limit the subleading term, on dimensional grounds, behaves as asx ∼r2f a0x
r→∞ ∼2κ2L4r4(A0t)2abx
r→∞, (3.41)
and thus asx ∼ O(ω0) which by equation (3.29) leads to a pole in the imaginary part of the conductivity proportional to the effective mass term4.
Infinite DC conductivity is also a characteristic of superconductors which we in-troduce in the next section.
In appendix A we present a different approach to compute the DC conductivity with holographic methods. It is based on the so-called Einstein relation which allows to express σDC entirely in terms of quantities characterising the thermody-namic equilibrium.
3.4. Holographic superconductivity 43
when χ≡0. However, as the temperature is lowered a second branch of solutions develops, the holographic dual of a superconducting state. To understand this let us look at the equation of motion for χ independently
χ00+ 4
r + f0 f + g0
g
χ0−
L2m2
r2f − L4e2A2t r4f2
χ= 0. (3.44) Note that of course the metric fieldsf andgand the Maxwell fieldAtare dynamical as well. However, the logic of the holographic superconductor forgoes without considering their dynamics. There are two effects which contribute to an arising instability of the scalar as temperature is decreased. Firstly, recall that a scalar in an AdS background is only stable above the BF bound (2.31). Thus a scalar mass M which satisfies that BF bound in the asymptotic AdS4 with radius Lgeometry can still violate the BF bound of the emerging AdS2 space with radiusL2 in the IR (3.23). This has the consequence that as temperature is lowered and the geometry resembles more and more the IR geometry (3.23) the scalar can become unstable5. The second effect is due to the coupling to the Maxwell field At which effectively reduces the mass of the scalar (3.44). At grows with decreasing temperature until eventually causes the effective mass to violate the BF bound. The instability results in a non-trivial profile of the scalar field χ(r) 6= 0. To summarise, we find the following allowed range for the scalar mass
−9
4 < L2m2 <−3
2 +3e2L6
4κ2 , (3.45)
where the upper bound is derived from the BF bound of AdS2 space and the contribution of the Maxwell field evaluated at T = 0. Choosing L2M2 = −2, in accordance with this range (3.45), the scalar’s boundary behaviour according to (2.28) is given by
χ= χ1 r +χ2
r2 +O r−3
. (3.46)
We do not want to source the non-trivial profile explicitly so we impose the bound-ary conditionχ0 = 0, switching off the non-normalisable mode. Nonetheless below a certain temperature, χ1 can be finite which leads to a spontaneous breaking of the U(1) gauge symmetry. The dual field theory operator has the interpretation
5With this logic it seems possible that there is a scalar instability even for a neutral scalar e= 0. This indeed was found and discussed in [17].
of a condensate ∆
∆≡χ2. (3.47)
Comparing the field theory grand potential (2.36) of the two branches of solutions it can be shown, that the dual of the solution with the scalar condensate indeed is energetically preferred over the dual of the AdS-RN solution. The transition to the new ground state behaves like a second order phase transition which goes in line with the mean field behaviour of the scalar condensate6
∆∼(1−T /Tc)1/2 (3.48)
close to the critical temperature Tc.
The development of the scalar profile can be thought of as a discharging process of the black hole due to pair production close to the horizon. The result goes as a ‘hair’ of the black hole. Note that this is not in contradiction with the ‘no-hair’-theorems, which state that black holes in flat space have to be uniquely characterised by a few charges, like their mass or electrical charge. In flat space, matter can only either fall into the black hole or radiate away to infinity. In AdS space however, the situation is different because the charged matter is trapped within the boundary but at the same time pushed away from the black hole by the electromagnetic force. As a result it is allowed to equilibrate in the vicinity of the horizon. This ‘work around’ the no-hair theorem lead to the discovery of many now black hole solutions.
It remains to show that the name giving feature of superconductor holds for this new type of solution: the infinite DC conductivity. Let us look at the equation of motion for the perturbationaxof the Maxwell field, analogously to the last section (3.32)7
r2f a0x0
+ω2
f ax =
2κ2L4(A0t)2+ 2e2|χ|2
ax+L2e2At|χ|2
r2f δgtx. (3.49) Comparing this equation to the one in the previous section it is clear that the scalar field term contributes to the effective mass of ax, in fact increases it. The
6The mean field behaviour is a consequence of the large-N limit [79].
7In contrast to the holographic setup in 3.3.2, it is now no longer possible to consistently switch on only the Maxwell field fluctuation. However, we use equation (3.49) at this point only on a conceptual level.
3.4. Holographic superconductivity 45
logic of (3.41) applies here despite the additional term proportional to the metric fluctuation δgtx. We can conclude that the presence of the condensate indeed contributes to the infinite DC conductivity and hence represents a superconducting phase of the dual field theory.
In addition to the infinite conductivity, a superconductor is also characterised by a gap in the fermion spectrum (3.11). This gap reappears in the frequency dependent behaviour of the electrical conductivity: Aside from the δ-peak at ω = 0, at low frequency the optical conductivity vanishes for frequencies smaller than the gap.
This can be thought of shifting the superconducting degrees of freedom at low energy into the δ-peak. Holographic superconductors do not show a true gap in the optical conductivity. However, it is suppressed exponentially and the order of magnitude of the gap is set by the condensate ∆ (3.47) [17], this may be called a pseudogap behaviour [14].
The situation where one part of the charge is still behind the horizon while the other part escaped and formed a condensate in the bulk corresponds to a state where part of the charge carriers is still in the normal phase while the other part forms a superconductor. Once again the microscopic physics of holographic super-conductor may be different from BCS theory, where charged quasiparticles form Cooper pairs. The systems with a holographic dual are truly quantum and have no quasiparticles.
At closer inspection we just discussed spontaneous symmetry breaking of the U(1) gauge symmetry in the bulk. However, according to the holographic dictionary, gauge symmetries of the gravity theory are dual to global symmetries of the quan-tum field theory [50]. The photon in holographic superconductors is thus not dynamical and the holographic superconductor really is a holographic superfluid.
Luckily this is not relevant for many of the properties of a real superconduc-tor [79, 80]. For example the Meissner-Ochsenfeld effect, where magnetic fields are expelled form the superconductor is captured by the holographic model as well [17].