Holographic metals at finite volume

TL;DR

This work analyzes a holographic metal at finite volume by constructing an electron star in asymptotically global AdS and examining its thermodynamic stability relative to RN-AdS black holes and Thermal AdS. Using a charged perfect fluid with Thomas–Fermi thermodynamics and a grand canonical ensemble, the authors map a phase diagram in the boundary variables $(T_ty, μ_ty)$, identifying a finite stable region for the electron star and a distinct first-order star–BH transition, with a zero-temperature quantum critical point at $μ_ty\approx1.435$ that organizes the phases. The zero-temperature analysis reveals a transition from TAdS$_4$ to Electron Star at $μ_ty=1$, and the overall phase structure is reminiscent of Hawking-Page physics but enriched by the star solution. These results provide a controlled platform to study finite-volume holographic metals and quantum criticality, with potential extensions to boundary correlators and holographic superconductivity on this background.

Abstract

We construct the electron star solution in asymptotically global AdS spacetime, and investigate its stability properties, both locally under perturbations and globally with respect to the Reissner-Nordström black hole and thermal AdS metrics. We interpret the resulting phase diagram as that of a holographic metal confined to a finite volume. We identify a quantum critical point at finite chemical potential, around which the different phases are organized.

Figures (4)

• Figure 1: Radial profiles of the local mass $M(r)$, charge $Q(r)$, and metric function $\chi(r)$ for the Electron Star, obtained via the shooting method by varying the central parameters $\mu_0$ and $T_0$. The colors of the curves correspond to the specific points marked in the phase space diagram (Fig. \ref{['fig:criticality']}). Solid lines represent solutions with positive central chemical potential $\mu_0$, while dashed lines correspond to $-\mu_0$ (with the same magnitude). Note that the mass and $\chi$ profiles coincide for both signs, illustrating the invariance of the geometry under charge conjugation.
• Figure 2: Top: (Left) The shaded region indicates the domain where the Electron Star is a stable solution according to the Katz criterion. This region is delimited by the red critical curve, which is constructed by varying the initial values $\mu_0$ and $T_0$ and identifying the critical point on the corresponding Katz curve. The colored curves represent families of solutions with fixed $\mu_0/T_0$, where circles and squares denote representative stable and unstable configurations, respectively. The solid black line marks the star-black hole phase transition derived in Sec. \ref{['sec:phase_transitions']}. (Right) Katz stability curves for the families indicated in the phase diagram. The vertical dashed red lines mark the critical point obtained for each curve. Bottom: Log-Log plor for the radial density profiles corresponding to the specific configurations marked in the top-left panel; the colors match the phase space trajectories. Solid lines represent stable solutions (circles in the top-left figure), characterized by an abrupt edge profile, while dashed lines correspond to unstable solutions (squares) exhibiting power-law behavior.
• Figure 3: Left: Phase diagram of the holographic electron star in the boundary plane $(\mu_\infty,T_\infty)$. The colored region indicates where the electron star is the dominant solution (minimal free energy), while the blue line and grey region correspond to TAdS and black hole dominance, respectively. The solid black lines denote first-order phase transition, and the dashed black line indicates where the Hawking-Page transition would occur in the absence of the star Chamblin:1999. The colored lines (green, brown, red) represent the slices along which we evaluate the free energies shown in the right panels. Right: Free energy comparisons for the electron star (orange), black hole (grey), and TAdS (blue) corresponding to the cuts in the phase diagram. The frame colors match the cuts in the left panel. The black dots mark the phase transition points. The red frame corresponds to the $T_\infty=0$ case, from which the quantum critical point is determined.
• Figure 4: Plots of the effective potential \ref{['eq:Veff']}, fixing the angular momentum $L=0$ (Left) and the energy $\epsilon=0.35$ (Right). Both plots has the horizon fixed (black dash line) at $M=10$ and $Q=4.5$. Plots of the effective potential \ref{['eq:Veff']}. Left: Fixed angular momentum $L=0$ with varying energy. Right: Fixed energy $\epsilon=0.35$ with varying angular momentum. Both plots assume a fixed background with horizon radius corresponding to $M=10$ and $Q=4.5$, and the charge and mass of the particle are $q=1$ and $m=0.2$ respectively.