Lectures on holographic methods for condensed matter physics

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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.

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Quantum critical phenomena

John Hertz (1976)

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Gravity Duals of Lifshitz-like Fixed Points

Shamit Kachru, Xiao Liu, Michael S Mulligan (2008)

We find candidate macroscopic gravity duals for scale-invariant but non-Lorentz invariant fixed points, which do not have particle number as a conserved quantity. We compute two-point correlation functions which exhibit novel behavior relative to their AdS counterparts, and find holographic renormalization group flows to conformal field theories. Our theories are characterized by a dynamical critical exponent \(z\), which governs the anisotropy between spatial and temporal scaling \(t \to \lambda^z t\), \(x \to \lambda x\); we focus on the case with \(z=2\). Such theories describe multicritical points in certain magnetic materials and liquid crystals, and have been shown to arise at quantum critical points in toy models of the cuprate superconductors. This work can be considered a small step towards making useful dual descriptions of such critical points.