Absorbing State Transitions with Discrete Symmetries

We study absorbing state transitions in one-dimensional classical systems for which the dark states are not unique due to an additional discrete symmetry. Specifically, we consider $Z_2$ symmetry for 2-state models (or bits) and $S_3$ or $Z_3$ symmetry for 3-state models. These considerations are motivated by the possibility of passive quantum error correction with specific errors since the same update rule applied in the qubits and qutrits will absorb into the logical space, which could potentially be a quantum memory. Specifically, we measure the domain walls and let them randomly walk by applying local feedback. If the feedback is perfect, the number of domain walls decays over time, and any initial state reaches an absorbing state with no domain walls within polynomial time. However, if the feedback is imperfect, it can cause branching of the domain walls, potentially driving the system into the opposing active phase. We first revisit the transition in the $Z_2$ symmetric local 2-state model and investigate both absorbing and active phases with a particular initial state, starting with a single domain wall. Then, we generalize this update rules for 3-state models, and find that the system always remains in the active phase unless the branching process is completely suppressed. We demonstrate the relevance of branching at the absorbing fixed point and argue that, unlike 2-state models, any finite branching rate in 3-state models always leads to the active phase. We further demonstrate that the absorbing phase can be stabilized against branching by still applying local feedback, but incorporating ``nonlocal information'' and biasing the walks of domain walls. By increasing the use of nonlocal information, we identify a transition from the active phase to the absorbing phase, which falls into a new universality class.