The dynamics of local subsystems in a thermodynamically large quantum many-body system can be understood as effectively open as the system produces its own effective bath. The action of this bath can be characterised in terms of the so-called influence matrices. In generic situations, the complexity of these objects grows unfavourably with time, however, there exist solvable cases where influence matrices can be characterised exactly even in the presence of non-trivial interactions. Here we show that Rule 201, a deterministic version of the Floquet-PXP model, is one of these solvable instances. Indeed, it admits influence matrices given by a finite-dimensional matrix-product operator (MPO) that solves a finite set of algebraic conditions. We provide the solution, and characterise multi-time autocorrelation functions.
The dynamics of local subsystems in a thermodynamically large quantum many-body system can be understood as effectively open as the system produces its own effective bath. The action of this bath can be characterised in terms of the so-called influence matrices. In generic situations, the complexity of these objects grows unfavourably with time, however, there exist solvable cases where influence matrices can be characterised exactly even in the presence of non-trivial interactions. Here we show that Rule 201, a deterministic version of the Floquet-PXP model, is one of these solvable instances. Indeed, it admits influence matrices given by a finite-dimensional matrix-product operator (MPO) that solves a finite set of algebraic conditions. We provide the solution, and characterise multi-time autocorrelation functions.
Studies of integrable systems have profoundly deepened the fundamental understanding of quantum many-body physics. While equilibrium properties such as ground states and thermodynamics can often be characterized efficiently, accurately characterizing nonequilibrium integrable dynamics remains a significant challenge. Here, we develop an influence matrix bootstrap approach for exactly solving nonequilibrium dynamics. This approach begins with an ansatz consisting of a set of local tensor relations, termed generalized zipper conditions, and uses finite-time data to construct exact solutions of local dynamics that extend to arbitrarily long times. We demonstrate this approach in the "Rule 201" quantum cellular automaton, an integrable Trotterization of the PXP Hamiltonian. This uncovers a rich landscape of nonequilibrium behavior exhibiting initial-state dependence. As an example, we investigate the fate of persistent oscillating dynamics under local non-integrable perturbations, and present analytical results for non-thermal relaxation constrained by conservation laws. We also obtain numerically exact results for entanglement growth across a broad class of initial states. Furthermore, from an information-theoretic perspective, we identify a refined structure of multitime correlations termed the hidden Markov orde}: the memory encoded in the dynamics separates into finite-length and long-range distributed components, which becomes transparent in an exact split-index matrix-product-state representation of the influence matrix. Our approach enables unified investigations of nonthermalizing and thermalizing regimes of nonequilibrium dynamics within a single analytically tractable model, and can be tested experimentally in state-of-the-art quantum simulators such as Rydberg atom arrays.
Exactly solvable models in quantum many-body dynamics provide valuable insights into many interesting physical phenomena and serve as platforms for rigorously investigating fundamental theoretical questions. Nevertheless, they are extremely rare and existing solvable models and solution techniques have serious limitations. In this paper, we present a family of exactly solvable unitary circuits that model quantum many-body dynamics in discrete space and time. Unlike many previous solvable models, one can exactly compute the full quantum dynamics initialized from any matrix product state in this family of models. The time evolution of local observables and correlations, the linear growth of Rényi entanglement entropy, spatiotemporal correlations, and out-of-time-order correlations are all exactly computable. A key property of these models enabling the exact solution is that any time-evolved local operator is an exact matrix product operator with a finite bond dimension, even at arbitrarily long times, which we prove using the underlying C * -(weak) Hopf algebra structure along with tensor network techniques. We lay down the general framework for construction and solution of this family of models, and give several explicit examples. In particular, we study in detail a model constructed out of a C * -weak Hopf algebra that is very close to a Floquet version of the PXP model, and the exact results we obtain may shed light on the phenomenon of quantum many-body scars, and more generally, Floquet quantum dynamics in constrained systems.
Influence matrices encode the action exerted on local subsystems by the rest of an extended quantum many-body system during their evolution. Thus, knowledge of the influence matrix facilitates computationally efficient simulations of local dynamics. Here we propose a new systematic approach to generating quantum circuits with complex dynamics for which the influence matrices can be written down exactly. In contrast to previous frameworks of this kind, such as dual-unitary circuits, the resulting influence matrices are non-Markovian, exhibiting nontrivial temporal correlations. We explicitly construct a broad family of circuits of this kind, based on dressing free-fermion (matchgate) circuits with appropriately chosen interaction terms. We show that, contrary to previous solvable instances, these circuits produce patterns of correlations that closely resemble that of typical many-body systems. Our approach can be directly interpreted in terms of an error correction scheme where the terms breaking the solvability of the influence matrices play the role of errors.
Recent experimental and theoretical works have made much progress toward understanding nonequilibrium phenomena in thermalizing systems, which act as thermal baths for their small subsystems, and many-body localized systems, which fail to do so.The description of time evolution in many-body systems is generally challenging due to the dynamical generation of quantum entanglement.In this work, we introduce an approach to study quantum many-body dynamics, inspired by the Feynman-Vernon influence functional.Focusing on a family of interacting, Floquet spin chains, we consider a Keldysh path-integral description of the dynamics.The central object in our approach is the influence matrix, which describes the effect of the system on the dynamics of a local subsystem.For translationally invariant models, we formulate a self-consistency equation for the influence matrix.For certain special values of the model parameters, we obtain an exact solution which represents a perfect dephaser (PD).Physically, a PD corresponds to a many-body system that acts as a perfectly Markovian bath on itself: at each period, it measures every spin.For the models considered here, we establish that PD points include dual-unitary circuits investigated in recent works.In the vicinity of PD points, the system is not perfectly Markovian, but rather acts as a bath with a short memory time.In this case, we demonstrate that the self-consistency equation can be solved using matrix-product states (MPS) methods, as the influence matrix temporal entanglement is low.A combination of analytical insights and MPS computations allows us to characterize the structure of the influence matrix in terms of an effective "statistical-mechanics" description.We finally illustrate the predictive power of this description by analytically computing how quickly an embedded impurity spin thermalizes.The influence matrix approach formulated here provides an intuitive view of the quantum many-body dynamics problem, opening a path to constructing models of thermalizing dynamics that are solvable or can be efficiently treated by MPS-based methods and to further characterizing quantum ergodicity or lack thereof.
We study operator dynamics in many-body quantum systems, focusing on generic features of systems that are ergodic, spatially extended, and lack conserved densities. Quantum circuits of various types provide simple models for such systems. We focus on Floquet quantum circuits, comparing their behavior with what has been found previously for circuits that are random in time. Floquet circuits, which have discrete time-translation symmetry, represent an intermediate case between circuits that are random in time and lack any symmetry, and systems with a time-independent Hamiltonian and continuous time-translation invariance. By making this comparison, one of our aims is to identify signatures of time-translation symmetry in Floquet operator dynamics. To characterize behavior we examine a variety of quantities in solvable models and numerically: operator autocorrelation functions; the partial spectral form factor; the out-of-time-order correlator (OTOC); and the paths in operator space that make the dominant contributions to the ensemble-averaged autocorrelation functions. Our most striking result is that ensemble-averaged autocorrelation functions show behavior that is distinctively different in Floquet systems compared to systems in which successive time-steps are independent. Specifically, while average autocorrelation functions decay on a microscopic timescale for circuits that are random in time, in Floquet systems they have a late-time tail with a duration that grows parametrically with the size of the operator support. In the simplest models this tail is separated from the initial decay by a minimum, so that the average autocorrelation function has an intermediate-time peak. The existence of these tails provides a way to understand deviations of the spectral form factor from random matrix behavior at times shorter than the Thouless time. In contrast to this feature in autocorrelation functions, we find no new aspects to the behavior of OTOCs for Floquet models compared to random-in-time circuits. We show that this difference between averaged autocorrelation functions and OTOCs can be understood in terms of the paths in operator space that contribute to the two quantities: paths for the former retain a limited support at late times, while paths for the latter are dominated by operator spreading.
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 broad applicability 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. Furthermore, by incorporating disorder-averaging into the formalism, we analyze discrete time-crystalline response using the IF of a bond-disordered kicked Ising chain. 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 to quantum statistical physics.