Logarithmic growth of operator entanglement in a clean non-integrable circuit

Abstract

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.

Figures (10)

• Figure 1: The brickwork dual-unitary circuit with periodic boundaries generated by the Floquet unitary \ref{['FloquetUnitary']} is mapped to a qutrit (red line) propagating in the ergodic direction and $\frac{L}{2}$ qubits (blue lines) propagating in the non-ergodic direction. The yellow lines are the identities in \ref{['PauliStringsNonZeroCorrelatorSpecialForm']} and do not participate in the dynamics. The black dots correspond to the scattering between the qutrit and a single qubit at each time step, described by the $6\times 6$ orthogonal matrix $A$\ref{['SixBySixMatrix']}, which corresponds to a single layer of unitaries $\mathscr{U}_e$ or $\mathscr{U}_o$. The above diagram is drawn for $L=6$.
• Figure 2: A heatmap of the difference between the Haar-averaged value and the time-averaged auto-correlator $\frac{1}{3}-\overline{C_{XX}}((\frac{L}{2})^2)$\ref{['TimeAveragedCorrelator']} evaluated at a time much larger than the system size $L$. The vertical axis is $\theta$ which parametrizes the single-site unitary $v_-$\ref{['SingleSiteErgodic']} while the horizontal axis is $J$ which parametrizes the entangling power of the two-site unitary. The plot on the left, (a), corresponds to $L=46$, while the plot on the right, (b), corresponds to $L=48$.
• Figure 3: (Main Plots) Plots of the auto-correlation functions $C_{XX}$ against the number of single layers of unitaries for $J=\frac{11\pi}{160}$, $\psi = \frac{1+\sqrt{5}}{2}\frac{\pi}{2}$, $\phi = (1+\sqrt{2})\frac{\pi}{2}$ and various system sizes $L$. The top plot (a) corresponds to a semi-ergodic point $\theta=\frac{21\pi}{80}$ while the bottom plot (b) corresponds to a non-semi-ergodic point $\theta=\frac{39\pi}{80}$. (Insets) The insets show the autocorrelators but only at time steps that are multiples of $L/2$. The horizontal axis is the number of multiples of $L/2$ unitaries that have been applied.
• Figure 4: Plots of the operator entanglement for a subsystem containing $l_A$ qubits and the one qutrit with an equal number of qubits on each side of the qutrit. The parameters are $J=\frac{11\pi}{160}$, $\psi = \frac{1+\sqrt{5}}{2}\frac{\pi}{2}$, $\phi = (1+\sqrt{2})\frac{\pi}{2}$ and the total system size is fixed at $L=50$. Plot (a) corresponds to the semi-ergodic point $\theta=\frac{21\pi}{80}$ while the plot (b) correspond to the non-semi-ergodic point $\theta=\frac{39\pi}{80}$. The plots are made with a logarithmic horizontal axes. The insets of these top two plots simply zoom in to the time when the different graphs start to diverge from one another. The x-axes for each graph is shifted by an offset that is the number of qubits in the subsystem over 2. The three thin gray lines in (a) are $\log(3\times2^2),\log(3\times2^4),\log(3\times2^6)$. A straight line has been drawn in both (a) and (b) as a visual guide to show that the operator entanglement grows at most logarithmically.
• Figure 5: These two plots (a) and (b) zoom into the little dips seen in the panels (a) and (b) in Fig. \ref{['OperatorEntanglement']}, respectively.