PaperlayerView exact revision on arXiv

Source-linked AI summary

Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles

F. Borgonovi, F. M. Izrailev, L. F. Santos, V. G. Zelevinsky

arXiv:1602.01874v2cond-mat.stat-mech

TL;DR

The review asks when interactions make thermalization possible in isolated finite quantum systems, where the relation between discrete, quasi-periodic dynamics and finite-time chaotic behavior remains central. It links thermalization to chaotic eigenstates and examines their structure, statistical signatures, and dynamics across models and realistic systems. It reports agreement among temperature definitions under specified conditions and describes entropy growth as a signal of strong chaos and statistical relaxation.

  • Problem

    The review asks under what conditions isolated quantum systems develop chaotic eigenstates and thermalize, including how quantum chaos can be relevant despite discrete spectra and quasi-periodic dynamics at t →∞.

    Source: §1 · p. 3

  • Method

    The review studies chaotic eigenstates in physically motivated bases and relates their structure, strength functions, and dynamics to thermalization across random-matrix models and realistic systems.

    Source: §1 · pp. 2–3

  • Results

    In finite systems with self-consistent mean fields and chaotic eigenfunctions, different thermometers agree, and linear Shannon-entropy growth signals the onset of strong chaos and statistical relaxation in the studied boson model.

    Source: §1 · pp. 6–7

  • Takeaways & Limitations

    The review presents chaotic eigenstate structure as a way to identify conditions for thermalization and statistical behavior in isolated systems.

    Source: §1 · p. 3, §4.1 · p. 23

    The review treats relatively small systems over finite time scales and does not discuss thermalization in the thermodynamic limit.

    Source: §8 · p. 62

Abstract

from arXiv · show

This review is devoted to the problem of thermalization in a small isolated conglomerate of interacting constituents. A variety of physically important systems of intensive current interest belong to this category: complex atoms, molecules (including biological molecules), nuclei, small devices of condensed matter and quantum optics on nano- and micro-scale, cold atoms in optical lattices, ion traps. Physical implementations of quantum computers, where there are many interacting qubits, also fall into this group. Statistical regularities come into play through inter-particle interactions, which have two fundamental components: mean field, that along with external conditions, forms the regular component of the dynamics, and residual interactions responsible for the complex structure of the actual stationary states. At sufficiently high level density, the stationary states become exceedingly complicated superpositions of simple quasiparticle excitations. At this stage, regularities typical of quantum chaos emerge and bring in signatures of thermalization. We describe all the stages and the results of the processes leading to thermalization, using analytical and massive numerical examples for realistic atomic, nuclear, and spin systems, as well as for models with random parameters. The structure of stationary states, strength functions of simple configurations, and concepts of entropy and temperature in application to isolated mesoscopic systems are discussed in detail. We conclude with a schematic discussion of the time evolution of such systems to equilibrium.

1. Introduction

This review links thermalization in isolated quantum systems to interaction-driven chaotic eigenstates, and surveys how the crossover can be identified and followed across models and physical systems. It connects eigenstate structure and strength functions to statistical properties, effective temperatures, and finite-time dynamics.

1. Introduction

  • The review extends the approach from random-matrix and nuclear and atomic applications to interacting fermion, boson, and spin models, including realistic systems.Examples include nuclei, atoms, trapped ions, cold atoms and molecules in optical lattices, and interacting spins.Source: §1 · pp. 2, 4
  • The approach focuses on chaotic eigenstate structure rather than local spectral statistics to identify conditions for relaxation, using physically motivated bases such as the mean-field basis.The review discusses Shannon and invariant entropy, component moments, and the energy shell as measures or guides to eigenstate complexity and delocalization.Source: §1 · pp. 3, 6
  • Thermalization can emerge at sufficiently high energy when residual interactions effectively mix simple states into chaotic eigenstates.The review frames this as a crossover from regular dynamics to behavior described by standard statistical mechanics on a finite time scale.Source: §1 · pp. 2, 4
  • A crossover in the strength function from Breit-Wigner to Gaussian shape accompanies eigenstates filling the energy shell and is used to estimate the onset of chaos and thermalization.The review also relates the Gaussian shape to fast statistical relaxation.Source: §1 · pp. 5–6
  • In finite systems with self-consistent mean fields and chaotic eigenfunctions, different thermometers agree, while isolated eigenstates can exhibit Fermi-Dirac or canonical occupation distributions under stated interaction conditions.The review reports that the single-particle and density-of-states thermodynamic temperatures essentially coincide when the single-particle thermometer is valid; in the TBRE, occupation distributions become close to Fermi-Dirac above a critical interaction strength.Source: §1 · pp. 6–7
  • The review connects strength-function shape and eigenstate structure to time evolution, including survival-probability decay regimes and entropy growth as a signal of strong chaos and statistical relaxation.It describes a crossover from periodic to linear-in-time Shannon entropy above a critical interaction in a one-dimensional interacting-boson model.Source: §1 · p. 7

2. Hamiltonians

The review organizes interacting-particle Hamiltonians around a mean-field basis and residual interactions, then compares random-interaction models with realistic systems. It describes how strong mixing and high level density produce complex eigenstates with signatures of quantum chaos.

2.1. Basic ideas

  • The mean-field representation splits the Hamiltonian into a regular one-body part H0 and residual interaction V, with interaction strength able to drive eigenstates from perturbative to chaotic structure.The mean field defines a natural basis; realistic systems can use phenomenological or self-consistent mean fields.Source: §2.1 · p. 8

2.2. Wigner Band Random Matrices

  • Wigner band random matrices model an unperturbed spectrum and random couplings restricted to a band, while later versions allow random unperturbed levels with constant average density.The banded structure is tied to a physically preferred mean-field basis and can reflect finite-range interactions or selection rules.Source: §2.2 · pp. 9–10

2.3. Two-body random interaction

  • Two-body random interactions yield sparse, band-like Hamiltonians with correlated nonzero elements, reflecting the two-body structure also found in physical systems.For few fermions in a single high-angular-momentum level, random rotationally invariant two-body interactions were associated with a predominance of J = 0 ground states.Source: §2.3 · pp. 10–11

2.4. Realistic models

  • The review applies its approach to deterministic realistic systems, including interacting spins and the nuclear shell model, whose properties can resemble those of random two-body Hamiltonians.The spin models have no random entries; the shell model uses two-body interactions constrained by conservation laws.Source: §2.4 · p. 11, §2.4.2 · p. 13

2.4.1. Spin-1 /2 models

  • The integrable spin model has a more regular Hamiltonian structure than the non-integrable model, although their eigenstate properties and dynamics can resemble those of two-body random interactions.The second model adds next-nearest-neighbor couplings; chaos emerges when their strength is comparable to nearest-neighbor couplings.Source: §2.4.1 · p. 12

2.4.2. Nuclear shell model

  • Only 63 two-body matrix elements conserving constants of motion are possible; taken from experimental data and renormalized for space truncation, they describe thousands of low-lying observables and transition rates; at high excitation energy, deterministic interacting-fermion systems reveal quantum-chaos signatures.Source: §2.4.2 · p. 13
  • At high excitation energy and level density, deterministic nuclear-shell-model eigenstates become complex combinations of single-particle excitations and show quantum-chaos signatures close to band random matrix ensembles.The matrix-element distributions need not be Gaussian.Source: §2.4.2 · p. 13

3. Strength Functions

This section examines strength functions as descriptions of how basis states spread over exact eigenstates, including how their energy profiles depend on interaction strength and bandwidth and relate to eigenstate structure and classical motion. It also covers chaotic enhancement in fission, saturation of spreading widths, and exponential tails used in ground-state and level-density calculations.

3.1. Definitions

  • The strength function is a smoothed energy-resolved projection of a basis state onto exact eigenstates, and its width reflects the number of principal components involved.In fully chaotic classical motion, its distribution corresponds to the projection of the unperturbed energy along trajectories of fixed total energy.Source: §3.1 · p. 14
  • In the four-spin example, strength functions and eigenstates occupy only fractions of their respective bases, with restricted spreading reflecting the banded interaction matrix.The component matrix can help predict conditions for chaos and thermalization.Source: §3.1 · p. 15

3.2. Standard model of strength functions

  • The Breit–Wigner approximation requires a dense, rigid background spectrum, weakly fluctuating uncorrelated couplings, and an intermediate interaction strength; it describes the central strength-function region but not remote wings.Its spreading width gives an approximate exponential survival decay, which cannot be exact because that would imply divergent energy dispersion.Source: §3.2 · pp. 17–18
  • For finite-band interactions, increasing perturbation strength drives the strength function toward a semicircle, while closed interacting systems can transition from Breit–Wigner to Gaussian near Γ ∼ σ_k.This transition is associated with the onset of strong chaos and statistical relaxation.Source: §3.2 · p. 18

3.3. Beyond the standard model

  • When coupling is much stronger than the background energy range, strength splits into two peaks through repulsion between the doorway state and a collective mode formed from background states.Without coherent collective-mode formation, the near-chaotic limit instead approaches a semicircle.Source: §3.3 · p. 19
  • For multiple giant-resonance overtones, convolution gives Γ_2 ≈ 2Γ_1 for Breit–Wigner strength functions and Γ_2 ≈ √2 Γ_1 for Gaussian ones.The comparison assumes phonon spreading widths change little with excitation energy, as supported by N-scaling.Source: §3.3 · p. 19
  • In heavy nuclei, N-scaling enhances parity violation beyond single-particle estimates: observed cross-section differences reach 10%, with chaotic mixing and increased level density contributing an enhancement of order √N.At neutron threshold, compound states can have N reaching 10^6 principal components.Source: §3.3 · p. 20
  • The asymmetry is the same across numerous fission decay channels with different fragment-mass and kinetic-energy distributions, indicating enhancement precedes channel selection.The passage locates the enhancement in the compound nucleus before the fissioning nucleus selects a decay channel.Source: §3.3 · p. 20
  • Chaotic enhancement also occurs in tunneling or mixing between configurations close in energy, including different shapes and long-lived isomers [105, 106].Source: §3.3 · p. 20
  • In the N-scaling regime, golden-rule numerator and denominator share approximately the same 1/N dependence, corresponding to spreading-width saturation in the chaotic limit.Isobaric analog resonances have spreading widths of similar order across nuclei with different spins, isospins, and excitation energies.Source: §3.3 · pp. 20–21
  • Strength-function tails become exponential beyond 2σ from the centroid, rather than following Breit-Wigner or Gaussian forms.This tail behavior underlies exponential convergence: progressively truncated Hamiltonian matrices can yield exact low-lying energies, especially the ground-state energy, by continuation in matrix dimension.Source: §3.3 · p. 21
  • Exponential convergence supports exact ground-state energies needed in astrophysical reactions and a spin- and parity-dependent level-density algorithm.The method starts with full diagonalization for a few truncation steps, then continues energies exponentially as the matrix dimension increases.Source: §3.3 · p. 21

4. Chaotic Eigenstates

This section shifts the account of quantum chaos from level statistics to individual many-body eigenstates, examining how their mixing and energy-shell structure relate to thermalization in isolated systems. It develops basis-dependent and invariant measures of eigenstate complexity, including correlational entropy, and discusses their scope and links to phase transitions, pairing, interaction strength, and spectral statistics in finite systems.

4.1. Qualitative discussion and a little of history

  • Level-spacing statistics are an inadequate many-body chaos criterion; the decisive feature is eigenstate structure, with chaotic states confined to an energy shell set by the interaction range.Level statistics can be experimentally inaccessible, appear early in chaotization, and cease evolving before some observables do; unlike billiard states, interacting-particle states need not fill the whole mean-field basis.Source: §4.1 · p. 21
  • In cerium, excited states of four valence electrons become random superpositions, with about Np.c. ∼100 components above 1 eV for J = 4; TBRI eigenstates resemble those of the deterministic atom.The comparison supports TBRI models as descriptions of generic properties of realistic systems, despite their random parameters.Source: Figure 7 · p. 22, §4.1 · p. 22
  • Individual chaotic eigenstates can have the macroscopic properties of equilibrium ensembles, while internal interactions drive equilibration in finite isolated systems without a heat bath.The onset conditions for chaotic states are the central issue; entropy and temperature can be introduced non-uniquely and may depend on the measuring device.Source: §4.1 · pp. 23–24

4.2. Information entropy

  • In 28Si, mean-field-basis information entropy or localization length grows smoothly with excitation toward the GOE limit; in 12C, it distinguishes spurious center-of-mass states.Information entropy depends on the chosen basis, but can quantify complexity in a mean-field basis; the 12C example shows states of similar complexity but different physical nature can be separated.Source: §4.2 · pp. 24–26, Figure 8 · p. 25, Figure 9 · p. 25

4.3. Moments of the distribution of components

  • Information entropy emphasizes small wave-function components, whereas the inverse participation ratio emphasizes large ones; both measures are basis-dependent and can mislead for coherent or overly mixed states.In degenerate models, an effectively very strong residual interaction can place states near the GOE complexity limit before chaotization develops, while collective nonstationary states may also have large localization lengths.Source: §4.3 · p. 26

4.4. Invariant correlational entropy

  • Averaging a single eigenstate’s density matrix over small parameter variations yields a basis-independent correlational entropy that retains phase relationships; uncorrelated chaotic mixing reduces it to information entropy.When neighboring states are strongly mixed and look alike, averaging over them instead can provide a thermodynamic description with corresponding entropy and temperature.Source: §4.4 · p. 27
  • Invariant correlational entropy can peak near a finite-system phase transition when interaction-parameter fluctuations mix wave functions and observables between phases.For 24Mg, the entropy map indicates a possible transition from isovector to isoscalar pairing if the latter interaction strength is about three times stronger; realistic isovector pairing affects only a few low-lying 0+0 states, while non-pairing interactions can weaken correlations and enhance chaotic trends.Source: §4.4 · p. 28, Figure 10 · p. 28
  • Pairing alone cannot produce full chaos: intermediate pairing strengths introduce partial chaos and thermalization trends, while a phase correlator distinguishes chaotic from collective states that are both delocalized.In small systems, regular ground and low-lying states can coexist with chaotic excited states; the phase correlator exceeds 1/N for correlated components and reaches 1 for the equal-amplitude, same-sign limit.Source: §4.4 · pp. 28–29

4.5. Eigenstates versus energy shell

  • Chaotic eigenstates require many principal components and weak component correlations, and are delocalized when they densely fill the interaction-defined energy shell.The scaling parameter βloc is the localization length divided by its maximum, which is set by the number of basis states within the shell; localization is defined relative to that shell, not the full basis.Source: §4.5 · p. 29
  • As interaction strength increases, strength functions cross from Breit-Wigner to Gaussian shapes as their width approaches the energy-shell width, marking the onset of delocalization.For small non-perturbative interactions, the Breit-Wigner half-width Γ is proportional to the square of the mean directly participating matrix element; at stronger interaction, the shell width σ defines the Gaussian shape.Source: §4.5 · p. 30
  • The rescaled localization length βloc numerically matches the level-spacing repulsion parameter β to within 1%, although the relation lacks a quantitative explanation.The parameter β is obtained by fitting the nearest-level-spacing distribution to the Brody distribution.Source: §4.5 · pp. 30–31
  • In spin models, energy-shell filling occurs above λcr ≈0.4 for non-integrable Model 2, while integrable Model 1 is chaotic-like but non-ergodic above µcr ≈0.5.Model 2 also shows a Poisson-to-Wigner-Dyson transition in level spacings, whereas Model 1 retains Poisson statistics; an analytical estimate based on coupling matrix elements reproduces the critical strengths.Source: §4.5 · pp. 31–32, Figure 12 · p. 31

5. Thermalization

This section examines how statistical relaxation and thermalization can emerge in isolated interacting systems, including deterministic two-particle models, as chaotic eigenstates support thermal descriptions and statistical occupation patterns. It also considers how finite size, energy, and dynamical correlations can affect agreement among inferred temperatures and distributions.

5.1. Emergence of thermalization in isolated systems

  • Interparticle interactions can act as a heat bath: above a critical strength or level density, chaotic many-body eigenstates enable relaxation to a steady-state statistical distribution.For finite systems, temperature definitions can differ, with differences increasing as particle number decreases; in the thermodynamic limit they converge.Source: §5.1 · pp. 32–33

5.2. Fermi-liquid description

  • In 28Si, single-particle occupancies evolve smoothly across the spectrum and correspond to effective temperatures, with thermodynamic and single-particle temperatures nearly coinciding except near spectral edges.Pairing calls for BCS-like occupations near the ground state; at high excitation, occupancies approach 1/2, while effective temperature varies across the spectrum.Source: §5.2 · pp. 33–34, Figure 13 · p. 33

5.3. Microcanonical versus canonical distribution

  • For small isolated systems, the fixed-energy occupation distribution is needed, while in large systems it approaches the canonical distribution.The canonical form becomes close when the thermal averaging function narrows toward a delta function at its maximum.Source: §5.3 · p. 37

5.4. Onset of the Fermi-Dirac distribution

  • In the TBRI model of four fermions on eleven orbitals, weak interaction yields a nonmonotonic ns distribution with fluctuations, while strong interaction with Npc ≫ 1 produces a smooth equilibrium distribution associated with Fermi–Dirac behavior.For strong interaction, the energy shift should be included to obtain correct results.Source: §5.4 · pp. 37–38, Figure 15 · p. 37
  • In Au25+, chaotic deterministic eigenstates yield Fermi-Dirac occupation distributions, and excited-state spectra connect to enhanced radiative recombination near the ionization threshold.The reported eigenstate has a Breit-Wigner energy shape; the recombination enhancement over direct radiative recombination explains experimental data.Source: §5.4 · p. 38

5.5. Di

  • Thermodynamic and canonical temperatures become equivalent at high excitation or with sufficiently many active particles, but can differ in finite systems such as atoms and nuclei.For example, the passage gives 4 active particles for cerium and 12 for 28Si.Source: §5.5 · p. 38
  • In the four-particle, eleven-orbital TBRI model, thermodynamic and canonical temperatures differ noticeably across the rescaled energy range.The model uses V = 0.12 and mean single-particle spacing d0 = 1.Source: §5.5 · p. 39, Figure 18 · p. 41
  • A detailed study finds thermalization and distinct temperatures even for two interacting particles modeled as coupled rotators.For this system, the thermodynamic inverse temperature differs from βBE, while βfit is close to βBE; Tcan does not reproduce the actual occupation distribution.Source: §5.5 · pp. 39, 41
  • In the two-particle model, high-energy eigenstates develop Bose-Einstein occupation distributions, while the ground and low-lying states retain specific, non-statistical forms.The basis is symmetrized, so occupations are compared with the Bose-Einstein distribution; at large quantum numbers it approaches the Boltzmann distribution.Source: §5.5 · pp. 40–41, Figure 19 · p. 42
  • Removing correlations between nonzero Hamiltonian matrix elements leaves βcan nearly unchanged but brings βfit and βBE closer to it, allowing canonical occupations when dynamical correlations are weak.The randomized model preserves the original matrix elements’ mean and variance while replacing them with random variables.Source: §5.5 · pp. 41–42
  • The thermodynamic inverse temperature differs from βcan but approaches it near the spectrum center; in the randomized model, βth approaches βfit at high temperature and deviates at low temperature.Near spectrum edges, the density of states is small and eigenstates are no longer chaotic.Source: §5.5 · pp. 42–43

6. Statistical relaxation

The section places statistical relaxation within non-equilibrium quantum dynamics, focusing on isolated systems whose Hamiltonian changes from H0 to H0 + V.

6. Statistical relaxation

  • The section relates diagonal entropy to information entropy and notes that adiabatic evolution times can be much shorter than the Weisskopf time.It hypothesizes that the Weisskopf time describes a wave packet’s closest return, while statistical equilibrium may not require probing every state in an energy window.Source: §6 · p. 43
  • Experiments with controllable interacting systems and newer numerical methods have enabled studies of coherent dynamics and large-system simulations.Platforms include NMR, ultracold atoms, and molecules in optical lattices; methods include matrix product states, DMRG, and linked-cluster approaches.Source: §6 · p. 43
  • Non-equilibrium quantum dynamics connects to quantum information, entanglement, quantum communication, transport, phase transitions, and thermalization.The passage links entanglement growth to short-time limits of time-dependent DMRG and describes applications in information transfer and quantum computation.Source: §6 · pp. 43–44
  • Quenched dynamics starts with an eigenstate of H0 and evolves under H = H0 + V after a perturbation much shorter than the system’s characteristic timescales.The section also considers initial states that are not eigenstates of H0.Source: §6 · p. 44

6.1. Survival probability: theoretical background

The survival probability is the Fourier image of the strength function, so its time dependence reflects that function’s shape. In the TBRI model, the decay crosses from exponential to Gaussian behavior as interaction strength increases, with a mixed regime between them.

6.1. Survival probability: theoretical background

  • The survival probability is the Fourier transform of the strength function, whose smooth energy dependence for chaotic eigenstates enables analytical evaluation of its time dependence.The survival probability is also called the return probability or quantum fidelity.Source: §6.1 · p. 44
  • Weak interactions yield exponential decay after a short quadratic stage, while strong interactions produce a long Gaussian-like decay; intermediate interactions show Gaussian then exponential behavior.Figure 21 illustrates these three regimes for the TBRI model.Source: §6.1 · pp. 45–46, Figure 21 · p. 46
  • For Np = 6 fermions and Ns = 12 single-particle states (924 many-body states), the exponential fit (6.9) gives Γ0 ≈ 0.97, compared with rough estimate Γ0 ≈ 1.03.The estimate is very close to the numerical data; Γ0 is measured in units d0/ℏ.Source: §6.1 · p. 46
  • In the intermediate regime, the observed crossover occurs near tc ≈ 0.3, while an estimate gives tc ≈ 0.5; the crossover time is only defined up to a factor of order unity.The Gaussian stage precedes exponential decay.Source: §6.1 · p. 46
  • For strong interactions, an exponential tail after Gaussian decay can lead to asymptotic decay W0(t) ≈ C exp(−Γ0t), with C potentially very large.The passage notes that exponential decay cannot continue indefinitely; the ultimate long-time behavior is set by the spectrum threshold.Source: §6.1 · p. 46

6.2. Survival probability: spin-1 /2 systems

In deterministic spin-1/2 systems, survival-probability decay is governed by the strength function and can take exponential, Gaussian, or faster-than-Gaussian forms. The examples show that decay depends on the initial state and energy-shell structure, and can be similar in integrable and chaotic models.

6.2. Survival probability: spin-1 /2 systems

  • For integrable and chaotic spin models alike, survival probability can decay exponentially, Gaussianly, or faster than Gaussian, and decay slows as the initial energy moves away from the spectrum center.The fastest decay is limited by the energy-time uncertainty relation.Source: §6.2 · p. 47

6.2.1. Integrable vs Chaotic Regime

  • As perturbation grows, both integrable and chaotic models pass from slow decay with a narrow strength function to exponential decay for Breit-Wigner functions and Gaussian decay for Gaussian functions.The reported survival-probability behavior is similar in the two models and continues until saturation.Source: §6.2.1 · pp. 47–48, Figure 22 · p. 47

6.2.2. Experimentally accessible initial states

  • The Néel-state width is independent of λ, while the domain-wall width is independent of system size; as λ increases, their energies move toward and away from the spectrum center, respectively.The Néel-state energy reaches zero at λ = (L − 1)/(L − 2).Source: §6.2.2 · pp. 48–49
  • The Néel state has a well-filled energy shell and Gaussian decay in both models, whereas the domain-wall state decays more slowly because its shell is narrow and poorly filled.For the Néel state, the curves overlap because its strength-function width is the same in both models.Source: §6.2.2 · p. 49, Figure 23 · p. 49, Figure 24 · p. 50

6.2.3. Decays faster than Gaussian

  • Faster-than-Gaussian decay is expected for very strong interactions or multimodal strength functions; a double-peaked function can produce decay at the energy-time uncertainty bound before saturation.After first reaching saturation, the double-peak case shows large oscillations with a Gaussian envelope.Source: §6.2.3 · pp. 49, 51
  • For a full random matrix, the strength function approaches a semicircle, and post-saturation survival-probability oscillations decay as t^-3.The full-random-matrix case is described as an extreme, less realistic limit.Source: §6.2.3 · pp. 49–50

6.3. Relaxation in the energy shell

The review links relaxation in the energy shell to the spreading of wave-packet populations among interaction-defined classes. In the TBRI model, entropy growth, survival probability, and occupation-number relaxation provide complementary descriptions of this process.

6.3. Relaxation in the energy shell

  • In the cascade picture, population spreads to the jth class around t = jτ, with τ = 1/Γ0; a finite number of classes leads to saturation and a steady-state distribution.For six particles and twelve levels, three interaction steps can populate all 924 basis states if eigenstates are fully delocalized.Source: §6.3 · pp. 52–53
  • Neglecting a smaller term gives linear entropy growth and an exponential increase in the number of principal components; for Gaussian strength functions, the linear growth is more pronounced than in the Breit-Wigner case.The proposed rate Γ0 ln K is described as a “quantum Lyapunov exponent.”Source: §6.3 · pp. 53–54
  • For six particles and twelve orbitals, the one-class entropy expression agrees well with numerical data, while the measured entropy departs from linear growth in the inset and is quadratic at short times.The model has 924 basis states and K = 261; its effective class count is estimated as nc ≈ 1.2.Source: §6.3 · pp. 53–54
  • In the Gaussian regime, Γ0 in the entropy expressions is replaced by the strength-function width ∆k0, and numerical data show strong evidence of linear entropy growth before saturation.The linear dependence is more pronounced than for the Breit-Wigner case.Source: §6.3 · p. 54, Figure 28 · p. 54
  • Occupation numbers begin thermalizing on τ ∼ 1/Γ0 in the Breit-Wigner regime or τ ∼ 1/∆k0 in the Gaussian regime; populating all classes takes longer, about ncτ.The first timescale describes initial thermalization, while later damped oscillations may occur during approach to complete equilibrium.Source: §6.3 · p. 55
  • Strong interactions produce a fast, monotonic approach of occupation numbers to thermal values with relatively small fluctuations, whereas weak interactions produce damped oscillations and larger equilibrium fluctuations.The weak-interaction case has a relatively small number of principal components.Source: §6.3 · p. 55

6.4. Linear entropy increase as an indicator of thermalization

Shannon entropy growth tracks the spread of evolving states and can indicate statistical relaxation, with distinct time patterns across interaction regimes. Analytical predictions describe the spin-model dynamics under stated initial-state conditions, while the boson model links linear growth to the Tonks-Girardeau crossover.

6.4. Linear entropy increase as an indicator of thermalization

  • For spin Models 1 and 2, entropy grows linearly after short-time quadratic behavior and agrees with analytical expressions for Breit-Wigner and Gaussian strength functions.The full dynamics also agrees with the semi-analytical expression from short times through saturation; the predictions require initial H0 eigenstates away from spectral edges.Source: §6.4 · pp. 55–56
  • The entropy-growth predictions depend on K and ∆k0 from the final Hamiltonian matrix in the H0 basis, allowing dynamics to be anticipated before diagonalizing H.The analytical expressions apply to initial Hamiltonians H0 and initial states far from spectral edges; predictions are not known for the non-H0 eigenstates discussed in Section 6.2.2 and Ref..Source: §6.4 · p. 56
  • The boson model starts with all particles in the m = 0 level on a one-dimensional ring, with interaction strength controlled by g and total angular momentum conserved.The calculations use finite Np and Ns chosen consistently to permit extrapolation to larger atom numbers.Source: §6.4 · pp. 56–57
  • For Np = 6 bosons, n/g ≫1 produces oscillating entropy, whereas n/g ≪1 yields linear growth followed by saturation.The crossover from oscillating saturation toward linear growth marks the onset of the Tonks-Girardeau regime and statistical relaxation to a steady-state momentum distribution.Source: §6.4 · p. 58, Figure 31 · p. 58

6.5. Relaxation of Few-Body Observables

Few-body observables generally relax after a transient and fluctuate around their infinite-time average. For chaotic initial states away from spectral edges, those fluctuations decrease exponentially with system size, a result also reported for an interacting integrable model.

6.5. Relaxation of Few-Body Observables

  • Observable dynamics is expressed using matrix elements Ok,k′ in the eigenstates of the Hamiltonian defining the initial state, with sums involving states beyond k0.The observables considered experimentally include magnetization, spin-spin correlations, and atom counts in a site or spatial region.Source: §6.5 · p. 58
  • After a transient, observables equilibrate probabilistically by fluctuating around their infinite-time average.The dynamics is generally very fast.Source: §6.5 · p. 58
  • For initial states with chaotic structure away from spectral edges and without excessive degeneracies, temporal fluctuations decrease exponentially with system size.This behavior was also found for interacting integrable Model 1, not only models with Wigner-Dyson level spacing.Source: §6.5 · p. 58

7. Thermal Equilibrium

The section asks when an isolated finite system’s infinite-time average agrees with the microcanonical ensemble. It identifies energy-shell and eigenstate conditions for approximate conventional thermalization, while noting that initial-state energy and integrability affect its viability.

7. Thermal Equilibrium

  • The central question is when the infinite-time average of an observable is close to its microcanonical average for a finite system.The review relates this question to conditions for conventional thermalization and the eigenstate thermalization hypothesis, which alone does not clarify when the equivalence should hold.Source: §7 · p. 59
  • Conventional thermalization requires chaotic eigenstates across an energy region, with strength-function width comparable to the energy-shell width, and a microcanonical window no wider than that shell.Under these conditions, some observables have nearly constant eigenstate expectation values uncorrelated with initial-state weights.Source: §7 · pp. 59–60
  • Thermalization depends on the initial state: it may fail near spectral edges or when the strength-function width is smaller than the energy shell.Ergodic filling of the energy shell can occur independently of whether level spacings follow Wigner-Dyson or Poisson statistics.Source: §7 · p. 60
  • For quenches to integrable final Hamiltonians, thermalization may occur when the initial state spans chaotic-like states; finite interacting integrable systems need an initial energy closer to spectrum center than chaotic systems.The initial state can sample different symmetry sectors of the integrable system.Source: §7 · p. 60
  • For integrable quenches, generalized Gibbs or microcanonical ensembles can describe equilibrium, but away from mid-spectrum their averages may deviate even in the thermodynamic limit.For interacting integrable systems, such as XXZ, which conserved quantities to include remains open.Source: §7 · p. 60

8. Concluding remarks

The review connects thermalization in isolated interacting quantum systems to the structure of chaotic many-body eigenstates and the basis used to characterize them. It summarizes signatures in strength functions and dynamics, while restricting its main analysis to few-particle systems on finite time scales.

8. Concluding remarks

  • The review situates many-body quantum chaos in the complicated structure of interacting-particle eigenstates and notes that a single typical wave function can yield conventional statistical distributions.An isolated atom’s emergence of a Fermi-Dirac distribution is cited as a demonstration.Source: §8 · pp. 60–61
  • Thermalization is linked to chaotic many-body eigenstates, whose emergence depends on interaction strength, eigenstate energy, and an appropriate mean-field basis.Energy-shell and strength-function concepts characterize the crossover from non-chaotic to chaotic eigenstates.Source: §8 · p. 61
  • The strength function typically crosses from Breit-Wigner to Gaussian-like form as interaction increases, and this crossover can identify chaotic eigenstates filling the energy shell.Its form also determines quench return-probability behavior: Breit-Wigner gives exponential decay beyond short-time Gaussian decay, while Gaussian strength functions sustain Gaussian decay longer; later decay becomes algebraic.Source: §8 · p. 61
  • For Gaussian strength functions, Shannon entropy increases linearly, so the number of excited many-body states grows exponentially in time, a reported fingerprint of strong quantum chaos.For interacting bosons in one-dimensional optical traps, the entropy crossover corresponds to the mean-field to Tonks-Girardeau crossover.Source: §8 · p. 62
  • The review’s analysis concerns relatively small isolated systems on finite time scales and does not treat thermalization in the thermodynamic limit.The cited discussion notes that the time for clear exponential relaxation increases strongly with particle number and tends to infinity in that limit.Source: §8 · p. 62

10. Further reading:

Three papers that came to the authors’ attention after acceptance address entropy production after interaction quenches, predictors of thermalization, and quasilocal charges in integrable systems.

10. Further reading:

  • A study examines fidelity decay and entropy production in many-particle systems after a random interaction quench.Source: §10 · p. 62
  • Another study considers powerlaw decay exponents as predictors of thermalization in many-body quantum systems [2].Source: §10 · p. 62
  • A third paper discusses quasilocal charges in integrable lattice systems, including integrals of motion of the XXZ model [3].Source: §10 · p. 62
Loading 1602.01874v2…