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Thermalization, Integrability Breaking, and Quantum Chaos

An isolated quantum system can equilibrate locally while its full state remains pure and its evolution reversible. Thermalization, integrability, operator growth, and spectral chaos explain different aspects of that behavior.

Enter this chapter

Section titled “Enter this chapter”

Begin with Equilibration, Thermalization, and Dephasing to distinguish local equilibration from thermalization and dephasing.

Helpful background. Equilibration, Thermalization, and Dephasing and Hilbert Positivity and Unitary Evolution develop the definitions used by the chapter’s opening pages.

Topics

Section titled “Topics”
  1. Equilibration, Thermalization, and Dephasing — Defines observable equilibration, diagonal ensembles, hydrodynamization, kinetic relaxation, thermal comparison, and recurrence without identifying them.
  2. Prethermalization, Floquet Dynamics, and Generalized Ensembles — Derives approximate conservation and explains why long-lived is not infinite-lived.
  3. Nonthermal Fixed Points and Wave Turbulence — Turns a scaling collapse into a flux-constrained test rather than a visual fit.
  4. Eigenstate Thermalization Hypothesis — States the diagonal and off-diagonal ansatz, derives its quench implication, and presents the thermalization evidence matrix.
  5. Quantum Scars, Hilbert-Space Fragmentation, and ETH Exceptions — Classifies rare-state, sector, constraint, and localization mechanisms by their measure in Hilbert space and stability.
  6. Out-of-Time-Order Correlators and Contour Regularization — Distinguishes the squared commutator from the several regularized four-point functions used to measure it.
  7. Lyapunov Growth and Chaos Bounds — States the analytic theorem behind the thermal bound and shows why a fitted exponential is not automatically its exponent.
  8. Operator Spreading and Scrambling — Separates microscopic causal support, butterfly fronts, broadening, and information-theoretic interpretations.
  9. Spectral Statistics, Form Factors, and Late-Time Evidence — Treats symmetry-resolved level correlations, disconnected subtraction, ramp and plateau windows, and finite-size recurrences.

How the ideas fit together

Section titled “How the ideas fit together”

Dephasing can suppress temporal variations of a few observables without selecting a Gibbs ensemble. Eigenstate thermalization supplies an additional mechanism relating individual many-body eigenstates to thermodynamics, whereas integrable systems retain more conserved data D’Alessio et al. 2016, §§ 2–7.

Spectral statistics, out-of-time-order correlators, and operator spreading test different structures and limits. Agreement with one diagnostic does not prove the others, nor does an early-time growth law determine the final ensemble. State, symmetry sector, size, and observation time are essential to the comparison.

Where to go next

Section titled “Where to go next”

Continue to Open QFT and Driven Dynamics to specify the subsystem and integrate out its environment.

References

Section titled “References”
  • D’Alessio, Luca, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. “From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics.” Advances in Physics 65 (2016): 239–362. doi:10.1080/00018732.2016.1198134. Open preprint.