Disordered systems and the replica method in AdS/CFT

1. Disordered systems and the replica method in AdS/CFT Yasuaki Hikida (KEK) Ref. Fujita, YH, Ryu, Takayanagi, JHEP12(2008)065 April 13, 2009@NTNU

2. 1. Introduction

3. Impurities Disordered systems Impurities may induce large effects • Disordered systems • Real materials • Spin glass systems • Quantum Hall effects Strongly coupled physics, AdS/CFT correspondence

4. AdS/CFTcorrespondence • [ Maldacena ] (d+1)-dim.gravity theory on AdS spacetime. d-dim. conformal field theory (CFT) living at the boundary(z=0) • Ex.4-dim. U(N) N=4supersymmetric gauge theory • Perturbation theory is well understood, however strongly coupled physics is difficult to analyze • Ex. Type IIB superstring theory on AdS5x S5 • Quantum aspects are not understood yet.

5. Strong coupling physics from AdS/CFT 4d N=4 U(N) SYM IIB string on AdS5 • Maps of coupling region Classical gravity Strongly coupled • Advantage to apply AdS/CFTcorrespondence • 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 ]

6. Exampels of AdS/CMP (I) • AdS/CFTsuperconductor [ Gubser, Hartnoll-Herzog-Horowitz, Maeda-Okamura, Herzog-Kovtun-son, ... ] • High-Tc superconductor • Conventional superconductor ⇒ BCStheory, Cooperpair • High-Tc superconductor ⇒ Poorly understood, strongly correlated • It might be useful to apply AdS/CFT correspondence • Dual gravity theory • Finite temperature ⇒ AdSblack hole • Condensation of a scalar dual to Cooperpair • 2nd order phase transition, infinite DCconductivity, energy gap, … Field theory Gravity theory Condensation of Cooper pair at finite temperature Condensation of a scalar field in AdS black hole

7. Examples of AdS/CMP (II) • Non-relativistic CFT • Schrödingergroup [Son, Balasubramanian-McGreevy, Sakaguchi-Yoshida, Herzog-Rangamani-Ross, Maldacena-Martelli-Tachikawa, Adams-Balasubramanian-McGreevy, Nakayama-Ryu-Sakaguchi-Yoshida, ...] • Galileansymmetry + scale invariance + special conformal (z=2) • Condensation of pair of fermionic gas(40K, 6Li) Tc~ 50 nK Strongly correlated, fermions at unitarity B: magnetic field BEC crossover BCS

8. Examples of AdS/CMP (III) • Non-relativistic CFT • Lifshitz-like model [ Kachru-Liu-Mulligan, Horava, Taylor ] • Time reversal symmetry, no extension to Schrödingergroup • RG flow to relativistic CFT • Quantum hall effects [ Keski-Cakkuri-Kraus, Davis-Kraus-Shah, Fujita-Li-Ryu-Takayanagi, YH-Li-Takayanagi ] • Chern-Simonstheory as an effective theory • Disordered systems [ Hartnoll-Herzog, Fujita-YH-Ryu-Takayanagi] • Supersymmetric method, replica method

9. Plan of talk • Introduction • The replica method • Field theory analysis • Holographic replica method • Conclusion • Appendix

10. 2. The replica method

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

12. 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

13. The replica method • Free energy • The replica method • Prepare n copies, take an average, then set n=0

14. Correlation functions • The effective action Relevant   Harris criteria • Correlation functions ( cf. the supersymmetric method )

15. 3. Field theory analysis

16. 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.

17. 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

18. Two point function • Anomalous dimension • RG flow equation

19. 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

20. RG flow • Beta functions • Flow of couplings

21. 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

22. 4. Holographic replica method

23. AdS/CFT dictionary d-dim. CFT at the boundary z=0 Gravity on (d+1)-dim.AdS • The map • Boundary behavior at z»0 • A scalar field satisfying KG eq. and the regularity at z=1 : a spin-less operator : a scalar field m : mass of the scalar BF bound Harris criteria Normalisability Unitarity bound Source to Legendre transform

24. Legendre transform [ Klebanov-Witten ] • Evaluation of action • Start from the (d+1) dim. action for the scalar • Insert the solution and partially integrate over z • Legendre transform

25. 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

26. 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

27. Summary of the method Two facts Replica method Prepare forｎ copies of QFT Deform by double-trace operator Finally take the limit of n = 0 The deformation of double-traceoperators 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.

28. 5. Conclusion

29. 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

30. 6. Appendix

31. Beta function • Perturbation from CFT • Shift of cut off length • UV cut off length l is shifted to l(1 + ) • Beta function

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