From black holes to strange metals

TL;DR

This work uses gauge/gravity duality to engineer non-Fermi liquids in which low-energy dynamics are controlled by an infrared fixed point associated with an AdS$_2$ throat. The emergent IR CFT induces a temporal, but not spatial, scaling of fermionic correlators, yielding a fermionic self-energy $\Sigma(\omega) \propto \omega^{2\nu_{k}}$ and a Fermi-surface structure whose quasiparticle content is dictated by $\nu_{k_F}$, producing regimes ranging from conventional quasiparticles ($\nu_{k_F}>1/2$) to marginal Fermi liquid behavior ($\nu_{k_F}=1/2$) to non-Fermi-liquid without sharp quasiparticles ($\nu_{k_F}<1/2$). The dc and optical conductivities mirror this structure, with $\sigma_{DC} \propto T^{-2\nu_{k_F}}$ and distinctive frequency dependences across the regimes, including a linear-in-$T$ resistivity at the marginal point. The results offer a holographic realization of strange-metal phenomenology and furnish a tractable framework to explore quantum criticality in strongly correlated systems, potentially illuminating the cuprates and heavy-fermion metals.

Abstract

Since the mid-eighties there has been an accumulation of metallic materials whose thermodynamic and transport properties differ significantly from those predicted by Fermi liquid theory. Examples of these so-called non-Fermi liquids include the strange metal phase of high transition temperature cuprates, and heavy fermion systems near a quantum phase transition. We report on a class of non-Fermi liquids discovered using gauge/gravity duality. The low energy behavior of these non-Fermi liquids is shown to be governed by a nontrivial infrared (IR) fixed point which exhibits nonanalytic scaling behavior only in the temporal direction. Within this class we find examples whose single-particle spectral function and transport behavior resemble those of strange metals. In particular, the contribution from the Fermi surface to the conductivity is inversely proportional to the temperature. In our treatment these properties can be understood as being controlled by the scaling dimension of the fermion operator in the emergent IR fixed point.

Figures (8)

• Figure 1: An illustration of the holographic approach to understanding non-Fermi liquids. The physics of a strongly-coupled many-body theory is exactly dual to that of a gravitational theory in a curved spacetime with one extra dimension. A finite density of charge in the many-body problem corresponds in the gravity theory to a black hole with an extra emergent conformal symmetry near its horizon: these correspond to the symmetries of two-dimensional Anti de-Sitter space. This space (when projected onto a two-dimensional plane) forms the basis of the M.C. Escher print Circle Limit IV, which is thus used here to represent the horizon of the black hole. Understanding fermionic response in this system corresponds to scattering bulk fermions off the black hole and shows excitations characteristic of Fermi or non-Fermi liquids.
• Figure 2: The geometry of an extremal $AdS_{4}$ charged black hole and the Fermi gas hovering outside it. The horizontal line denotes the boundary spacetime, which has $d=2+1$ dimensions, and the vertical axis denotes the radial direction $r$ of the black hole, which is the direction extra to the boundary spacetime. The boundary lies at $r=\infty$. The horizon here is a plane, ${\mathbb{{R}}}^2$. As will be discussed in section \ref{['sec:IIIA']}, the coordinate $r$ can be interpreted as the energy scale of the boundary theory. The black hole spacetime asymptotes to that of AdS$_4$ near the boundary and factorizes into AdS$_2 \times {\mathbb{{R}}}^2$ near the horizon, with AdS$_2$ including the $r,t$ directions, and ${\mathbb{{R}}}^2$ comprised by the spatial $x,y$ directions.
• Figure 4: Conductivity from gravity. The current-current correlator in \ref{['kubo']} can be obtained from the propagator of the gauge field $A_x$ with endpoints on the boundary. Wavy lines correspond to gauge field propagators and the dark line denotes the bulk propagator for the $\psi$ field. The left diagram is the tree-level propagator for $A_x$, while the right diagram includes the contribution from a loop of $\psi$ quanta. The contribution from the Fermi surface associated with boundary fermionic operator ${{\mathcal{O}}}$ can be extracted from the diagram on the right.
• Figure 5: Density plot of the spectral function $A (\omega,k)$ as a function of $\omega$ and $k$. We have chosen units so that $\mu = 1$ and in all plots $m=0$. Left: $q=1$ with Fermi momentum $k_F= 0.53$ and $\nu_{k_F} = 0.24 < {{\frac{1}{2}}}$ for which there is no sharp quasiparticle at the Fermi surface and the peak dispersion is nonlinear (see equation \ref{['peaD']}). Middle: $q = 1.56$ with Fermi momentum $k_F = 0.952$ and $\nu_{k_F} = 0.500$ which corresponds to a marginal Fermi liquid. Right: $q=2$ with Fermi momentum $k_F = 1.315$ and $\nu_{k_F} = 0.73 > {{\frac{1}{2}}}$, for which there are stable quasi-particles at the Fermi surface.
• Figure 6: The imaginary part of the current-current correlator \ref{['kubo']} receives its dominant contribution from diagrams in which the fermion loop goes into the horizon. This also gives an intuitive picture that the dissipation of current is controlled by the decay of the particles running in the loop, which in the bulk occurs by falling into the black hole.