Introduction to Holographic Superconductors

1. Introduction to Holographic Superconductors Bin Wang Fudan University

2. 为什么在LHC要出成果的前夜，高能物理和引力物理学家要来思考能标比他们通常研究的量级低很多的物理现象？为什么在LHC要出成果的前夜，高能物理和引力物理学家要来思考能标比他们通常研究的量级低很多的物理现象？ 第一，AdS/CFT对应性是对强耦合场理论进行研究的特殊方法，运用这个方法强耦合场的一些问题可以被计算处理，概念也变得更清晰。凝聚态物理中有很多强场系统，这些系统对基于弱相互作用和对称破缺理论的传统凝聚态理论是个挑战。 第二，凝聚态系统也许能提供一个舞台从实验上来实现很多有趣的高能物理的理论想法。标准模型的拉氏量和它假定的完备性在我们的宇宙中是特别的。在凝聚态物理中有很多有效哈密顿，更多的哈密顿能被诱导出来。也许最终能产生和已知的AdS对偶的新兴场论，在实验上实现AdS/CFT对应性。 第三，AdS/CFT对应性改变了对自然的看法，传统物理中按照场的能级分类变得不是很重要了。如果量子引力理论能对偶于很多和量子极端电子相同的特性，那么哪个更基本的问题就没有意义了。 取而代之的是强调两者之间的对偶更有意义。这个观点有实际的效果，比如寻找超导表述的对偶，人们意识到黑洞的无毛定理也许有漏洞，人们也许能找到新的黑洞解。 There is a long long long way to go…

3. Outline • PART I: 1) Introduction to superconductivity 2) Simple model for a holographic superconductor 3) Probe limit in SAdS, RNAdS backgrounds (condensate and conductivity) • PART II: signature of the phase transition • PART III: Probe limit GBAdS background • PART IV: magnetic field discussion • PART V: more on phase transition

4. References • Review articles: 1) S. Hartnoll, 0903.3246 2) C. Herzog, 0904.1975 3) G. Horowitz, 1002.1722 • Our works: • Q. Pan, B. Wang, et al, arXiv:0912.2475 PRD(10) • X He, B Wang, R-G Cai, C-Y Lin, arXiv:1002.2679 PLB(10) • X-H Ge, Bin Wang, et al, arXiv:1002.4901 JHEP(10) • R-G Cai, Z-Y Nie, B Wang, H-Q Zhang, arXiv:1005.1233 • Q. Pan, B. Wang, 1005.4743

5. Superconductivity 101 • In conventional superconductors, when T<Tc • Electrical resistivity->0 • Meissner effect, magnetic field is expelled • Landau-Ginzburg: superconductivity is a second order phase transition with a complex scalar field ϕ as order parameter, Free energy: T>Tc, F(min) is at ϕ=0, T<Tc, F(min) at ϕ not 0 • BCS theory: pairs of elections with opposite spin can bind to form a charged boson called a Cooper pair. Below a critical temperature Tc, there is a second order phase transition and these bosons condense. microscopic theory of superconductivity

6. It was once thought that the highest Tc for a BCS superconductor was around 30K. But in 2001, MgB2 was found to be superconducting at 40K and is believed to be described by BCS. Some people now speculate that BCS could describe a superconductor with Tc = 200K. The new high Tc superconductors were discovered in 1986. These cuprates (e.g. YBaCuO) are layered and superconductivity is along CuO2 planes. Highest Tc today (HgBaCuO) is Tc = 134K. Another class of superconductors discovered March 08 based on iron and not copper FeAs(…) Highest Tc = 56K. The pairing mechanism is not well understood. Unlike BCS theory, it involves strong coupling. AdS/CFT is an ideal tool to study strongly coupled field theories.

7. Sachdev, 0907.0008

8. Gravitational Dual • How do we go about constructing a holographic dual for a superconductor? Superconductor ------ temperature Gravity {black hole} ---- temperature (Hawking) Gauge/Gravity duality, BH T ~ dual field theory T • Why do we need AdS space? Gauge/Gravity dual in AdS space AdS BH is thermodynamically stable C>0 Confining box in AdS BH

10. Why not “dilatonic” black holes with scalar hair? • a result of a coupling • Historic steps to have scalar hair condensation: • Hertog (2006) showed that for a real scalar field with arbitrary potential V(φ), neutral AdS black holes have scalar hair if AdS is unstable. • Gubser (2008) argued that a charged scalar field around a charged black hole would have the desired property. Consider This does not work for asymptotically flat spacetimes

13. Will cause the scalar hair to form at low temperature

14. Current remains finite at the horizon

16. Local gauge symmetry in the bulk In the bulk

17. condensation • Similar to BCS theory observed in many materials, condensate rises quickly as the system is cooled below the critical temperature and goes to a constant as T->0. • Near Tc, (Landau-Ginsberg) Scalar hair is developed

18. condensation • The qualitative behavior is the same as before. There is a critical Tc above which the condensate is zero. Near Tc, . • In all but one case, T->0, condensate grows. If the condensate is big, the probe limit breaks.

25. The limit of the electric filed in the bulk is the electric field on the boundary Expectation value of the induced current

26. Real part of the conductivity T>Tc, the conductivity is constant.

27. Damping term Charge carries mass m, charge e, number density n in normal conductor satisfying : Relaxation time Current J=env Number density in normal conductor

32. Always true??

33. Part 2 signature of the phase transition • Second order transition between a non-superconducting state at high T and a superconducting state at low T. • Whether the QNM can be an effective probe of this phase transition? X. He, B.Wang, RG Cai, CY Lin, PLB(10), 1002.2679. RG Cai, ZY Nie, B Wang, HQ Zhang, 1005.1233 • It has been argued that the QNM can reflect the black hole phase transition G. Koutsoumbas, S. Musiri, E. Papantonopoulos, and G. Siopsis, JHEP 0610, 006 (2006). J. Shen, B. Wang, R. K. Su, C.Y. Lin, and R. G. Cai, JHEP 0707, 037 (2007). X. Rao, B. Wang, and G. H. Yang, Phys. Lett. B 649, 472 (2007).

34. the Einstein-Maxwell field interacting with a charged scalar field with a minimal Lagrangian The electrically charged black hole in d-dimensional AdS space is described by the metric The Hawking temperature: The gauge field ansatz is Considering the scalar field ψ perturbing the RN-AdS BH:

35. The radial part of the equation can be separated by setting the radial wave equation can be expressed as the effective potential Tortoise coordinate Negative will cause V negative Choosing scaling symmetry ,, change the value of L corresponds to vary the temperature of the black hole, L big, T small.

36. Fix q, change L Fix L, change q • Increase L, lower T, the spacetime is easier to be destroyed (for fixed coupling to EM field) • Increase q, the spacetime is easier to be destroyed (for fixed BH temperature)

37. 3. the topology of the spacetime has the influence on evolution of the perturbation The spherical background is the easiest to be destroyed due to the perturbation the spacial topology influence on the condensation?? 4. the same black hole parameters (r+,Q,L), the stability of the black hole spacetime can be broken easier in the high dimensions. the scalar hair can be formed easier in the higher dimensional background! Stability of charged fermions perturbation in a Reissner-Nordstrom-anti-de Sitter black hole spacetime has been studied in RG Cai, ZY Nie, B Wang, HQ Zhang, 1005.1233

38. Part 3 Probe limit in GB AdS BH • Motivation to study the GB AdS BH: 1. Examine the Mermin-Wagner (MW) theorem in holographic superconductors. The MW theorem forbids continuous symmetry breaking in (2+1)-d because of large fluctuations in lower dimensions. It is possible that fluctuations in holographic superconductors in 2+1d are suppressed because classical gravity corresponds to the large N limit. If this is true, then higher curvature corrections should suppress condensation. 2. Examine the so called universal relation 3. Examine the dimensional influence on condensation Gregory et al, JHEP(09) QY Pan, B Wang et al, PRD(10) 0912.2475

39. The background solution of a neutral GB AdS black hole is in the asymptotic region We can define the effective asymptotic AdS scale the temperature of the CFT: In the background of the d-dimensional GB AdS black hole, we consider a Maxwell field and a charged complex scalar field with the action

40. In probe limit: Taking the ansatz only function of r the equations of motion for Boundary conditions: at the horizon at the asymptotic region AdS/CFT correspondence, We can impose boundary conditions that either vanishes

41. The condensations of these operators are subjected to curvature corrections The four lines from bottom to top correspond to increasing mass, The scalar mass influence on the condensation: a. for the same the condensation gap becomes larger if m is less negative b. The difference caused by the influence of the scalar mass will become smaller when there is higher curvature correction in the AdS background.

42. The influence the dimensionality on the scalar condensation. For the same scalar mass, we see: as the spacetime dimension increases, the condensation gap becomes smaller for the same the scalar hair can be formed easier in the higher-dimensional background. the difference caused by the curvature corrections are reduced when the spacetime dimension becomes higher.

43. The condensation for the scalar operator for different choices of the mass of the scalar field has completely different behavior as is changing. Selecting the value of we get the same qualitative dependence of the condensation for

44. the high curvature corrections really change the expected universal relation in the gap frequency. Pan, Wang et al PRD(10)

45. Part 4 Including magnetic field The holographic superconductor in the presence of the external magnetic field (Probe limit) In Superconductor: Meissner effect, magnetic field is expelled From the Ginzburg-Landau theory, the upper critical magnetic field How about in holographic superconductor??

46. We begin with the 4-dimensional Schwarzschild AdS black hole the metric can be rewritten introduce a charged, complex scalar field into the 4-dimensional Einstein-Maxwell action with negative cosmological constant In the probe approximation, the Maxwell and scalar field equations obey

47. To solve these equations analytically we will follow the logic used by Abrikosov • consider the weak magnetic field limit, and , obtain the spatially • independent condensate solutions. To the sub-leading order, we can • treat the magnetic field as a small perturbation. • 2. consider that the magnetic field is strong enough. We will regard the • scalar field as a perturbation and examine its behavior in the presence • of strong magnetism.

48. Weak magnetic field limit: In the weak magnetic field limit, the equations of motion reduce Condensations are similar to those found for electric field.

49. Strong magnetic field limit We will regard the scalar field as a perturbation and examine its behavior in the neighborhood of the upper critical magnetic field the scalar field AdS/CFT The vacuum expectation values at the asymptotic AdS boundary bulk coordinate boundary coordinates To the leading order, it is consistent to set the ansatz

50. The equation of motion for then becomes separating the variables, we have at the horizon z = 1 near the AdS boundary z -> 0 Requiring the solutions to be connected smoothly, we got the condensation of the scalar field at critical Tc, and Tc~Bc Ge,Wang et al (10)