Disordered systems and the replica method in AdS/CFT

Yasuaki Hikida (KEK)

Ref. Fujita, YH, Ryu, Takayanagi, JHEP12(2008)065

April 13, 2009@NTNU

1. Introduction

Disordered systems• Impurities

Impurities may induce large effects

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

Strongly coupled physics, AdS/CFT correspondence

AdS/CFT correspondence

(d+1)-dim. gravity theory on AdS spacetime.

d-dim. conformal field theory (CFT)

• Ex. 4-dim. U(N) N=4 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.

[ Maldacena ]

living at the boundary (z=0)

Strong coupling physics from AdS/CFT

IIB string on AdS5

Classicalgravity

• Maps of coupling region

4d N=4 U(N) SYM

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 ]

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, …

Condensation of Cooper pair at finite temperature

Condensation of a scalar field in AdS black hole

Field theory Gravity theory

Examples of AdS/CMP (II)• Non-relativistic CFT

– Schrödinger group[Son, Balasubramanian-McGreevy, Sakaguchi-Yoshida, Herzog-

Rangamani-Ross, Maldacena-Martelli-Tachikawa, Adams-Balasubramanian-McGreevy, Nakayama-Ryu-Sakaguchi-Yoshida, ... ]• Galilean symmetry + scale invariance + special conformal (z=2)• Condensation of pair of fermionic gas (40K, 6Li)

B: magnetic fieldBCSBEC crossover

Strongly correlated, fermions at unitarity

Tc ~ 50 nK

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

Plan of talk

1. Introduction2. The replica method3. Field theory analysis4. Holographic replica method5. Conclusion6. Appendix

2. The replica method

Disordered systems• Types of disorder– Annealed disorder• Impurities are in thermal equilibrium.

– Quenched disorder• Impurities are fixed.

• An example: Random bond Ising model

Set up• Prepare a d-dim. quantum field theory– Ex. U(N) N=4 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

The replica method• Free energy

• The replica method– Prepare n copies, take an average, then set n=0

Correlation functions

• The effective action

Relevant Harris criteria

• Correlation functions

( cf. the supersymmetric method )

3. Field theory analysis

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

Conformal dimension Unitarity

Harris criteria

Higher point functions can be neglected.

Double trace deformation• Perturbation by a double trace operator– A simpler case for an exercise

– Beta function

Non-trivial fixed point

One-loop exact in large N limit

Two point function• Anomalous dimension

• RG flow equation

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

RG flow• Beta functions

• Flow of couplings

Two pint function• Redefinition of operators

• Two point functions1. In hated basis of replicated theory2. In the original basis3. In the limit of

Unitrity bound is violated

4. Holographic replica method

AdS/CFT dictionary• The map

• Boundary behavior at z»0– A scalar field satisfying KG eq. and the regularity at z=1

d-dim. CFT at the boundary z=0

: a spin-less operatorm : mass of the scalar

BF bound NormalisabilityUnitarity bound Harris criteria

Source to Legendre transform

: a scalar field

Gravity on (d+1)-dim. AdS

Legendre transform• Evaluation of action– Start from the (d+1) dim. action for the scalar

– Insert the solution and partially integrate over z

– Legendre transform

[ Klebanov-Witten ]

Double trace deformation• 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

[ Witten ]

Dual gravity computation• 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

EOM for i

( cf. Aharony-Clark-Karch, Kiritsis, Kiritsis-Niarchos )

[ Fujita-YH-Ryu-Takayanagi ]

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.

5. Conclusion

Summary and discussions• Summary– Disordered systems and the replica method

• Prepare n QFTs, introduce disorder, then take n->0 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(N|N) or U(N|N) supergroup structure

6. Appendix

Beta function• Perturbation from CFT

• Shift of cut off length– UV cut off length l is shifted to l(1 + )

• Beta function

Anomalous dimension• Perturbation from CFT

• Wave function renormalization– Two point function

– Shift of UV cut off

– Anomalous dimension