Nonequilibrium Many-Body Dynamics

An isolated quantum system can evolve unitarily while a local observable settles near a steady value. That apparent relaxation can arise from dephasing, transport, or constrained equilibration, and it need not imply thermalization. Driving, disorder, and approximate conservation laws introduce further possibilities.

This chapter relates preparation and dynamics to the observables used to distinguish those behaviors. Begin with Nonequilibrium Overview for the initial-value problem and relevant time scales.

Required background. Time-Evolution Operator supplies unitary propagation and time-ordering. Thermodynamic Limit supplies controlled size sequences, recurrence and finite-size cautions, and order-of-limits discipline.

Helpful background. Density operators prepare mixed initial states; Quantum Statistical Mechanics prepares comparison ensembles; Correlation Functions prepares two-time probes; Many-Body Entanglement prepares subsystem and operator-growth diagnostics. Open-system evolution is a separate branch whenever an environment is explicit.

Local relaxation can coexist with global unitary evolution

For a time-independent isolated Hamiltonian,

$$\rho(t)=e^{-iHt/\hbar}\rho(0)e^{iHt/\hbar}.$$

Unitary evolution preserves the eigenvalues of $\rho$, and therefore its purity and global von Neumann entropy. Local reduced states can nevertheless change because correlations between subsystems evolve. A globally pure state can have a nearly thermal reduced state on a small region.

In a finite system with a discrete spectrum, the infinite-time average is

$$\overline\rho=\sum_E P_E\rho(0)P_E,$$

where $P_E$ projects onto the full eigenspace at energy $E$. Coherences between different energies average away, while coherences within a degenerate eigenspace remain. This average is not a claim that the instantaneous state converges to $\overline\rho$; oscillations and recurrences can persist.

To test equilibration, specify an observable, a time window, and the size of fluctuations around its average. To test thermalization, additionally compare suitable local observables with a thermal ensemble matched to the conserved quantities, and check the initial energy distribution and system-size dependence. If additional conserved quantities constrain the dynamics, a generalized ensemble may be required.

Choose a dynamical question

Different preparations and observations lead to different branches.

1. Declare the common protocol. Begin with Nonequilibrium Overview to separate state preparation, generator, probe, clocks, and late-time claims.
2. Take the quench branch. Quantum Quenches develops sudden many-body preparation and post-quench observables. Relaxation and Thermalization then separates dephasing, equilibration, ensemble agreement, and loss of local memory.
3. Test the thermalizing mechanism. Eigenstate Thermalization Hypothesis connects symmetry-resolved eigenstate matrix elements to local thermal behavior. It is neither a definition of thermalization nor a theorem for every interacting system.
4. Take a constrained-memory branch. Integrability and Generalized Gibbs Ensembles Preview treats extensive conserved structure. Many-Body Localization Preview treats disorder-supported memory with explicit dimensional, finite-size, and stability cautions.
5. Take a long-lived intermediate branch. Prethermalization Preview applies when scale separation or high-frequency driving produces a controlled plateau before later evolution.
6. Separate chaos from information spreading. Quantum Chaos Preview develops symmetry-resolved spectral statistics and dynamical chaos diagnostics. Scrambling and OTOCs Preview develops operator growth and recovery tests. Neither is a universal prerequisite for the other.
7. Take the return-amplitude branch. Loschmidt Echo and Dynamical Phase Transitions Preview treats return amplitudes and thermodynamic rate-function singularities without turning them into equilibrium phase transitions.
8. Take the driven branch. Driven Many-Body Systems develops generic energy absorption and response beyond linear response. Floquet Systems Preview specializes periodic drives, quasienergy, micromotion, heating, and prethermal or localized exceptions.

Example: a plateau after a quench

Suppose a local observable approaches a size-stable plateau after a quench while the global state remains pure and entanglement continues to grow. This is consistent with local equilibration, not yet thermalization. Compare several local observables with a Gibbs ensemble matched to the conserved energy and charges; resolve symmetries; test system-size, subsystem-size, and time windows; and exclude integrable, localized, prethermal, hydrodynamic, and finite-size explanations. Global purity is compatible with locally thermal reduced states. Report which observables agree, to what accuracy, with which ensemble, and over which time and size range.

Continue with the observable and evolution you need

Correlations and Response defines two-time probes; Many-Body Entanglement and Information supplies state and operator entanglement diagnostics. Locality explains restrictions on propagation.

For numerical errors and accessible time scales, use Computational Many-Body QM. An explicit environment requires Open Quantum Systems; closed-system dephasing is not energy loss to a bath. The Schwinger–Keldysh bridge introduces a field-theory description of real-time questions.

Common pitfalls

“A plateau proves thermalization.” It may show finite-time equilibration of one observable. Thermalization additionally requires agreement with the appropriate ensemble across a declared observable class and limit.

“Dephasing is dissipation.” A closed system can dephase under unitary evolution without losing global purity or energy to an environment.

“The diagonal ensemble is the long-time state.” It gives infinite-time averages under stated spectral conditions; instantaneous finite-time states continue evolving and can recur.

“ETH is thermalization.” ETH is an eigenstate-structure mechanism that can support local thermal behavior after suitable preparations. Symmetry sectors, conserved charges, observable class, and finite-size scaling matter.

“Integrability means no relaxation.” Local observables can equilibrate to a generalized constrained ensemble while retaining information absent from a Gibbs description.

“A prethermal plateau is a steady phase.” It is an intermediate regime with a lifetime and drift controlled by scale separation; its eventual fate must be stated.

“Level statistics, OTOC decay, and scrambling are equivalent chaos tests.” They probe different structures and require distinct normalization, symmetry, temperature, and recovery controls.

“A Loschmidt-rate cusp is an equilibrium phase transition.” It is a nonanalyticity in a dynamical rate function under a specified protocol and limit.

“Floquet energy is conserved.” Quasienergy is defined modulo the drive quantum; generic interacting driven systems can absorb energy even when a one-period operator is exactly unitary.

“Finite-size localization proves stable many-body localization.” Drift with size, dimension, disorder ensemble, rare regions, boundary conditions, and observation time must be tested; frontier stability claims remain qualified.

Exercises

Exercise 1: Classify four late-time observations

Route (a) a local plateau matching a Gibbs prediction after a quench, (b) a plateau fixed by many conserved charges, (c) a long high-frequency-drive plateau with slow heating, and (d) a growing operator front with OTOC decay. State the strongest initial claim for each.

Solution

For (a), use Quenches → Relaxation and Thermalization and add ETH only as a tested mechanism; one observable supports local equilibration, while thermalization needs broader agreement with a matched ensemble. For (b), use Integrability and Generalized Gibbs Ensembles and verify the charge set and local predictions. For (c), use Driven Systems → Floquet → Prethermalization, identifying the effective generator, control parameter, lifetime, and heating beyond the plateau. For (d), use Scrambling and OTOCs plus Locality; operator growth does not alone establish spectral chaos, state thermalization, or information loss.

Exercise 2: Repair a thermalization claim

A finite spin chain shows a stable central-site magnetization over the final third of one simulation, and the value is near a canonical prediction. What must be added before claiming thermalization?

Solution

Declare the initial energy distribution, Hamiltonian, boundaries, symmetry sector, exact charges, observable normalization, and matched ensemble. Test multiple local observables and subsystems, several sizes and time windows, temporal fluctuations and recurrence scales, numerical and truncation errors, and alternative diagonal or generalized ensembles. Check integrability, localization, prethermal drift, and hydrodynamic slow modes. Report finite-time agreement for the tested observable unless these additional comparisons support a stronger thermalization conclusion.

References

• L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, “From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics,” Advances in Physics 65, 239–362 (2016).
• C. Gogolin and J. Eisert, “Equilibration, Thermalisation, and the Emergence of Statistical Mechanics in Closed Quantum Systems,” Reports on Progress in Physics 79, 056001 (2016).
• T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, “Thermalization and Prethermalization in Isolated Quantum Systems,” Journal of Physics B 51, 112001 (2018).
• R. Nandkishore and D. A. Huse, “Many-Body Localization and Thermalization in Quantum Statistical Mechanics,” Annual Review of Condensed Matter Physics 6, 15–38 (2015).
• A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, “Colloquium: Nonequilibrium Dynamics of Closed Interacting Quantum Systems,” Reviews of Modern Physics 83, 863–883 (2011).