Entanglement Growth after Quenches

After a quench, subregion entanglement can show a regulator-scale transient, an extended growth regime, finite-region saturation, and finite-volume recurrence. These stages are physical only after the initial state and subtraction are frozen; a straight segment in lattice units is not yet a continuum entanglement velocity.

Required background. Equilibration and dephasing supplies the closed-system relaxation picture, closed-time-path evolution supplies real-time observables, and the quench contract fixes the regulated problem.

Helpful background. Thermal and excited-state entanglement supplies the candidate saturation scale.

Growth regimes

For a pure global state and region $A$, compute

$$\Delta S_A(t)=S\!\left(\rho_A(t)\right)-S\!\left(\rho_A(0)\right).$$

Subtracting $S_A(0)$ removes a shared leading area divergence for a fixed cut, but state-dependent ultraviolet changes still require convergence checks. In a one-dimensional global quench, a common sequence is:

• $t$ comparable to the preparation or cutoff scale: nonuniversal curvature;
• preparation scale $\ll t\ll\ell/(2v)$: approximately linear or model-specific growth;
• $t\sim\ell/(2v)$: crossover toward a volume-law value;
• $t$ comparable to $L/v$: finite-size return or recurrence.

The relevant $v$ depends on the mechanism. In an integrable model it may be a weighted set of quasiparticle group velocities; in a chaotic system a coarse-grained $v_E$ may emerge.

Mass-quench extraction

For the Gaussian mass quench defined on the preparation page, calculate $S_\ell(t)$ for several physical interval lengths. Plot $\Delta S_\ell(t)/\ell$ against $t/\ell$ only after holding $a$, $L$, and masses in physical units. A credible collapse requires:

1. the early interval excluded by a stated cutoff criterion;
2. agreement of slopes across several $\ell$ before saturation;
3. lattice refinement at fixed $t$ and $\ell$;
4. volume refinement before the earliest return.

Calabrese and Cardy obtained the characteristic linear-to-saturation behavior in controlled one-dimensional quench settings using a boundary-state description; see Calabrese and Cardy 2005, §§ 2–4. The detailed slope and crossover are not universal for arbitrary interactions or preparations.

The entropy curve is a direct diagnostic. Quasiparticle or membrane language becomes explanatory only after it predicts that curve across regions and resolutions. The map is schematic.

Interpreting saturation

For a small region in a large system, saturation near a thermodynamic entropy density can reflect local equilibration while the global state remains pure. For half of a finite pure system, purity enforces $S_A=S_{\bar A}$ and modifies the plateau. Integrability can lead to a generalized-equilibrium entropy rather than an ordinary Gibbs value. A plateau caused by bond-dimension truncation is numerical failure, not equilibration.

Fitting the cutoff transient, a finite-size crossover, or a truncation plateau can produce a stable but false growth law. Vary each relevant scale before naming a rate. The map is schematic.

Common pitfalls

Using the raw entropy across cutoffs. Compare a subtraction or finite information measure at fixed physical geometry.

Calling the first visible slope asymptotic. Exclude the preparation transient and demonstrate a widening fit window under refinement.

References

• Calabrese, Pasquale, and John Cardy. “Evolution of Entanglement Entropy in One-Dimensional Systems.” Journal of Statistical Mechanics: Theory and Experiment 2005 (2005): P04010. DOI.