Quantum chaos does not mean a quantum trajectory literally becomes a classically chaotic path. It means that spectral and dynamical signatures of a sufficiently complex quantum system lose the regular structure of integrable motion and begin to resemble universal random-matrix statistics after exact symmetries are removed.
Guide 46 ended with an integrable free-fermion spin chain. Its many conserved quasiparticle occupations strongly constrain dynamics. Generic interacting many-body Hamiltonians usually lack that extensive integrable structure. Their nearby energy levels repel, local observables often satisfy eigenstate thermalisation, and initially local quantum information spreads into increasingly nonlocal degrees of freedom.
This guide develops symmetry resolution, spectral unfolding, Poisson versus Wigner–Dyson spacing statistics, adjacent-gap ratios, random-matrix symmetry classes, spectral form factors, eigenstate thermalisation, operator growth, out-of-time-order correlators and scrambling. None of these diagnostics alone defines chaos universally; they are complementary probes that must be matched to the model and symmetry sector.
Resolve symmetries → order energy levels → remove smooth density → test local spectral correlations → compare eigenvectors and dynamics → diagnose thermalisation and scrambling.
1. Integrable versus chaotic is not simply solvable versus unsolvable
An integrable quantum many-body system possesses an extensive family of independent conserved quantities that strongly constrains motion.
A nonintegrable or quantum-chaotic model generally has only the conserved quantities required by obvious symmetries such as energy, particle number, momentum or parity.
Numerical diagonalisation can solve a finite chaotic Hamiltonian exactly at small size. “Chaotic” therefore does not mean “no eigenvalues can be computed”. It describes statistical structure and dynamical complexity as system size grows.
2. Why symmetry sectors must be separated first
Suppose H commutes with a symmetry operator Q. The Hilbert space splits into invariant sectors labelled by Q eigenvalues.
Levels from different sectors are allowed to cross because they do not repel one another through matrix elements forbidden by symmetry.
If spectra from several sectors are mixed, those independent crossings can falsely make a chaotic system look Poisson-like.
Therefore level-statistics analysis should be performed within one irreducible symmetry block whenever possible: fixed particle number, momentum, parity, total spin and any other exact quantum numbers.
3. Ordered spectrum and raw gaps
Within one symmetry sector, order the energies
E₁≤E₂≤…≤E_D.
Define nearest-neighbour gaps
δ_n=E_{n+1}−E_n.
Raw δ values are not directly comparable across the spectrum because the mean density of states varies with energy. Random-matrix predictions concern local fluctuations after that smooth density variation is removed.
4. Spectral unfolding
Let N(E) count the number of levels below E. Decompose it into a smooth part N̄(E) plus fluctuations.
Map each energy to an unfolded coordinate
ε_n=N̄(E_n).
The unfolded mean spacing is approximately one. Then
s_n=ε_{n+1}−ε_n
can be compared with universal spacing distributions.
Unfolding itself can bias results if the smooth fit is too rigid or too flexible. Local ratio statistics are popular partly because they avoid explicit unfolding.
5. Poisson level statistics
For many integrable spectra after symmetry resolution, nearby levels behave approximately independently.
Normalised spacings then follow
P_P(s)=e^{-s}, s≥0.
The probability density is finite at s=0. There is no level repulsion.
Poisson statistics are a common integrability signature, not a theorem that every integrable finite-size model must fit perfectly at every energy.
6. Wigner–Dyson level repulsion
For time-reversal-invariant chaotic systems in the Gaussian orthogonal ensemble class, the Wigner surmise is
P_GOE(s)=(π/2)s exp(−πs²/4).
Near s=0, P(s)∝s. Small spacings are suppressed: levels repel.
For other symmetry classes, the small-spacing behaviour is P(s)∝s^β with Dyson index β=1,2 or 4 for the standard orthogonal, unitary and symplectic ensembles.
7. Why random matrices appear
Random-matrix theory does not claim the physical Hamiltonian matrix entries are literally independent random numbers.
The claim is universality: after coarse model-specific information such as mean level density and exact symmetries are removed, local spectral correlations of complex systems can match those of the random-matrix ensemble sharing the same antiunitary symmetry class.
This is analogous to critical universality in Guide 45: detailed microscopic structure can disappear from appropriately rescaled long-range or statistical observables.
8. GOE, GUE and GSE
- GOE, β=1: real-symmetric class, commonly associated with time-reversal symmetry squaring to +1 after other symmetries are resolved.
- GUE, β=2: complex-Hermitian class, commonly associated with broken time-reversal symmetry.
- GSE, β=4: symplectic class, associated with time-reversal symmetry squaring to −1 and Kramers structure.
Physical antiunitary symmetries, not the visual appearance of a matrix in an arbitrary basis, determine the appropriate ensemble class.
9. Adjacent-gap ratio
Define
r_n=min(δ_n,δ_{n+1})/max(δ_n,δ_{n+1}).
Then 0≤r≤1.
Because r compares neighbouring raw gaps, the local density of states largely cancels and no explicit unfolding is required.
Common large-sample benchmarks are approximately
- Poisson:
⟨r⟩≈0.386; - GOE:
⟨r⟩≈0.53; - GUE:
⟨r⟩≈0.60.
Finite-size values drift and depend on which spectral window is selected.
10. Derive the Poisson mean ratio
For two independent exponential gaps x,y, define r=min(x,y)/max(x,y).
Using symmetry between x and y,
⟨r⟩=2∫_0^∞dy∫_0^y dx e^{-x-y}(x/y).
Evaluating the integral gives
⟨r⟩=2ln2−1≈0.386294.
This exact Poisson result provides a useful calibration for numerical spectra.
11. Worked ratio sequence
Suppose four consecutive gaps are
0.8, 1.1, 0.9, 1.0.
The adjacent ratios are
0.8/1.1≈0.7273;0.9/1.1≈0.8182;0.9/1.0=0.9.
The sample mean is about 0.815. That value is higher than GOE, but three ratios are far too few for a chaos diagnosis.
Random-matrix benchmarks describe distributions over many levels and often many disorder realisations or nearby spectral windows.
12. Long-range spectral statistics
Nearest-neighbour spacings probe only local spectral correlations.
Longer-range diagnostics include number variance, spectral rigidity and correlation functions of the density of states.
Chaotic random-matrix spectra are more rigid than independent Poisson points: fluctuations in the number of levels inside a long interval grow more slowly.
This long-range rigidity becomes especially visible in the spectral form factor.
13. Spectral form factor
Define the finite-temperature spectral form factor
K(β,t)=|Z(β+it)|²
where
Z(β+it)=Tr[e^{-(β+it)H}].
At β=0, K(t)=|Tr(e^{-iHt})|².
After suitable smoothing or ensemble averaging, chaotic spectra often display a dip, then a random-matrix “ramp” from correlated levels, then a late-time plateau set by the discrete spectrum.
14. Why averaging matters for the form factor
A single finite deterministic spectrum produces large oscillations in K(t).
To expose universal structure one may average over disorder realisations, a narrow energy window, a time window or a family of Hamiltonians.
Different averaging procedures can change the apparent ramp and plateau. A plot should state what was averaged and how the smooth density contribution was handled.
15. Eigenvectors carry thermalisation information
Spectral statistics alone do not tell us how observables behave in individual eigenstates.
The eigenstate thermalisation hypothesis (ETH) states, schematically, that for a few-body observable O in a chaotic many-body system, matrix elements in the energy basis behave like
O_mn=Ō(E)δ_mn+e^{-S(E)/2} f(E,ω)R_mn.
Here E=(Em+En)/2, ω=Em−En, S(E) is thermodynamic entropy, f is smooth and Rmn is an irregular order-one variable.
ETH is an ansatz supported broadly in nonintegrable systems, not a universal theorem for every Hamiltonian.
16. Diagonal ETH
The diagonal part says
⟨E_n|O|E_n⟩≈Ō(E_n)
with exponentially small fluctuations between nearby eigenstates in a large system.
Thus one individual high-energy eigenstate can reproduce thermal expectation values of local observables at the corresponding energy density.
This is much stronger than saying a thermal mixture averages many eigenstates.
17. Off-diagonal ETH and relaxation
For an initial state
|ψ(0)⟩=Σ_n c_n|E_n⟩,
the observable evolves as
⟨O(t)⟩=Σ_{m,n}c_m* c_n e^{i(E_m−E_n)t}O_mn.
Irregular energy differences dephase off-diagonal contributions. ETH makes their typical size small and smooth in energy, leaving a long-time value close to the microcanonical prediction when the initial energy distribution is narrow.
18. Integrable systems thermalise differently
An integrable system retains many conserved mode occupations.
A standard Gibbs ensemble depending only on total energy can therefore forget information that remains dynamically constrained.
After quenches, local observables may relax instead toward a generalised Gibbs ensemble containing the additional conserved quantities.
This is why Guide 46’s free-fermion chain is an ideal contrast case for ETH.
19. Many-body localisation is another exception
Strong quenched disorder can prevent generic interacting systems from thermalising over very long times and, in idealised models, support a many-body-localised phase with emergent quasi-local integrals of motion.
Level statistics in such regimes tend toward Poisson rather than Wigner–Dyson despite interactions.
Modern work has refined the stability limits of many-body localisation, especially in dimensions above one and in the presence of baths. It should not be treated as a universal outcome of “large disorder”.
20. Operator growth
In the Heisenberg picture, a local operator W evolves as
W(t)=e^{iHt}We^{-iHt}.
For an interacting local Hamiltonian, W(t) develops support on increasingly distant degrees of freedom.
Operator growth is constrained by Lieb–Robinson-type light cones but can still convert a simple local perturbation into a highly nonlocal many-body operator.
21. Out-of-time-order correlator
Choose local operators W and V that initially commute because they are spatially separated.
One scrambling diagnostic is the squared commutator
C(t)=−⟨[W(t),V]²⟩
for Hermitian/unitary choices where the sign makes C nonnegative in common conventions.
An equivalent out-of-time-order correlator is
F(t)=⟨W(t)VW(t)V⟩.
If W(t) still effectively commutes with V, F remains near its initial value. As the operator front reaches V, F changes and C grows.
22. Scrambling is not the same as ordinary decoherence
A closed many-body system can scramble information under perfectly unitary evolution.
The information has not disappeared into an environment. It has become encoded in complicated nonlocal correlations that are inaccessible to simple local measurements.
Decoherence, by contrast, describes entanglement with uncontrolled environmental degrees of freedom after tracing them out.
Scrambling can occur without decoherence, and decoherence can occur without strong internal chaos.
23. Butterfly velocity
In spatially local chaotic systems, the OTOC front can propagate approximately ballistically.
At separation x, one often writes a front scale
x≈v_B t
where vB is the butterfly velocity.
vB is model and temperature dependent and need not equal the Lieb–Robinson velocity, sound velocity or quasiparticle group velocity.
24. Lyapunov-like growth
In systems with an appropriate semiclassical or large-N regime, an OTOC can show an intermediate window
C(t)∼ε e^{λ_L t}
before saturating.
λL is called a quantum Lyapunov exponent by analogy with classical sensitivity.
Not every quantum-chaotic system has a clean exponential window, and finite spin chains often show front growth better described by operator spreading than by one universal λL.
25. Chaos bound
Maldacena, Shenker and Stanford proved, under analyticity, factorisation and thermal assumptions appropriate to their framework, the bound
λ_L≤2πk_BT/ℏ.
Black-hole and certain strongly interacting large-N systems can saturate this bound.
The bound is not a statement that every chaotic finite-dimensional system possesses an exponential OTOC with this rate.
26. Loschmidt echo
A fidelity-based sensitivity diagnostic compares forward evolution under H with imperfect reversal under H+δH:
M(t)=|⟨ψ|e^{i(H+δH)t}e^{-iHt}|ψ⟩|².
Rapid decay indicates sensitivity to Hamiltonian perturbations, but the decay regime depends on perturbation strength, density of states and classical/quantum correspondence.
Loschmidt echo, OTOCs and level statistics probe related but distinct aspects of quantum complexity.
27. Entanglement growth after a quench
Starting from a low-entanglement product state, a chaotic local Hamiltonian typically produces entanglement entropy that grows approximately linearly in time for a subsystem before finite-size saturation.
The entanglement velocity can differ from the butterfly velocity and Lieb–Robinson velocity.
Fast entanglement growth is one reason tensor-network simulation becomes difficult for generic real-time chaotic dynamics even when one-dimensional ground states were tractable.
28. Random circuits as a controlled chaos model
Local random unitary circuits discard detailed Hamiltonian conservation laws while retaining locality.
They provide analytically tractable models of operator spreading, entanglement growth and unitary-design formation.
Because gates are random by construction, they are not literal microscopic models of materials. Their value is universality: many coarse scrambling features can be separated from Hamiltonian-specific details.
29. Common misconception: level repulsion proves a system is chaotic in every sense
Wigner–Dyson statistics are strong evidence for random-matrix spectral correlations after symmetry resolution. They do not by themselves prove ETH, fast scrambling or any specific transport law.
30. Common misconception: Poisson statistics always means integrability
Unresolved symmetries, localisation, mixed sectors or insufficient spectral windows can also produce weak level repulsion. Integrability should be supported by conserved quantities or other structural evidence.
31. Common misconception: random-matrix universality means the Hamiltonian is random
A deterministic local Hamiltonian can show random-matrix fluctuation statistics. Universality concerns rescaled spectral correlations, not literal microscopic randomness.
32. Worked synthesis problem
A numerical spin-chain calculation produces 4000 eigenvalues in one fixed momentum/parity sector near the centre of the spectrum. The adjacent-gap-ratio mean is 0.528.
Step 1: Compare benchmarks. 0.528 is close to the GOE scale ≈0.53 and far from the Poisson value ≈0.386.
Step 2: Symmetry check. Verify that all remaining exact unitary and antiunitary symmetries have been treated correctly; otherwise the comparison is unreliable.
Step 3: Window check. Repeat in nearby energy windows and multiple system sizes to test stability.
Step 4: Eigenvector check. Plot diagonal matrix elements of a local observable against energy. A narrow smooth ETH band would add independent evidence of thermalising chaos.
Step 5: Dynamics check. Use a local quench or OTOC to test operator spreading. The final conclusion should combine spectral and dynamical diagnostics rather than relying on one statistic.
33. Practice set
- Why must exact symmetry sectors be separated before level statistics?
- What is spectral unfolding?
- State the Poisson spacing distribution.
- What feature distinguishes Wigner–Dyson spacing near s=0?
- What physical information selects GOE versus GUE?
- Define the adjacent-gap ratio r.
- What is the exact Poisson mean r?
- What does the spectral form factor probe?
- State the schematic ETH ansatz.
- Why can a single eigenstate look thermal under ETH?
- What does an OTOC diagnose?
- Why is scrambling different from decoherence?
Answers
- Independent symmetry sectors can cross without repulsion and fake Poisson statistics when mixed.
- Removing the smooth density-of-states variation so local mean level spacing is normalised.
e^{-s}.- Level repulsion: P(s) vanishes as a power of s.
- The system’s antiunitary/time-reversal symmetry class after other exact symmetries are resolved.
min(δ_n,δ_{n+1})/max(δ_n,δ_{n+1}).2ln2−1≈0.386294.- Long-range correlations of energy levels through the Fourier transform of spectral density correlations.
O_mn=Ō(E)δ_mn+e^{-S(E)/2}f(E,ω)R_mn.- Its local observable expectation values vary smoothly with energy and agree with thermodynamic values at that energy density.
- The growth of noncommutativity/operator support between initially separated local operators.
- Scrambling redistributes information into nonlocal correlations under unitary dynamics; decoherence loses local coherence to an uncontrolled environment.
Sources and further study
[1] Oriol Bohigas, Marie-Joya Giannoni and Charles Schmit, Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws, Physical Review Letters 52, 1 (1984). The foundational quantum-chaos/random-matrix conjecture.
[2] V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature. Introduced adjacent-gap-ratio diagnostics in many-body localisation/chaos studies.
[3] Marcos Rigol, Vanja Dunjko and Maxim Olshanii, Thermalization and its mechanism for generic isolated quantum systems. A central modern demonstration of eigenstate thermalisation in nonintegrable systems.
[4] Juan Maldacena, Stephen H. Shenker and Douglas Stanford, A bound on chaos. The thermal quantum Lyapunov bound under its stated assumptions.
[5] Brian Swingle, Unscrambling the physics of out-of-time-order correlators. A review of OTOCs, operator growth and scrambling diagnostics.
Continue through Quantum Mathematics
Guide 45: Quantum Many-Body Hamiltonians, Ground States, Correlation Functions and Phase Transitions supplies gaps, correlations and locality. Guide 46: Quantum Spin Chains, Jordan–Wigner Transformation, Free Fermions and Criticality is the integrable comparison model. Guide 48: Quantum Thermodynamics, Gibbs States, Free Energy, Work and Resource Theories formalises equilibrium and nonequilibrium resource constraints.