Holographic Superconductors

TL;DR

The paper constructs and analyzes a holographic model of a 2+1D superconductor via its AdS/CFT dual, incorporating backreaction beyond the probe limit and examining magnetic responses. Using a bulk action with gravity, a Maxwell field, and a charged scalar, it demonstrates a superconducting phase below a critical temperature with a hairy black-hole geometry, and computes electric, thermal, and thermoelectric conductivities, revealing a finite gap and a delta-function at zero frequency due to translational invariance. It further shows type II behavior with superconducting droplets in a magnetic field, derives the London equation and a finite photon mass when gauging, and discusses the relation to Landau-Ginzburg theory as a strong-coupling, microscopic-inspired framework. The results provide a robust holographic platform for exploring strongly coupled superconductivity, quantum criticality, and magnetic response, with implications for embedding in string theory and for understanding non-BCS pairing mechanisms.

Abstract

It has been shown that a gravitational dual to a superconductor can be obtained by coupling anti-de Sitter gravity to a Maxwell field and charged scalar. We review our earlier analysis of this theory and extend it in two directions. First, we consider all values for the charge of the scalar field. Away from the large charge limit, backreaction on the spacetime metric is important. While the qualitative behaviour of the dual superconductor is found to be similar for all charges, in the limit of arbitrarily small charge a new type of black hole instability is found. We go on to add a perpendicular magnetic field B and obtain the London equation and magnetic penetration depth. We show that these holographic superconductors are Type II, i.e., starting in a normal phase at large B and low temperatures, they develop superconducting droplets as B is reduced.

Figures (8)

• Figure 1: The value of the condensate as a function of temperature, with the charge density held fixed, for the two different boundary conditions: a) from bottom to top, the various curves correspond to ${q}=1$, 3, 6, and 12; b) from top to bottom, the curves correspond to ${q} = 3$, 6, and 12. The value $q=1$ gives a much larger condensate in this case, achieving $\sqrt{q \langle {\mathcal{O}}_2 \rangle}/T_c \approx 21$ so we have not plotted it. Note that the large $q$ limit is approached in opposite directions in the two cases.
• Figure 2: The blue solid line is the critical temperature as a function of ${q}$. The dashed red line is the probe limit (naively extrapolated to all ${q}$). a) The dimension one case where the probe limit corresponds to $0.2255 \sqrt{{q}}$; b) the dimension two case with the probe limit $0.1173 \sqrt{{q}}$.
• Figure 3: The dashed red line is the real part of the conductivity at $T=T_c$ (for $q=3$). The blue lines are the same conductivities at successively lower temperature: a) The dimension one operator with $T/T_c = 0.810$, 0.455 and 0.201; b) the dimension two operator with $T/T_c = 0.651$ and 0.304. There is a delta function at the origin in all cases.
• Figure 4: We plot the real part of the conductivity as a function frequency normalized by the condensate, either ${q} \langle {\mathcal{O}}_1 \rangle$ or $\sqrt{ {q} \langle {\mathcal{O}}_2 \rangle}$ as appropriate. This data was taken at low temperature, $T = 0.03 \ {q} \langle {\mathcal{O}}_1 \rangle$ and $T = 0.03 \sqrt{ {q} \langle {\mathcal{O}}_2 \rangle}$ for a variety of charges ${q} = 1, 3, 6$ and $12$. The curves with steeper slope correspond to larger ${q}$. There is a delta function at the origin.
• Figure 5: In order to prevent the flux from penetrating the superconductor, of area $R^2$, the currents would have to do enough work to expel the field from a volume of size $R^3$, as shown in the left hand figure. This work cannot be supplied by the free energy gain of superconducting on the thin film. Therefore, the flux always penetrates the film, as shown in the right hand figure.