Quasiparticle Pictures of Entanglement Growth

The quasiparticle picture explains entanglement growth when a quench produces entangled excitations that subsequently propagate approximately ballistically. It becomes quantitative only after specifying the production entropy, species, dispersion, and scattering assumptions. A light-cone sketch alone is not a derivation.

Required background. Entanglement growth after quenches supplies the direct entropy data to be predicted.

Pair formula

For an interval of length $\ell$ in a homogeneous one-dimensional integrable system, a representative formula is

$$S_\ell(t)-S_\ell(0) \simeq \sum_\alpha\int dk\,s_\alpha(k)\, \min\!\left(2|v_\alpha(k)|t,\ell\right),$$

up to normalization conventions for $dk$. Here $\alpha$ labels quasiparticle species, $v_\alpha(k)=\partial_k\varepsilon_\alpha(k)$ is the group velocity, and $s_\alpha(k)$ is the entanglement contribution fixed by the post-quench mode occupations. Pairs separated by less than $\ell$ contribute while one member lies inside and one outside the interval.

The pair construction and its conformal realization are derived by Calabrese and Cardy 2005, §§ 2–4. In interacting integrable systems, velocities are dressed by the stationary state and the entropy density is the thermodynamic Yang–Yang contribution. Alba and Calabrese 2017, Eqs. (1)–(4), pp. 7947–7949 test this generalized formula against integrable spin-chain dynamics.

Exact covariance benchmark

For the free scalar mass quench, determine Bogoliubov occupations $n_k$, mode entropy $s(k)$, and lattice group velocity

$$v_k=\frac{\partial\omega_k(m_f)}{\partial k}.$$

Compute the exact interval entropy from the time-dependent covariance matrix. Without fitting $s(k)$, insert the independently calculated $s(k)$ and $v_k$ into the quasiparticle integral. Compare several $\ell$ and both early growth and the saturation crossover. Agreement only after an arbitrary overall rescaling does not validate the production rule.

The strongest check changes the dispersion while preserving the initial occupation profile as closely as possible. The predicted front and growth rate should change through $v_k$ in the manner seen in the exact covariance result.

The quasiparticle picture is an effective middle layer. Production entropies and dressed velocities must come from the microscopic quench, then predict the direct entropy diagnostic. The map is schematic.

Where the picture fails

Generic interactions scatter and broaden quasiparticles, confinement can bind the presumed pairs, and unstable particles decay. Multiparticle entanglement may not admit a pair decomposition. Even in an integrable model, using the bare maximum velocity in place of the full entropy-weighted distribution can mispredict the slope.

Transferring a free-pair formula to a confining, decaying, or generic chaotic regime is a change of dynamical class. The direct entropy comparison must decide whether the picture survives. The map is schematic.

Exercises

Suppose all produced pairs move with speed $v$ and carry total entropy density $s$. Evaluate the formula before and after $t=\ell/(2v)$.

Solution

The integral reduces to $s\min(2vt,\ell)$. Entropy grows linearly as $2svt$ and saturates at $s\ell$.

References

• Alba, Vincenzo, and Pasquale Calabrese. “Entanglement and Thermodynamics after a Quantum Quench in Integrable Systems.” Proceedings of the National Academy of Sciences 114 (2017): 7947–7951. DOI.
• Calabrese, Pasquale, and John Cardy. “Evolution of Entanglement Entropy in One-Dimensional Systems.” Journal of Statistical Mechanics: Theory and Experiment 2005 (2005): P04010. DOI.