Holographic thermalization with Lifshitz scaling and hyperscaling violation

TL;DR

The paper investigates holographic thermalization in backgrounds with Lifshitz scaling and hyperscaling violation by modeling gravitational collapse via a Vaidya-type hvLif geometry. It develops analytic and numerical results for the time evolution of holographic entanglement entropy for strip and spherical boundary regions, both in static hvLif spacetimes and during the collapse, and identifies three growth regimes (initial power-law, linear, and saturation) that generalize previous relativistic results. By imposing null energy condition constraints, the authors map allowed exponents $(\zeta,\theta)$ and derive thin-shell matching conditions that yield explicit expressions for the entanglement entropy's time dependence, including the linear growth velocity $v_E$ governed by $d_\theta+\zeta$. Their findings show how the combination $d_\theta+\zeta$ controls early-time dynamics, while saturation times scale with region size and are largely universal with respect to $(\theta,\zeta)$ in certain limits. The work extends Liu and Suh to hvLif theories, providing a framework to study thermalization near non-relativistic quantum critical points with potential applications to strongly correlated systems exhibiting anisotropic scaling.

Abstract

A Vaidya type geometry describing gravitation collapse in asymptotically Lifshitz spacetime with hyperscaling violation provides a simple holographic model for thermalization near a quantum critical point with non-trivial dynamic and hyperscaling violation exponents. The allowed parameter regions are constrained by requiring that the matter energy momentum tensor satisfies the null energy condition. We present a combination of analytic and numerical results on the time evolution of holographic entanglement entropy in such backgrounds for different shaped boundary regions and study various scaling regimes, generalizing previous work by Liu and Suh.

Figures (15)

• Figure 1: The grey area is the region of the $(\zeta,\theta)$ plane defined by (\ref{['eq:hsnec1']}) and (\ref{['eq:hsnec2']}), obtained from the Null Energy Condition, and also (\ref{['eq:asympt']}). The panels show $d=2,3,4$. The red dots denote $AdS_{d+2}$ and the horizontal dashed lines indicate the critical value $\theta =d-1$. The blue lines denote the upper bound defined by the condition (\ref{['eq:lineargro3']}).
• Figure 2: The profiles $z(x)$ of the extremal surfaces for a strip with $\ell=8$ for different boundary times: $t=0$ (hvLif regime, red curve), $t=3.6$ (intermediate regime, when the shell is crossed, blue curve) and $t=5$ (black hole regime, black curve). The final horizon is $z_h=1$. These plots have $d=2$, $\theta =2/3$ and $\zeta = 1.5$. The left panel shows the situation in the thin shell limit ($a=0.01$), while in the right panel $a=0.5$.
• Figure 3: Strip and $a=0.01$ (thin shell). Regularizations (\ref{['eq:area1']}), (\ref{['eq:area2']}) and (\ref{['eq:area3']}) of the area for $d=1$ (dashed red), $d=2$ (blue) and $d=3$ (green) with $\theta=d-1$ and $\zeta=2-1/d$. Left panels: areas as functions of $\ell/2$ for fixed $t=1.5$ (bottom curves) and $t=2.5$ (upper curves). Right: area as functions of the boundary time $t$ with fixed $\ell=3$ and $\ell=5$. The latter ones are characterized by larger variations.
• Figure 4: Regularized area (\ref{['eq:area3']}) for the strip in the thin shell regime ($a=0.01$) for the critical value $\theta =d-1$ and $\zeta=2-1/d$ (continuous curves) compared with the corresponding cases without hyperscaling $\theta = 0$ (dashed curves). We plot $d=1$ (red), $d=2$ (blue) and $d=3$ (green). Left panel: plots at fixed $t=1.5$ (bottom curves) and $t=2.5$ (upper curves). Right panel: plots at fixed $\ell=3, 5$ (larger strips have larger variations for $A^{(3)}_{\textrm{\tiny reg}}$). Strips with smaller $\ell$ thermalize earlier.
• Figure 5: Regularized area (\ref{['eq:area3']}) for the strip with $a=0.5$. These plots should be compared with Fig. \ref{['fig:A3dim']}, because the parameters $d$, $\theta$ and $\zeta$ and the color code are the same.