Absorbing state transitions with discrete symmetries

TL;DR

This work investigates absorbing-state transitions in (1+1)D classical stochastic systems with discrete symmetries, focusing on $\ ext{Z}_2$ and $Q>2$ state models realized via local update rules. Domain-wall dynamics, parity conservation, and branching/annihilation describe the transition between absorbing and active phases, with the $ ext{Z}_2$ case yielding a DP2-class transition at $p_c$ and exponents $\\theta$ and $\\nu_{\\parallel}$ consistent with that universality. For $Q=3$, pure local feedback renders branching a relevant perturbation, preventing a robust absorbing phase, but introducing minimally nonlocal information biases produces a new absorbing-state transition with distinct critical exponents and an apparent new universality class. The results bridge classical absorbing-state dynamics and quantum-circuit implementations, suggesting LOCC-inspired strategies for passive error correction and highlighting the role of nonlocal information in stabilizing absorbing phases. Overall, discrete symmetries and targeted nonlocal feedback fundamentally shape out-of-equilibrium phase structure and point to practical routes for error-protected dynamics in quantum architectures.

Abstract

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.

Figures (9)

• Figure 1: (a) Depiction of a local $\mathbb{Z}_2$-symmetric model with two-site gates that exhibits an absorbing state transition. The channel first measures the presence or absence of a domain wall ($\hat{Z}_r\hat{Z}_{r+1}$) on each bond and then applies feedback that branches (by acting with $\hat{X}_r\hat{X}_{r+1}$) or diffuses (via $\hat{X}_r$, $\hat{X}_{r+1}$, or $\hat{\mathbf{1}}_r\hat{\mathbf{1}}_{r+1}$) the domain wall if it is present. (b) Schematic representation of the active phase, in which the active domain grows ballistically over time after initializing with only one domain wall. (c) Illustration of the absorbing phase, where the dynamics of a single domain wall can be coarse-grained into multi-domain-wall "bubbles".
• Figure 2: Minimal "bubble" for (a) the $\mathbb{Z}_2$-symmetric two-state model, and (b) the $\mathbb{S}_3$- or $\mathbb{Z}_3$-symmetric three-state model. After a single branching event, the probability distribution of the bubble lifetime $\tau_B$ scales as $\tau_B^{-5/2}$ and $\tau_B^{-3/2}$ for the two-state and three-state local models, respectively.
• Figure 3: (a) Schematic of update rules using nonlocal information and local gates. The domain wall of interest separates a "filled" domain $\mathcal{L}$ on the left from a "striped" domain $\mathcal{R}$ on the right. We examine the next-nearest domain on each side: if the next left domain is striped, we record the distance $r_{\mathrm{left}}$, and if the next right domain is filled, we record $r_{\mathrm{right}}$. If the next-nearest domain does not match the desired type, the corresponding distance is recorded as infinity. The biased walk is then directed towards the side with the shorter distance. (b) To avoid boundary effects, we use periodic boundary conditions with two walls separated by a distance larger than any characteristic length scale achievable within our runtime. Representative spatial configurations of domains as a function of time are shown (c) in the active phase ($q<q_c$), (d) at the critical point ($q=q_c$), and (e) in the absorbing phase ($q>q_c$).
• Figure 4: (a) Number of domain walls $N_{dw}$ in the local $\mathbb{Z}_2$-symmetric model over time plotted on a log-log scale, for various branching rates $p$. (b) Effective exponent $\theta(t)$ as a function of $\log(1/t)$; the critical branching rate is when $\theta(t)$ saturates to a constant value as $\log (1/t)\rightarrow0$. (c) Data collapse after rescaling $N_{dw} t^{-\theta}$ against $t|q-q_c|^{\nu_\parallel}$ with $p_c\approx0.24$, $\theta\approx0.29$ and $\nu_\parallel\approx3.4$. (d) A density plot of $\log_{10} \mathcal{Q}$, where $\mathcal{Q}$ is a metric for the quality of the data collapse scaling, shows the optimal values of $\theta$ and $\nu_\parallel$ for the scaling ansatz. We average over $2\times 10^4$ samples.
• Figure 5: Numerical results for the local $\mathbb{Z}_2$-symmetric model in (a,b) the active phase with $p=0.6$, and (c) the absorbing phase with $p\in \{0.05,0.1,0.2\}$. (a) The number density profile $\langle n \rangle$ is displayed on a rescaled axis $x/t$ for various $t$ (labels), after averaging over $10^3$ samples. The perfect collapse demonstrates that the active domain initiated from a single domain wall ballistically spreads out, resulting in a uniform profile for the number density. (b) The profile near the edge of the ballistic front shows the sharp boundaries on the scale of $\mathcal{O}(1)$. The origin of the $x$-axis is shifted by $x_{\mathrm{left}}$, the position of the leftmost site of the domain walls. (c) The cumulative distribution of the bubble lifetime showing an asymptotic tail scaling as $P_S(\tau_B>t)\sim t^{-3/2}$.

Theorems & Definitions (2)

• Lemma A.1
• Lemma A.2