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

超导： 二个元素 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！

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 Outline:

1) Building a holographic superconductor i) Breaking an Abelian gauge symmetry near a black hole horizon S. Gubser, 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/ ) two Abelian gauge fields and a non-normalizable coupling of the scalar to one of them.

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.

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.

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

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.

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

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.

Flat case: Here require 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.

ii)Building a holographic superconductor S. Hartnoll, C.P. Herzog and G. Horowitz, arXiv: PRL 101, (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:

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

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 source current Conductivity:

frequency dependence Superfluid density

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.

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

Consider m^2=0, -9/4 in d=3 case m^2=0, -3, -4 in d=4 case. 1. when m^2 >= -d^2/4+1, only the “+” mode is normalizable. 2. when –d^2/4 <=m^2 <-d^2/4+1, both modes normalizable. 3. 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:

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.

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

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/

T/T_c~0.1 Massless case BF

Massless case! BF

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?

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

The conductivity is directly related to the reflection coefficient, with the Frequency simply giving the incident energy. Therefore: 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.

2) Backreaction and others Holographic superconductors 3H’s, arXiv: Extend in two directions: 1)any charge of scalar field, back reaction. 2) background magnetic field. The model:

The metric ansatz: Choosing corresponding to a conformal scalar.

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

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

The critical temperature: The probe limit

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

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:

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

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

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. 1)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.

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.

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,

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

3) Zero temperature limit i) Zero temperature limit of Holographic superconductors G. Horowitz and M. Roberts, arXiv: ii) Low temperature behavior of the Abelian Higgs Model in AdS Space S. Gubser and A. Nellore, arXiv: Ground States of holographic superconductors arXiv: * 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.

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).

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

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

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.

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

(i) Superconductors from superstrings S. Gubsers et al, arXiv: , PRL(2009)

The model: For small eta.

Up lift to 10 dim. The self-dual five-form:

The phase transition: high temperature phase: Low temperature phase: hairy black hole with eta \=0.

The pressure difference between the unbroken and broken phases. Near T_0, Second order phase transition!

(ii) Holographic superconductivity in M theory J.P. Gaunntlett et al. arXiv: , PRL (2009) Quantum Criticality and Holographic superconductors in M theory arXiv: Any SE_7 metric can, locally, be written as a fibration over a six-dim Kahler-Einstein space, KE_6: Here eta is the one-form dual to the Reeb Killing vector satisfying d\eta= 2J where J is the Kahler form of KE_6.

The D=4 equation of motion admits a vacuum solution with vanishing matter fields, which uplifts to the D=11 solution: 1.When epsilon=1, this AdS_4 x SE_7 solution is supersymmetric and describes M2-branes sitting at the apex of the Calabi-Yau four-fold (CY_4) cone whose base space is given by the SE_7. 2. When epsilon=-1, the solution is a “skew-whiffed” AdS_4 X SE_7 solution, which describes anti-M2-branes sitting at the apex of the CY_4 cone. These solutions break all of the supersymmetriy, except for the special case when the SE_7 is the round 7-sphere, S^7, in which case it is maximally supersymmetric.

3H’s paper L=1/2, q=2 In the case 2, one has the action with scaling dimension \triangle=1 or 2. RN-AdS

5) P-wave superconductors and back reaction S. Gubser and S. Pufu, arXiv: M. Ammon, et al., arXiv: The order parameter is a vector! The model is

Near horizon: Far field: The total and normal component charge density: Defining superconducting charge density:

For the equations of motion, there is a one-parameter family of solutions. Each point along the contour represents a solution. Points on the curve labeled by “ superconducting” breaking the abelian symmetry generated by U(1)_3. T

Electromagnetic perturbations:

Back reaction: second/first order phase transition depends on the coupling alpha, the ration of the five dimensional gravitational constant to the Yang-Mills coupling. Second order: in the probe limit, alpha=0. First order: alpha_c> Model:

Gauge field: Metric: Equations of motion:

There are four scaling symmetries: One exact solution: w=0, RN-AdS black hole. But we are interested in the solutions with w=\ 0 Thermodynamics:

alpha=0.316, T/T_c= 0.45 The results show: second order phase transition as alpha <alpha_c=0.365 first order phase transition as alpha >alpha_c

alpha=0.032, alpha=0.316, alpha=0.447, 160 (1-T/T_c)^(1/2) Note that: difference scales for different curves.

alpha=0.316 alpha=0.447

The critical temperature vs alpha: Some intuition: in dual CFT, increasing alpha means increasing the ratio of charged degrees of freedom to total degree of freedoms.

6) D-wave superconductor J.W. Chen, et al., arXiv: The model: Symmetric traceless tensor Note that it is not trivial to construct a higher spin (interacting) field model even in a flat spacetime.

In the probe limit:

The order parameter: m^2=-1/4 The critical exponent:

The conductivity: T decreases

7) Open questions? (G.T. Horowitz, arXiv: )

Thank You !