A closed quantum many-body system that is suddenly driven out of equilibrium evolves unitarily, yet its local observables can relax as if the system were coupled to a heat bath. This problem, crystallized by Cramer et al. in their study of bosons in optical superlattices (Phys. Rev. Lett. 101, 063001 (2008)), asks how local relaxation and entanglement growth coexist after interacting quantum quenches, through three linked questions: (1) to what interaction strengths the remarkably simple limiting pictures — free bosons at U=0U=0, hard-core bosons at large UU, and the thermal (canonical Gibbs) local description corroborated by thermal perturbation theory at large UU — remain essentially valid for the locally relaxed state, in what regimes genuinely different relaxation dynamics can be identified, and on what timescales; (2) which global observables retain memory of the initial state while all local reduced states appear relaxed, and for how long; and (3) how the late-time regime, inaccessible to matrix-product-state simulation because bipartite entanglement grows linearly in time, can be accessed, classically or experimentally. Later work cast the ensemble question in terms of the eigenstate thermalization hypothesis and generalized Gibbs ensembles, settled the integrable and strongly interacting limits empirically, and directly measured entanglement-driven thermalization, but a rigorous characterization of thermalization for generic interacting systems, a quantitative theory of memory timescales near integrability, and access beyond the entanglement barrier remain open.

Background

A quantum quench is the simplest way of driving an isolated many-body system far from equilibrium: the system, prepared in a simple state (often the ground state) of one Hamiltonian, is suddenly subjected to a different Hamiltonian and then evolves unitarily. The paradox of relaxation is that although the full state remains pure and retains all information about the initial condition, expectation values of observables restricted to small subsystems can relax to stationary values that look thermal. Two maximum-entropy descriptions compete for this stationary regime. The Gibbs (canonical) ensemble assigns the subsystem the density matrix ρ∝e−βH\rho\propto e^{-\beta H} fixed only by the temperature-like parameter β\beta; the generalized Gibbs ensemble (GGE) instead maximizes entropy subject to the expectation values of all conserved quantities, and is expected when the dynamics is integrable, meaning the model possesses an extensive set of conserved quantities (for example the Bose-Hubbard model in its non-interacting or hard-core limits). For generic nonintegrable systems, the prevailing conjecture is the eigenstate thermalization hypothesis (ETH): individual many-body eigenstates already carry thermal values of few-body observables, so relaxation needs no mechanism beyond dephasing between eigenstates. Reviews by Polkovnikov et al. and by D'Alessio et al. survey this framework; Gogolin and Eisert review the rigorous partial results.

The concrete setting of this problem is the one-dimensional Bose-Hubbard model,

H=−J∑i(b^i+1†b^i+h.c.)+U2∑in^i(n^i−1)+Δ2∑i(−1)in^i,H=-J\sum_{i}\left(\hat b_{i+1}^{\dagger}\hat b_i+\mathrm{h.c.}\right)+\frac{U}{2}\sum_{i}\hat n_i(\hat n_i-1)+\frac{\Delta}{2}\sum_{i}(-1)^i\hat n_i,

where b^i\hat b_i annihilates a boson on lattice site ii, JJ is the hopping amplitude, UU the on-site repulsion, and Δ\Delta the strength of a period-2 superlattice potential — in the laboratory, a standing-wave optical lattice (a light-shift periodic potential for ultracold atoms) with doubled period. Cramer et al. proposed preparing the product state ∣1,0,1,0,…⟩|1,0,1,0,\ldots\rangle (one atom on each odd site, none on even sites) and quenching to the homogeneous model. Using time-dependent DMRG — the time-dependent variant of the density-matrix renormalization group, in which the wavefunction is represented as a matrix product state, a chain of contracted local tensors whose bond dimension χ\chi bounds the bipartite entanglement entropy SES_E (the von Neumann entropy of a subsystem's reduced state, SE≤log⁡χS_E\le\log\chi) — they found local relaxation at all interaction strengths studied. At large UU the relaxed nearest-neighbor coherence matches thermal perturbation theory (∼J/U\sim J/U), supporting a Gibbs-like local state; at U=0U=0, where the model reduces to free bosons, they proved that the quasi-momentum distribution S(q,t)S(q,t) (the momentum-space occupation, measurable by time-of-flight expansion, in which the trap is released and momentum maps to position) does not relax at all — local relaxation coexisting with a global observable that keeps complete memory of the initial state.

The same entanglement that drives local relaxation also blocks its simulation. The quench launches ballistically propagating excitations whose entanglement builds up at a roughly constant rate, so the required bond dimension grows exponentially in time; Cramer et al. could simulate only about six hopping times (t≈6t\approx6 in units where J=1J=1, with up to 5000 states and L≤32L\le32). Kim and Huse later showed that entanglement spreads ballistically even in a nonintegrable spin chain whose energy transport is diffusive, so this linear growth — and the exponential matrix-product-state cost it implies, the entanglement barrier reviewed by Schollwöck and by Paeckel et al. — is generic, not an artifact of special models. Whether the late-time regime beyond this barrier can be reached, by classical algorithms, by quantum simulation, or by direct experimental probes of entanglement, is the third strand of the problem. The three strands are one problem because in a closed system the entanglement-building dynamics is simultaneously the mechanism of local relaxation, the storage of global memory, and the obstruction to classical simulation.

Problem Statement

The problem is to give a complete account of post-quench relaxation in closed interacting quantum many-body systems — in the concrete setting of the quench of the one-dimensional Bose-Hubbard model from the density-wave product state ∣1,0,1,0,…⟩|1,0,1,0,\ldots\rangle (one boson on each odd site, prepared in a period-2 superlattice potential that is then switched off) and in interacting quenches generally — by answering all three of the following linked questions, which are mandatory components of the single objective — the concrete Bose-Hubbard quench above being the required deliverable to which all three apply, the broader class of interacting quenches serving as the motivating context — since the same unitary entanglement-building dynamics produces the local relaxation, stores the initial-state memory, and drives the simulation cost:

  1. Local state and interaction regimes. (a) Determine whether the stationary values of local observables after the quench at finite interaction strength are those of the thermal (canonical Gibbs) state; (b) to what interaction strengths the remarkably simple limiting pictures — free bosons at U=0U=0, hard-core bosons with free-fermion behavior at large UU, and the thermal-perturbation-theory description (∼J/U\sim J/U nearest-neighbor coherence) that corroborates it — remain essentially valid, meaning an accurate description of the locally relaxed state to within the accuracy of Cramer et al.'s own comparisons; (c) in what regimes genuinely different relaxation dynamics can be identified, meaning a qualitatively distinct stationary value or decay law rather than a shifted limiting parameter; and (d) characterize the associated relaxation timescales.

  2. Memory structure. (a) Identify which global observables — such as the quasi-momentum distribution S(q,t)S(q,t) (the momentum-space occupation distribution), which provably never relaxes at U=0U=0 — retain information about the initial state while all sufficiently local reduced states (subsystems whose size is held fixed as the system size grows) appear relaxed at generic finite interaction strength; (b) quantify how long that memory persists; and (c) determine whether it constitutes true nonrelaxation rather than very slow relaxation — nonrelaxation meaning the time-averaged observable does not approach its stationary value within the accessible window up to the finite-size recurrence time, and very slow relaxation meaning it does approach that value on a timescale long compared with the initial relaxation of local observables — as a function of interaction strength and of the degree of integrability breaking (for the stated model the two are linked, the dynamics being integrable at U=0U=0 and in the hard-core limit U→∞U\to\infty).

  3. Late-time access. Since the bipartite entanglement entropy grows approximately linearly in time and matrix-product-state simulation cost grows exponentially with it: (a) provide access to the late-time dynamics beyond this entanglement barrier — the window the matrix-product-state simulations of Cramer et al. could not reach (t≈6t\approx6 in units where J=1J=1), extending toward the very-large-time recurrence regime where the finite size of the quantum system becomes visible — and (b) the finite-size scaling of relaxation, by classical methods, by quantum-simulation experiments running longer than matrix-product-state algorithms can track, or by direct experimental measurement of the growing entanglement, sufficient to test items 1 and 2 in that late-time window.

A complete account requires, for each sub-ask listed under items 1-3, an explicit answer — a determination, a range, a classification, or a timescale, as that sub-ask requires; an account that settles only some of the sub-asks is a partial result, not a completion of the task.

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