Membrane, Hydrodynamic, and Coarse-Grained Entanglement Laws

In chaotic locally equilibrating systems, late-time entanglement can admit a coarse-grained variational description: an entanglement membrane minimizes an effective action subject to the final entangling surface and initial-state data. This is a hydrodynamic-scale law, not a microscopic equation and not a universal description of integrable or pre-equilibrium evolution.

Required background. Entanglement growth after quenches supplies the entropy profiles that the effective law must reproduce.

Helpful background. Quasiparticle growth supplies a contrasting integrable mechanism.

Variational law

In one spatial dimension, a useful form for an entropy profile is

$$S(x,t)=\min_y\left[ S(y,0)+s_{\rm eq}\,t\, \mathcal E\!\left(\frac{x-y}{t}\right) \right].$$

$s_{\rm eq}$ is the equilibrium entropy density and $\mathcal E(v)$ is a model-dependent membrane tension. Convexity and causality constrain its shape; $\mathcal E(0)$ controls homogeneous entanglement production, while endpoint data connect to the fastest allowed influence on the coarse-grained entropy surface. The exact normalization varies across conventions, so infer $\mathcal E$ and $s_{\rm eq}$ together.

Random-unitary dynamics provides a controlled setting in which entanglement growth maps to a minimal-cut or growth problem, as shown by Nahum et al. 2017, §§ II–IV. Mezei and Stanford 2017, §§ 2–4 relate entanglement spreading and velocity constraints in chaotic systems.

Fit one cut, predict another

Choose a regulated chaotic chain or field truncation after local equilibration. From a half-space or single-cut entropy curve, estimate $s_{\rm eq}\mathcal E(0)$. Use inhomogeneous initial entanglement profiles or several interval geometries to constrain $\mathcal E(v)$. Then freeze the tension and predict:

• a second interval length;
• an off-center cut;
• a weakly inhomogeneous initial entropy profile;
• the onset of finite-region saturation.

Parameter uncertainty must propagate into every prediction. A flexible tension fitted separately to each curve has no explanatory content.

The membrane is a coarse-grained effective layer for locally equilibrated chaotic dynamics. It earns that role by predicting multiple entropy geometries from one tension function. The map is schematic.

Velocity and regime boundaries

The entanglement velocity derived from $\mathcal E(0)$ need not equal the butterfly velocity or microscopic causal speed. Hydrodynamic conserved modes may add slow corrections. At times comparable to the local equilibration time, the membrane tension need not be defined. In integrable systems, persistent quasiparticle occupations generally require a different description.

The characteristic failures are fitting prehydrodynamic transients, importing a chaotic tension into an integrable model, and identifying $v_E$ with a different front velocity. The map is schematic.

Evidence boundary

This description reflects results available through 10 August 2026. Membrane laws are well developed in random circuits, holographic models, and coarse-grained chaotic many-body settings. Their use for a specific continuum QFT remains a model-dependent scaling claim requiring independent cutoff and window control.

References

• Mezei, Márk, and Douglas Stanford. “On Entanglement Spreading in Chaotic Systems.” Journal of High Energy Physics 2017, no. 5 (2017): 065. DOI.
• Nahum, Adam, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. “Quantum Entanglement Growth under Random Unitary Dynamics.” Physical Review X 7 (2017): 031016. DOI.