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 ${\tau }_{\mathrm{th}}$—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 ${\tau }_{\mathrm{th}}$ with the system size observed for systems with and without the conservation laws.

Figure 1

Deviations of GSFF $K_g(t)$ from the RMT prediction for two different gates with no splittings, $b=d=0$. Symbols denote the exact numerical results for up to 18 sites ($L=9$). Solid color line depicts $K_g^{(1)}(t)$ computed according to Eqs. (9) and (13). Notice that the ${\tau }_{\mathrm{th}}$ does not scale with $L$. The solid black lines show the asymptotic from Eq. (16). We wrote the gate’s parameters in Table sm-1 of Sec. VII of the Supplemental Material [45].

Figure 2

Deviations of the GSFF $|K_g^{(m)}(t)-1|$ with conservation laws. We show the results for different magnetization sectors at $L=7$ (14 sites) and $J=0.3$. The black line is the prediction (21).

Figure 3

Symbols show the numerical results for deviations of the spectral form factor $|K(t)-1|$ with allowed splittings and mergers. Solid lines show the decay of the skeleton correlation functions ${\lambda }^t$ from Eq. (12) and the true decay of the two-point correlation function ${\lambda }_{\max }^t$. In case 2, the two-point correlation function is well described by the skeleton contributions (see Ref. [43] for when this holds). In contrast, case 1 exhibits a slower decay of the deviations than given by (12), agreeing with the slower decay of the correlation functions. We obtained ${\lambda }_{\max }$ from direct numerical evaluation of the two-point correlation functions in infinite volume. The data shown are for the last two gates in Table sm-1 of Sec. VII of the Supplemental Material [45], $L=8$.