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Localization and Glassy Dynamics Of Many-Body Quantum Systems
Giuseppe Carleo, Federico Becca, Marco Schiró, Michele Fabrizio
TL;DR
The paper asks whether strongly interacting, isolated lattice bosons can fail to explore many-body configuration space and thereby avoid ordinary thermalization. Using numerical dynamics, Lanczos-chain mappings, and time-dependent variational Monte Carlo, it finds threshold-dependent localization and long-lived metastable states associated with slowed density excitations. These results connect many-body localization-like behavior with a glassy dynamical arrest, while leaving finite-size and variational-scope questions open.
Problem
The paper investigates whether isolated strongly interacting bosons can become dynamically trapped instead of exploring the available many-body configuration space and thermalizing.
Method
The study combines numerical Bose-Hubbard dynamics, a Lanczos-basis mapping to an effective tight-binding chain, and time-dependent variational Monte Carlo for quenches and higher-dimensional systems.
Results
Above an interaction or energy threshold, density excitations slow sharply and the system develops long-lived inhomogeneous or correlated metastable states.
Takeaways & Limitations
The results suggest that strong-interaction constraints and effective doublon attraction can produce glassy dynamical arrest and hinder thermalization in many-body bosonic systems.
Takeaways & Limitations
The localization evidence may reflect finite-size spectra, while the variational approach assumes that quantum dynamics is captured by far fewer collective variables than the full Hilbert space.
Abstract
from arXiv · showhide
When classical systems fail to explore their entire configurational space, intriguing macroscopic phenomena like aging and glass formation may emerge. Also closed quanto-mechanical systems may stop wandering freely around the whole Hilbert space, even if they are initially prepared into a macroscopically large combination of eigenstates. Here, we report numerical evidences that the dynamics of strongly interacting lattice bosons driven sufficiently far from equilibrium can be trapped into extremely long-lived inhomogeneous metastable states. The slowing down of incoherent density excitations above a threshold energy, much reminiscent of a dynamical arrest on the verge of a glass transition, is identified as the key feature of this phenomenon. We argue that the resulting long-lived inhomogeneities are responsible for the lack of thermalization observed in large systems. Such a rich phenomenology could be experimentally uncovered upon probing the out-of-equilibrium dynamics of conveniently prepared quantum states of trapped cold atoms which we hereby suggest.
Results
Strong interactions can trap initially inhomogeneous bosonic states above a dynamical threshold, producing extremely slow density relaxation. A Lanczos-chain mapping attributes this behavior to localization near an effective potential well in many-body configuration space.
- Inhomogeneous initial states and dynamical localization: At density n = 1, large U/J keeps empty-and-doubly-occupied initial patterns near their starting profiles, unlike the rapid homogenization at small U/J.The equilibrium target is the homogeneous configuration (... 1, 1, 1, 1, ...).
- Inhomogeneous initial states and dynamical localization: The relaxation time τR is defined as the first time local density approaches its homogeneous value, and its inverse shows a sharp threshold step.The step becomes sharper as system size increases.
- Inhomogeneous initial states and dynamical localization: Above (U/J)_dyn^c, density profiles remain dynamically trapped in long-lived inhomogeneous configurations.The relaxation-time increase sharpens with system size.
- Mechanism: Strong-U dynamics is slowed because effective attraction makes doublon aggregates difficult to break apart, so cluster dissociation matters more than single-doublon decay.Small clusters can freeze the evolution, while annihilation and recombination are faster at small U.
- Many-body-space mapping: In the Lanczos basis, the Bose-Hubbard dynamics becomes a particle moving along a semi-infinite chain of many-body states with onsite energies and nearest-neighbor hoppings.This representation uses the initial state as the chain origin.
- Many-body-space mapping: For small U/J the effective particle diffuses across the chain, whereas above a critical interaction it remains localized near the origin.A deep edge potential well and effectively random onsite energies hinder hybridization and escape at large U/J.
- Scope: The observed localization may be a finite-size effect or may indicate genuine localization that survives in the thermodynamic limit.The supplied results do not resolve this distinction.
- Density dependence: For n = 2/3, relaxation crosses over more smoothly with U/J and shows no evidence of increasing relaxation times with system size.This contrasts with the sharp threshold behavior at unit density.
Homogeneous initial states and Quantum Quenches
Quantum quenches from homogeneous states reveal long-lived density correlations when the final interaction is much stronger than the initial one. The resulting metastability is linked to constrained density excitations and resembles dynamical arrest in glassy systems.
- Homogeneous initial states and Quantum Quenches: For Uf ≪ Ui, the density autocorrelation rapidly decays to zero, whereas for Uf ≫ Ui it remains on a finite long-lived plateau before eventual decay.For finite systems, C(t) ultimately approaches zero.
- Homogeneous initial states and Quantum Quenches: The relaxation time extracted from C(t) increases dramatically above a threshold final interaction strength.This parallels the threshold behavior found for inhomogeneous initial states.
- Interpretation: The finite plateau C⋆ may indicate excess double occupancies without an available relaxation channel.Interaction-induced dynamical constraints severely slow density excitations.
- Interpretation: The resulting long-lived metastable states and abrupt density timescales closely resemble the phenomenology of glassy materials.The comparison is based on dynamical arrest and long-lived inhomogeneous states.
Variational description, lack of thermalization and higher dimensions
The study uses time-dependent variational Monte Carlo to describe damping and relaxation after interaction quenches, finding non-thermal quasi-steady behavior at strong interactions in one and two dimensions.
- Variational description: The simulations use exact diagonalization for small homogeneous systems and variational calculations for larger one- and two-dimensional systems.Figure 4 uses N = 12 exact diagonalization, while Figure 5 includes N = 200 observables compared with grand-canonical thermal averages.
- Variational description: Time-dependent variational Monte Carlo extends a Jastrow-like wave function with time-dependent correlations to describe damping and relaxation of local observables.The approach is benchmarked against t-DMRG and uses a time-dependent Jastrow factor depending on site occupancies.
- Lack of thermalization: After quenches from Ui = 2J, weak final interactions produce damped potential energy approaching a quasi-steady value consistent with thermal behavior.Thermal averages are computed in the grand-canonical ensemble at an effective temperature fixed by the initial state's average energy.
- Lack of thermalization: At large Uf, potential energy relaxes on a fast scale τD ∼ 1/Uf toward a non-thermal quasi-steady state, while the much longer τR controls eventual escape toward equilibrium.The long relaxation scale is associated with the delayed approach to thermal equilibrium.
- Higher dimensions: In two dimensions, strong repulsion likewise separates time averages from thermal averages, indicating nonergodic dynamics similar to the one-dimensional case.The authors attribute the behavior to high-energy incoherent excitations that lack efficient relaxation channels.
Discussion
Strong interactions and sufficient initial energy can trap lattice bosons in inhomogeneous metastable states. The dynamics also maps onto a particle moving on a semi-infinite tight-binding chain, revealing a localization transition in many-body space.
- Discussion: Strong interactions trap repulsive lattice bosons in long-lived metastable states lacking translational symmetry when the initial energy exceeds a threshold.The slowing of density excitations and long-lived inhomogeneities resemble a glass transition.
- Discussion: Self-induced attraction among doublons is identified as a major process that freezes density dynamics on long time scales.The mechanism is connected to dynamical arrest in time-dependent density correlations.
- Discussion: The many-body evolution maps to a particle leaving the edge of a semi-infinite tight-binding chain whose sites represent many-body wave functions.Both the on-site energies and hoppings vary along the chain, with an edge potential well generated by the initial state's high-energy content.
- Discussion: A delocalization-localization transition occurs when the effective edge-well depth becomes sufficiently large, preventing diffusion across the many-body chain.The authors present this analogy as evidence worth further investigation for ergodicity breaking in many-body space.
Exact Diagonalization
The exact evolution can be expressed through the Hamiltonian spectrum, but large Hilbert spaces require iterative short-time propagation. The same framework also computes time-dependent correlation functions while monitoring conservation laws for numerical accuracy.
- The exact quantum evolution is determined by the Hamiltonian eigenstates, eigenvalues, and initial-state overlaps.
- For large Hilbert spaces, truncated Taylor expansion replaces full diagonalization by repeated short-time applications of H.The series is summed over powers of H and iterated to reach long times.
- Time-dependent correlation functions are obtained by evolving both the original state and the state produced by applying operator A.
- Conservation of total energy provides a numerical check, remaining effectively constant over the longest simulated times.
Lanczos Method
The Lanczos method converts the many-body Hamiltonian into a reduced tridiagonal representation generated recursively from an initial state. This enables dynamics to be interpreted as motion on an effective semi-infinite chain.
- The Lanczos method constructs L orthonormal states in which the full Hamiltonian becomes tridiagonal.
- The basis is generated recursively by repeatedly applying the Hamiltonian to an initial vector |0⟩.
- The tridiagonal matrix is specified by its diagonal and nearest-neighbor coefficients.
- After construction, evolution is mapped to a particle initially localized at |0⟩ moving under the reduced Hamiltonian H_L.
- Finite-precision effects are assessed because accumulated truncation errors can destroy orthogonality among recently generated Lanczos vectors.Calculations tested floating-point precision up to 10^4 bits.
Time-Evolving Block decimation
Time-evolving block decimation represents many-body states with local tensors and bond matrices, then advances them through repeated Suzuki–Trotter time steps. Accuracy is controlled by time-step and entanglement truncation errors.
- TEBD represents a generic state in a Hilbert space of dimension D^N using local tensors and bond matrices.
- Time evolution applies the real-time propagator e^-iH∆t repeatedly within a Suzuki–Trotter approximation.
- The method incurs systematic errors from both finite time steps and the finite entanglement retained by the representation.Convergence was checked against the time step and entanglement cutoff, reaching χ = 400.
Real-time variational Monte Carlo
Real-time variational Monte Carlo reduces many-body dynamics to time-dependent variational parameters and evaluates observables through sampling. The approach targets damping and relaxation while being benchmarked against t-DMRG.
- Real-time variational Monte Carlo avoids an intrinsic sign or phase problem because its probability density is strictly non-negative.The method is presented as an out-of-equilibrium extension of variational techniques.
- The approach assumes that relevant dynamics can be captured by far fewer collective variables than the full Hilbert-space dimension.This reduction is identified as the key requirement for an efficient time-dependent variational method.
- Excitation operators include density-density and doublon-related correlations, with additional correlation types needed for large quenches.
- The time-dependent Jastrow-like ansatz is designed to capture damping and relaxation that simpler approaches cannot describe, and its accuracy is benchmarked against t-DMRG.
- The variational wave function uses a time-independent uncorrelated state and time-dependent complex parameters θ(t) and α_k(t).
- The variational dynamics maps the Hamiltonian evolution onto equations for collective parameters associated with the excitation operators.
- For an N = 100 periodic chain initialized at Ui = 2J, variational Monte Carlo results are compared with t-DMRG data from an N = 64 open chain.
- The equations preserve the wave-function norm and energy, while observables are evaluated from the sampled probability distribution over configurations.
Inhomogeneous states dynamics at n = 2/3
At filling n = 2/3, inhomogeneous states show a smoother interaction-driven relaxation crossover, with no observed system-size increase in relaxation times. This contrasts with the sharper dynamical constraints reported at unit filling.
- Interaction-dependent relaxation: At n = 2/3, the initial profile (. . . 2, 0, 0, 2, 0, 0, . . .) exhibits a smoother crossover in density relaxation times as U/J increases.The corresponding exact evolution is shown for an N = 15 chain, with final interactions U/J = 0, 1, . . . 10.
- Finite-size behavior: No evidence appears for relaxation times increasing with system size at filling n = 2/3.The inset compares inverse relaxation times for lattice sizes N = 9, 12, and 15.
- Comparison with unit filling: The weaker finite-size dependence suggests that effective interaction constraints among doublons are stronger at unit filling than at n = 2/3.The strongest dynamical-arrest manifestation is expected at n = 1, where a sharp relaxation-time crossover occurs.
- Cluster dependence: Larger doublon clusters are expected to have longer relaxation times even away from unit filling.The possibility of non-thermal behavior at non-unit filling is not excluded.
Lanczos-basis analysis of an integrable model
The integrable hard-core-boson model restricts motion to a finite fraction of the Hilbert space, yet the Lanczos-basis wave packet fully delocalizes in the thermodynamic limit. Thus, Lanczos-basis localization is not necessarily caused by integrability.
- Lanczos-state structure: In the integrable hard-core-boson model, integrability reduces the number of allowed Lanczos states relative to the total Hilbert space.The Lanczos hopping decreases strongly with iteration and approaches zero after a finite number ν(N) of states.
- Wave-packet dynamics: Despite this restriction, the Lanczos particle’s average position increases, indicating full delocalization in the thermodynamic limit.Figure 9 compares the effective-chain hopping with the time-dependent wave-packet position; horizontal lines mark ν(N)/2.
- Interpretation: The analysis shows that localization in the Lanczos basis is not necessarily implied by integrability.The hard-core-boson dynamics provides an integrable comparison in which delocalization still occurs.