Schrdinger symmetry and AdS /NRCFT correspondence Dept. of Phys. - PowerPoint PPT Presentation

Informal Seminar @ KEK Jun. 8, 2009 Schrödinger symmetry and AdS /NRCFT correspondence Dept. of Phys. Kyoto Univ. Kentaroh Yoshida Partly based on Alg. M. Sakaguchi, K.Y, arXiv:0805.2661, 0806.3612 Y. Nakayama, S. Ryu, M. Sakaguchi, K.Y, arXiv:0811.2461 CSM Y. Nakayama, M. Sakaguchi, K.Y, arXiv:0812.1564, arXiv:0902.2267 Sean Hartnoll, K.Y, arXiv:0810.0298, Gravity Sakura Schäfer Nameki, Masahito Yamazaki, K.Y, arXiv:0903.4245 1

1. INTRODUCTION AdS/CFT correspondence Quantum gravity, non-pert. def. of string theory Gravity (string) on AdS space CFT From classical gravity to strongly coupled theory Application of AdS/CFT: quark-gluon plasma (hydrodynamics) condensed matter systems (superfluid) (quantum) critical point (strongly coupled) CFT A new arena to study AdS/CFT 2

Phenomenological approach Critical phenomena Einstein gravity (critical pts. in condensed matter systems) EX phase transition instability • New physics in gravitational theory • New analytical method in condensed matter physics Embedding into string theory One may expect something new in string theory 3

Holographic condensed matter physics AdS/CMP EX 1. Superconductor [Gubser, Hartnoll-Herzog-Horowitz] 2. Quantum Hall effect [Davis-Kraus-Shar] [Fujita-Li-Ryu-Takayanagi] 3. Unitary Fermions [Son, Balasubramanian-McGreevy] 4. Lifshitz field theory [Kachru-Liu-Mulligan] app. to gravity Horava-Lifshitz gravity [Horava] NOTE Most of condensed matter systems are non-relativistic. 4

Main subject Holographic duals for non-relativistic (NR) CFTs ? An example of NRCFTs: Unitary fermions Schrödinger symmetry (one of the NR scaling symm.) What is the gravity dual ? [Son] Let’s discuss AdS/NRCFT with Schrödinger symmetry as a keyword Today: An overview of AdS/NRCFT based on Schrödinger symmetry Tomorrow: A detailed explanation of my works on gravity duals for NRCFTs 5

Plan of the talk 1. Introduction (finished) 2. Unitary fermions - BCS-BEC crossover 3. Schrödinger symmetry - How to realize Schrödinger symmetry in AdS/CFT - 4. DLCQ description of Schrödinger symmetry 5. NR limits of Chern-Simons matter systems 6. Summary and Discussion 6

1. Unitary fermions - BCS-BEC crossover - 7

Cold atoms Magneto-Optical Trap (MOT) laser (evaporative) cooling # of atoms = 10 3 - 10 6 EX 4 He, 3 He, 23 Na, 6 Li, 40 K, etc. Critical temperature Velocity Liquid 4 He 2.17 K ~ 10 m/s 2 x 10 -3 K ~ mK Liquid 3 He ~ m/s Bose alkali gases 10 -7 - 10 -5 K ~ μK ~ cm/s Fermi alkali gases 10 -6 K ~ μK ultracold 8

Advantages of cold atoms Designability of the system 1) Optical lattice 2) Feshbach resonance Optical lattice Cooling trap A lattice developed by laser beam EX Comparison to 1D Hubbard model Exact agreement A possibility that cold atoms give a new laboratory to test AdS/CFT Tabletop AdS/CFT ! 9

Fermions at unitarity (an example of the systems realized by using cold atom techniques) Superfluidity of the atomic gas of 6 Li and 40 K (fermions) [2004] [Regal et.al, Zwierlein et.al] By varying the external magnetic field, the interaction between the atoms is BEC Unitary BCS tunable (Feshbach resonance) BCS-BEC crossover For 40 K [Regal-Jin, 2003] 10 10

Scattering length & s-wave function Weak attractive Strong attractive Massless bound state No bound state Bound state dimer Resonance 11

BCS-BEC crossover [Carlos A.R. Sa de Melo, Physics Today Oct. 2008] 12

NRCFT? Quantum critical region (QCR) and crossover Described by CFT T crossover QCR BEC BCS B Quantum critical pt. (Feshbach resonance at zero temp.) [P.Nikolic-S.Sachdev, cond-mat/0609106] 13

3. Schrödinger symmetry 14

What is Schrödinger algebra ? Non-relativistic analog of relativistic conformal algebra Conformal Poincare Galilei [Hagen, Niederer,1972] Schrödinger algebra = Galilean algebra + dilatation + special conformal Dilatation (in NR theories) EX Free Schrödinger eq. (scale invariant) 15

Special conformal trans. a generalization of mobius tras. The generators of Schrödinger algebra = Galilean algebra (Bargmann alg.) C has no index 16

The Schrödinger algebra Galilean algebra Dilatation Dynamical exponent Special conformal SL (2) subalgebra 17

Interpretation from scaling dimension Jacobi id. c.f Normalization of dilatation op. is fixed. 18

Algebra with arbitrary z Galilean algebra + Dilatation Dynamical exponent • M is not a center any more. • conformal trans. C is not contained. 19

How to realize the Schrödinger symmetry 2 possible ways: 1. A subalgebra of a relativistic conformal group (NOT as IW contraction!) DLCQ description A geometric realization (gravity) [Son, Balasubramanian-McGreevy] 2. A non-relativistic limit of a field theory EX 1+2 D relativistic CSM 1+2 D NR CSM NR ABJM (N=6 CSM) gravity dual? [Nakayama-Sakaguchi-K.Y] 20

4. DLCQ description of Schrödinger symmetry 21

FACT A Schrödinger algebra in d+1 D is embedded into a ``relativistic’’ conformal algebra in (d+1)+1 D as a subalgebra. EX. Schrödinger algebra in 2+1 D can be embedded into SO(4,2) in 3+1 D A relativistic conformal algebra in ( d+1) +1 D The generators: 22

The embedding of the Schrödinger algebra in d+1 dim. spacetime LC combination: A light-like compactification of Klein-Gordon eq. (d+1)+1 D KG eq. with Sch. eq. d+1 D The difference of dimensionality Rem: This is not the standard NR limit of the field theory Remember the light-cone quantization 23

Application of the embedding to AdS/CFT CFT The field theory is compactified on the light-like circle: DLCQ description NRCFT = LC Hamiltonian Gravity with -compactification [Goldberger,Barbon-Fuertes ] Symmetry is broken from SO(2,d+2) to Sch(d) symmetry But the problem is not so easy as it looks. What is the dimensionally reduced theory in the DLCQ limit? 24

Progress in AdS/NRCFT based on the DLCQ description 1. Deformation of the DLCQ AdS background [Son, Balasubramanian-McGreevy] AdS space deformation This metric satisfies the e.o.m of Einstein gravity with a massive gauge field Coset construction is possible [S.Schäfer Nameki, M. Yamazaki, K.Y] 2. String theory embedding [Herzog-Rangamani-Ross] [Maldacena-Martelli-Tachikawa] a) null Melvin twist [Adams-Balasubramanian-McGreevy] b) brane-wave [Hartnolll-K.Y] Super Schrödinger inv. background (SUSY embedding) Generalization of our work : [Donos-Gauntlett] [O Colgain-Yavartanoo] [Bobev-Kundu] [Bobev-Kundu-Pilch] [Ooguri-Park] 25

3. Super Schrödinger algebra in AdS/CFT Schrödinger in 2+1 D SO(4,2) U U U Super Schrödinger in 2+1 D PSU(2,2|4) U Some of super Schrödinger algebras have been found [Sakaguchi-KY] The maximal number of supercharges is 24 24 = 16 supertranslations + 8 superconformal symmetries AdS 5 x S 5 with - compactification [Maldacena-Martelli-Tachikawa] Tomorrow’s seminar: 1. Coset construction of Schrodinger inv. metric 2. brane-wave deformation 26

Unitary Fermions? 27

5. NR limits of Chern-Simons matter systems - Another direction to AdS/NRCFT - 28

What is the origin of difficulty? DLCQ interpretation If we start from the embedding of the Schrödinger group into the relativistic conformal group, then we have to confront a difficulty of DLCQ. Another approach • Start from the well known example of NRCFT • Consider the usual NR limit in the context of AdS/CFT But there are a few examples of NRCFT: NL Schrödinger, Jackiw-Pi model (NR CSM), SUSY extensions (1+2 D) Jackiw-Pi model: Schrödinger invariant The JP model is obtained by taking the usual NR limit of a relativistic CSM 29

First of all, need more examples. Super Sch. inv. field theory? Gravity dual? NR super Chern-Simons matter systems N=2 NR Chern-Simons matter system [Leblanc-Lozano-Min, hep-th/9206039] N=3 NR Chern-Simons matter system [Nakayama-Ryu-Sakaguchi-KY, 0811.2461] [Nakayama-Sakaguchi-KY,0902.2204] N=6 NR Chern-Simons matter system (NR ABJM) and their cousins (depending on the matter contents) NOTE Interacting SUSY singlet is possible [Nakayama-Sakaguchi-KY, 0812.1564] There is no direct analog of the Coleman-Mandula theorem for NR SUSY. NR SUSY itself is interesting Bose-Fermi mixture (realized by using cold atoms) Another approach to AdS/CMP 30

NR limit of N =2 Chern-Simons matter system N=2 relativistic Chern-Simons matter system [Lee-Lee-Weinberg] complex scalar 2-comp.complex fermion Expanding the potential Mass NR limit: Field expansion: particle anti-particle particle anti-particle Here we keep particles only Take limit 31

[Leblanc-Lozano-Min,1992] NR action Pauli int. The CS term is not changed even after the NR limit. Super Schrödinger invariant The second comp. of the fermion has been deleted by using the e.o.m. Note When we set , the JP model is reproduced 32