We show numerically that a finite isolated cluster of interacting spins 1/2 exhibits a surprising nonthermal statistics when subjected to a series of small nonadiabatic perturbations by an external magnetic field. The resulting occupations of energy eigenstates are significantly higher than the thermal ones on both the low and the high ends of the energy spectra. This behavior semiquantitatively agrees with the statistics predicted for the so-called "quantum microcanonical" ensemble, which includes all possible quantum superpositions with a given energy expectation value. Our findings also indicate that the eigenstates of the perturbation operators are generically localized in the energy basis of the unperturbed Hamiltonian. This kind of localization possibly protects the thermal behavior in the macroscopic limit.
We show numerically that a finite isolated cluster of interacting spins 1/2 exhibits a surprising nonthermal statistics when subjected to a series of small nonadiabatic perturbations by an external magnetic field. The resulting occupations of energy eigenstates are significantly higher than the thermal ones on both the low and the high ends of the energy spectra. This behavior semiquantitatively agrees with the statistics predicted for the so-called "quantum microcanonical" ensemble, which includes all possible quantum superpositions with a given energy expectation value. Our findings also indicate that the eigenstates of the perturbation operators are generically localized in the energy basis of the unperturbed Hamiltonian. This kind of localization possibly protects the thermal behavior in the macroscopic limit.
In classic statistical physics, an isolated system corresponds to a constant energy shell in the phase space, which can be described by the microcanonical ensemble. While, for an isolated quantum system, the conventional treatment is to subject the system to a narrow energy window in the Hilbert space instead of the energy shell in classical phase space, and then confine the participating eigen states of system wave function in the narrow window, so that the microcanonical ensemble can be recovered in the framework of quantum mechanics. Apart from the traditional theory, there is a more self-consistent description for the isolated quantum system, that is, the quantum microcanonical (QMC) ensemble. The QMC ensemble abandons the narrow energy window assumption, and allows all the eigen states to contribute to the system wave function on condition that the system average energy is fixed at a given value. At the same time, the total occupation probability of these eigen states is conserved to unity. The most probable probability distribution obtained in the Hilbert space for an isolated quantum system according to the constraints specified above is called the QMC statistics. There is a clear difference between the QMC distribution and the traditional Gibbs distribution having an exponential form. Through the external periodic drives, an isolated quantum system may produce the QMC distribution, which is a consequence of the interplay between internal origins and external drives. In this paper, we investigate the conditions for the formation and suppression of QMC distribution by using the exact diagonalization method based on the one-dimensional Ising model. We start with the one-dimensional Ising model and focus on three different cases of periodic drives: systems under vertical (along the z axis), horizontal (along the x axis), horizontal magnetic field together with random internal (along the y axis) magnetic field. For all these three cases, the external magnetic fields are set to be ordinary rectangular pulses and the Gibbs distributions are taken as the initial states. We then study the evolutions and their asymptotic tendencies to the QMC distributions of the eigen state occupation probability under the effect of external periodic magnetic field. The results show that under the vertical magnetic field, the eigen state occupation probability does not change, and the system cannot produce the QMC distribution; under the horizontal magnetic field, the system tends to display a QMC distribution, but only partly; under horizontal and random internal magnetic fields at the same time, the transition to QMC distribution can be fully realized, and finally the system is almost completely thermalized. In order to clarify the different behaviors of the Ising model in the three cases, we also calculate the information entropy of the eigen state of Floquet operator in the eigen representation of the unperturbed Hamiltonian. We find that as the information entropy of the Floquet eigen state increases, the convergence to the QMC distribution in the Hilbert space is improved. We also notice that the mechanism for the emergence of QMC distribution is closely related to the thermalization effect of the isolated quantum system. Our analyses show that when the magnetic field is vertical, it cannot trigger the thermalization of the system. When the magnetic field is horizontal, the system becomes partly, but not completely, thermalized. When we add a horizontal periodic magnetic field and a random internal magnetic field at the same time, the system can be completely thermalized to infinite temperature. Thus, the asymptotic behavior towards the QMC statistics is a reflection of the fact that the isolated quantum system is thermalizable under periodic drives.
This work describes the statistics for the occupation numbers of quantum levels in a large isolated quantum system, where all possible superpositions of eigenstates are allowed provided all these superpositions have the same fixed energy. Such a condition is not equivalent to the conventional microcanonical condition because the latter limits the participating eigenstates to a very narrow energy window. The statistics is obtained analytically for both the entire system and its small subsystem. In a significant departure from the Boltzmann-Gibbs statistics, the average occupation numbers of quantum states exhibit in the present case weak algebraic dependence on energy. In the macroscopic limit, this dependence is routinely accompanied by the condensation into the lowest-energy quantum state. This work contains initial numerical tests of the above statistics for finite systems and also reports the following numerical finding: when the basis states of large but finite random matrix Hamiltonians are expanded in terms of eigenstates, the participation of eigenstates in such an expansion obeys the newly obtained statistics. The above statistics might be observable in small quantum systems, but for the macroscopic systems, it rather re-enforces doubts about self-sufficiency of nonrelativistic quantum mechanics for justifying the Boltzmann-Gibbs equilibrium.
Eigenstate thermalization is widely accepted as the mechanism behind thermalization in generic isolated quantum systems. Using the example of a single magnetic defect embedded in the integrable spin-1/2 XXZ chain, we show that locally perturbing an integrable system can give rise to eigenstate thermalization. Unique to such setups is the fact that thermodynamic and transport properties of the unperturbed integrable chain emerge in properties of the eigenstates of the perturbed (nonintegrable) one. Specifically, we show that the diagonal matrix elements of observables in the perturbed eigenstates follow the microcanonical predictions for the integrable model, and that the ballistic character of spin transport in the integrable model is manifest in the behavior of the off-diagonal matrix elements of the current operator in the perturbed eigenstates.
There is a dichotomy in the nonequilibrium dynamics of quantum many-body systems. In the presence of integrability, expectation values of local operators equilibrate to values described by a generalized Gibbs ensemble, which retains extensive memory about the initial state of the system. On the other hand, in generic systems such expectation values relax to stationary values described by the thermal ensemble, fixed solely by the energy of the state. At the heart of understanding, this dichotomy is the eigenstate thermalization hypothesis (ETH): individual eigenstates in nonintegrable systems are thermal, in the sense that expectation values agree with the thermal prediction at a temperature set by the energy of the eigenstate. In systems where ETH is violated, thermalization can be avoided. Thus, establishing the range of validity of ETH is crucial in understanding whether a given quantum system thermalizes. Here, we study a simple model with confinement, the quantum Ising chain with a longitudinal field, in which ETH is violated. Despite an absence of integrability, there exist rare (nonthermal) states that persist far into the spectrum. These arise as a direct consequence of confinement: pairs of particles are confined, forming new ``meson'' excitations whose energy can be extensive in the system size. We show that such states are nonthermal in both the continuum and in the low-energy spectrum of the corresponding lattice model. We highlight that the presence of such states within the spectrum has important consequences, with certain quenches leading to an absence of thermalization and local observables evolving anomalously.
Quantum systems with strong long-range interactions are thought to robustly resist thermalization because their energy spectra organize into a discrete number of multiplets. We show that this nonthermal behavior is surprisingly fragile to an experimentally ubiquitous perturbation: a staggered magnetic field. Using self-consistent mean-field theory and exact diagonalization, we reveal that the energy spectrum, while composed of discrete subspaces or multiplets, collectively forms a dense spectrum. The equilibration time is independent of system size and depends only on the fluctuations in the initial state. For initial states at low to intermediate energies, the long-time average aligns with the S z sector-weighted microcanonical ensemble. However, for states in the middle of the spectrum, the long-time average depends on the initial state due to quantum scarlike eigenstates localized at unstable points in classical phase space. Our results can be readily tested on a range of experimental platforms, including Rydberg atoms or optical cavities.
Ergodic isolated quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH), i.e. the expectation values of local observables in the system's eigenstates approach the predictions of the microcanonical ensemble.However, the ETH does not specify what happens to expectation values of local observables within an energy window when the average over disorder realizations is taken.As a result, the expectation values of local observables can be distributed over a relatively wide interval and may exhibit nontrivial structure, as shown in Aramthottil et al (2021 Phys.Rev. B 104 214201) for a quasiperiodic disordered system for site-resolved magnetization.We argue that the non-Gaussian form of this distribution may falsely suggest non-ergodicity and a breakdown of ETH.By considering various types of disorder, we find that the functional forms of the distributions of matrix elements of the site-resolved magnetization operator mirror the distribution of the onsite disorder.We argue that this distribution is a direct consequence of the local observable having a finite overlap with moments of the Hamiltonian.We then demonstrate how to adjust the energy window when analyzing expectation values of local observables in disordered quantum many-body systems to correctly assess the system's adherence to ETH, and provide a link between the distribution of expectation values in eigenstates and the outcomes of quench experiments.