Integrable versus Chaotic Entanglement Dynamics

Integrable, chaotic, and localized systems can all begin with similar-looking entropy curves. Linear growth is not a chaos test: stable integrable quasiparticles can produce it, and a chaotic membrane can produce it for entirely different reasons. Localized slow dynamics is a third comparator rather than one endpoint of a single integrable-to-chaotic axis. Classify the dynamics with independent spectral, transport, and relaxation evidence before using that class to explain entanglement.

Required background. Entanglement growth supplies the direct curves being classified.

Helpful background. Quasiparticle growth supplies the integrable scaling theory, and membrane entanglement supplies the chaotic coarse-grained theory.

Establish the dynamical class independently

For a regulated finite system, first resolve every exact symmetry: total charge, momentum, spatial parity, spin inversion, gauge sector, and any antiunitary class. Mixing independent blocks produces artificial near-degeneracies and can make a chaotic spectrum look Poissonian.

Within one symmetry block, order energy levels $E_n$, define spacings $s_n=E_{n+1}-E_n$, and use the unfolding-free folded ratio

$$\widetilde r_n=\frac{\min(s_n,s_{n-1})}{\max(s_n,s_{n-1})}.$$

For uncorrelated Poisson spacings,

$$\langle\widetilde r\rangle_{\rm P}=2\log2-1 \approx0.386294.$$

For the time-reversal-invariant Gaussian orthogonal ensemble, the $3\times3$ Wigner-like surmise is

$$4-2\sqrt3\approx0.535898,$$

while large-matrix numerics are closer to $0.5307$. The distinction matters when a quoted benchmark is used as an exact unit test. Atas et al. 2013, Eqs. (1), (2), and (7), Table I derive the ratio distributions and report the ensemble values.

Level statistics alone are insufficient. Add local-observable relaxation, conserved-charge transport, eigenstate matrix elements, response functions, and finite-size trends. In a continuum QFT, the finite-volume regulator and energy window are part of this classification.

Integrable quasiparticle baseline

For a homogeneous low-entangled initial state in a one-dimensional integrable system, the leading interval entropy in the scaling limit $t,\ell\to\infty$ at fixed $t/\ell$ is

$$\begin{aligned} S_A(t)=\sum_n\Bigg[ &2t\int_{2|v_n(\lambda)|t<\ell} d\lambda\,|v_n(\lambda)|s_n(\lambda)\\ &+\ell\int_{2|v_n(\lambda)|t>\ell} d\lambda\,s_n(\lambda) \Bigg]+o(\ell). \end{aligned}$$

$n$ labels stable quasiparticle species, $s_n$ is the stationary Yang–Yang entropy weight, and $v_n$ is the dressed post-quench velocity. Define

$$s_{\rm stat}=\sum_n\int d\lambda\,s_n(\lambda), \qquad j_S=\sum_n\int d\lambda\,|v_n(\lambda)|s_n(\lambda).$$

Then the early leading term is $S_A=2tj_S$ and the plateau is $\ell s_{\rm stat}$. Alba and Calabrese 2017, Eqs. (1)–(3) test this result in interacting integrable spin chains.

The evidence for this explanation is not the linear term. It is the successful prediction from independently determined stationary occupations, entropy weights, and dressed velocities across several interval sizes and initial states. Bound states and every stable species must be included. The formula is a powerful result in its scaling regime, not a theorem covering arbitrary integrable QFT preparations.

For the single-speed benchmark $s_{\rm stat}=\log2$, $|v|=1$, and $\ell=10$,

$$S_A(t)=\log2\,\min(2t,10).$$

Thus $S_A(3)=6\log2\approx4.158883$, while $S_A(7)=10\log2\approx6.931472$.

Chaotic coarse-grained baseline

After local equilibration in a short-range chaotic system, a membrane tension or local entropy-growth function can govern the leading large-scale entropy. The diagnostic test is complementary to the integrable one: determine $s_{\rm eq}$ independently, infer a constrained tension from one family of profiles, and predict new geometries without refitting.

Random unitary circuits provide controlled results in which the mean entropy grows linearly and noise-induced fluctuations in one spatial dimension fall in the KPZ class. In the setting of Nahum et al. 2017, §§ II–V, height fluctuations grow as $t^{1/3}$ and spatial correlations extend over $t^{2/3}$. Those exponents rely on the noisy random-unitary ensemble; a deterministic chaotic Hamiltonian need not share the same stochastic fluctuation law.

Likewise, a successful chaotic membrane does not mean the system has no quasiparticle-like spectral features at short times. The claim is that after local equilibration, a state-independent coarse tension predicts the leading entropy geometry better than a persistent integrable occupation formula. The crossover time and corrections remain model dependent.

Localized and slow comparators

Noninteracting Anderson localization suppresses particle transport and generally leads to bounded entanglement growth after a product-state quench. Interactions in a many-body localized regime change the story: localized degrees of freedom dephase one another and can generate unbounded but logarithmically slow entanglement while transport remains strongly suppressed Bardarson, Pollmann, and Moore 2012.

In an effective local-integral-of-motion picture, an interaction between degrees of freedom separated by $r$ behaves as

$$V(r)\sim V_0e^{-r/\xi}.$$

They dephase on

$$t_{\rm deph}(r)\sim\frac{\hbar}{V_0}e^{r/\xi}.$$

Solving for the distance reached by time $t$ gives

$$r(t)\sim\xi\log\!\left(\frac{V_0t}{\hbar}\right),$$

and hence the leading expectation

$$S_A(t)\sim s_\infty\xi \log\!\left(\frac{V_0t}{\hbar}\right)$$

over the dephasing window. Serbyn, Papić, and Abanin 2013, Eqs. (1)–(5) develop this mechanism. The coefficient, onset, and saturation depend on the state, disorder, and finite system.

Long prethermal plateaus, rare-region subdiffusion, and finite-size slowdowns can imitate localization. A bath can ultimately destroy isolation. Report localized behavior only with size growth, longer-time stability, transport suppression, disorder statistics, and environmental controls.

Weak breaking and crossover collapse

Add an integrability-breaking perturbation,

$$H(g)=H_{\rm int}+gV.$$

For small $g$, dressed quasiparticle behavior may survive until a scattering or thermalization time $t_{\rm cross}(g)$. A golden-rule regime often motivates $t_{\rm cross}\propto g^{-2}$, but the exponent is not universal. Measure it rather than building it into every fit.

A strong crossover analysis:

1. determines the integrable prediction at $g=0$ without entropy refitting;
2. compares data at fixed physical $t$, $\ell$, and energy density;
3. extracts $t_{\rm cross}$ from several observables;
4. tests collapse against $t/t_{\rm cross}(g)$;
5. extends beyond the prethermal window at increasing size;
6. resolves symmetry sectors again after $V$ is added.

Failure of the integrable formula at late rescaled time and success of an out-of-sample chaotic description support a crossover. A drifting finite-size window does not.

Matched three-class comparison

Use preparations with matched energy density, initial correlation length, region geometry, interaction range, regulator, and numerical tolerance. Record:

• entropy curves for several intervals and Rényi orders;
• stationary entropy density or generalized-ensemble entropy;
• charge and energy transport;
• level-ratio statistics within exact symmetry blocks;
• operator or OTOC fronts with a declared regularization;
• out-of-sample quasiparticle and membrane predictions;
• disorder distributions and rare-event sensitivity.

Linear entropy growth may appear in both the first two columns; logarithmic growth may appear in localization or a long crossover. The joint pattern, not one exponent, classifies the regime.

The chapter orientation map treats integrable, chaotic, and localized evolution as distinct dynamical-class inputs. Its failure controls require weak breaking and window variation, while the diagnostic comparison keeps the entropy inference separate from transport and scrambling.

Common pitfalls

Calling linear growth chaotic. Both integrable quasiparticles and chaotic membranes can produce it. Test their independent inputs and predictions.

Mixing symmetry sectors in level statistics. Resolve exact blocks before comparing with Poisson or random-matrix values.

Calling a long transient localized. Increase size, time, disorder sampling, and isolation controls; search for crossover scales.

Exercises

Evaluate the single-speed quasiparticle benchmark at $t=3$ and $t=7$ for $s_{\rm stat}=\log2$, $v=1$, and $\ell=10$.

Solution

At $t=3$, $2t=6<10$, so $S=6\log2\approx4.158883$. At $t=7$, the interval sector has saturated, so $S=10\log2\approx6.931472$.

Compute the folded ratios for spacings $(s_1,s_2,s_3)=(1,2,1/2)$ and compare their mean with the Poisson and GOE reference values.

Solution

The two ratios are

$$\widetilde r_2=\frac12, \qquad \widetilde r_3=\frac{1/2}{2}=\frac14,$$

so their mean is $3/8=0.375$. It lies near the Poisson mean $0.386294$ and far from the GOE range near $0.53$, but two samples are nowhere near enough for classification; finite-size distributions and symmetry resolution are essential.

Derive logarithmic entanglement growth from $V(r)=V_0e^{-r/\xi}$.

Solution

Dephasing becomes appreciable when $V(r)t/\hbar\sim1$. Therefore

$$r(t)\sim\xi\log(V_0t/\hbar).$$

If each newly dephased unit length contributes an entropy density $s_\infty$, then $S(t)\sim s_\infty r(t)$, giving the logarithmic law. This reasoning fixes the functional form but not a universal coefficient.

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.
• Atas, Y. Y., E. Bogomolny, O. Giraud, and G. Roux. “Distribution of the Ratio of Consecutive Level Spacings in Random Matrix Ensembles.” Physical Review Letters 110 (2013): 084101. DOI.
• Bardarson, Jens H., Frank Pollmann, and Joel E. Moore. “Unbounded Growth of Entanglement in Models of Many-Body Localization.” Physical Review Letters 109 (2012): 017202. DOI.
• Nahum, Adam, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. “Quantum Entanglement Growth under Random Unitary Dynamics.” Physical Review X 7 (2017): 031016. DOI.
• Serbyn, Maksym, Z. Papić, and Dmitry A. Abanin. “Universal Slow Growth of Entanglement in Interacting Strongly Disordered Systems.” Physical Review Letters 110 (2013): 260601. DOI.