The eigenstate thermalization hypothesis (ETH) attempts to bridge the gap between quantum mechanical and statistical mechanical descriptions of isolated quantum systems. Here, we define unbiased measures for how well the ETH works in various regimes, by mapping general interacting quantum systems on regular lattices onto a single particle living on a high-dimensional graph. By numerically analyzing deviations from ETH behavior in the nonintegrable Ising model, we propose a quantity that we call the n-weight to democratically characterize the average deviations for all operators residing on a given number of sites, irrespective of their spatial structure. It appears to have a simple scaling form, which we conjecture to hold true for all nonintegrable systems. A closely related quantity, which we term the n-distinguishability, tells us how well two states can be distinguished if only n-site operators are measured. Along the way, we discover that complicated operators on average are worse than simple ones at distinguishing between neighboring eigenstates, contrary to the naive intuition created by the usual statements of the ETH that few-body (many-body) operators acquire the same (different) expectation values in nearby eigenstates at finite energy density. Finally, we sketch heuristic arguments that the ETH originates from the limited ability of simple operators to distinguish between quantum states of a system, especially when the states are subject to constraints such as roughly fixed energy with respect to a local Hamiltonian.
Many-body localization (MBL) is well characterized in Fock space. To quantify the degree of this Fock-space localization, the multifractal dimension
Dq is employed; it has been claimed that
Dq shows a jump from the delocalized value
Dq=1 in the eigenstate thermalization hypothesis (ETH) phase to a smaller value
0<Dq<1 at the ETH-MBL transition, yet exhibiting a conspicuous discrepancy from the fully localized value
Dq=0, which indicates that multifractality remains inside the MBL phase. Here, to better quantify the situation, we employ, instead of the commonly used computational basis, the one-particle density matrix (OPDM) and use its eigenstates (natural orbitals) as a Fock state basis for representing many-body eigenstates $|\ensuremath{\psi}\ensuremath{\rangle}$ of the system. Using this basis, we compute
Dq and other indices quantifying the Fock-space localization, such as the local purity
S, which is derived from the occupation spectrum ${{n}_{\ensuremath{\alpha}}}$ (eigenvalues of the OPDM). We highlight the statistical distribution of Hamming distance ${x}_{\ensuremath{\mu}\ensuremath{\nu}}$ occurring in the pairwise coefficients $|{a}_{\ensuremath{\mu}}{|}^{2}{|{a}_{\ensuremath{\nu}}|}^{2}$ in
S, and compare this with a related quantity considered in the literature.
Quantum many-body scars (QMBS) represent a mechanism for weak ergodicity breaking, characterized by the coexistence of atypical non-thermal eigenstates within an otherwise thermalizing many-body spectrum. In this work, we revisit the spin-
1 XY model on a periodic chain and construct several new families of exact scar eigenstates embedded within its extensively degenerate manifolds that owe their origins to an interplay of
U(1) magnetization conservation and chiral symmetries. We go beyond previously studied towers of states and first identify a novel set of interference-protected eigenstates resembling Fock space cage states, where destructive interference confines the wave function to sparse subgraphs of the Fock space. These states exhibit subextensive entanglement entropy, and when subjected to a transverse magnetic field, form equally spaced ladders whose coherent superpositions display long-lived fidelity oscillations. We further reveal a simpler organizing principle behind these nonthermal states by using the commutant algebra framework, in particular by showing that they are simultaneous eigenstates of non-commuting local operators. Moreover, in doing so, we uncover two more novel families of exact scars: a tower of volume-entangled states, and a set of mirror-dimer states with some free local degrees of freedom. Our results illustrate the power and interplay of interference-based and algebraic mechanisms of non-ergodicity, offering systematic routes to identifying and classifying QMBS in generic many-body quantum systems.