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Magnetic Vortices in Gauge/Gravity Duality

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Magnetic Vortices in Gauge/Gravity Duality Magnetic Vortices in Gauge/Gravity Duality Dissertation by Migael Strydom Magnetic Vortices in Gauge/Gravity Duality Dissertation an der Fakult¨atf¨urPhysik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Migael Strydom aus Pretoria M¨unchen, den 20. Mai 2014 Dissertation submitted to the faculty of physics of the Ludwig{Maximilians{Universit¨atM¨unchen by Migael Strydom supervised by Prof. Dr. Johanna Karen Erdmenger Max-Planck-Institut f¨urPhysik, M¨unchen 1st Referee: Prof. Dr. Johanna Karen Erdmenger 2nd Referee: Prof. Dr. Dieter L¨ust Date of submission: 20 May 2014 Date of oral examination: 18 July 2014 Zusammenfassung Wir untersuchen stark gekoppelte Ph¨anomene unter Verwendung der Dualit¨at zwischen Eich- und Gravitationstheorien. Dabei liegt ein besonderer Fokus einer- seits auf Vortex L¨osungen, die von einem magnetischem Feld verursacht werden, und andererseits auf zeitabh¨angigen Problemen in holographischen Modellen. Das wichtigste Ergebnis ist die Entdeckung eines unerwarteten Effektes in einem ein- fachen holografischen Modell: ein starkes nicht abelsches magnetisches Feld verur- sacht die Entstehung eines Grundzustandes in der Form eines dreieckigen Gitters von Vortices. Die Dualit¨at zwischen Eich- und Gravitationstheorien ist ein m¨achtiges Werk- zeug welches bereits verwendet wurde um stark gekoppelte Systeme vom Quark- Gluonen Plasma in Teilchenbeschleunigern bis hin zu Festk¨orpertheorien zu be- schreiben. Die wichtigste Idee ist dabei die der Dualit¨at: Eine stark gekoppelte Quantenfeldtheorie kann untersucht werden, indem man die Eigenschaften eines aus den Einsteinschen Feldgleichungen folgenden Gravitations-Hintergrundes be- stimmt. Eine der Gravitationstheorien, die in dieser Arbeit behandelt werden, ist ei- ne Einstein{Yang{Mills Theorie in einem AdS{Schwarzschild Hintergrund mit SU(2)-Eichsymmetrie. Der Ansatz fur¨ das Eichfeld ist so gew¨ahlt, dass die zu- geh¨orige Quantenfeldtheorie einem externen Magnetfeld ausgesetzt ist. Oberhalb eines kritischen Magnetfeldes wird die Konfiguration instabil und zeigt einen Pha- senubergang¨ zu einem Supraleiter. Die Instabilit¨at wird mit zwei Ans¨atzen untersucht. Zum einen werden Fluk- tuationen des Hintergrunds betrachtet und die Quasinormalmoden analysiert. Zum anderen zeigt die numerische Analyse der Bewegungsgleichungen, dass das effektive Schr¨odinger-Potential mit st¨arker werdendem Magnetfeld sich so lange ver¨andert, bis ein gebundener Zustand m¨oglich wird. Der sich ergebende supra- leitende Grundzustand ist durch ein dreieckiges Vortexgitter gegeben, wie eine st¨orungstheoretische Entwicklung uber¨ einem kleinen Parameter proportional zur Gr¨oße des Kondensats zeigt. Zur Bestimmung des energetisch bevorzugten Zu- stands wird mithilfe der holographischen Ubersetzungsvorschrift¨ die Gesamtener- gie verschiedener L¨osungen berechnet. Hierfur¨ wird die L¨osung der Bewegungsglei- chungen zur dritten Ordnung in oben genanntem Parameter berechnet. Zus¨atzlich wird gezeigt, dass dieses Ergebnis auch fur¨ den Fall einer AdS{hard wall Geometrie gilt, also einer Feldtheorie mit Confinement. Als n¨achstes erweitern wir das einfache Gravitationsmodell um ein chemisches Potential und wiederholen die Untersuchung. Sind das chemische Potential, das magnetische Feld oder beide groß genug, so befindet sich das System in einer su- praleitenden Phase. Wir berechnen das Phasendiagramm des Systems numerisch. Der Grundzustand der supraleitenden Phase nahe dem Phasenubergang¨ ist ein dreieckiges Vortexgitter, wobei der Gitterabstand nur von der St¨arke des magne- tischen Feldes abh¨angt. Die Relevanz dieser Ergebnisse wird im Zusammenhang mit inhomogenen Grundzust¨anden in holographischen Supraleitern diskutiert, ei- nem Themengebiet welches in letzter Zeit viel Interesse auf sich gezogen hat. Die erhaltenen Resultate sind nicht nur aufgrund der vorher unbekannten inhomoge- nen L¨osung der Gravitationstheorie mit Schwarzem Loch neuartig. Es ist auch interessant, dass ein großes magnetisches Feld die Vortexstruktur im Grundzu- stand induziert anstatt sie zu unterdrucken.¨ i Des Weiteren untersuchen wir zeitabh¨angige Ph¨anomene in einer holographi- schen Erweiterung des Kondomodells. Letzteres beschreibt ein einfaches Modell in der Festk¨orperphysik, in welchem eine magnetische Verunreinigung stark an ein Elektronenreservoir koppelt. Die holographische Beschreibung erfordert Techniken der numerischen Relativit¨atstheorie und erlaubt uns die Entwicklung des Systems nach einem pl¨otzlichen Sprung in der Kopplungskonstante zu simulieren. Diese Doktorarbeit basiert auf Ergebnissen, die der Autor w¨ahrend des Studi- ums am Max-Planck-Institut-fur-Physik¨ in Munchen,¨ Deutschland unter der Be- treuung von PD Dr. J. K. Erdmenger von August 2011 bis Mai 2014 erreicht hat. Die relevanten Ver¨offentlichungen sind: [1] M. Ammon, J. Erdmenger, P. Kerner, and M. Strydom, \Black Hole Instability Induced by a Magnetic Field," Phys.Lett. B706 (2011) 94{ 99, arXiv:1106.4551 [hep-th], [2] Y.-Y. Bu, J. Erdmenger, J. P. Shock, and M. Strydom, \Magnetic field induced lattice ground states from holography," JHEP 1303 (2013) 165, arXiv:1210.6669 [hep-th]. ii Abstract We study strongly-coupled phenomena using gauge/gravity duality, with a par- ticular focus on vortex solutions produced by magnetic field and time-dependent problems in holographic models. The main result is the discovery of a counter- intuitive effect where a strong non-abelian magnetic field induces the formation of a triangular vortex lattice ground state in a simple holographic model. Gauge/gravity duality is a powerful theoretical tool that has been used to study strongly-coupled systems ranging from the quark-gluon plasma produced at particle colliders to condensed matter theories. The most important idea is that of duality: a strongly coupled quantum field theory can be studied by investi- gating the properties of a particular gravity background described by Einstein's equations. One gravity background we study in this dissertation is AdS{Schwarzschild with an SU(2) gauge field. We switch on the gauge field component that gives the field theory an external magnetic field. When the magnetic field is above a critical value, we find that the system is unstable, indicating a superconducting phase transition. We find the instability in two ways. Firstly, we do a quasinormal mode analysis, studying fluctuations about the background. Secondly, we rewrite the equations in Schr¨odingerform and numerically find that, as the magnetic field is increased, the potential deepens until it is capable of supporting a bound state. Next we show that the resulting superconducting ground state is a triangular vortex lattice. This is done by performing a perturbative expansion in a small parameter proportional to the condensate size. After solving the equations to third order, we use the holographic dictionary to calculate the total energy of different lattice solutions and identify the minimum energy state. In addition, we show that the result holds in an AdS{hard wall model as well, which is dual to a confining theory. Next we extend the simple gravity model to include a chemical potential and repeat the analysis. When the chemical potential, magnetic field or both are large, the system is in a superconducting phase. We calculate the precise phase diagram numerically. The ground state in the superconducting phase near the phase tran- sition line is shown to be a triangular vortex lattice with lattice spacing depending only on the magnetic field strength. We comment on the relevance of the results to the study of inhomogeneous ground states in holographic superconductors, a topic in which there has been much interest recently. Our results are novel not only because of the previously unknown inhomogeneous black hole solution, but also because of the effect of a large magnetic field inducing rather than inhibiting the vortex lattice ground state in a holographic model. We also study time-dependent phenomena in a holographic generalisation of the Kondo model, a simple condensed matter model of a magnetic impurity cou- pled strongly to a sea of electrons. This requires techniques from numerical rel- ativity and allows us to determine the response of the system to a quench in the coupling. This dissertation is based on work the author did during a PhD fellowship under the supervision of PD Dr. J. K. Erdmenger at the Max-Planck-Institut f¨urPhysik in Munich, Germany from August 2011 to May 2014. The relevant publications are: iii [1] M. Ammon, J. Erdmenger, P. Kerner, and M. Strydom, \Black Hole Instability Induced by a Magnetic Field," Phys.Lett. B706 (2011) 94{ 99, arXiv:1106.4551 [hep-th], [2] Y.-Y. Bu, J. Erdmenger, J. P. Shock, and M. Strydom, \Magnetic field induced lattice ground states from holography," JHEP 1303 (2013) 165, arXiv:1210.6669 [hep-th]. iv Samevatting Ons bestudeer sterk gekoppelde fenomene deur die gebruik van dualiteit tus- sen ykteorie¨e en gravitasieteorie¨e. Ons fokus spesifiek op vorteks oplossings wat deur magnetiese velde voortgebring word, asook tyd-afhanklike probleme in holo- grafiese modelle. Die belangrikste resultaat is die ontdekking van 'n onverwagte effek waar sterk nie-abelse magnetiese velde 'n driehoekige vorteksrooster grond- toestand uitlok in 'n holografiese model. Die dualiteit tussen ykteorie¨e en gravitasie is 'n nuttige instrument wat al ge- bruik is om sterk-gekoppelde stelsels te bestudeer wat wissel van die kwark-gluon plasma, wat geproduseer is by deeltjieversnellers, tot gekondenseerde materie teo- rie¨e. Die belangrikste
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