Entanglement-driven thermalization in isolated many-body systems: system-size scaling, integrability, localization, and subsystem-ensemble sensitivity

Two-copy interference in a quantum gas microscope directly measured subsystem Rényi entropies of a quenched six-site Bose-Hubbard system (Kaufman et al., Science 353, 794 (2016)) and found local observables consistent with thermal ensembles in the presence of a near-volume entanglement law. The authors proposed studying how thermalization depends on total system size and subsystem-to-system ratio, how integrable Hamiltonians fail to thermalize, the entanglement dynamics of many-body localized regimes, and the sensitivity of subsystem ensembles to the traced-out reservoir. The surviving open core is the systematic dependence of subsystem thermalization on system size and subsystem fraction — including the boundary where the grand-canonical description fails — a direct Rényi-versus-ETH/generalized-Gibbs comparison in an interacting integrable lattice platform, and the finite-temperature characterization of which operator classes distinguish a thermalized pure state from a thermal ensemble.

Background

When an isolated quantum many-body system is driven far from equilibrium — for example by a quantum quench, an abrupt change of its Hamiltonian — it evolves unitarily and remains in a pure state, yet experiments and simulations show that expectation values of local observables often relax to values predicted by statistical-mechanical ensembles. The eigenstate thermalization hypothesis (ETH) explains this: in non-integrable systems, individual excited eigenstates are so highly entangled that their reduced density matrices on small subsystems coincide with thermal ones, so no fine knowledge of the quench is needed. The mechanism is entanglement: the thermal entropy of a subsystem is its entanglement entropy with the rest of the system, which for highly excited states grows in proportion to subsystem volume (a volume law), while the global state stays pure. The entanglement entropy $S$ of a subsystem is the von Neumann entropy of its reduced density matrix; the second Rényi entropy $S_2=-\log \mathrm{Tr}\rho_A^2$ (with $\rho_A$ the subsystem density matrix and $\mathrm{Tr}\rho_A^2$ the purity) is an operationally accessible proxy, measurable in systems of itinerant particles by interfering two identical copies of the state: a beam-splitter tunneling operation between the copies converts purity into measurable single-site parity oscillations. Kaufman et al. (Science 353, 794 (2016)) implemented exactly this in a quantum gas microscope — an apparatus that images individual atoms of a $^{87}$Rb Bose-Hubbard lattice with single-site resolution — using two identical copies of a six-site chain, six bosons each (a 462-dimensional Hilbert space). After a quench, they observed entanglement growth toward a near-volume law, vanishing mutual information between complementary subsystems (mutual information being the total correlation shared by two subsystems, which thermalization drives to zero), and single-site and few-site number statistics in excellent agreement with microcanonical and canonical ensembles, while a grand-canonical ensemble (which allows particle-number fluctuations of the subsystem against a particle-number-conserving reservoir) reproduced the data only as long as the subsystem was small compared to the full system. They also noted that only a small set of operators, such as the measured global purity or other fine-tuned state projectors, could distinguish the thermalized pure state from a genuine thermal mixture. Because thermal ensembles differ precisely in how they trade system size against conserved-quantity constraints, three neighboring regimes sharpen the question of how general this entanglement-driven thermalization is. Integrable systems, possessing extensively many conserved quantities beyond energy, do not obey ETH; they relax instead to generalized Gibbs ensembles — maximum-entropy ensembles that fix the expectation value of every conserved quantity, not energy alone — and their quench entanglement grows ballistically as entangled quasiparticle pairs (pairs of elementary excitations propagating in opposite directions) spread out, rather than saturating to thermal values. Disordered interacting systems can enter a many-body localized (MBL) phase, where thermalization is suppressed and entanglement grows only logarithmically in time — the slow dephasing signature that distinguishes genuine many-body localization from non-interacting Anderson localization. Finally, subsystem statistical mechanics has its own finite-size theory: which ensemble (microcanonical, canonical, grand-canonical, or generalized) describes a subsystem depends on the size and structure of the traced-out remainder, a question addressed by canonical typicality (the fact that nearly all pure states in a fixed-energy shell have near-thermal subsystems), subsystem ETH, and the recent deep-thermalization framework of projected ensembles (the ensemble of pure conditional states obtained by measuring the complementary subsystem).

Problem Statement

With the aim of characterizing how thermalization emerges from entanglement in an isolated interacting many-body system — and where it fails — answer, for systems equipped with subsystem-resolved entanglement measurements (such as two-copy interference or randomized-measurement Rényi protocols) and local-observable readout, the following related questions left open by Kaufman et al. (Science 353, 794 (2016)):

(1) System-size and subsystem-fraction scaling: how do the dynamics of entanglement growth after a quench, the approach to the volume law (including the finite-size back-bending near half system size and the vanishing of mutual information), and the quantitative agreement of local observables with microcanonical, canonical, and grand-canonical ensembles change as the total system size is increased and as the ratio of subsystem size to system size is varied — in particular, where and how does the grand-canonical description of a number-constrained subsystem break down, and with what finite-size scaling?

(2) Integrability and conserved quantities: in integrable Hamiltonians or systems with additional conserved quantities, how does the failure of thermalization manifest in subsystem Rényi entropies and in local observables when compared directly with eigenstate-thermalization and generalized-Gibbs predictions — i.e., what is the direct experimental relation between the set of conserved quantities and the equilibrium ensemble of a subsystem?

(3) Long-time behavior and localization: what are the long-time entanglement dynamics and thermalization signatures in regimes exhibiting many-body localization, including the characteristic logarithmic-in-time entanglement growth and the suppression of thermalization, and how does this dynamics behave asymptotically and under coupling to a small thermal inclusion?

(4) Reservoir and ensemble sensitivity: how sensitive are the thermal properties of a subsystem to the effective size and structure of the traced-out reservoir, and which classes of operators — beyond fine-tuned state projectors such as the global purity — remain able to distinguish a globally pure thermalized state from a thermal ensemble, in the finite-temperature, particle-number-conserving regime?

These four questions form one coherent experimental and theoretical objective: a quantitative map of the validity boundary of ensemble descriptions of pure states, in the plane of system size, subsystem fraction, conservation laws, and localization. Questions (1), (2), and (4) are to be answered in full. For question (3), answer only the direct Rényi-entropy measurement of the localized entanglement dynamics and its long-time asymptotics; behavior under coupling to a small thermal inclusion (avalanches) is out of scope. An answer need not use the original $^{87}$Rb six-site apparatus, but it must address the stated dependences on isolated interacting lattice systems and must not replace them with a single-platform, single-parameter proxy.

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