalar fields in the bulk with strong self-
couplings, together with gravity and a Maxwell field. The strong self-coupling allows a
probe approximation in which the backreaction of both the gravity and the bulk gauge
field can be ignored, as in [13]. It is important to examine the effects of backreaction.
First, as we will see below, the zero mode is present only in the scalar sector - not for the
gauge field or the metric perturbations. It is therefore of interest to know whether the
critical dynamics of the gauge field and metric also simplifies and lead to scaling properties
of the charge density and energy-momentum tensor in the boundary theory. Perhaps more
importantly, it is interesting to know whether the system thermalizes in any sense at late
times. This requires a complete treatment of the dynamics of the bulk metric. For a slow
driving far away from any critical point, the evolution is essentially adiabatic. If we start
from the ground state, as in the zero temperature cases of [8] and [9], this means that
there is no collapse into a black hole. If the quench crosses a critical point, the system
gets excited and it would be interesting to know what happens in the bulk.
In this paper, we take the first step in incorporating backreaction by addressing the
first question above. We will find that even though there is no zero mode in the gauge and
gravity sectors, the scalar zero mode feeds in through nonlinearities and leads to a break-
down of adiabatic evolution of the gauge and gravity fields. In the critical region there is
again an expansion in fractional powers of v. The scalar dynamics is dominated by the
zero mode and the nonlinear coupling with the gauge field and the metric leads to scaling
solutions for all the fields. The AdS/CFT dictionary then yields scaling functions for the
expectation value of the order parameter, the charge density and the energy momentum
tensor. We will not address the question of late time behavior and thermalization : this
would require detailed numerical work which we postpone to a later investigation.
For this purpose, we consider the holographic superfluid model of [14], subsequently
studied by [15]. The model considered in [8] is a variation of this model : the scalar
has a self-coupling in addition to minimal coupling to the gauge field and the metric.
One of the boundary space directions is compact with some radius R. Quantum quench
3
is performed by introducing a time dependent boundary condition which corresponds to
a time dependent source for the order parameter in the boundary field theory. In [8]
non-linearity arose from self coupling of the scalar. In this paper, however, we set the
self-coupling to zero - as in the work of [14].
Now the backreaction of the gauge field cannot be ignored. However, when the charge
of the field is large, there is a probe approximation where the backreaction of gravity can
be ignored (which was used in [14]). We first consider this probe approximation. We
determine the equilibrium exponents, and then proceed to examine the breakdown of the
adiabatic expansion. We show that the zero mode of the scalar field leads to a breakdown
of adiabaticity in both the scalar and the gauge sector. The time of breakdown is the
same for both the fields - this serves as a consistency check on the calculation. We
then examine the dynamics in the critical region closely following [7] - [9]. In a way
analogous to these works we find that there is a consistent small-v expansion in fractional
powers of v. To leading order of this expansion, the zero mode of the scalar dominates
the dynamics. While there is no zero mode for the gauge field, the equations of motion
determine the dependence of the gauge field in the AdS radial direction in terms of the
scalar zero mode, which leads again to decoupling of modes. The resulting leading order
dynamics then exhibits scaling behavior like (1.1), and the expansion in fractional powers
of v provides a way to calculate the corrections to scaling.
We then proceed beyond the probe approximation and consider the backreaction of
the metric and show the breakdown of adiabaticity, the existence of a small-v expansion
in fractional powers of v and the emergence of scaling solutions are quite similar to the
gauge field case.
In Section (2) we describe the basic setup. Section (3) deals with the quench dynamics
in the probe approximation. In section (4) we incorporate the backreaction of gravity.
Section (5) contains conclusions and discussions.
2 The Basic Setup
The bulk action in d + 2 dimensions is given by
S =
∫
dd+2x√g
[ 1
2κ2
(
R + d(d + 1)
L2
)
− 1
4Fµν F µν − (|∂µΦ − iqAµΦ|2 − m2|Φ|2)]
,
(2.1)
where Φ is a complex scalar field with charge q and Aµ is an abelian gauge field, and
the other notations are standard. Henceforth we will use L = 1 units. One of the
spatial directions, which we will denote by θ will be considered to be compact. The radial
4
a time dependent source for the order parameter in the boundary field theory. In [8]
non-linearity arose from self coupling of the scalar. In this paper, however, we set the
self-coupling to zero - as in the work of [14].
Now the backreaction of the gauge field cannot be ignored. However, when the charge
of the field is large, there is a probe approximation where the backreaction of gravity can
be ignored (which was used in [14]). We first consider this probe approximation. We
determine the equilibrium exponents, and then proceed to examine the breakdown of the
adiabatic expansion. We show that the zero mode of the scalar field leads to a breakdown
of adiabaticity in both the scalar and the gauge sector. The time of breakdown is the
same for both the fields - this serves as a consistency check on the calculation. We
then examine the dynamics in the critical region closely following [7] - [9]. In a way
analogous to these works we find that there is a consistent small-v expansion in fractional
powers of v. To leading order of this expansion, the zero mode of the scalar dominates
the dynamics. While there is no zero mode for the gauge field, the equations of motion
determine the dependence of the gauge field in the AdS radial direction in terms of the
scalar zero mode, which leads again to decoupling of modes. The resulting leading order
dynamics then exhibits scaling behavior like (1.1), and the expansion in fractional powers
of v provides a way to calculate the corrections to scaling.
We then proceed beyond the probe approximation and consider the backreaction of
the metric and show the breakdown of adiabaticity, the existence of a small-v expansion
in fractional powers of v and the emergence of scaling solutions are quite similar to the
gauge field case.
In Section (2) we describe the basic setup. Section (3) deals with the quench dynamics
in the probe approximation. In section (4) we incorporate the backreaction of gravity.
Section (5) contains conclusions and discussions.
2 The Basic Setup
The bulk action in d + 2 dimensions is given by
S =
∫
dd+2x√g
[ 1
2κ2
(
R + d(d + 1)
L2
)
− 1
4Fµν F µν − (|∂µΦ − iqAµΦ|2 − m2|Φ|2)]
,
(2.1)
where Φ is a complex scalar field with charge q and Aµ is an abelian gauge field, and
the other notations are standard. Henceforth we will use L = 1 units. One of the
spatial directions, which we will denote by θ will be considered to be compact. The radial
4
direction will be denoted by r. The mass of the scalar is chosen in the range
m2
BF < m2 < m2
BF + 1, (2.2)
where m2
BF = −(d + 1)2/2 is the Breitenholer-Freedman bound.
The boundary theory has a finite chemical potential µ, so that
Limr→∞(At) → µ. (2.3)
The temperature vanishes.
Let us first set Φ = 0 (which is always a solution). As shown in [14], there is a value
of the chemical potential µ = µ0 such that for µ < µ0 the preferred solution to Einstein
equation is an AdS soliton
ds2 = dr2
r2h(r) + r2
(
−dt2 +
d−1∑
i=1
dx2
i
)
+ r2h(r)dθ2 ,
(2.4)
h(r) = 1 −
( r0
r
)d+1
,
(2.5)
At = µ , (2.6)
with constant parameters µ and r0. The periodicity of θ in this solution is
θ ∼ θ + 4π
(d + 1)r0
, (2.7)
µ0 is given by
µ0 = r0(d + 1)(2d) d−1
2(d+1)
(d − 1) d
d+1 (d + 1)1/2 . (2.8)
For µ > µ0 the preferred background is an extremal black brane. We will consider the
soliton phase.
In the remainder of the paper we will rescale all distances to set r0 = 1.
3 The Probe Approximation
We now consider the effect of the scalar field. In this section we consider the regime
q2 ≫ κ2, (3.9)
5
m2
BF < m2 < m2
BF + 1, (2.2)
where m2
BF = −(d + 1)2/2 is the Breitenholer-Freedman bound.
The boundary theory has a finite chemical potential µ, so that
Limr→∞(At) → µ. (2.3)
The temperature vanishes.
Let us first set Φ = 0 (which is always a solution). As shown in [14], there is a value
of the chemical potential µ = µ0 such that for µ < µ0 the preferred solution to Einstein
equation is an AdS soliton
ds2 = dr2
r2h(r) + r2
(
−dt2 +
d−1∑
i=1
dx2
i
)
+ r2h(r)dθ2 ,
(2.4)
h(r) = 1 −
( r0
r
)d+1
,
(2.5)
At = µ , (2.6)
with constant parameters µ and r0. The periodicity of θ in this solution is
θ ∼ θ + 4π
(d + 1)r0
, (2.7)
µ0 is given by
µ0 = r0(d + 1)(2d) d−1
2(d+1)
(d − 1) d
d+1 (d + 1)1/2 . (2.8)
For µ > µ0 the preferred background is an extremal black brane. We will consider the
soliton phase.
In the remainder of the paper we will rescale all distances to set r0 = 1.
3 The Probe Approximation
We now consider the effect of the scalar field. In this section we consider the regime
q2 ≫ κ2, (3.9)
5
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