Thermalization, Integrability Breaking, and Quantum Chaos

An isolated quantum field can lose memory of phases long before it is described by a thermal ensemble, can remain for parametrically long times in a prethermal state, or can display self-similar transport without approaching equilibrium at all. Quantum chaos is likewise not one observable: eigenstate statistics, out-of-time-order correlators, operator fronts, and late-time spectral correlations test different structures. This chapter supplies the distinctions and cross-checks needed to decide which claim is warranted.

Helpful background. Equilibration, thermalization, and dephasing establishes the chapter’s vocabulary. Hilbert positivity and unitary evolution explains why an isolated finite system never literally relaxes as a density operator under exact unitary evolution.

Enter this chapter

Begin by naming the observable, initial state, symmetry sector, system size, and order of the long-time and thermodynamic limits. Then ask which mechanism is actually being tested.

Observed behavior	First diagnostic	What it can establish	What remains separate
Local values become nearly stationary	temporal mean and variance	equilibration of the chosen observables in a stated window	agreement with a thermal ensemble
A long-lived plateau precedes heating	approximate charge or Floquet generator	controlled prethermal dynamics while the small parameter remains effective	eventual thermalization and resonance proliferation
Correlators collapse under rescaling	conserved flux and exponent relation	self-similar transport in a resolved scaling window	attraction to a universal fixed point
Individual eigenstates look microcanonical	diagonal and off-diagonal matrix elements	finite-size evidence for ETH in one sector and observable class	a theorem for all states or operators
Selected states revive or sectors disconnect	overlap, connectivity, and perturbation tests	a scar, fragmented sector, or other bounded exception	failure of thermalization in the rest of Hilbert space
A squared commutator grows	fully specified contour and operator pair	operator noncommutativity on that contour	universal chaos or information loss
A front propagates	velocity and broadening fit	spatial operator growth in a defined regime	a Lyapunov exponent or transport coefficient
Level correlations match random matrices	symmetry-resolved unfolding and form factor	spectral-chaos evidence over resolved scales	local thermalization for a chosen initial state

These routes overlap, but none is a synonym for another. The review by D’Alessio et al. 2016, §§2–7 gives a systematic bridge between ETH, random-matrix diagnostics, quenches, and finite-size evidence.

Choose a route

Goal	Route	Exit capability
Separate relaxation notions	equilibration	state exactly which observable equilibrates, to which ensemble if any, and in which limit
Analyze a long plateau	prethermalization	identify an approximate generator and estimate its failure time
Test far-from-equilibrium scaling	nonthermal fixed points	perform a collapse and check its conservation-law exponent
Test generic eigenstates	ETH	turn matrix-element scaling into a bounded quench prediction
Diagnose weak or strong ergodicity breaking	ETH exceptions	distinguish rare states from extensive disconnected sectors
Build a real-time chaos diagnostic	OTOCs → growth bounds	specify the contour, find a legitimate exponential window, and check every theorem hypothesis
Track spatial growth	operator spreading	separate causal support, butterfly propagation, front broadening, and saturation
Evaluate late-time evidence	spectral statistics	symmetry-resolve, unfold or filter, finite-size scale, and compare independent diagnostics

Three limit orders that change the answer

Let a finite system with Hamiltonian $H$ start in $\lvert\psi_0\rangle=\sum_n c_n\lvert n\rangle$. For an observable $A$,

$$\langle A(t)\rangle =\sum_{m,n}c_m^*c_n e^{i(E_m-E_n)t}A_{mn}.$$

The infinite-time mean removes nondegenerate energy gaps and gives the diagonal ensemble,

$$\overline{\langle A\rangle} =\sum_n\lvert c_n\rvert^2A_{nn},$$

but the state remains pure and the evolution remains quasiperiodic. A thermal statement needs an additional comparison between this diagonal value and the appropriate Gibbs or generalized ensemble. A continuum or thermodynamic claim also needs a controlled order of limits:

$$\lim_{V\to\infty}\lim_{t\to\infty}, \qquad \lim_{t\to\infty}\lim_{V\to\infty}, \qquad t,V\to\infty\ \text{with a fixed scaling variable}$$

need not agree. Recurrences survive at every finite volume, while kinetic, hydrodynamic, and self-similar regimes usually concern times that grow more slowly than the recurrence time.

Guide to the nine pages

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.

Chapter conventions and evidence ceiling

The default thermal state is $\rho_\beta=Z^{-1}e^{-\beta H}$. Every OTOC states its operator ordering and the placement of density-matrix factors; no bare symbol $F(t)$ is portable across regularizations. Every spectral statistic is computed inside an irreducible symmetry sector, with degeneracies, unfolding or filtering, and ensemble averaging declared. Velocities are defined from a chosen front marker and do not exceed the relativistic causal speed in a local relativistic QFT, though lattice Lieb–Robinson velocities are regulator-dependent bounds.

The chapter uses the following evidence ladder:

1. exact finite-system identity or theorem under stated hypotheses;
2. controlled expansion or parametrically separated time window;
3. convergence across size, time, operator, sector, and fitting choices;
4. agreement of diagnostics that do not share the same failure mode; and
5. a bounded physical interpretation.

The thermalization and chaos evidence matrix applies that ladder to the principal claims. A single small-system fit never occupies the last rung.

Review the chapter

Distinguish the claims. A local density approaches its Gibbs value while the global pure state remains at unit trace distance from the Gibbs state. This is compatible with local thermalization: the claim concerns a restricted observable algebra, not convergence of the full state.

Test an OTOC exponent. A semilog plot is linear for two time points before saturation. A valid chaos claim still needs a specified regularization, a parametrically broad interval between dissipation and scrambling, small connected correction, analytic hypotheses, finite-size control, and independent evidence.

Diagnose an exception. Ten atypical eigenstates occur among exponentially many thermal-looking states and strongly overlap one prepared state. This supports weak ergodicity breaking for that preparation; it does not establish Hilbert-space fragmentation or invalidate ETH for typical states.

Compare spectral and dynamical evidence. Wigner–Dyson spacings after symmetry resolution and a regularized OTOC front probe different correlations. Agreement strengthens a bounded chaos interpretation; disagreement is diagnostic rather than a reason to discard one result.

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.
• Moudgalya, Sanjay, B. Andrei Bernevig, and Nicolas Regnault. “Quantum Many-Body Scars and Hilbert Space Fragmentation: A Review of Exact Results.” Reports on Progress in Physics 85 (2022): 086501. doi:10.1088/1361-6633/ac73a0. Open preprint.
• Mori, Takashi, Tatsuhiko N. Ikeda, Eriko Kaminishi, and Masahito Ueda. “Thermalization and Prethermalization in Isolated Quantum Systems: A Theoretical Overview.” Journal of Physics B 51 (2018): 112001. doi:10.1088/1361-6455/aabcdf. Open preprint.