Eigenstate thermalization hypothesis
For a specified deterministic nonintegrable local Hamiltonian, prove quantitative ETH estimates in a fixed symmetry sector and finite-energy-density window for a stated class of local observables. Distinguish all-eigenstate (strong) ETH, vanishing-fraction exc
Problem statement
For a specified deterministic nonintegrable local Hamiltonian, prove quantitative ETH estimates in a fixed symmetry sector and finite-energy-density window for a stated class of local observables. Distinguish all-eigenstate (strong) ETH, vanishing-fraction exceptions (weak ETH), and statistical off-diagonal estimates.
Lean 4
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Background
ETH explains how isolated quantum systems thermalize: it posits that matrix elements of local observables in the energy eigenbasis take the Srednicki form, with smooth diagonal parts (equal to the microcanonical average) and pseudo-random, exponentially-small-in-entropy off-diagonal parts. If ETH holds, individual eigenstates already look thermal and expectation values relax to Gibbs values.
What is known: Numerical evidence supports ETH in many nonintegrable chains, and controlled random-matrix/circuit or specially structured models admit rigorous results. Canonical typicality concerns typical vectors in a subspace, not automatically the actual eigenbasis of a deterministic Hamiltonian. Weak ETH theorems under specific assumptions do not imply strong ETH; some integrable models can satisfy weak local statements while failing the stronger thermalization picture. Rare scars can violate strong ETH without disproving weak ETH.
Deriving ETH from the structure of a concrete, non-random local Hamiltonian — checkable sufficient conditions guaranteeing the diagonal and off-diagonal ansatz for its actual eigenstates — is open. This is the incremental, rigorous heart of quantum thermalization. Related: B5, B7, M4, U3.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Deutsch, 'Quantum statistical mechanics in a closed system', PRA 43, 2046 (1991) link
- [2]Srednicki, 'Chaos and quantum thermalization', PRE 50, 888 (1994) link
- [3]D'Alessio, Kafri, Polkovnikov & Rigol, 'From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics', Adv. Phys. 65 (2016) link
Recorded progress
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