Thermalization with a chemical potential from AdS spaces

TL;DR

This work investigates how a finite chemical potential affects thermalization in strongly coupled field theories using holography. The authors implement a RNAdS-Vaidya model with a collapsing shell of charged matter, and probe the evolution with renormalized geodesic lengths and minimal-area surfaces, corresponding to equal-time two-point functions and Wilson loops, across dimensions $d=3$–$6$ and a full range of $\mu/T$. They find that increasing $\mu/T$ delays thermalization, with larger delays for bigger probes, while UV modes thermalize first in a top-down pattern; swallow-tail features appear for large probes, indicating multiple competing bulk solutions. These results extend the understanding of equilibration in strongly coupled plasmas with finite density and offer cross-dimensional insights relevant to holographic QGP and related condensed matter systems.

Abstract

The time-scale of thermalization in holographic dual models with a chemical potential in diverse number of dimensions is systematically investigated using the gauge/gravity duality. We consider a model with a thin-shell of charged dust collapsing from the boundary toward the bulk interior of asymptotically anti-de Sitter (AdS) spaces. In the outer region there is a Reissner-Nordström-AdS black hole (RNAdS-BH), while in the inner region there is an anti-de Sitter space. We consider renormalized geodesic lengths and minimal area surfaces as probes of thermalization, which in the dual quantum field theory (QFT) correspond to two-point functions and expectation values of Wilson loops, respectively. We show how the behavior of these extensive probes changes for charged black holes in comparison with Schwarzschild-AdS black holes (AdS-BH), for different values of the black hole mass and charge. The full range of values of the chemical potential over temperature ratio in the dual QFT is investigated. In all cases, the structure of the thermalization curves shares similar features with those obtained from the AdS-BH. On the other hand, there is an important difference in comparison with the AdS-BH: the thermalization times obtained from the renormalized geodesic lengths and the minimal area surfaces are larger for the RNAdS-BH, and they increase as the black hole charge increases.

Figures (4)

• Figure 1: Evolution of the shell of dust described by the AdS-Vaidya metric with $r_h=1$, compared with a shell of charged dust described by the RNAdS Vaidya type metric also with $r_h=1$ and $Q/M=1/2$. In both cases the separation of the boundary field theory operator pair is $\ell=2.6$. The black hole horizon is indicated by a dashed horizontal red line. The horizonal dashed blue lines indicate the position of the shell in each case. The time differences in each pair of figures is due to the fact that they have been obtained numerically, and therefore, there are minor differences.
• Figure 2: Thermalization of the renormalized geodesic lengths for a RNAdS Vaidya type metric with $d+1=4, 5, 6, 7$ and the boundary separation $\ell=2,3,4$. The first curve in each picture (fluor green) indicates the case $Q=0$, i.e. the Schwarzschild-AdS case. The last curve in each case corresponds to the extremal case where $\mu/T\rightarrow\infty$. For $\ell=4$ swallow tails appear before the thermalization takes place. They are shown with more detail in the corresponding insets.
• Figure 3: Thermalizacion of the renormalized minimal area surfaces for RNAdS Vaidya type metric with $d+1=4, 5, 6, 7$, while the boundary separation is $\ell=1, 1.5, 2$. In each picture the first curve (fluor green) indicates the case $Q=0$, i.e. the Schwarzschild-AdS case. The rest of the curves in each figure correspond to $Q=0.50, 0.75, \cdots, \sqrt{\frac{d}{d-2}}$. The last case is the extremal where $\mu/T\rightarrow\infty$. Insets allow to show the swallow tails in detail. Note that in order to make easier the comparison among different figures the vertical and horizontal scales are kept fixed in all figures.
• Figure 4: Three different solutions that appear as part of the swallow tail in thermalization of expectation values of Wilson loops in the extremal RNAdS Vaidya geometry with $d=3$. Time corresponds to $t_0=1.778$ in all three cases and separation length is $\ell=2$. The red dashed line corresponds to the event horizon ($z_h=1$), while the blue dashed one is the position of the shell. In the first two cases, the shell is located inside the event horizon but still have solutions inside. In the insets we take a closer look at the region near the horizon and consider the shell thickness ($-0.02\leq v \leq 0.02$) in blue.