Lectures on Holographic Superfluidity and Superconductivity

TL;DR

This work surveys how the AdS/CFT correspondence can illuminate strongly coupled condensed-matter phenomena, focusing on quantum phase transitions and the quantum critical region. It develops a field-theory framework for transport via Green's functions and Ward identities, then uses a simple Einstein–Maxwell gravity dual to compute transport coefficients and hydrodynamic behavior in 2+1 dimensions, including a cyclotron resonance at finite $T$, $B$, and $n$. The second half introduces holographic superconductors by coupling a charged scalar or SU(2) gauge field in the bulk, yielding a boundary order parameter below $T_c$, a London-like relation for the superfluid density, and a gapped conductivity with a calculable second-sound mode. Together, these results illustrate how holography can provide universal or semi-universal insights into transport and phase transitions in strongly interacting systems, while also highlighting limitations and directions for modeling real materials. $\,$

Abstract

Four lectures on holography and the AdS/CFT correspondence applied to condensed matter systems. The first lecture introduces the concept of a quantum phase transition. The second lecture discusses linear response theory and Ward identities. The third lecture presents transport coefficients derived from AdS/CFT that should be applicable in the quantum critical region associated to a quantum phase transition. The fourth lecture builds in the physics of a superconducting or superfluid phase transition to the simple holographic model of the third lecture.

Figures (9)

• Figure 1: A typical phase diagram involving a second order quantum critical point.
• Figure 2: Resistivity of thin films of bismuth versus temperature. The different curves correspond to different thicknesses, varying from a 4.36 Å film that becomes insulating at low temperatures, to a thicker 74.27 Å film that becomes superconducting. The figure is reproduced from ref. LiuHavilandGoldman.
• Figure 3: A cartoon phase diagram for a superconductor such as La$_{2-x}$Sr$_{x}$CuO$_4$. AF stands for anti-ferromagnetic and SC for superconducting.
• Figure 4: A third conjectural axis has been added to our phase diagram for a high $T_c$ superconductor. This figure was taken from ref. SubirReview.
• Figure 5: A contour plot for the large Nernst effect measured in La$_{2-x}$Sr$_x$CuO$_4$. The Nernst coefficient $\nu = \theta_{xy} / B$ is plotted in units of nV / K T. $T_{onset}$ is defined as the temperature at which $\nu$ begins to differ substantially from its high temperature behavior. In the dark blue region, the material is superconducting. This figure is described in ref. Ong in more detail.