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Koushik Balasubramanian Submitted to the Department of Physics in Partial Fulfillment of the Requirements for the Degree of ARCHVES Doctor of Philosophy at The

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Koushik Balasubramanian Submitted to the Department of Physics in Partial Fulfillment of the Requirements for the Degree of ARCHVES Doctor of Philosophy at The Holographic View of Non-relativistic Physics by Koushik Balasubramanian Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of ARCHVES Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY , me- September 2012 © Massachusetts Institute of Technology 2012. All rights reserved. Author .......................................... j .................. Department of Physics June 15, 2012 C ertified by ........................................... ..... ........... John McGreevy Associate Professor Thesis Supervisor - /1 Accepted by ....... ................. .; ror .. ... .. .. ... .. ..-. Krishna Rajagopal Chairman, Department Committee on Graduate Theses Holographic View of Non-relativistic Physics by Koushik Balasubramanian Submitted to the Department of Physics on June 15, 2012, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Motivated by the AdS/CFT correspondence for relativistic CFTs, it seems natural to gen- eralize it to non-relativistic CFTs. Such a dual description could provide insight into strong coupling phenomena observed in condensed matter systems. Scale invariance can be realized in non-relativistic theories in many ways. One freedom is the relative scale dimension of time and space, called the dynamical exponent z. In this thesis, we will mainly focus on the case where z = 2, however gravity duals for other values of z have also been found. In the first part of the thesis, we study NRCFTs that are Galilean invariant. Discrete light cone quantization (DLCQ) of V= 4 super Yang-Mills theory is an example of such a system with z = 2 scaling symmetry. A more realistic example of a system with the same set of symmetries is a system of cold fermions at unitarity. These non-relativistic systems respect a symmetry algebra known as the Schr6dinger algebra. We propose a gravity dual that realizes the symmetries of the Schr6dinger algebra as isometries. An unusual feature of this duality is that the bulk geometry has two extra dimensions than the CFT, instead of the usual one. The additional direction is a compact direction and shift symmetry along this direction corresponds to the particle number transformation. This solution can be embedded into string theory by performing a set of operations (known as the Null-Melvin twist) on AdS 5 x S' solution of type IIB supergravity. This method also provides a way of finding a black hole solution which has asymptotic Schr6dinger symmetries. The field theory dual of these gravity solutions happens to be a modified version of DLCQ V = 4 super Yang-Mills theory. The thermodynamics of these theories is very different from that of cold atoms. This happens to be a consequence of realizing the entire Schr6dinger group as isometries of the spacetime. We give an example of a holographic realization in which the particle number symmetry is realized as a bulk gauge symmetry. In this proposal, the Schr6dinger algebra is realized in the bulk without the introduction of an additional compact direction. Using this proposal, we find a confining solution that describes a non-relativistic system at finite density. We use the holographic dictionary to compute the conductivity of this system and it is found to exhibit somewhat unusual behavior. In the second part of the thesis we study gravity duals of Lifshitz theories. These are non-relativistic scale invariant theories that are not boost invariant. These theories do not have a particle number symmetry unlike the boost invariant NRCFTs. We present solutions of 1OD and 111D supergravity theories that are dual to Lifshitz theories. We present a black hole solution that is dual to a strongly interacting Lifshitz theory at finite temperature. We show that the finite temperature correlators in the interacting theories do not exhibit ultra-local behavior which was observed in free Lifshitz theories. Thesis Supervisor: John McGreevy Title: Associate Professor Dedication To My Loving Parents for their Limitless Love and Endless Encouragement. Acknowledgments First, I would like to thank my advisor John McGreevy for his guidance and motivation, and for being very patient with me all these years. His intuition and knowledge has been a constant source of inspiration. I am greatly indebted to him for giving me the opportunity to work with him. I thank Allan Adams for the numerous illuminating and equally entertaining physics discussions. I thank him and Alan Guth for agreeing to be a part of my thesis committee. I thank all the members and graduate students at the Center for Theoretical Physics for tolerating me for five years. I would like to thank Ethan Dyer, Nabil Iqbal, Vijay Kumar, Loganayagam, Mark Mezei, Mike Mulligan, Jae Hoon Lee, Josephine Sunok Suh, David Vegh, and Sho Yaida for numerous fun-filled discussions. A special thanks to Daniel Park (my coffee buddy) for all the interesting physics and philosophical discussions. I would also like to thank Joyce Berggren, Catherine Modica, Scott Morley, Charles Suggs, and Crystal Young for all the help they have provided and their support. I would like to thank Hong Liu for many motivating and illuminating physics discussions. I also tiarnk Krishnan Naravan for the collaboration from which I learned much. I would like to thank R. I. Sujith and Suresh Govindarajan for their support and encouragement. I would like to thank all the members of the MIT cricket club for all the fun-filled and thrilling moments on the playground. A special thanks to Manikandan and Srinivasan for making important umpiring decisions. I would like to thank my relatives, especially my sister-in-law for their support during tough times. I would also like to thank Sivaram Ambikasaran, Vishal Srinivas, Arvind Shankar, and A. V. Varun for all the enjoyable and motivating conversations I have had with them. Words are not enough to thank Sapta for the constant support and motivation. Finally, I would like to thank my parents, and my brother for more than I can ever say. Contents 1 Introduction 11 1.1 An alternate view of strongly interacting systems . ... ... ... ... .11 1.2 A phenomenological approach to AdS/CFT . .. ....... 13 1.3 A braney picture of the correspondence . ... ... .. .... ..... ... 14 1.4 Holographic view of RG flows . ... ... ... .. ....... ... 17 1.5 Synopsis ... .... .... .... .... .... ..... .... ..... 18 2 Gravity Duals of Non-relativistic Conformal Field Theories 20 2.1 Introduction ... ... ... .. ... .... .. 20 2.2 Geometry .. .. ... ... .. .. ... .. .. 24 2.3 Correspondence ... ... ... ... .. .. .. 27 2.4 Discussion .. .... .... .. .. .... ... 29 3 Hot spacetime for cold atoms 31 3.1 Introduction . ... .. ... .. .. ... .. .. ... .. .. .. .. ..... 31 3.2 Embedding in String Theory . .. ... .. .. ... .. .. ... .. ..... 33 3.2.1 The Null Melvin Twist . .. .. .. ... .. .. .. .. .. .. .. 34 3.2.2 Rampaging Melvin Eats Extremal D3-brane, Spits Out Sch5 ..... 35 3.2.3 Solutions with Finite Temperature and Chemical Potential . .... 38 3.2.4 In Search of QCP: Finite Density at T = 0 .. .. .. .. .. 41 3.2.5 The Null Melvin Twist with finite L as a modified DLCQ . 43 3.2.6 The DLCQO of N= 4 SYM .. .. ... .. .. .. .. .. 44 6 3.3 Black Hole Thermodynamics in Schrbdinger Spacetimes . .. 46 3.3.1 E ntropy .. .. .. .. .. .. ... .. .. .. .. .. 46 3.3.2 A comment on the correspondence for the stress tensor . 47 3.3.3 Expectation values of the stress tensor . .. .. .. .. 48 3.3.4 Comments on chemical potential . ... .. 51 3.3.5 Comments on the first law of thermodynamics . .. 52 3.3.6 Thermodynamics in physical variables . .. .. 53 3.4 V iscosity . .. ... .. ... .. ... ... .. ... .. .. 54 3.4.1 Scalar wave equation in the finite-temperature solution . .. .. 55 3.5 D iscussion . ... .. ... .. ... ... .. ... .. .. .. 58 3.A Details of Melvinization ... .. .. .. ... .. .. .. .. .. 59 3.A.1 Buscher Rules and Conventions .. .. .. .. .. 59 3.A.2 The Hopf Vector on p2 .. ... .. ... .. .. 59 3.A.3 Constructing the finite temperature solution . .. .. 60 3.B Frame (in)dependence of the viscosity calculation . .. .. .. 63 3.C Comments on reduction to five dimensions ... .. .. .. 64 4 On the particle number in Galilean holography 66 4.1 Introduction .. ... .. ... .. .. ... .. .. .. .. .. .. .. .. 66 4.2 Getting rid of the direction . .. .. .. .. .. .. .. .. .. .. 68 4.2.1 Dimensional Reduction of SchQOT o . .. .. .. .. .. 69 4.2.2 Symmetry Generators .. .. .. .. .. .. .. .. .. .. .. 71 4.2.3 Wave equation ... .. ... .. ... .. .. .. .. .. .. .. .. 74 4.3 Black hole solution .. .. .. .. .. ... .. .. .. .. .. .. .. .. 74 4.3.1 Thermodynamics .. ... .. ... .. .. .. .. .. .. .. .. .. 75 4.4 A Holographic Mott Insulator? .. .. .. .. .. .. .. .. .. .. .. 76 4.4.1 Lift to eleven dimensions and mass gap . ... .. .. ... .. ... 77 4.4.2 What is a translation-invariant insulator? . .. .. .. .. .. .. .. 80 4.5 Examples with multiple species . .. ... .. .. ... .. .. ... .. .. 82 7 4.5.1 Symmetries .. ... .... .... .... ...... 84 4.5.2 z - 2 ... .... .... .... ... .. ... 85 4.6 Comments on the superfluid state ....... .. ..... 85 4.7 Conclusions ..... ...... ...... .... ...... 86 4.A Black hole solution in a system with two scalars . ...... 87 4.B Uplifting to M-theory ..... ..... ..... ...... 88 5 Lifshitz solutions of 1OD and 11D supergravity 90 5.1 Introduction . ... ... ... ... ... ... ... 90 5.2 Null deformations of AdS .. ... .. .. ... .. ...... 95 5.3 Dimensional reduction to Lifshitz spacetimes . .. .. 98 5.3.1 Dimensional reduction, more rigorously .
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