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Introduction to Holographic Superconductors

Introduction to Holographic Superconductors. Rong-Gen Cai （ 蔡荣根 ）. Institute of Theoretical Physics Chinese Academy of Sciences. Xidi, Anhui, May 28-June 6, 2010. 引力： 它不仅是一个基本相互作用， is a theory of everything? AdS/CFT is not only a correspondence, 也是一个工具！.

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Introduction to Holographic Superconductors

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  1. Introduction to Holographic Superconductors Rong-Gen Cai （蔡荣根） Institute of Theoretical Physics Chinese Academy of Sciences Xidi, Anhui, May 28-June 6, 2010

  2. 引力： 它不仅是一个基本相互作用， is a theory of everything? AdS/CFT is not only a correspondence, 也是一个工具！ 超导： 二个元素

  3. Outline: 1) Building a holographic superconductor 2) Back reaction and something else 3) Zero-temperature limit 4) Realization in String theory 5) p-wave superconductors & back reaction 6) d-wave superconductor 7) Open questions

  4. 1)Building a holographic superconductor i) Breaking an Abelian gauge symmetry near a black hole horizon S. Gubser, 0801.2977 Coupling an Abelian Higgs model to gravity plus a negative cosmological constant leads to black hole which spontaneously break the gauge invariance via a charged scalar condensate slightly outside their horizon. This suggests that black holes can superconduct. (S. Gubser, phase transitions near black hole horizons, hep-th/0505189) two Abelian gauge fields and a non-normalizable coupling of the scalar to one of them.

  5. Consider the following action: i) Comparing the usual Abelian Higgs model, a term phi^4 is missed ii) Comparing the usual Ginzburg-Landau story, even for the case of m^2>=0, the symmetry breaking can still happen. The charged black hole makes an extra contribution to the scalar potential which makes the phi=0 solution unstable, provided that q is large enough and that m^2 is not too positive, and that the black hole sufficiently highly Charged and sufficiently cold.

  6. The effective potential for the scalar field: A negative m_{eff}^2 produces a unstable mode in phi. * Seek for a marginally stable linearized perturbations around these Solutions breaking the u(1) symmetry.

  7. A self-consistent solution: RN-AdS black hole Now consider the perturbation around the RN-AdS black hole. Scaling symmetry: X’s charge alpha

  8. which lead to the simplification: r_+=Q=1 and remaining two parameters k and Q; A marginally stable perturbation is one where phi depends only on r and is infinitesimally small (that is, it doesn’t back-react on the other field. The simplest possibility is for the appearance of this mode to signal a second order phase transition to an ordered state where phi is not-zero. Assume that phi is real everywhere, because phase oscillations in the r direction would only raise the energy of the mode, making it less likely become unstable.

  9. Boundary value problem: (1) At the horizon r=1. (2) At asymptotically infinity.

  10. Two solutions: （1）if m^2L^2 > -5/4, which is not permitted by unitary (2) if -9/4 <m^2L^2 <-5/4, either solution is permitted.

  11. Here require Flat case: As a result, if phi is stable at infinity, then is still stable at the horizon. Special case: If m^2 < 4 q^2, unstable, super-radiation.

  12. Building a holographic superconductor • S. Hartnoll, C.P. Herzog and G. Horowitz, arXiv: 0803.3295 • PRL 101, 031601 (2008) Temperature: black hole Condensate : charged scalar field Black hole solution with scalar hair at low temperature, no hair at high temperature. At high temperature:

  13. Consider the case m^2L^2=-2, which corresponds to a conformal coupled scalar. In the probe limit and A_t= Phi The condensate of the scalar operator O_i in the field theory At the large r limit:

  14. Conductivity The Maxwell equation at zero spatial momentum and with a time dependent of the form Exp [-i w t]: At the horizon: ingoing wave condition At the infinity: AdS/CFT current source Conductivity:

  15. frequency dependence Superfluid density

  16. Summary: • The CFT has a global abelian symmetry corresponding a • massless gauge field propagating in the bulk AdS space. • 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. • 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.

  17. iii) Holographic superconductors with various condensates G.T. Horowitz and M. Roberts, arXiv: 0810.1077 Different mass cases between m^2=0 and m^2_{BF} in 2+1 Dim and 3+1 Dim Consider the action: Background:

  18. Consider m^2=0, -9/4 in d=3 case m^2=0, -3, -4 in d=4 case. • when m^2 >= -d^2/4+1, only the “+” mode is normalizable. • when –d^2/4 <=m^2 <-d^2/4+1, both modes normalizable. • when m^2 = -d^2/4, there is a logarithmic branch, unstable unless • it acts as a source. On the other hand, the asymptotic behavior of varphi:

  19. The condensate tends to increase with lambda. • lambda > lambda_{BF}, (3/2 for d=3, 2 for d=4), the condensate • quickly saturates a fixed value; lambda=lambda_{BF}, approaches • to a fixed value roughly linearly; and lambda <lambda_{BF}, • it appears to diverge.

  20. Conductivity: this is related to the retarded current-current two-point function for our global U(1) symmetry The linearized equation of motion for The retard current Green function for the gauge field perturbation: (D. Son and A. Starinets, hep-th/0205051)

  21. d=3: d=4: The logarithmic divergence can be removed with a boundary counterterm in the gravity action. But it breaks the conformal invariance. After that, one has hep-th/0002125

  22. BF Massless case T/T_c~0.1

  23. BF Massless case!

  24. with deviations of less than 8%, while the corresponding one is 3.5 in BCS theory. * The case of lambda=lambda_{BF} looks strange? Is it true?

  25. Reformulation of the conductivity Introducing : ( -infinity, 0) lambda=2 q=2 V(0) =0 lambda>1 =const lambda=1 =diverges, if ½ <lambda <1 T=0 T=T_c

  26. Therefore: The conductivity is directly related to the reflection coefficient, with the Frequency simply giving the incident energy. Spike in the case of lambda=lambda_{BF}: at low frequency, the incoming wave from the right is almost entirely reflected. If the potential is high enough, one can raise the frequency so that about one wavelength fits between the potential and z=0. In this case, the reflected wave can interfere with the incident and cause its amplitude at z=0 to be exponentially small. This produces a spike in the conductivity. If one can raise the frequency so that two wave-lengths fits between the potential and z=0, one gets the second spike.

  27. 2)Backreaction and others Holographic superconductors 3H’s, arXiv: 0810.1563 • Extend in two directions: • any charge of scalar field, back reaction. • 2) background magnetic field. The model:

  28. The metric ansatz: Choosing corresponding to a conformal scalar.

  29. The most important feature is that in all cases there is a critical temperature T_c below which a charged condensate form T>T_c: RN-AdS black hole solution

  30. q=1,3,6,12 q=3,6,12 Full back reacting system cures the divergence!

  31. The critical temperature: The probe limit

  32. T=T_c, q=3 Conductivity T/T_c=0.810,0.455,0.201 T/T_c=0.651, 0.304

  33. Critical magnetic fields: Type I: there is a first order phase transition at H=H_c, above which magnetic field lines penetrate uniformly. Type II: vortices start to form at H=H_{c1}. In the vortex core, the materials reverts to its normal state and magnetic field lines are allowed to penetrate. The vortices become more dense as the magnetic field is increased and at an upper critical field strength H=H_{c2}, the material ceases to superconduct. The starting point is the dyonic black hole solution:

  34. Working in polar coordinates: Scalar equation: Zero mode: instability Set: The lowest mode n=1: c_1=0 Near horizon:

  35. q=12,6,3 q= 12,6,3 The critical magnetic field: B_{c2} At lower temperature a superconductor can support a larger magnetic field.

  36. The London equation in low temperatures The London equation: , valid for small w and k. which explains both the infinite conductivity and the Meissner effect. • * One important and subtle issue is that the two limits w->0 and k->0 • do not always commute. • In the limit k=0 and w->0, •  infinite DC conductivity. • 2) In the limit w=0 and k->0, • together with the Maxwell • equation , the other limit of the London equation • implies that magnetic field lines are excluded from superconductors.

  37. How to get the London equation? Take the form: Take k=0, there is a pole of Im[sigma] at w=0. Let the residence of the pole be n_s, the superfluid density.

  38. Correlation length: By the AdS/CFT, the retarded Green function for J_x is Defining a correlation length Solving the equation, the including k dependence,

  39. Vortices: the holographic superconductor vortex M. Montull et al, 0906.2396 PRL (2009) In the probe limit:

  40. 3) Zero temperature limit i) Zero temperature limit of Holographic superconductors G. Horowitz and M. Roberts, arXiv: 0908.3677 ii) Low temperature behavior of the Abelian Higgs Model in AdS Space S. Gubser and A. Nellore, arXiv: 0810.4554 Ground States of holographic superconductors arXiv: 0908.1972 * The extremal limit has zero charge inside the horizon. This is expected consequence of the horizon having zero horizon area. * The near horizon behavior of the zero temperature solution depends on the mass and charge of the bulk scalar field. Here discusses two cases, In both cases, one can solve for the solutions analytically near r=0.

  41. 1) m^2=0: this corresponds to a marginal, dimension three operator developing a nonzero expectation value in the dual superconductor. To determine the leading order behavior near r=0, make an ansatz: One has from the equations of motion: Then one can now numerically integrate this solution to large radius and adjust alpha so that the solution for phi is normalizable. One Finds it is possible provided q^2>3/4. (this is required by stability).

  42. The value of alpha depends weakly on q. In all cases |alpha| < 0.3 . Zero temperature, lambda=3, q=1 solution T

  43. 2) The ansatz near r=0: T=0

  44. The horizon at r=0 has a mild singularity. The scalar field diverges Logarithmically and the metric takes the form: The Poincare invariance is restored near the horizon, but not the full conformal invariance. If introduce a new radial coordinate then the metric becomes: near the horizon which is located at .

  45. 4) Realization in string/M theory • Superconductors from superstrings • S. Gubsers et al, arXiv: 0907.3510, PRL(2009) • (ii) Holographic superconductivity in M theory • J.P. Gaunntlett et al. arXiv: 0907.3796, PRL (2009) • Quantum Criticality and Holographic superconductors in M theory • arXiv: 0912.0512

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