Holographic Quenches and Entanglement Growth

After a homogeneous holographic quench, a large region can exhibit an intermediate regime of entropy growth proportional to its boundary area and elapsed time. The coefficient defines an entanglement velocity only after the state, region shape, spacetime dimension, bulk action, and normalization are fixed. It is not a universal material speed, and the leading extremal-surface answer omits finite-$N$ entropy corrections.

Required background. Time-Dependent Geometries and Holographic Thermalization supplies the collapse geometry and probe dependence.

Helpful background. Membrane, Hydrodynamic, and Coarse-Grained Entanglement Laws supplies the effective description; Entanglement, Correlation, and Hydrodynamic Fronts distinguishes the relevant velocities.

Extremal surfaces through a shell

For a boundary region $A$ at time $t$, the leading covariant prescription is

$$S_A(t)=\frac{\operatorname{Area}(\gamma_A)}{4G_N}, \qquad \gamma_A\ \text{extremal},\quad \partial\gamma_A=\partial A.$$

This covariant extremal-surface prescription is due to Hubeny, Rangamani, and Takayanagi 2007. Its homology and boundary-anchoring conditions remain part of the problem in a time-dependent geometry.

In the thin-shell metric, parameterize a strip surface by $z(x)$ and $v(x)$. Translation invariance along the transverse directions reduces the area to

$$\mathcal A =L^{d-1}A_\perp \int dx\,z^{-(d-1)} \sqrt{1-f(v,z)v'^2-2v'z'}.$$

The Euler–Lagrange equations are solved on both sides of the shell, with continuity and refraction conditions from varying the crossing point. Subtracting the vacuum area removes the common ultraviolet divergence.

Growth and saturation

For a sufficiently large strip or ball, the intermediate result can be written

$$\frac{dS_A}{dt} =s_{\mathrm{eq}}A_{\partial A}v_E +O(A_{\partial A}T^{d-2}),$$

where $s_{\mathrm{eq}}$ is the final thermal entropy density. In an Einstein black brane

$$v_E =z_h^{\,d-1} \max_{z>z_h} \frac{\sqrt{-f(z)}}{z^{d-1}},$$

with the maximum taken behind the horizon in these coordinates. The saturation time scales as $t_{\mathrm{sat}}\sim R/v_E$ only in the regime where the linear front picture applies. The entanglement-tsunami analysis and its geometric velocity were derived by Liu and Suh 2014.

As the first application, integrate the strip equations across a Vaidya shell, extract the early analytic growth, fit the intermediate slope, and locate saturation. Report $R T$, shell thickness, dimension, subtraction scheme, and whether saturation is continuous. Small regions can skip the linear regime entirely.

Velocity adversary

Carry the fitted $v_E$ to a different dimension, shape, charged state, or higher-derivative action. The extremal functional and the maximizing geometry change, so the coefficient generally changes. Finite-$N$ bulk entropy also changes the prescription beyond the area term.

The stable claim is a state- and model-specific front coefficient in a controlled large-region regime. Causal speed, butterfly velocity, energy velocity, and entanglement velocity need not coincide; a comparison must use their own operational definitions.

Limits and handoff

Take $N\to\infty$ and strong coupling before using one classical extremal surface. Take $R T\gg1$ before extracting a membrane-like velocity. Reversing either limit can expose quantum corrections or finite-size saturation rather than a clean linear regime.

Quantum Information in QFT owns entanglement dynamics and bounds, Thermal and Nonequilibrium QFT owns quenches, and the later entropy chapter owns quantum extremal-surface corrections.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Hubeny, Veronika E.; Rangamani, Mukund; and Takayanagi, Tadashi. “A Covariant Holographic Entanglement Entropy Proposal.” Journal of High Energy Physics 2007, 062 (2007). doi:10.1088/1126-6708/2007/07/062.
• Liu, Hong, and S. Josephine Suh. “Entanglement Tsunami: Universal Scaling in Holographic Thermalization.” Physical Review Letters 112, 011601 (2014). doi:10.1103/PhysRevLett.112.011601.