Disordered systems and the replica method in AdSCFT

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Disordered systems and the replica method in
AdS/CFT

• Yasuaki Hikida (KEK)
• Ref. Fujita, YH, Ryu, Takayanagi, JHEP12(2008)065
• April 13, 2009_at_NTNU

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1. Introduction
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Disordered systems

• Impurities

Impurities may induce large effects

• Disordered systems
• Real materials
• Spin glass systems
• Quantum Hall effects

Strongly coupled physics, AdS/CFT correspondence
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AdS/CFT correspondence
Maldacena
(d1)-dim. gravity theory on AdS spacetime.
d-dim. conformal field theory (CFT)
living at the boundary (z0)

• Ex. 4-dim. U(N) N4 supersymmetric gauge theory
• Perturbation theory is well understood, however
  strongly coupled physics is difficult to analyze

• Ex. Type IIB superstring theory on AdS5 x S5
• Quantum aspects are not understood yet.

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Strong coupling physics from AdS/CFT

• Maps of coupling region

IIB string on AdS5
4d N4 U(N) SYM
Classical gravity
Strongly coupled

• Advantage to apply AdS/CFT correspondence
• An alternative approach to strong coupling
  physics
• Lattice gauge theory
• Geometrical understanding, analytical
  computation,
• Time-dependence
• Quark gluon plasma
• Shear viscosity
• Comparison to experiments at RHIC

Kovtun-Son-Starinets
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Exampels of AdS/CMP (I)

• AdS/CFT superconductor
• Gubser, Hartnoll-Herzog-Horowitz,
  Maeda-Okamura, Herzog-Kovtun-son, ...
• High-Tc superconductor
• Conventional superconductor ? BCS theory, Cooper
  pair
• High-Tc superconductor ? Poorly understood,
  strongly correlated
• It might be useful to apply AdS/CFT
  correspondence
• Dual gravity theory
• Finite temperature ? AdS black hole
• Condensation of a scalar dual to Cooper pair
• 2nd order phase transition, infinite DC
  conductivity, energy gap,

Field theory
Gravity theory
Condensation of Cooper pair at finite temperature
Condensation of a scalar field in AdS black hole
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Examples of AdS/CMP (II)

• Non-relativistic CFT
• Schrödinger group
• Son, Balasubramanian-McGreevy,
  Sakaguchi-Yoshida, Herzog-Rangamani-Ross,
  Maldacena-Martelli-Tachikawa, Adams-Balasubramania
  n-McGreevy, Nakayama-Ryu-Sakaguchi-Yoshida, ...
• Galilean symmetry scale invariance special
  conformal (z2)
• Condensation of pair of fermionic gas (40K, 6Li)

Tc 50 nK
Strongly correlated, fermions at unitarity
B magnetic field
BCS
BEC
crossover
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Examples of AdS/CMP (III)

• Non-relativistic CFT
• Lifshitz-like model
• Kachru-Liu-Mulligan, Horava, Taylor
• Time reversal symmetry, no extension to
  Schrödinger group
• RG flow to relativistic CFT
• Quantum hall effects
• Keski-Cakkuri-Kraus, Davis-Kraus-Shah,
  Fujita-Li-Ryu-Takayanagi, YH-Li-Takayanagi
• Chern-Simons theory as an effective theory
• Disordered systems
• Hartnoll-Herzog, Fujita-YH-Ryu-Takayanagi
• Supersymmetric method, replica method

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Plan of talk

• Introduction
• The replica method
• Field theory analysis
• Holographic replica method
• Conclusion
• Appendix

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2. The replica method
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Disordered systems

• Types of disorder
• Annealed disorder
• Impurities are in thermal equilibrium.
• Quenched disorder
• Impurities are fixed.
• An example Random bond Ising model

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Set up

• Prepare a d-dim. quantum field theory
• Ex. U(N) N4 4d SYM
• Perturb the theory by a operator
• Ex. a single trace operator
• Take an average over the disorder

The disorder configuration depends on x
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The replica method

• Free energy

• The replica method
• Prepare n copies, take an average, then set n0

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Correlation functions

• The effective action

Relevant ?
? Harris criteria

• Correlation functions

( cf. the supersymmetric method )
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3. Field theory analysis
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Set up

• Original theory without disorder
• d-dim. conformal field theory in the large N
  limit
• Our disordered system
• Deform the theory by a singlet operator
• n copies of CFT with double trace deformation

Harris criteria
Unitarity
Conformal dimension
Higher point functions can be neglected.
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Double trace deformation

• Perturbation by a double trace operator
• A simpler case for an exercise
• Beta function

One-loop exact in large N limit
Non-trivial fixed point
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Two point function

• Anomalous dimension

• RG flow equation

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Large N disordered system

• Replica theory
• n CFTs CFT1 CFT2 ... CFTn
• Single trace operators
• Double trace deformation

Regularization with ?
Start with a CFT with deformation ?, then
introduce the disorder
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RG flow

• Beta functions
• Flow of couplings

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Two pint function

• Redefinition of operators
• Two point functions
• In hated basis of replicated theory
• In the original basis
• In the limit of

Unitrity bound is violated
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4. Holographic replica method
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AdS/CFT dictionary

• The map
• Boundary behavior at z0
• A scalar field satisfying KG eq. and the
  regularity at z1

d-dim. CFT at the boundary z0
Gravity on (d1)-dim. AdS
a spin-less operator
? a scalar field
m mass of the scalar
BF bound
Normalisability
Harris criteria
Unitarity bound
Source to
Legendre transform
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Legendre transform
Klebanov-Witten

• Evaluation of action
• Start from the (d1) dim. action for the scalar
• Insert the solution and partially integrate over
  z
• Legendre transform

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Double trace deformation
Witten

• A simpler case with one CFT
• Deformation by a double trace operator
• The deformed action in the gravity side
• Two point function

EOM for ?
reproduces the field theory result
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Dual gravity computation
Fujita-YH-Ryu-Takayanagi

• Set up
• n CFTs with
• n AdS spaces sharing the same boundary
• coupled to each other by boundary conditions for
  ?i
• The deformed action in the gravity side

( cf. Aharony-Clark-Karch, Kiritsis,
Kiritsis-Niarchos )
EOM for ?i
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Summary of the method

• Two facts
• Replica method
• Prepare for n copies of QFT
• Deform by double-trace operator
• Finally take the limit of n 0
• The deformation of double-trace operators
• In the dual gravity theory the boundary condition
  for Á is changed.
• Holographic replica method
• Prepare for n copies of AdS space
• Each AdS space share its boundary
• Deform the boundary condition for scalars Ái in
  AdS
• Each AdS space interacts each other though the
  boundary
• Finally take the limit of n 0
• We compute the two point function and find
  agreement.

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5. Conclusion
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Summary and discussions

• Summary
• Disordered systems and the replica method
• Prepare n QFTs, introduce disorder, then take
  n-gt0 limit
• RG flow and the two point function
• Conformal perturbation theory
• Holographic replica method
• Multiple AdS spaces coupled though the boundary
• Open problems
• Quantum disordered system
• Dual geometry is AdS black hole
• Other quantities
• E.g. two point function of currents
• Holographic supersymmetric method
• OSP(NN) or U(NN) supergroup structure

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6. Appendix
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Beta function

• Perturbation from CFT
• Shift of cut off length
• UV cut off length l is shifted to l(1 ?)
• Beta function

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Anomalous dimension

• Perturbation from CFT
• Wave function renormalization
• Two point function
• Shift of UV cut off
• Anomalous dimension