Anomalous subdiffusion from subsystem symmetries

We introduce quantum circuits in two and three spatial dimensions which are classically simulable, despite producing a high degree of operator entanglement. We provide a partial characterization of these “automaton” quantum circuits and use them to study operator growth, information spreading, and local charge relaxation in quantum dynamics with subsystem symmetries, which we define as overlapping symmetries that act on lower-dimensional submanifolds. With these symmetries, we discover the anomalous subdiffusion of conserved charges; that is, the charges spread slower than diffusion in the dimension of the subsystem symmetry. By studying an effective operator hydrodynamics in the presence of these symmetries, we predict the charge autocorrelator to decay (i) as $ln\left(t\right)/\sqrt{t}$ in two dimensions with a conserved U(1) charge along intersecting lines and (ii) as $1/t^{3/4}$ in three spatial dimensions with intersecting planar U(1) symmetries. Through large-scale studies of automaton dynamics with these symmetries, we numerically observe charge relaxation that is consistent with these predictions. In both cases, the spatial charge distribution is distinctly non-Gaussian and reminiscent of the diffusion of charges along a fractal surface. We numerically study the onset of quantum chaos in the spreading of local operators under these automaton dynamics and observe power-law broadening of the ballistically propagating fronts of evolving operators in two and three dimensions and the saturation of out-of-time-ordered correlations to values consistent with quantum chaotic behavior.

Figure 1

Linelike and planar subsystem symmetries. (Top) We study 2D square lattices with linear subsystem symmetry where charge is conserved on each row and column of the lattice. We also look at 3D cubic lattices with planar subsystem symmetry. In this case, each plane contains a conserved U(1) charge. (Bottom) The four-site unitary gates act nontrivially only on the above two configurations. In three dimensions, such a four-site gate can act along any plane of the cube which intersects four spins.

Figure 2

One time step of the 2D automaton circuit. (Left) The three types of gates which are applied to the 2D circuit. The displayed gates act on layers $\ell =1,5$, and 11. For the next-nearest-neighbor rectangular gates, the unitary acts trivially on the middle two sites of the rectangle. (Right) A single period of the circuit consists of 16 layers, applied as shown in the spacetime cross section. The red line indicate the causal cone for the first site.

Figure 3

Charge subdiffusion in 2D. The autocorrelation function $G(0,t)$ for the 2D automaton circuit with line symmetries, which decays like $ln\left(t\right)/\sqrt{t}$. (Inset) Comparison of $ln\left(t\right)/\sqrt{t}$ and $1/\sqrt{t}$ decay, showing a clear deviation from normal diffusion.

Figure 4

Spatial charge distribution in 2D. The simulated evolution, with $N={396}^2$ sites, of $G_{\mathit{r},t}$ along the line $\mathit{r}=(x,0)$, which shows a dramatic deviation from the usual Gaussian behavior of normal diffusion. Notice the distinctive cusp near $x=0$ which is a sign of fractional subdiffusion. (Insets) The simulated evolution of the 2D difference equation (29). For $x\lesssim \sqrt{t},G(x,t)\sim ln(t/x^2)/\sqrt{t}$. For $x\gg \sqrt{t}$, we have $G(x,t)\sim e^{-x^2/\sqrt{t}}$

Figure 5

Charge subdiffusion with planar subsystem symmetries in 3D. The autocorrelation function $G(0,t)$. We numerically fit this (using the intermediate times only) to the function $G\left(t\right)=c{t}^{-z}$ and find $z=0.741\left(6\right)$ and $c=0.046\left(1\right)$. This is close to the analytical prediction, that $G\left(t\right)\sim t^{-0.75}$ as $N,t\to \infty$.

Figure 6

Dynamics with planar subsystem symmetries in 3D. (Top) The space-time correlation function $G(\mathit{r},t)$ for a 2D slice of the 3D cubic lattice at fixed time $t=200$ for a system with $N={120}^3$ sites. Notice that the charge has higher density along the intersection of two planes which contain the charge. (Bottom) 1D slices of the same data, plotted for different y positions. Notice that again $G(\mathit{r},t)$ has a cusp near the origin.

Figure 7

XX OTOC for the 2D automaton circuit. We measure the XX OTOC along the line $\mathit{r}=(x,0)$ for the 2D automaton circuit. In (a) and (b), we observe a clear light cone, for a system with $N={396}^2$ sites, whereby $F(\mathit{r},t)\approx 1$ for $\left|\mathit{r}\right|>v_{B}t$ and $F(\mathit{r},t)\approx 0$ for $\left|\mathit{r}\right|<v_{B}t$, as well as a broadening of the ballistically propagating front of the operator. In (c), we observe that at a fixed spatial position, the OTOC decays exponentially to zero. Shown are exponential curves $\sim exp(-ct)$ where $c\sim 1.53–1.16$ depending on the position $x$. For this plot, we simulated a smaller system with $N={96}^2$ sites in order to resolve this exponential decay with greater accuracy. In all plots, $T^{*}=L/\left(2v_B\right)$ is the time for the light cone to spread across the entire lattice.

Figure 8

Scaling collapse of the OTOC for the 2D and 3D automaton circuits. (Top) Shown is a scaling collapse of $F_{XX}(x,t)$ in 2D to the functional form in Eq. (43), from which we observe that the operator front broadens as $\sim t^{\alpha }$ with $\alpha =0.308\left(18\right)$. (Bottom) The scaling collapse of the front of $F_{ZX}(x,t)$ in 3D with $\alpha =0.220\left(5\right)$. These should be compared with the theoretically predicted values of $\alpha =1/3$ and 0.24 for the 2D and 3D cases respectively. Again, we take $T^{*}=L/\left(2v_B\right)$. Both OTOC's are calculated along the $x$ axis.

Figure 9

The ZX OTOC. The OTOC $F_{ZX}(\mathit{r},t)$ is shown along the $x$ axis, for both the 2D (top) and 3D (bottom) automaton circuits considered previously. Here we can see a clear tail behind the leading OTOC wave front. In the 2D circuit, this tail appears to have the same functional form as a subleading tail for a 1D circuit with a global U(1) symmetry. Fitting the tail at the latest times to $F_{ZX}\sim (v_{B}t-{\left|\mathit{r}\right|)}^{-\beta }$ gives $\beta =0.50\left(1\right)$. For the 3D circuit, the tail of the OTOC (when $v_{B}t\gg \left|\mathit{r}\right|)$ takes the form $F_{ZX}\sim (v_{B}t-{\left|\mathit{r}\right|)}^{-\beta }$ with $\beta =1.02\left(7\right)$. Again, we take $T^{*}=L/\left(2v_B\right)$.

Figure 10

Recurrence times. The number of time steps that the 2D circuit is run before an initial product state $|n\rangle$ returns to its initial value $U^t|n\rangle =|n\rangle$. The average number of time steps needed to see such a recurrence grows exponentially with the volume of the system.