1
The Holographic Superconductor Johar M. Ashfaque The holographic superconductor has been around for a long time. But the most important contributions came only in 2008. The holographic superconductor is an example of the phenomenological AdS/CF T where we do not start with a well-defined string theory duality but rather we assume that there is an AdS/CF T duality that would describe the system of interest and introduce the necessary ingredients to define it without ever deriving them from the fundamental description. I.
HOLOGRAPHIC SUPERCONDUCTOR INGREDIENTS
In order to define a holographic superconductor, we will need the following ingredients: 1. We need an AdS background, since we want to describe an approximately conformal field theory (conformal at zero temperature). Moreover, the superconductors that are not well understood are the high Tc ones, which are non-Fermi liquids (can not be described by the standard Fermi liquid theory), that are at strong coupling, and so can admit a weak coupling gravitational description in the dual. Also, many of them are layered like the cuprates and the organic superconductors and therefore can be approximated as being 2 + 1-dimensional. 2. We want to describe the charge transport, so we need to describe a conserved U (1) current Jµ in the field theory that couples in the gravity dual to a gauge field Aµ (Aµ is a source for the operator R Jµ through a coupling J µ (x)Aµ (x)). 3. We want to describe a superconductor, therefore we would like to have symmetry breaking. We will consider at least one composite operator O to have non-zero VEV in the superconducting phase. This operator will couple to the field φ in the bulk. 4. Finally, we want to describe a theory at finite temperature which boils down to introducing a black hole in the gravity dual. With these ingredients, the Lagrangian for the gravity dual is found to be given by 1 d(d − 1) 1 2 L= 2 R+ − |(∂µ − iqAµ )ψ|2 − m2 |ψ|2 − V (|ψ|). − 2 Fµν 2κ L2 4g Here d is the dimension of the boundary field theory, d = 3 in the case of AdS4 and d(d − 1) ≡ −Λ L2 is the cosmological constant. Also V (|ψ|) is the potential. We will be interested in the case where V = 0 and m2 corresponding to the field ψ is stable at infinity. In Minkowski space, stability is simple: a field with m2 ≥ 0 is stable since that means that the curvature of the potential is positive so the field can not slip away to infinity whereas m2 < 0 is unstable and the field can slip away to infinity, lowering its energy. However, in the AdS background, Breitenlohner and Freedman found that there is a modification of this rule due to the extra terms from the curved background in the Klein-Gordon equation and we have stability even if m2 < 0 satisfying the Breitenlohner-Freedman (BF) bound. For scalars, the bound is m2 ≥ −
d2 4L2
for AdSd+1 . For AdS4 this gives 9 m2 L2 ≥ − . 4 Since we want to describe a superconducting system, it means that for T < Tc we must have a composite operator which is called the condensing operator hOi = 6 0. But since O couples to ψ through the coupling
R
Oψ. This means that hOi = 6 0 can happen only if ψ 6= 0.
2 II.
SUPERCONDUCTING BLACK HOLES A.
Ansatz
The needed ansatz for the holographic dual to the superconductor is then ds2 = gtt (r)dt2 + grr (r)dr2 + ds22 (r) Aµ dxµ = Φ(r)dt,
ψ = ψ(r).
In this ansatz, the terms involving ψ in the Langraian become −|(∂µ − iqAµ )ψ|2 − m2 |ψ|2 → −g rr |∂r ψ|2 − (m2 + g tt q 2 Φ2 )|ψ|2 . Since g tt < 0, we note that the effective mass squared m2ef f that appears in front of |ψ|2 is smaller than the mass squared m2ef f = m2 + g tt q 2 Φ2 < m2 .
B.
Background
In fundamental AdS/CF T , the probe approximation is usually consider one brane or Nf N branes as probes in the background created by other N branes and in that case one can consider it as an approximation in Nf /N . As the background we can either consider the AdS4 -Reissner-Nordstrom black hole or the neutral black hole. The AdS-RN background is written like all black holes in terms of a function appearing in gt t and in g rr as ds2 = −f (r)dt2 +
f (r) = k −
Φ(r) =
dr2 + r2 dΩ22(k) , f (r)
Q2 r2 2M + 2 + 2, r 4r R
Q Q − , r rH
ψ = 0,
dΩ2k=1 = dθ2 + sin2 θdφ2 , dΩ2k=0 = dx2 + dy 2 . We note here that k = 1 is the usual black hole, which in the extremal case has the horizon of the type AdS2 × S 2 but we also have the k = 0 black hole with horizon of the type AdS2 × R2 which is what we are interested in. Also note that an extra constant term has been added to Φ besides the usual electric potential to have Φ(r = rH ) = 0. The neutral black hole background is obtained by putting Q = 0 but also considering k = 0 so dΩ22 = dx2 + dy 2 and f (r) =
2M r2 − . R2 r
3 C.
Probe Approximation
The equations of motion in the probe approximation are the following: • The scalar ψ equation on the ansatz √ 1 √ ∂r −gg r r∂r ψ + m2ef f ψ = 0 −g which explicitly becomes ψ 00 +
m2 f0 2 Φ2 + ψ0 + 2 ψ − ψ = 0. f r f f
• The equation of motion for Φ in the background is 2 2ψ 2 Φ=0 Φ00 + Φ0 − r f which is the same equation which one would expect for a massive vector field. We are primarily interested in the case of m2 = −2/R2 which is obtained for either ∆ = 1 or ∆ = 2. Then at infinity we have ψ(r) '
ψ (2) ψ (1) + 2 + ... r r
Φ(r) ' µ −
ρ + .... r
It so happens that ψ (1) and ψ (2) are normalizable and therefore correspond to the scalar VEVs for the composite operators √ hOi i = 2ψ (i) , i = 1, 2. We can choose either one of the scalar modes to vanish and calculate the VEV generated by the other. Putting µ = 0, and considering only ρ 6= 0, we can calculate Tc , ψ (i) and hence hOi i. Numerically one finds the results shown below:
FIG. 1. (a) Condensate 1 as the function of temperature. (b) Condensate 2 as a function of temperature.
Moreover, near T = Tc one finds a square root vanishing with 1/2 T hO1 i ' 9.3Tc 1 − , Tc 1/2 T hO2 i ' 144Tc 1 − . Tc One also finds numerically the relation between the critical temperature Tc and the charge density ρ √ Tc (µ = 0) ' 0.118 ρ.
4 III.
BREITENLOHNER-FREEDMAN BOUND AND SUPERCONDUCTOR
One can calculate m2ef f at the horizon in the extremal AdS4 -RN black hole and obtain m2ef f = m2 −
γ 2 q2 < m2 2R2
where q is the charge and γ2 =
g 2 2R2 . κ2N,4
On the other hand, to obtain the instability at the horizon, the effective mass square needs to violate the BF bound at the horizon. We can conclude that the ingredients necessary for holographic superconductivity of the gravity dual are: • We need to have a BF bound that is weaker at the horizon than at infinity. • Due to the coupling of the scalar to A0 , the effective mass squared is lowered at the horizon. IV.
SUMMARY
• For a holographic high Tc layered superconductor, we need an AdS4 background with a bulk gauge field, bulk charged scalar and a black hole. • For the scalar to be superconducting, we need that ψ = 0 to be thermodynamically unstable towards a hairy solution with ψ 6= 0 for T < Tc , where the scalar is stable at infinity but unstable at the horizon. Then m2ef f < m2 because of the coupling to the gauge field A0 . • The background in which we can put the scalar and the electric potential is an AdS4 -ReissnerNordstrom black hole or a neutral AdS4 black hole with planar horizon. • The existence of a Tc requires non-zero chemical potential µ or charge density ρ. √ • For ∆ = 1 or ∆ = 2, we obtain hOi i ∝ (1 − T /Tc )1/2 as T → Tc and Tc ∝ ρ. • There is a bound on the dimensions of operators that superconduct, 3 + 2∆(∆ − 3) ≤ q 2 γ 2 . Here, we made use of the fact that the mass m of the scalar field φ is related to the dimension ∆ of the operator dual to it by m2 R2 = ∆(∆ − 3). • To superconduct, we need a BF bound at the horizon that is weaker than at infinity, allowing for a field stable at infinity, but unstable at the horizon and the coupling of ψ with A0 .