Introduction to Holographic Superconductors

TL;DR

This work surveys how gauge/gravity duality can model superconductivity via a charged AdS black hole developing hair, linking bulk scalar condensation to a boundary superconducting order parameter. It analyzes both the probe limit and full backreaction, showing a second‑order superconducting transition with a nonzero condensate and a delta function in the optical conductivity, alongside a robust gap‑to‑Tc ratio. Magnetic fields reveal type II behavior with a vortex lattice and a London–type response, while zero‑temperature limits and string‑theory embeddings illuminate IR structure and potential extensions to P/W/D‑wave and lattice phenomena. The study underscores both the explanatory power and the open, technically rich questions surrounding holographic superconductors and their connection to real strongly coupled materials.

Abstract

These lectures give an introduction to the theory of holographic superconductors. These are superconductors that have a dual gravitational description using gauge/gravity duality. After introducing a suitable gravitational theory, we discuss its properties in various regimes: the probe limit, the effects of backreaction, the zero temperature limit, and the addition of magnetic fields. Using the gauge/gravity dictionary, these properties reproduce many of the standard features of superconductors. Some familiarity with gauge/gravity duality is assumed. A list of open problems is included at the end.

Figures (10)

• Figure 1: The condensate as a function of temperature. The critical temperature is proportional to the chemical potential.
• Figure 2: Condensates with different dimension, $\lambda$, as a function of temperature. The condensate tends to increase with $\lambda$. Figure is taken from Horowitz:2008bn.
• Figure 3: The formation of a gap in the real part of the conductivity as the temperature is lowered below the critical temperature. The curves describe successively lower temperatures. There is also a delta function at $\omega = 0$. Figure is for the dimension two condensate and is taken from Hartnoll:2008vx.
• Figure 4: The low temperature limit of the optical conductivity for the dimension two condensate. The solid line denotes the real part and the dashed line denotes the imaginary part. Figure is taken from Horowitz:2008bn.
• Figure 5: The low temperature limit of the conductivity for the dimension 3/2 condensate. Note the extra spike that appears inside the gap. Figure is taken from Horowitz:2008bn.