Absorbing state transitions with discrete symmetries

Robust phases of matter, which remain stable under small perturbations, are of fundamental importance in statistical physics and quantum information. Recent advances in interactive quantum dynamics have led to renewed interest in out-of-equilibrium dynamical phases and associated phase transitions in both classical and quantum many-body systems. Motivated by these developments, we investigate whether a stable absorbing phase can exist in one-dimensional classical stochastic systems, with local update rules, in the presence of fluctuations. We study models with multiple absorbing states related by discrete symmetries, such as Z2 for two-state systems, and Z3 or S3 for three-state systems. In these models, domain walls perform random walks and coarsen under local rules, which, if perfect, eventually bring the system to an absorbing state in polynomial time. However, imperfect feedback can cause domain walls to branch, potentially leading to an opposing active phase. While two-state models exhibit a well-known transition between absorbing and active phases as the branching rate increases, in three-state models with only local dynamics, branching is a relevant perturbation, ruling out a robust absorbing phase under purely local rules. However, we discover that by incorporating nonlocal information into the feedback, the absorbing phase can be stabilized, with the transition between the active and absorbing phases belonging to a new universality class. Finally, we outline how these classical rules can be implemented using deterministic quantum circuits and discuss their connections to passive error correction.

💡 Research Summary

The paper investigates absorbing‑state phase transitions in one‑dimensional classical stochastic systems that possess discrete symmetries (Z₂, Z₃, S₃) and therefore multiple absorbing configurations. The authors first revisit the well‑studied Z₂‑symmetric two‑state model, where a domain wall (the boundary between ↑ and ↓ spins) is treated as a particle. At each discrete time step a particle either branches with probability p (creating two new walls on adjacent bonds) or performs a symmetric random walk with probability 1‑p. After the update, pairs of particles on the same bond annihilate. This dynamics can be implemented exactly by a deterministic quantum circuit consisting of a Z⊗Z measurement followed by conditional X⊗X (branching) or X‑type diffusion operations. Numerical simulations reveal a critical point at p_c≈0, with critical exponents θ≈0.29 and ν_∥≈3.4, matching the directed‑Ising (DP2) universality class. In the active phase (p>p_c) branching dominates, producing expanding active domains whose width grows diffusively (∝t^{1/2}). In the absorbing phase (p<p_c) annihilation dominates, and the average number of walls decays as t^{-1/2}. Starting from a single wall, the system always retains an odd number of particles, so it never reaches the vacuum. The authors analyze “bubbles” – temporary clusters generated by a single branching event – and find the bubble‑lifetime distribution P(τ_B>t)∼t^{-3/2}, implying a finite mean lifetime. Consequently, branching is an irrelevant perturbation at the absorbing fixed point, allowing a robust absorbing phase for small but non‑zero p.

The paper then turns to three‑state (Q=3) models, where each site can be 0, 1, or 2. Two types of domain walls exist: R (increment by +1 mod 3) and L (decrement by –1 mod 3). The global constraint is that the difference n_R−n_L is conserved modulo 3, corresponding to a Z₃ or, if R↔L symmetry is present, an S₃ symmetry. The minimal reaction scheme includes annihilation (R+L→∅), coagulation (R+R→L, L+L→R) and branching (R→R+L+L, L→L+R+R) with rates α, γ_R/L, λ_R/L respectively. Numerical experiments show that any non‑zero branching rate drives the system into the active phase; the absorbing phase is absent. The authors explain this by a renormalization‑group argument: in one dimension the branching process is a relevant perturbation at the λ=0 fixed point. The bubble‑lifetime distribution now decays as P(τ_B>t)∼t^{-5/2}, giving a divergent mean lifetime ⟨τ_B⟩, so a single branching event can sustain activity indefinitely. Thus, purely local update rules cannot stabilize an absorbing phase for Q>2.

To overcome this limitation, the authors introduce non‑local feedback. The global configuration of domain walls is measured (or classically communicated), and the subsequent local updates are conditioned on this global information. For example, branching is suppressed when the conserved quantity n_R−n_L (mod 3) is zero, and allowed otherwise. This scheme is analogous to classical LOCC protocols used in quantum error correction, where syndrome information is shared non‑locally before local recovery operations are applied. Simulations of the non‑local model reveal a new critical point p_c^{nl}>0 and critical exponents that do not belong to DP2, indicating a distinct universality class. The non‑local information effectively renders the branching perturbation irrelevant, thereby stabilizing a robust absorbing phase.

Finally, the authors map both the Z₂ and the three‑state models onto deterministic quantum circuits. Two‑site gates implement the Z⊗Z measurement and conditional X‑type operations; for qutrits, analogous three‑level gates and multi‑outcome measurements realize the R/L dynamics. They discuss how these circuits naturally implement a form of passive error correction: the manifold of absorbing states corresponds to a set of logical codewords (e.g., the three uniform product states), and any error creates a domain wall. The feedback rules then automatically annihilate or coagulate walls, restoring the logical state without active syndrome decoding. This establishes a concrete link between classical absorbing‑state transitions with discrete symmetries and fault‑tolerant quantum information processing.

In summary, the work shows that (i) Z₂‑symmetric two‑state models exhibit a DP2 absorbing‑active transition that can be realized with purely local rules; (ii) for Q≥3, branching is a relevant perturbation that destroys the absorbing phase under local dynamics; (iii) incorporating non‑local feedback stabilizes an absorbing phase and defines a new universality class; and (iv) these stochastic rules can be embedded in deterministic quantum circuits, offering a route toward passive error‑correcting dynamics. The paper thus bridges concepts from non‑equilibrium statistical mechanics, renormalization group theory, and quantum information science.