On the integrability structure of the deformed rule-54 reversible cellular automaton

Abstract

We study quantum and stochastic deformations of the rule-54 reversible cellular automaton (RCA54) on a 1+1-dimensional spatiotemporal lattice, focusing on their integrability structures in two distinct settings. First, for the quantum deformation, which turns the model into an interaction-round-a-face brickwork quantum circuit (either on an infinite lattice or with periodic boundary conditions), we show that the shortest-range nontrivial conserved charge commuting with the discrete-time evolution operator has a density supported on six consecutive sites. By constructing the corresponding range-6 Lax operator, we prove that this charge belongs to an infinite tower of mutually commuting conserved charges generated by higher-order logarithmic derivatives of the transfer matrix. With the aid of an intertwining operator, we further prove that the transfer matrix commutes with the discrete-time evolution operator. Second, for the stochastic deformation, which renders the model into a Markov-chain circuit, we investigate open boundary conditions that couple the system at its edges to stochastic reservoirs. In this setting, we explicitly construct the non-equilibrium steady state (NESS) by means of a staggered patch matrix ansatz, a hybrid construction combining the previously used commutative patch-state ansatz for the undeformed RCA54 with the matrix-product ansatz. Finally, we propose a simple empirical criterion for detecting integrability or exact solvability in a given model setup, introducing the notion of digit complexity.

Figures (16)

• Figure 2: Graphical representation of the dynamics \ref{['Ucircuit']}. Time runs bottom-up.
• Figure 3: Graphical representation of the dynamics \ref{['bigUopen']}. Time runs bottom-up.
• Figure 4: Non-zero entries of the matrix representation of the Lax operator $\check{\mathcal{L}}(u)$ are shown. From its structure, it is clear that the matrix is diagonal with respect to the first and last (sixth) qubits, hence it has the structure of a face tensor.
• Figure 5: Graphical representation of the equation \ref{['RLLcheck']}. The orange and golden boxes represent range 6 operators $\check{\mathcal{L}}(u)$ and $\check{\mathcal{L}}(v)$, respectively, acting in the vertical direction (over qubits as indicated by thin vertical lines). $\check{R}$ denotes the range 8 2-parameter operator $\check{R}(v,u)$. Note that all operators are diagonal in the first and the last qubit they act upon. In other words, if a corner of the tensor-box touches the thin vertical line, the corresponding qubit participates only as a control, and its state cannot change.
• Figure 6: Graphical proof that $[t(u),t(v)]=0$. The orange box represents $\check{\mathcal{L}}(u)$, the gold box represents $\check{\mathcal{L}}(v)$, the light yellow box represents $\check{R}(v,u)$, and the light blue box represents $\check{R}(v,u)^{-1}$. For simplicity, we present the proof for $N=10$, as the generalization to arbitrary $N$ is immediate. We remark that we consider a system with periodic boundary conditions.