The occupation number is a key observable for diagnosing thermalization, as it connects directly to standard statistical laws such as Fermi-Dirac, Bose-Einstein, and Boltzmann distributions. In the context of spin systems, it represents the population of the sublevels of the magnetization in the z direction. We use this quantity to probe the onset of thermalization in an isolated one-dimensional quantum spin-1 Ising model with transverse and longitudinal fields and in its classical counterpart. Thermalization is achieved when the long-time average of the occupation number converges to the microcanonical prediction as the chain length L increases, consistent with the emergence of ergodicity. However, the finite-size scaling analysis in the quantum model is challenged by the exponential growth of the Hilbert space with L. To overcome this limitation, we turn to the classical model, which enables access to much larger system sizes. By tracking the dynamics of individual spins on their three-dimensional Bloch spheres and employing tools from random matrix theory, we establish a quantitative criterion for classical ergodicity in interacting spin systems. We find that deviations from classical ergodicity decay algebraically with system size. This power-law scaling then provides a quantitative bound on the approach to thermal equilibrium in the quantum model.
Abstract A study of quantum-classical correspondence for the onset of thermalization in an isolated system of four interacting particles is presented. Provided that classical dynamics of the system is chaotic, thermalization is present and fits the expected statistical mechanics results. Quantum classical correspondence is established for the classical analogue of the single particle distributions of occupation numbers. The latter quantity is useful to test thermalization and it allows to obtain the microcanonical Boltzmann factor in the quantum and semi-classical cases. The correspondence between them and the statistical mechanics results are surprisingly good considering that we have only four interacting particles. The resulting Boltzmann factor for the occupation number distribution depends on the eigenfunction width, which is a semi-classical parameter explicitly calculated and consistent with the quantum one.
We investigate the relation between the classical ergodicity and the quantum eigenstate thermalization in the fully connected Ising ferromagnets. In the case of spin-1/2, an expectation value of an observable in a single-energy eigenstate coincides with the long-time average in the underlying classical dynamics, which is a consequence of the Wentzel-Kramers-Brillouin approximation. In the case of spin-1, the underlying classical dynamics is not necessarily ergodic. In that case, it turns out that, in the thermodynamic limit, the statistics of the expectation values of an observable in the energy eigenstates coincides with the statistics of the long-time averages in the underlying classical dynamics starting from random initial states sampled uniformly from the classical phase space. This feature seems to be a general property in semiclassical systems, and the result presented here is crucial in discussing equilibration, thermalization, and dynamical transitions of such systems.
Permutation symmetry plays a central role in the understanding of collective quantum dynamics. By introducing power law couplings that algebraically decay with the distance between the spins
r as
1/rα, we break this symmetry with a non-zero
α. This allows us to probe the emergence of new dynamical behaviors, including thermalization in an otherwise permutation symmetric Hamiltonian with all-to-all spin interactions along
x direction subjected to periodic kicks in transverse direction. As we increase
α, the system interpolates from an infinite range spin system at
α=0 exhibiting permutation symmetry, to a short range integrable model as
α→∞ where this permutation symmetry is absent. We focus on this change in the behavior of the system as
α is tuned, using dynamical quantities like total angular momentum and von Neumann entropy. Starting from the chaotic limit of the permutation symmetric Hamiltonian at
α=0, for the finite system sizes considered, we find that for small
α, the steady state values of these quantities remain close to the permutation symmetric subspace values corresponding to
α=0. At intermediate
α values, these show signatures of thermalization exhibiting values corresponding to that of random states in full Hilbert space. On the other hand, the large
α limit approaches the values corresponding to integrable kicked Ising model. In addition, we also study the dependence of thermalization on the driving period
τ, with results indicating the onset of thermalization for smaller values of
α when
τ is large, thereby extending the thermalizing window in the intermediate range of
α. We further confirm these results using effective dimension and spectral statistics.
From the analysis of the relaxation process of isolated lattice many-body quantum systems quenched far from equilibrium, we deduce a criterion for predicting when they are certain to thermalize. It is based on the algebraic behavior $\ensuremath{\propto}{t}^{\ensuremath{-}\ensuremath{\gamma}}$ of the survival probability at long times. We show that the value of the power-law exponent $\ensuremath{\gamma}$ depends on the shape and filling of the weighted energy distribution of the initial state. Two scenarios are explored in detail: $\ensuremath{\gamma}\ensuremath{\ge}2$ and $\ensuremath{\gamma}<1$. Exponents $\ensuremath{\gamma}\ensuremath{\ge}2$ imply that the energy distribution of the initial state is ergodically filled and the eigenstates are uncorrelated, so thermalization is guaranteed to happen. In this case, the power-law behavior is caused by bounds in the energy spectrum. Decays with $\ensuremath{\gamma}<1$ emerge when the energy eigenstates are correlated and signal lack of ergodicity. They are typical of systems undergoing localization due to strong onsite disorder and are found also in clean integrable systems.
Ergodicity in quantum many-body systems is---despite its fundamental importance---still an open problem. Many-body localization provides a general framework for quantum ergodicity and may therefore offer important insights. However, the characterization of many-body localization through simple observables is a difficult task. In this article, we introduce a measure for distances in Hilbert space for spin-
21 systems that can be interpreted as a generalization of the Anderson localization length to many-body Hilbert space. We show that this many-body localization length is equivalent to a simple local observable in real space, which can be measured in experiments of superconducting qubits, polar molecules, Rydberg atoms, and trapped ions. By using the many-body localization length and a necessary criterion for ergodicity that it provides, we study many-body localization and quantum ergodicity in power-law-interacting Ising models subject to disorder in a transverse field. Based on the nonequilibrium dynamical renormalization group, numerically exact diagonalization, and an analysis of the statistics of resonances, we find a many-body localized phase at infinite temperature for small power-law exponents. Within the applicability of these methods, we find no indications of a delocalization transition.