1. Holographic Superconductors from Gauss-Bonnet Gravity Rong-Gen Cai Institute of Theoretical Physics Chinese Academy of Sciences (May 7, 2012) 2012 海峡两岸粒子物理和宇宙学研讨会, 重庆 ， 5.7-5.11

2. Contents: 1.Introduction 2.Holographic superconductors 3.S-wave superconductors in Gauss-Bonnet gravity 4.P-wave superconductors in Gauss-Bonnet gravity 5.Some analytical results 6. Conclusions Based on a few works: arXiv:1007.3321 (PRD82:066007, 2010) arXiv:1012.5559 (PRD83:066013,2011) arXiv:1103.2833 (JHEP1104:028,2011) arXiv:1103.5568 (PRD83:126007,2011) arXiv:1105.5000 (PRD84:046001,2011) ******

3. 1950, Landau-Ginzburg theory 1957, BCS theory: interactions with phonons Superconductor ： Vanishing resistivity (H. Onnes, 1911) Meissner effect (1933) 1980’s: cuprate superconductor 2000’s: Fe-based superconductor 1. Introduction

4. AdS/CFT correspondence (J. Maldacena, 1997) “Real conceptual change in our thinking about Gravity.” (E. Witten, Science 285 (1999) 512) IIB superstring theory on AdS 5 x S 5 N=4 SYM Theory

5. How to build a holographic superconductor model ？ CFT AdS/CFT Gravity global symmetry abelian gauge field scalar operator scalar field temperature black hole phase transition high T/no hair ； low T/ hairy BH

6. No-hair theorem? S. Gubser, 0801.2977

7. Building a holographic superconductor S. Hartnoll, C.P. Herzog and G. Horowitz, arXiv: 0803.3295 PRL 101, 031601 (2008) High Temperature （ black hole without hair): 2. Holographic superconductors

8. Consider the case of m^2L^2=-2 ， like a conformal scalar field. In the probe limit and A _t = Phi At the large r boundary:Scalar operator condensate O_i:

10. Boundary conduction: at the horizon: ingoing mode at the infinity: AdS/CFT source: Conductivity: Conductivity Maxwell equation with zero momentum : current

11. A universal energy gap: ~ 10%  BCS theory: 3.5  K. Gomes et al, Nature 447, 569 (2007)

12. Summary: 1.The CFT has a global abelian symmetry corresponding a massless gauge field propagating in the bulk AdS space. 2.Also require an operator in the CFT that corresponds to a scalar field that is charged with respect to this gauge field.. 3. Adding a black hole to the AdS describes the CFT at finite temperature. 4.Looks for cases where there are high temperature black hole solutions with no charged scalar hair, but below some critical temperature black hole solutions with charged scalar hair and dominates the free energy.

13. P-wave superconductors S. Gubser and S. Pufu, arXiv: 0805.2960 The order parameter is a vector! The model is

14. The ratio of the superconducting charge density to the total charge density. Vector operator condensate

15. 3. S-wave superconductors in GB gravity Holographic Superconductors with Higher Curvature Corrections R. Gregory et al., JHEP0910, 010 (2009) Holographic Superconductors with various condensates in Einstein-Gauss-Bonnet gravity Q. Pan et al. PRD 81, 106007 (2010) The model: R.G. Cai, PRD65, 084014 (2002)

16. Motivation:  To see the effect of high derivative terms on the holographic phase transition.  What is universal? What is not universal? S-wave: critical exponent =1/2?

17. At the boundary: mass increase

18. alpha increase

19. The energy gap is no longer universal.

20. 4. P-wave superconductors in GB gravity

23. 5. Some analytic results Analytic studies on holographic superconductors in GB gravity H.F. Li, R.G. Cai and H.Q. Zhang, JHEP1104, 028 (2011), 1103.2833 The background: P-wave

24. Equations of motion:

26. Operator condensate:

27. S-wave cases: matching method: R. Gregory et al, JHEP 0910, 010 (2009) The analytic method is powerful, also works well in various cases, For instance, insulator/superconductor, RGC et al, PRD 83, 126007 (2011), 1103.5568.

28. Holographic insulator/superconductor transition The model: The AdS soliton solution T. Nishioka et al, JHEP 1003,131 (2010)

29. The ansatz: The equations of motion: The boundary: both operators normalizable if

30. Near r_0: Impose b=B=0.

31. Near the critical point: Consider the case: m^2=-15/4 ( in this case, numerical results) the dimension of the operator: 3/2; 5/2, both normalizable. 1)

32. The minimal eigenvalues: numerical result

33. 2)Numerical result 3) General case:

34. Operator condensate and charge density

35. P-wave insulator/superconductor The model: Ansatz:

36. The boundary condition: Near the critical point: A.Akhawa et al arXiv:1011.6158

37. Operator condensation and charge density: in complete agreement with numerical calculations

38. 6. Conclusions 1) AdS/CFT is a powerful tool to understand superconductors 2) The high curvature terms make the phase transition harder. 3) Near the critical point, some analytic results. 4) Is it helpful to figure out the mechanism for high T_c?

39. Thanks !