General Relativity and the Cuprates

TL;DR

This work extends a basic holographic superconductor by introducing a periodic lattice and computes the optical conductivity in the superconducting state. It reveals a two-fluid response with a Drude-like normal component, a superconducting delta function, and a robust mid-infrared power law of exponent -2/3 that matches cuprate measurements, along with a gap Δ ≈ 4 Tc and significant uncondensed spectral weight at low temperatures. The Ferrell-Glover-Tinkham sum rule is satisfied only when high-frequency contributions are included, and the results suggest a degree of universality in strong-coupling transport captured by holography, while highlighting limitations such as the s-wave nature of the model and the need to explore d-wave generalizations.

Abstract

We add a periodic potential to the simplest gravitational model of a superconductor and compute the optical conductivity. In addition to a superfluid component, we find a normal component that has Drude behavior at low frequency followed by a power law fall-off. Both the exponent and coefficient of the power law are temperature independent and agree with earlier results computed above $T_c$. These results are in striking agreement with measurements on some cuprates. We also find a gap $Δ= 4.0\ T_c$, a rapidly decreasing scattering rate, and "missing spectral weight" at low frequency, all of which also agree with experiments.

Figures (10)

• Figure 1: For each $T/\mu$ we plot the charge of the scalar field for which $T$ would be the critical temperature. This is repeated for several values of the amplitude of the lattice. From the top down the lines represent: $A_0 = 0.0, 0.4, 0.8, 1.2, 1.6, 2.0, 2.4$. Setting $e =2$ we read off the critical temperatures used in this paper.
• Figure 2: The mean value of the condensate as a function of temperature for $e = 2$ and various values of the lattice amplitude. From the inner to outer curves: $A_0 = 0.0, 0.4, 0.8, 1.2, 1.6, 2.0, 2.4$. The colors agree with Fig. \ref{['fig:charge']}.
• Figure 3: The condensate as a function of $x$ for $e=2$, $A_0 = 2$, and $T/T_c = .2, .7, .9, .99$ from top down. Note that by $T/T_c = .7$ the condensate has almost reached its low temperature limit.
• Figure 4: The real and imaginary parts of the conductivity for $A_0 =2, k_0 = 2$, and $T/T_c = .71$.
• Figure 5: The low frequency part of the conductivity for $T/T_c = 1.0$ (blue circles), .97 (red squares), .86 (yellow diamonds), and .70 (green triangles). The vertical red line in Re$(\sigma)$ denotes the zero frequency delta function.