Spectral probes of the holographic Fermi groundstate: dialing between the electron star and AdS Dirac hair

TL;DR

This work analyzes finite-density fermions in AdS/CFT by showing that the electron star (TF/adiabatic) and AdS Dirac hair (single-particle) are two limits of the same charged Fermi gas in AdS, connected through the effective constituent charge $q_{\text{eff}}=qL/\kappa$. Through fermion spectral functions computed in electron-star backgrounds and a Schrödinger/WKB framework for the $\omega=0$ problem, the authors demonstrate how the number of normalizable modes (Fermi surfaces) decreases as $q_{\text{eff}}$ grows, approaching the Dirac hair limit where a single mode dominates. They also show that increasing adiabaticity (decreasing $\kappa/L$) increases the number of bound states, linking the multi-shell structure of the electron star to its holographic Fermi-groundstate. The result suggests a best-of-both-worlds groundstate that blends features of both limits, offering a more complete holographic description of finite-density, strongly coupled Fermi systems with tails extending to the AdS boundary.

Abstract

We argue that the electron star and the AdS Dirac hair solution are two limits of the free charged Fermi gas in AdS. Spectral functions of holographic duals to probe fermions in the background of electron stars have a free parameter that quantifies the number of constituent fermions that make up the charge and energy density characterizing the electron star solution. The strict electron star limit takes this number to be infinite. The Dirac hair solution is the limit where this number is unity. This is evident in the behavior of the distribution of holographically dual Fermi surfaces. As we decrease the number of constituents in a fixed electron star background the number of Fermi surfaces also decreases. An improved holographic Fermi groundstate should be a configuration that shares the qualitative properties of both limits.

Figures (10)

• Figure 1: Electron star metric for $z=2,\hat{m}=0.36$, $c\simeq1.021, \hat{M}\simeq3.601, \hat{Q}\simeq2.534, \hat{\mu}\simeq 2.132$ compared to pure AdS. Shown are $f(r)/r^2$ (Blue), $r^2 g(r)$ (Red) and $h(r)$ (Orange). The asymptotic AdS-RN value of $h(r)$ is the dashed blue line. For future use we have also given $\mu_{\mathtt{loc}} = h/\sqrt{f}$ (Green) and $\mu_{q_\mathtt{eff}}= \sqrt{g^{ii}} h/\sqrt{f}$ (Red Dashed) At the edge of the star $r_s \simeq4.253$ (the intersection of the purple dashed line setting the value of $m_{\mathtt{eff}}$ with $\mu_{\mathtt{loc}}$) one sees the convergence to pure AdS in the constant asymptotes of $f(r)/r^2$ and $r^2 g(r)$.
• Figure 2: Electron star MDF spectral functions with multiple peaks as a function of $k$ for $\omega=10^{-5}, z=2, \hat{m}=0.36$. The blue curve is for $\kappa=0.091$; the red curve is for $\kappa=0.090$. Note that the vertical axis is logarithmic. Visible is the rapidly decreasing spectral weight and increasingly narrower width for each successive peak as $k_F$ increases.
• Figure 3: (A) Electron-star MDF spectral functions as a function of $\kappa$ for $z=2, \hat{m}=0.36, \omega=10^{-5}$. Because the peak height and weights decrease exponentially, we present the adjacent ranges $k\in [0.017,0.019]$ and $k\in[0.019,0.021]$ in two different plots with different vertical scale. (B1/B2) Locations of peaks of spectral functions as a function of $\kappa$: comparison between the electron star (B1) for $z=2, \hat{m}=0.36, \omega=10^{-5}$ (the dashed gray line denotes the artificial separation in the 3D representations in (A)) and AdS-RN (B2) for $m=0$ as a function of $q$ in units where $\mu=\sqrt{3}$ These two Fermi-surface 'spectra' are qualitatively similar.
• Figure 4: The behavior of the Schrödinger potential $V(s)$ for $z_+$ when $k$ is negative. Such a potential has no zero-energy bound state. The potential is rescaled to fit on a finite range. As $|k|$ is lowered below $k_{max}$ for which the potential is strictly positive, a triple pole appears which moves towards the horizon on the left (Fig A. The Blue,Red,Orange,Green curves are decreasing in $|k|$). The pole hits the horizon for $k=0$ and disappears. Fig B. shows the special case $m_{\mathtt{eff}}=0$ where two zeroes collide with two of the triple poles to form a single pole.
• Figure 5: The Schrödinger potential $V(s)$ for the fermion component $z_+$ of in the ES background $\hat{m}=0.36, z=2, c_0=0.1$. Fig. A. shows the dependence on the momentum $k= 0.0185$ (Purple), $k = 5$ (Blue), $k = 10$ (Red) for $\kappa=0.092$. Fig. B. shows the dependence on $\kappa=0.086$ (Purple), $\kappa=0.092$ (Blue), $\kappa=0.1$ (Red) for $k=0.0185$. Recall that $s=0$ is the AdS boundary and $s=-\infty$ is the near-horizon region.