Lectures on holographic non-Fermi liquids and quantum phase transitions

TL;DR

The notes survey holographic methods for studying finite-density quantum matter without quasiparticles, using charged AdS black holes to model strongly coupled field theories. A central construct is the semi-local quantum liquid (SLQL), an emergent IR fixed point with AdS$_2$-like scaling that yields universal, momentum-dependent scaling dimensions and ω-dependent spectral functions, governing non-Fermi liquid behavior and quantum criticality. The authors derive a master relation for retarded Green functions, identify conditions for Fermi surfaces and scalar instabilities, and show how SLQL can hybridize with a Landau-Ginzburg sector to produce a variety of quantum critical points, including bifurcating, hybridized, and marginal types with rich, non-Landau dynamics. They connect these holographic phases to phenomena such as Marginal Fermi Liquid behavior and linear-T resistivity, and discuss how SLQLs can eventually order into Fermi liquids, superconductors, or antiferromagnetic-like states, offering a universal intermediate-energy picture for strongly correlated systems. The work emphasizes the role of UV/IR data separation, the importance of AdS$_2$ holography, and the potential relevance of these intermediate phases to real materials, including cuprates and heavy fermions, with broader implications for non-Fermi liquids and quantum criticality.

Abstract

In these lecture notes we review some recent attempts at searching for non-Fermi liquids and novel quantum phase transitions in holographic systems using gauge/gravity duality. We do this by studying the simplest finite density system arising from the duality, obtained by turning on a nonzero chemical potential for a U(1) global symmetry of a CFT, and described on the gravity side by a charged black hole. We address the following questions of such a finite density system: 1. Does the system have a Fermi surface? What are the properties of low energy excitations near the Fermi surface? 2. Does the system have an instability to condensation of scalar operators? What is the critical behavior near the corresponding quantum critical point? We find interesting parallels with those of high T_c cuprates and heavy electron systems. Playing a crucial role in our discussion is a universal intermediate-energy phase, called a "semi-local quantum liquid", which underlies the non-Fermi liquid and novel quantum critical behavior of a system. It also provides a novel mechanism for the emergence of lower energy states such as a Fermi liquid or a superconductor.

Figures (23)

• Figure 1: Left: a cartoon picture of the phase diagram of cuprate superconductors; Right: a cartoon picture of linear temperature dependence of the resistivity in the strange metal phase.
• Figure 2: A cartoon picture of the geometry of an extremal charged black hole. The horizon lies at an infinite proper distance away and in the near horizon region the warp for the spatial part approaches a constant, while that for the time direction shrinks to zero at the horizon.
• Figure 3: Charged black hole with nonzero gauge field is dual to field theory state with nonzero chemical potential; in the infrared there is an emergent conformal symmetry corresponding to the AdS$_2 \times \mathbb{R}^{d-1}$ part of the geometry.
• Figure 4: At a finite chemical potential, a CFT$_d$ flows in the IR to SLQL. On the gravity side this is realized geometrically via the flow of the AdS$_{d+1}$ near the boundary to AdS$_2 \times \mathbb{R}^{d-1}$ near the horizon.
• Figure 5: A cartoon picture: the system separates into domains of size $\xi \sim {1 \over \mu}$. Within each domain a conformal quantum mechanics governs dynamics in the time direction with a power law correlation (i.e. infinite relaxation time).