Constructing the AdS dual of a Fermi liquid: AdS Black holes with Dirac hair

TL;DR

This work establishes a holographic dictionary for finite-density Fermi liquids by linking the boundary Fermi surface pole strength to bulk Dirac data in AdS/CFT. Using Migdal's theorem, the authors show that the occupation discontinuity $ riangle n_F$ is encoded in the normalizable part of a bulk Dirac wavefunction, and that a nonzero bulk fermion density corresponds to a phase with Dirac hair on an AdS black hole. Thermodynamic and spectral analyses reveal a first‑order transition from a charged RN black hole to a Dirac‑hair ground state at low temperature, indicating the holographic dual of a Fermi liquid with finite bulk fermion occupancy. The finite-density phase exhibits FL‑like quasiparticle poles in the fermion spectral function and suggests that fully backreacted geometries should asymptote to Lifshitz‑type backgrounds with reduced entropy, offering a concrete holographic realization of a Fermi liquid in strongly coupled systems.

Abstract

We provide new evidence that the holographic dual to a strongly coupled charged Fermi Liquid has a non-zero fermion density in the bulk. We show that the pole-strength of the stable quasiparticle characterizing the Fermi surface is encoded in the spatially averaged AdS probability density of a single normalizable fermion wavefunction in AdS. Recalling Migdal's theorem which relates the pole strength to the Fermi-Dirac characteristic discontinuity in the number density at $\ome_F$, we conclude that the AdS dual of a Fermi liquid is described by occupied on-shell fermionic modes in AdS. Encoding the occupied levels in the total probability density of the fermion field directly, we show that an AdS Reissner-Nordström black hole in a theory with charged fermions has a critical temperature, at which the system undergoes a first-order transition to a black hole with a non-vanishing profile for the bulk fermion field. Thermodynamics and spectral analysis confirm that the solution with non-zero AdS fermion-profile is the preferred ground state at low temperatures.

Figures (6)

• Figure 1: (A) Temperature dependence of the Fermi liquid occupation number discontinuity $\Delta n_F$ and operator $I$ for a fermionic field of mass $m=-1/4$ dual to an operator of dimension $\Delta=5/4$. We see a large density for $T/\mu$ small and discontinuously drop to zero at $T\approx 0.05\mu$. At this same temperature, the proxy free energy contribution per particle (the negative of $I$) vanishes. (B) The free energy $F=F^{fermion}+F^{Maxwell}$ (Eq. (\ref{['eq:6']})) as a function of $T/\mu$ ignoring the contribution from the gravitational sector. The blue curve shows the total free energy $F=F^{Maxwell}$, which is the sum of a bulk and a boundary term. The explicit fermion contribution $F_{fermion}$ vanishes, but the effect of a non-zero fermion density is directly encoded in a non-zero $F^{Maxwell}_{bulk}$. The figure also shows this bulk $F^{Maxwell}_{bulk}$ and the boundary contribution $F^{Maxwell}_{bulk}$ separately and how they sum to a continuous $F_{total}$. Although formally the explicit fermion contribution $F_{f}\sim I$ in equation (\ref{['eq:7']}) vanishes, the bulk Maxwell contribution is captured remarkably well by its value when the cut-off is kept finite. The light-green curve in the figure shows $F_f$ for a finite $z_0 \sim 10^{-6}$. For completeness we also show the total charge density, Eq. (\ref{['eq:12']}). The dimension of the fermionic operator used in this figure is $\Delta=1.1$.
• Figure 2: The boundary behaviour of $J_-(0)$ in for a generic solution (blue) to Eqs. \ref{['ieq']} and a normalizable Dirac-hair solution (red) for $m=-1/4$ in the background of an AdS-RN black hole with $\mu/T=128.8$. The dotted lines show the scaling $z^{11/2}$ and $z^{4}$ of the leading and subleading terms in an expansion of $J_-^0(z)$ near $z=0$; the dashed line shows the scaling $z^{5/2}$ of the subsubleading expansion whose coefficient is $|B_-(\omega_F,k_F)|^2$. That the Dirac hair solution (red) scales as the subsubleading solution indicates that the current $J_-^0$ faithfully captures the density of the underlying normalizable Dirac field.
• Figure 3: (A) Approximate power-law scaling of the Fermi liquid characteristic occupation number discontinuity $\Delta n_F/\mu^{2\Delta} \sim T^{-\delta}$ as a function of $T/{\mu}$ for $\Delta=5/4$. This figure clearly shows the saturation of the density at very low $T/\mu$. The saturation effect is naturally interpreted as the influence of the characteristic Fermi energy. (B) The scaling exponent $\delta$ for different values of the conformal dimension $\Delta$. There is a clear correlation, but the precise relation cannot be determined numerically. The scaling exponent of the current $I/\mu^{2\Delta+1}\sim T^{-1/\nu}$ obeys $\nu=2$ with great accuracy, on the other hand (Inset).
• Figure 4: (A) The radial electric field $-E_z=\partial \Phi/\partial z$, normalized to the midpoint value $E_z(z)/E_z(1/2)$ for whole interior of the finite fermion density AdS-RN solution (upper) and near the horizon (lower). One clearly sees the soft, log-singularity at the horizon. The colors correspond to increasing temperatures from $T=0.04\mu$ (lighter) to $T=0.18\mu$ (darker), all with $\Delta=1.1$. (B) The occupation number jump $\Delta n_F$ and free energy contribution $I$ as a function of temperature in AdS-Schwarzschild. We see the jump $\Delta n_F$ saturate at low temperatures and fall off at high $T$. An exponential fit to the data (red curve) shows that in the critical region the fall-off is stronger than exponential, indicating that the transition is first order. The conformal dimension of the fermionic operator is $\Delta=1.1$. (C) The radial electric field $-E_z=\partial \Phi/\partial z$, normalized to the midpoint value ($E_z(z)/E_z(1/2)$) for the finite fermion density AdS-Schwarzschild background. The divergence of the electric field $E_z$ is again only noticeable near the horizon and can be neglected in most of the bulk region.
• Figure 5: The single-fermion spectral function in the probe limit of pure AdS Reissner-Nordström (red/yellow) minus the spectrum in the finite density system (blue). The conformal dimension is $\Delta=5/4$, the probe charge $g=2$, and $\mu/T=135$. We can see two quasiparticle poles near $\omega=0$, a non-FL pole with $k_F^{probe}\simeq 0.11\mu$ and $k_F^{\Delta n_F}\simeq 0.08\mu$ respectively and a FL-pole with $k_F^{probe}\simeq 0.18\mu$ and $k_F^{\Delta n_F}\simeq 0.17\mu$. The dispersion of both poles is visibly similar between the probe and the finite density backgroudnd. At the same time, the non-FL pole has about $8$ times less weight in the finite density background, whereas the FL-pole has gained about $6.5$ times more weight.