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Schrödinger symmetry and

AdS /NRCFT correspondence

Informal Seminar @ KEKJun. 8, 2009

Kentaroh YoshidaPartly based on

Dept. of Phys. Kyoto Univ.

M. Sakaguchi, K.Y, arXiv:0805.2661, 0806.3612

Y. Nakayama, S. Ryu, M. Sakaguchi, K.Y, arXiv:0811.2461

Y. Nakayama, M. Sakaguchi, K.Y, arXiv:0812.1564, arXiv:0902.2267

Alg.

CSM

Sean Hartnoll, K.Y, arXiv:0810.0298,

Sakura Schäfer Nameki, Masahito Yamazaki, K.Y, arXiv:0903.4245

Gravity

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Application of AdS/CFT: 　 quark-gluon plasma (hydrodynamics)

condensed matter systems (superfluid)

Gravity (string) on AdS space CFT

Quantum gravity, non-pert. def. of string theory

From classical gravity to strongly coupled theory

1. INTRODUCTION

AdS/CFT correspondence

(quantum) critical point (strongly coupled) CFT

A new arena to study AdS/CFT

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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

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[Gubser, Hartnoll-Herzog-Horowitz]

[Davis-Kraus-Shar] [Fujita-Li-Ryu-Takayanagi]

EX 1. Superconductor

Holographic condensed matter physics

2. Quantum Hall effect

3. Unitary Fermions [Son, Balasubramanian-McGreevy]

4. Lifshitz field theory [Kachru-Liu-Mulligan]

Horava-Lifshitz gravity [Horava]

app. to gravity

NOTE Most of condensed matter systems are non-relativistic.

AdS/CMP

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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 ?

Main subject

Today: An overview of AdS/NRCFT based on Schrödinger symmetry

Tomorrow: A detailed explanation of my works on gravity duals for NRCFTs

Let’s discuss AdS/NRCFT with Schrödinger symmetry as a keyword

[Son]

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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

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1. Unitary fermions

- BCS-BEC crossover -

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Cold atoms

4He, 3He, 23Na, 6Li, 40K, etc.

Bose alkali gases

Fermi alkali gases

Liquid 3He

Liquid 4He

Critical temperature

2.17 K

2 x 10-3 K ~ mK

10-7 - 10-5 K ~ μK

10-6 K ~ μK

Magneto-Optical Trap (MOT) laser (evaporative) cooling

EX

Velocity

~ cm/s

~ m/s

~ 10 m/s

# of atoms = 103 - 106

ultracold

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Advantages of cold atoms

Designability of the system

Tabletop AdS/CFT !

EX Comparison to 1D Hubbard model Exact agreement

1) Optical lattice 2) Feshbach resonance

A possibility that cold atoms give a new laboratory to test AdS/CFT

Cooling trapA lattice developed by laser beam

Optical lattice

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Fermions at unitarity

Superfluidity of the atomic gas of 6Li and 40K (fermions)

By varying the external magnetic field,

the interaction between the atoms is

tunable (Feshbach resonance)

[2004]

BCS-BEC crossover

[Regal et.al, Zwierlein et.al]

[Regal-Jin, 2003]For 40K

BEC BCSUnitary

(an example of the systems realized by using cold atom techniques)

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Scattering length & s-wave function

Strong attractive

Bound state

Massless bound state

dimer

Weak attractive

No bound state

Resonance

BCS-BEC crossover

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[Carlos A.R. Sa de Melo, Physics Today Oct. 2008]

NRCFT?

Quantum critical region (QCR) and crossover

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T

QCR

Quantum critical pt.

crossover

B

Described by CFT

BEC BCS

(Feshbach resonance at zero temp.)

[P.Nikolic-S.Sachdev, cond-mat/0609106]

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3. Schrödinger symmetry

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What is Schrödinger algebra ?

Non-relativistic analog of relativistic conformal algebra

Conformal 　　　　 Poincare Galilei

Schrödinger algebra

= Galilean algebra + dilatation + special conformal

EX

Free Schrödinger eq. (scale invariant)

Dilatation (in NR theories)

[Hagen, Niederer,1972]

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Special conformal trans.

The generators of Schrödinger algebra

C has no index

= Galilean algebra

a generalization of mobius tras.

(Bargmann alg.)

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The Schrödinger algebra

Dynamical exponent

Galilean algebra

SL(2) subalgebra

Dilatation

Special conformal

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Interpretation from scaling dimension

Jacobi id.

c.f

Normalization of dilatation op. is fixed.

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Algebra with arbitrary z

Dynamical exponent

Galilean algebra

Dilatation

+

• M is not a center any more.

• conformal trans. C is not contained.

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How to realize the Schrödinger symmetry

2 possible ways:

1. A subalgebra of a relativistic conformal group

(NOT as IW contraction!)

2. A non-relativistic limit of a field theory

DLCQ description

A geometric realization (gravity) [Son, Balasubramanian-McGreevy]

EX 1+2 D relativistic CSM 1+2 D NR CSM

NR ABJM (N=6 CSM) gravity dual?

[Nakayama-Sakaguchi-K.Y]

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4. DLCQ description of

Schrödinger symmetry

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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

FACT

A relativistic conformal algebra in (d+1)+1 D

The generators:

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A light-like compactification of Klein-Gordon eq.

with

The difference of dimensionality

Rem: This is not the standard NR limit of the field theory

(d+1)+1 D

d+1 D

The embedding of the Schrödinger algebra in d+1 dim. spacetime

LC combination:

KG eq.

Sch. eq.

Remember the light-cone quantization

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Application of the embedding to AdS/CFT

The field theory is compactified on the light-like circle:

with -compactification

[Goldberger,Barbon-Fuertes ]

DLCQ description

But the problem is not so easy as it looks.

What is the dimensionally reduced theory in the DLCQ limit?

CFT

Gravity

Symmetry is broken from SO(2,d+2) to Sch(d) symmetry

NRCFT = LC Hamiltonian

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Progress in AdS/NRCFT based on the DLCQ description

1. Deformation of the DLCQ AdS background

deformation AdS space

This metric satisfies the e.o.m of Einstein gravity with a massive gauge field

[Son, Balasubramanian-McGreevy]

Coset construction is possible [S.Schäfer Nameki, M. Yamazaki, K.Y]

2. String theory embedding

a) null Melvin twist [Herzog-Rangamani-Ross] [Maldacena-Martelli-Tachikawa]

[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]

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Some of super Schrödinger algebras have been found [Sakaguchi-KY]

The maximal number of supercharges is 24

24 = 16 supertranslations + 8 superconformal symmetries

Schrödinger in 2+1 D SO(4,2)

PSU(2,2|4)Super Schrödinger in 2+1 D

UU

UU

AdS5 x S5 with - compactification [Maldacena-Martelli-Tachikawa]

3. Super Schrödinger algebra in AdS/CFT

Tomorrow’s seminar:

1. Coset construction of Schrodinger inv. metric

2. brane-wave deformation

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Unitary Fermions?

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5. NR limits of Chern-Simons

matter systems

- Another direction to AdS/NRCFT -

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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)

The JP model is obtained by taking the usual NR limit of a relativistic CSM

Jackiw-Pi model:

Schrödinger invariant

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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]

N=6 NR Chern-Simons matter system [Nakayama-Sakaguchi-KY,0902.2204]

(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.

First of all, need more examples. Super Sch. inv. field theory? Gravity dual?

NR SUSY itself is interesting Bose-Fermi mixture (realized by using cold atoms)

Another approach to AdS/CMP

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NR limit of N=2 Chern-Simons matter system

N=2 relativistic Chern-Simons matter system

complex scalar 2-comp.complex fermion

MassExpanding the potential

Field expansion:

Take limitHere we keep particles only

particle anti-particle particle anti-particle

[Lee-Lee-Weinberg]

NR limit:

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[Leblanc-Lozano-Min,1992]

The second comp. of the fermion has been deleted by using the e.o.m.

Pauli int.

NR action

The CS term is not changed even after the NR limit.

Note When we set , the JP model is reproduced

Super Schrödinger invariant

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NR SUSY

1 st SUSY

2 nd SUSY

Expand in terms of 1/c

N=2 relativistic SUSY

kinematical

dynamical

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N=2 super Schrödinger algebra

super Schrödinger algebra with 6 SUSY and U(1) R-symmetry

2 2 2

The bosonic Schrödinger algebra +

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N=3 NR CSM system [Nakayama-Ryu-Sakaguchi-KY]

NR action

2 sets of a complex scalar field and a 2-comp. complex fermion

When , N=2 CSM is reproduced.

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Characteristic of N ≥ 3 SUSY

SUSY parameters in SUSY transformation are not separated.

SUSY seems to be enhanced in the NR limit

Actually, only 1st SUSY survives and 2nd SUSY is broken due to the potential.

(SUSY is enhanced in the free theory limit )

Other charcterisitcs:

• # of 1st SUSY ≥ # of 2nd SUSY

• # of 2nd SUSY = 2 ?

• 2 1st SUSY, 2 2nd SUSY, 2 conformal SUSY form a multiplet.

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N=3 super Schrödinger algebra

super Schrödinger algebra with 8 SUSY and U(1)2 R-symmetry

The bosonic Schrödinger algebra +

2 2 2 2

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Other NR limits

In the usual NR limit, only the particles are kept and anti-particles are discarded by hand.

But there is no reason to exclude anti-particles.

NR limits of N=3 CSM system

[Nakayama-Sakaguchi-KY]

[Nakayama-Ryu-Sakaguchi-KY]

Exotic cases

LLM

P: particle

AP: anti-particle

B: both

N: none

: after the action has been improved by adding 4-fermi int.

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NR limit of ABJM model (N=6 CSM system) [Nakayama-Sakaguchi-KY] [Lee3]

After taking a mass deformation to the ABJM model, we can take a NR limit

1st SUSY: 10, 2nd SUSY: 2, conformal SUSY: 2

R-symmetry: SU(2) x SU(2) x U(1)

The original ABJM: 12 SUSY + 12 conformal SUSY, SO(6) = SU(4) R-symmetry

Here we shall discuss the SUSY only.

A massive ABJM: 12 SUSY + 0 conformal SUSY, SU(2) x SU(2) x U(1)

(maximal) SUSY of NR ABJM

Mass deformation

NR limit with all particles (no anti-particle)[Hosomichi-Lee3-Park]

It is possible to consider other matter contents.

[Aharony-Bergman-Jafferis-Maldacena]

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R-symmetry of NR ABJM

5 complex SUSY charges

R-symmetry: SU(2) x SU(2) x U(1)

14 SUSY charges of NR ABJM

= 2 dynamical SUSY + 2 conformal SUSY + 10 kinematical SUSY

(m, n = 1,2,3,4)singlet under SU(2) x SU(2) SU(2) x SU(2)

U(1) R-symmetry:

=

dynamical SUSY & conformal SUSY are also singlet under SU(2) x SU(2)

SU(4)

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1. What is the gravity dual to a mass deformed ABJM ?

1. What is the dual to the NRABJM?

What is the gravity dual to NR ABJM theory ?

The gravity dual to the ABJM theory = AdS4 X CP3

Questions

Bena-Warner geometry

5D Son x M6

Fix from the R-symmetry structure of NRABJM

Generalization of Hartnoll-K.Y. dynamical SUSY + conformal SUSY

[Donos-Gauntlett] [O Colgain-Yavartanoo] [Bobev-Kundu]

[Bobev-Kundu-Pilch] [Ooguri-Park]

[in working]

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6. Summary and Discussion

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Summary

• BCS-BEC crossover - unitary Fermions

• Schrödinger symmetry

2. Examples of super Schrödinger invariant field theories

N=3 NR CSM, NR ABJM [Nakayama-Ryu-Sakaguchi-KY][Nakayama-Sakaguchi-KY]

Gravity dual to NR ABJM ?

1. AdS/NRCFT based on DLCQ description

How to realize the Schrödinger symmetry in AdS/CFT

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Difficulties in AdS/NRCFT

1. If start from gravity (with the embedding of the Schrödinger into rel. conformal)

Difficulty of DLCQ (including interactions)

What is the substance of the theory in DLCQ limit ?

2. If start from the well-known NRCFTs (with the conventional NR limit)

What is the gravity solution?

There may be a limit in the gravity side, which corresponds to the NR limit

in the CFT side, if AdS/CFT works well even in the NR limit

DiscussionThere is no concrete example of AdS/NRCFT

where both sides are clearly understood

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Some relations to condensed matter physics ?

1. NR SUSY in cold atoms

Bose-Fermi mixture EX 87Rb and 40K

NR superconformal symmetry ?

2. Feedback to string theory

Condensed matter physics string theory

EX Jackiw-Pi model (NR) ABJM

Soliton sols. [Kawai-Sasaki]

Condensed matter physics in ABJM[Hikida-Li-Takayanagi]

[Fujita-Li-Ryu-Takayanagi]

dim. reduction of NR CSM theories

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Thank you!Thank you!

Crossover region

The singular part of the free energy density transforms

crossover exponent

The behavior is close to a critical point

Quantum critical region

2 parameters are relevant

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The embedding for an arbitrary z case:

LC combination:

The relativistic case is also included.

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DLCQ and deformation

1

2

3

DLCQ (x- -cpt.) pp-wave def.

Sch. symm.

[Son, BM]

[Goldberger et.al]

2002- in the context of pp-wave

Historical order

-compactification is important for the interpretation as NR CFT

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A list of super Schrödinger algebras (Not completed)

coincides with the symm. of N=2 NR Chern-Simons-matter system

# susy R-symm. superconformal

sch(2) 24 su(4) N=4, (3+1)-dim psu(2,2|4)

sch(2) 12 su(2)2×u(1) N=2*, (3+1)-dim psu(2,2|4)

sch(2) 12 su(2)×u(1) N=2, (3+1)-dim su(2,2|2)

sch(2) 6 u(1)3 N=1*, (3+1)-dim psu(2,2|4)

sch(2) 6 u(1) N=1, (3+1)-dim su(2,2|1)

sch(1) 24 so(8) N=8, (2+1)-dim osp(8|4)

sch(4) 24 sp(4) N=2, (5+1)-dim osp(8*|4)

[Leblanc-Lozano-Min]

1) sch(d) implies that the Schrödinger algebra in d spatial dimensions

Look for NRCFTs and gravity sols. corresponding to the above algebras.

2) Each of the above algebras contains smaller super Schrödinger algebras

[Sakaguchi-KY]

DLCQ

DLCQ

DLCQ

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The number of supercharges for other NR limits

Q1: 1st SUSY, Q2: 2nd SUSY, S: conformal SUSY

P: particle, AP: anti-particle