Holographic metals at finite temperature

TL;DR

This work extends holographic models of strongly interacting 2+1D fermions to finite temperature by constructing static solutions in which an electron cloud sits above a charged black brane in AdS, with inner and outer edges and back-reaction. The analysis shows that at low $T$ the bulk fluid forms an electron-star-like interior with Lifshitz scaling, while at high $T$ the cloud cannot be supported and the geometry reduces to AdS-RN; a third-order phase transition occurs at a critical temperature $T_c$ (with $T_c/\mu \approx 0.0589$) signaling the disappearance of the cloud. Free energy calculations confirm the cloud is preferred when present and smoothly connect to the electron-star result as $T\to 0$, while finite-temperature conductivity smoothly interpolates between electron-star and AdS-RN behavior, being governed by bulk fermions in a finite radial band. The results offer a controlled holographic framework for finite-temperature transport and phase structure in low-dimensional, strongly coupled fermionic systems, with implications for Fermi-surface physics and Lifshitz scaling in the deep interior.

Abstract

A holographic dual description of a 2+1 dimensional system of strongly interacting fermions at low temperature and finite charge density is given in terms of an electron cloud suspended over the horizon of a charged black hole in asymptotically AdS spacetime. The electron star of Hartnoll and Tavanfar is recovered in the limit of zero temperature, while at higher temperatures the fraction of charge carried by the electron cloud is reduced and at a critical temperature there is a second order phase transition to a configuration with only a charged black hole. The geometric structure implies that finite temperature transport coefficients, including the AC electrical conductivity, only receive contributions from bulk fermions within a finite band in the radial direction.

Figures (8)

• Figure 1: The auxiliary function $r(u)$ in (\ref{['ucondition']}), used in the construction of the electron cloud solution, is plotted for $\hat{m}=1$ (dashed red) and $\hat{m}=0.55$ (solid blue).
• Figure 2: The radial profiles of the fluid variables $(\hat{\sigma},\hat{\rho},\hat{p})$ for $\hat{m}=0.55$, $\hat{\beta}=10$, and $\hat{q}^2=4.49$.
• Figure 3: The curvature scalar $R$ versus the proper distance $s$ measured from the outer edge of the electron cloud for $\hat{m}=.55\,, \hat{\beta}=10$ and for $T/\mu$ values $.55, .22, 9.4 \times 10^{-2}, 9 \times10^{-4}$ and $1 \times 10^{-5}$. The curves extend further to the right with decreasing temperature. The value for $R$ in the Lifshitz region deep inside an electron star with the same $\hat{m}$ and $\hat{\beta}$ is shown as a horizontal line for reference.
• Figure 4: On the left we plot the horizon radius $v_0^{-1}$ as a function of the temperature, $T$. On the right we show the charge parameter of the inside black brane solution \ref{['rnsolution']}, $\hat{q}^2$, versus $T$ for $\hat{m}=.55$ and $\hat{\beta}=10$. For reference we plot the extremal value $\hat{q}^2=6$.
• Figure 5: The free energy densities of AdS-RN black brane and electron cloud solutions for $\hat{\beta}=10$ and $\hat{m}=.55$. In addition the free energy for an extremal black hole and the electron star solution of Hartnoll:2010gu are shown with a red box and a green dot, respectively.