Ergodic and Nonergodic Dual-Unitary Quantum Circuits with Arbitrary Local Hilbert Space Dimension

Dual-unitary quantum circuits can be used to construct $1+1$ dimensional lattice models for which dynamical correlations of local observables can be explicitly calculated. We show how to analytically construct classes of dual-unitary circuits with any desired level of (non-)ergodicity for any dimension of the local Hilbert space, and present analytical results for thermalization to an infinite-temperature Gibbs state (ergodic) and a generalized Gibbs ensemble (nonergodic). It is shown how a tunable ergodicity-inducing perturbation can be added to a nonergodic circuit without breaking dual unitarity, leading to the appearance of prethermalization plateaux for local observables.

Figure 1

Correlation functions $c_{\rho \sigma }(t,t)$, where $\rho$, $\sigma \in {\mathbb{C}}^{q\times q}$ are randomly generated matrices with $\mathrm{tr}(\sigma )=0$ leading to a thermal value $c_{\rho \sigma }(t,t)\to 0$. Local Hilbert space dimension $q=6$ and 4 different operators are considered. After an initial transient regime, in the ergodic models the correlations either exponentially decay to zero (stationary) or oscillate around zero (nonstationary) with period $q$, whereas in the nonergodic models the correlations decay to a nonzero value (stationary) with possible oscillations around these nonzero values with a tunable period (nonstationary).

Figure 2

Correlation function $c_{\rho \sigma }(t,t)$ at short and long time scales for a randomly generated traceless $\sigma$ and a randomly generated density matrix $\rho$ with $\mathrm{tr}(\rho \sigma )=1$ for a nonergodic gate $U$ on top of which an ergodicity-inducing perturbation is added with strength $\epsilon$. Local Hilbert space dimension $q=6$. Vertical dashed lines mark $t=\log (2)/(1-|\lambda |)\propto {\epsilon }^{-2}$, with $\lambda$ the dominant nontrivial eigenvalue of $\mathcal{M}$, and horizontal lines denote the thermal and GGE values.