Abstract We rigorously examine, in generality, the ergodic properties of quantum lattice models with short range interactions, in the
C∗ C ∗ algebra formulation of statistical mechanics. Ergodicity results, in the context of group actions on
C∗ C ∗ algebras, assume that the algebra is asymptotically abelian, which is not generically the case for time evolution. The Lieb-Robinson bound tells us that, in a precise sense, the spatial extent of any time-evolved local operator grows linearly with time. This means that the algebra of observables is asymptotically abelian in a space-like region, and implies a form of ergodicity outside the light-cone. But what happens within it? We show that the long-time limit of the n-th moment of a ray-averaged observable, along space-time rays of almost every speed, converges to the n-th power of its expectation in the state (i.e. its ensemble average). Thus ray averages do not fluctuate in the long time limit. This is a statement of ergodicity, and holds in any state that is invariant under space-time translations and that satisfies weak clustering properties in space. The ray averages can be performed in a way that accounts for oscillations, showing that ray-averaged observables cannot sustainably oscillate in the long time limit. We also show that in the GNS representation of the algebra of observables, for any KMS state with the above properties, the long-time limit of the ray average of any observable converges (in the strong operator topology) to the ensemble average times the identity, again along space-time rays of almost every speed. This is a strong version of ergodicity, and indicates that, as operators, observables get “thinner” almost everywhere within the light-cone. A similar statement holds under oscillatory averaging.
We study a so-called semi-ergodic brickwork dual-unitary circuits where, in the infinite volume limit, the two-point correlation functions of single-site operators exhibit ergodic behavior along one light ray and non-ergodic behavior along the other light ray. Here, however, we study intermediate and long-time dynamics of a system in a finite, large volume. Under such dynamics, the Heisenberg evolution of a single traceless single-site operator lies within a restricted subspace, and this time evolution can be mapped to a simpler problem of a single qutrit scattering with a bunch of qubits sequentially. Despite the model being non-integrable and free from any quenched disorder, the operator entanglement grows at most logarithmic in time, contrary to prior expectations. The auto-correlation function can be written in terms of a sum of products of
SO(3) matrices, allowing for a random matrix prediction for the auto-correlation function at late times. The operator size distribution also becomes bimodal at certain times, displaying intermediate behavior between chaotic and free systems.
A physical theory of quantum mechanics can be based on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space-time reflection symmetry (
PT symmetry).Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data.One would think that a non-Hermitian Hamiltonian would give a quantum theory that violates unitarity.However, if
PT symmetry is not broken, it is possible to use a physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite.This construction is general and works for any
PT -symmetric Hamiltonian.The dynamics is governed by unitary time evolution.This formulation does not conflict with the requirements of conventional quantum mechanics.There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics.
Feynman-Vernon influence functional (IF) was originally introduced to describe the effect of a quantum environment on the dynamics of an open quantum system. We apply the IF approach to describe quantum many-body dynamics in isolated spin systems, viewing the system as an environment for its local subsystems. While the IF can be computed exactly only in certain many-body models, it generally satisfies a self-consistency equation, provided the system, or an ensemble of systems, are translationally invariant. We view the IF as a fictitious wavefunction in the temporal domain, and approximate it using matrix-product states (MPS). This approach is efficient provided the temporal entanglement of the IF is sufficiently low. We illustrate the versatility of the IF approach by analyzing several models that exhibit a range of dynamical behaviors, from thermalizing to many-body localized. In particular, we study the non-equilibrium dynamics in the quantum Ising model in both Floquet and Hamiltonian settings. We find that temporal entanglement entropy may be significantly lower compared to the spatial entanglement and analyze the IF in the continuous-time limit. We simulate the thermodynamic-limit evolution of local observables in various regimes, including the relaxation of impurities embedded in an infinite-temperature chain, and the long-lived oscillatory dynamics of the magnetization associated with the confinement of excitations. By incorporating disorder-averaging into the formalism, we analyze discrete time-crystalline response using the IF method. In this case, we find that the temporal entanglement entropy scales logarithmically with evolution time. The IF approach offers a new lens on many-body non-equilibrium phenomena, both in ergodic and non-ergodic regimes, connecting the theory of open quantum systems theory to quantum statistical physics.