Lectures on holographic methods for condensed matter physics

TL;DR

Hartnoll surveys how holographic duality can model strongly coupled condensed matter phenomena, emphasizing quantum criticality, transport, spectral functions, and holographic superconductivity. The notes develop a minimal AdS/CFT toolkit—Schwarzschild and RN–AdS black holes, Lifshitz/Schrödinger geometries, and Einstein–Maxwell–scalar systems—to compute thermodynamics, linear response, and order-parameter dynamics. Concrete results include conductivities, diffusion, cyclotron resonances, impurity-induced momentum relaxation, and emergent superconductivity from scalar condensation, with qualitative parallels to graphene and heavy-fermion cuprates. The discussion also clarifies the strengths and limits of holographic approaches, highlighting their value as tractable laboratories for universal strongly coupled physics while acknowledging gaps to real materials and the large-N framework.

Abstract

These notes are loosely based on lectures given at the CERN Winter School on Supergravity, Strings and Gauge theories, February 2009 and at the IPM String School in Tehran, April 2009. I have focused on a few concrete topics and also on addressing questions that have arisen repeatedly. Background condensed matter physics material is included as motivation and easy reference for the high energy physics community. The discussion of holographic techniques progresses from equilibrium, to transport and to superconductivity.

Figures (15)

• Figure 1: Typical temperature and coupling phase diagram near a quantum critical point. The two low temperature phases are separated by a region described by a scale-invariant theory at finite temperature. The solid line denotes a possible Kosterlitz-Thouless transition. Figure taken from reference sachdev2.
• Figure 2: At large $g$, the dashed couplings are weaker ($J/g$) than the solid ones ($J$). This favours pairing into spin singlet dimers as shown. Figure taken from reference sachdev2.
• Figure 3: Phase diagram of CePd$_2$Si$_2$ as a function of temperature and pressure. The bottom left phase is antiferromagnetically ordered whereas the bottom middle phase is superconducting. Figure taken from reference heavyAF.
• Figure 4: Schematic temperature and hole doping phase diagram for a high-$T_c$ cuprate. There are antiferromagnetic and a superconducting ordered phases. Figure taken from kitpblog.
• Figure 5: Eternal black hole in AdS together with an initial spacelike slice for evolution in the right asymptotic region. Straight blue lines denote the asymptotically AdS boundaries and the future and past event horizons. Black lines are singularities. The red line denotes a possible time slice ending on the future event horizon.