A model of a Fermi liquid using gauge-gravity duality

TL;DR

The paper develops a holographic model for a density driven crossover from a 2+1D conformal critical point to a confining Fermi liquid by using an AdS4 geometry terminated with a hard-wall IR boundary. A bulk QED description with a classical gauge field and quantum fermions yields a self-consistent Gauss law that enforces a Luttinger relation between the boundary charge density and the Fermi surface area. An explicit mean-field solution shows a finite density state with a small number of Fermi surfaces and a spectrum E_l(k) = +/- sqrt(k^2 + M_l^2) with M_l fixed by IR boundary conditions, while beyond mean-field the Ward identity and bulk Green's function analysis support a Landau Fermi liquid in the boundary theory. The work highlights how confinement and finite density shape the low energy excitations and discusses limitations related to the ad hoc IR termination and the prospect of μ driven deconfinement scaling in future research.

Abstract

We use gauge-gravity duality to model the crossover from a conformal critical point to a confining Fermi liquid, driven by a change in fermion density. The short-distance conformal physics is represented by an anti-de Sitter geometry, which terminates into a confining state along the emergent spatial direction. The Luttinger relation, relating the area enclosed by the Fermi surfaces to the fermion density, is shown to follow from Gauss's Law for the bulk electric field. We argue that all low energy modes are consistent with Landau's Fermi liquid theory. An explicit solution is obtained for the Fermi liquid for the case of hard-wall boundary conditions in the infrared.

Figures (3)

• Figure 1: Dispersion spectrum of the fermions. The blue lines correspond to the spectrum Eq. (\ref{['diracspec']}, at $m=1$, $z_m = 3$, and $\mu=0$. The red lines are at $q\mu=1.7$ and $q^2 e^2 =3$. The horizontal red line is at $q\mu$, and the shaded region shows the additional states filled by the chemical potential. Note that the two sets of band dispersions are offset even at large momenta and energies: this arises from a Hartree shift in the energies due to the added density of particles. The wavefunctions of the states at large momenta are not modified by this shift. At smaller momenta, both the $k$ dependence and wavefunctions are different between the two sets.
• Figure 2: Electric field and electrochemical potential for a non-zero fermion density. Parameters as in Fig. \ref{['fig:disp']}.
• Figure 3: Wavefunctions at $k=k_F$ with parameters as in Fig. \ref{['fig:disp']}. The blue lines show the zero density state in Eq. (\ref{['bessel']}), plotted at the wavevector with eigenenergy 1.7. The red lines are the state at the Fermi level in the finite density fermion system with $\mu=1.7$. The differences in the two lines are the consequences of the self-consistent electric field generated by the fermion density.