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1
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
2
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
3
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
4
[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
5
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]
6
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
8
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
9
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
10
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)
10
11
Scattering length & s-wave function
Strong attractive
Bound state
Massless bound state
dimer
Weak attractive
No bound state
Resonance
NRCFT?
Quantum critical region (QCR) and crossover
13
T
QCR
Quantum critical pt.
crossover
B
Described by CFT
BEC BCS
(Feshbach resonance at zero temp.)
[P.Nikolic-S.Sachdev, cond-mat/0609106]
15
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]
16
Special conformal trans.
The generators of Schrödinger algebra
C has no index
= Galilean algebra
a generalization of mobius tras.
(Bargmann alg.)
17
The Schrödinger algebra
Dynamical exponent
Galilean algebra
SL(2) subalgebra
Dilatation
Special conformal
19
Algebra with arbitrary z
Dynamical exponent
Galilean algebra
Dilatation
+
• M is not a center any more.
• conformal trans. C is not contained.
20
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]
22
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:
23
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
24
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
25
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]
26
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
29
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
30
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
31
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:
32
[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
34
N=2 super Schrödinger algebra
super Schrödinger algebra with 6 SUSY and U(1) R-symmetry
2 2 2
The bosonic Schrödinger algebra +
35
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.
36
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.
37
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
38
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.
39
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]
40
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)
41
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]
43
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
44
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
45
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
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
49
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
50
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