Operator Entanglement in Interacting Integrable Quantum Systems: The Case of the Rule 54 Chain

In a many-body quantum system, local operators in the Heisenberg picture $O(t)=e^{iHt}O{e}^{-iHt}$ spread as time increases. Recent studies have attempted to find features of that spreading which could distinguish between chaotic and integrable dynamics. The operator entanglement—the entanglement entropy in operator space—is a natural candidate to provide such a distinction. Indeed, while it is believed that the operator entanglement grows linearly with time $t$ in chaotic systems, we present evidence that it grows only logarithmically in generic interacting integrable systems. Although this logarithmic growth has been previously established for noninteracting fermions, there has been no progress on interacting integrable systems to date. In this Letter we provide an analytical upper bound on operator entanglement for all local operators in the “Rule 54” qubit chain, a cellular automaton model introduced in the 1990s [Bobenko et al., CMP 158, 127 (1993)], and recently advertised as the simplest representative of interacting integrable systems. Physically, the logarithmic bound originates from the fact that the dynamics of the models is mapped onto the one of stable quasiparticles that scatter elastically. The possibility of generalizing this scenario to other interacting integrable systems is briefly discussed.

Figure 1

Numerical results for the OE $S\mathbf{(}O(t)\mathbf{)}$ for bipartition $A=(-\infty ,x]$, $B=(x,\infty )$, in three interacting integrable models. (Left) Growth of $S\mathbf{(}O(t)\mathbf{)}$ at $x=0$: in all three models the growth appears to be logarithmic for local operators: (top) $O=S_x^{+}$ in the Rule 54 chain, (middle) $O=S_x^z$, $S_x^{+}$ and $S_x^z{S}_{x+1}^z$ in the $XXZ$ chain at $\mathrm{\Delta }=0.4$, (bottom) $O=S_x^z$ in the spin-1 Babujan-Takhtajan chain. (Right) Profile of the OE for $O=S_x^z$ at different times in the same three models. The operator spreading is clearly visible in all cases.

Figure 2

Operator spreading in the Rule 54 chain. (a) Example of the dynamics generated by Eq. (2) acting on the qubit chain, here drawn as a staggered lattice: qubits in the state “1” (“0”) are drawn as black (white) squares. Red lines superimposed on the black squares show the left and right moving solitons. The dashed box highlights a scattering event, where two solitons get time delayed. The mapping from qubits to solitons is illustrated at the bottom: left and right moving solitons correspond to nearest-neighbor black sites, while a scattering pair corresponds to a single black site surrounded by two white ones. (b) Spacetime picture of a typical evolution. (c) Spreading of a diagonal operator $O=|0\check{1}0⟩⟨0\check{1}0|$: the forward and backward light cones are present. After folding one is back to the situation (c). The solitonic algorithm is able to tell whether a soliton configuration $|s⟩⟨s|$ contributes or not to $O(t)$. (d) Spreading of off-diagonal operators $O=|0\check{1}0⟩⟨0\check{0}0|$. Folding the forward and backward light cones, one sees that the configurations $|s⟩$, $|s^{\prime }⟩$ coincide for $x<x_1$ or $x>x_2$, while $|s^{\prime }⟩$ it can be obtained from $|s⟩$ inside the interval $(x_1,x_2)$ simply by applying a time shift of one unit time.