Building an AdS/CFT superconductor

TL;DR

The paper demonstrates a holographic model in which a planar AdS black hole coupled to a charged scalar in the bulk yields a 2+1D superconductor on the boundary, with a second-order phase transition and a divergent DC conductivity. By analyzing Maxwell fluctuations in the bulk, the authors show that below Tc the AC conductivity develops a gap and a delta function at zero frequency, from which a superfluid density is extracted; the normal component decays exponentially with a pairing energy Δ that relates to the condensate via ⟨O_i⟩/2. They identify 2Δ as the charged-particle gap, consistent with strong pairing, and discuss coherence-factor–dependent signatures for the two boundary operators, along with sum-rule constraints. The work highlights how a relatively simple AdS/CFT setup can capture key superconducting features in a strongly coupled, large-N regime and outlines several natural extensions (backreaction, magnetic fields, broader operators) to deepen the holographic understanding of pairing mechanisms.

Abstract

We show that a simple gravitational theory can provide a holographically dual description of a superconductor. There is a critical temperature, below which a charged condensate forms via a second order phase transition and the (DC) conductivity becomes infinite. The frequency dependent conductivity develops a gap determined by the condensate. We find evidence that the condensate consists of pairs of quasiparticles.

Figures (4)

• Figure 1: The condensate as a function of temperature for the two operators ${\mathcal{O}}_1$ and ${\mathcal{O}}_2$. The condensate goes to zero at $T=T_c \propto \rho^{1/2}$.
• Figure 2: The formation of a gap in the real, dissipative, part of the conductivity as the temperature is lowered below the critical temperature. Results shown for both the ${\mathcal{O}}_1$ operator (left) and the ${\mathcal{O}}_2$ operator (right). There is also a delta function at $\omega = 0$. The rightmost curve in each plot corresponds to $T/T_c = .0066$ (left) and $T/T_c = .0026$ (right).
• Figure 3: The gap at small $T/T_c$, with the frequency normalised in terms of the condensate. On the right the gap is finite, but since $\langle {\mathcal{O}}_1 \rangle$ becomes large at small $T$, the gap on the left is also becoming large. The dashed curve on the left plot is (\ref{['eq:sqrtlaw']}).
• Figure 4: The imaginary part of the conductivity at small $T/T_c$, with the frequency normalised in terms of the condensate. The analytic expression (\ref{['eq:sqrtlaw']}) is also shown on the left, but is indistinguishable from the numerics.