Chaos and Ergodicity in Extended Quantum Systems with Noisy Driving

We study the time-evolution operator in a family of local quantum circuits with random fields in a fixed direction. We argue that the presence of quantum chaos implies that at large times the time-evolution operator becomes effectively a random matrix in the many-body Hilbert space. To quantify this phenomenon, we compute analytically the squared magnitude of the trace of the evolution operator-the generalized spectral form factor-and compare it with the prediction of random matrix theory. We show that for the systems under consideration, the generalized spectral form factor can be expressed in terms of dynamical correlation functions of local observables in the infinite temperature state, linking chaotic and ergodic properties of the systems. This also provides a connection between the many-body Thouless time tau(th)-the time at which the generalized spectral form factor starts following the random matrix theory prediction-and the conservation laws of the system. Moreover, we explain different scalings of tau(th) with the system size observed for systems with and without the conservation laws.

Automated summary

In this study, we investigate the time-evolution operator in a local quantum circuit with random fields, finding that quantum chaos leads to it effectively behaving as a random matrix in many-body quantum space at large times. By analyzing the squared magnitude of the trace of the evolution operator, known as the generalized spectral form factor, we compare our findings with random matrix theory predictions. Furthermore, we establish a connection between chaotic and ergodic properties of the systems through the realization that the generalized spectral form factor can be expressed in terms of dynamical correlation functions of local observables in the infinite temperature state. Additionally, we explore the relationship between the many-body Thouless time tau(th) and the conservation laws of the system, explaining the different scalings observed for systems with and without conservation laws.