Absorbing-State Transition

• Absorbing-State Transition is a nonequilibrium phase transition where activity ceases once the system enters an absorbing configuration with no escape under local rules.
• The concept is applied in classical lattice models, granular systems, reaction-diffusion models, and quantum circuits to explore criticality and dynamic scaling.
• Key measurements include order parameter decay, diverging correlation lengths, and universality classes such as directed percolation and the Manna class under different dynamical constraints.

An absorbing-state transition is a nonequilibrium phase transition between a fluctuating active phase and a dynamically inert absorbing phase, defined by the property that once the system falls into an absorbing configuration, dynamics halts and escape from that sector is impossible under local rules. Such transitions are observed in classical stochastic lattice systems, driven particulate media, reaction-diffusion models, interacting quantum circuits, and open quantum systems, and their universality and dynamical scaling properties have been a central topic in nonequilibrium statistical mechanics.

1. Definition and Conceptual Framework

The defining characteristic of an absorbing state is that, once reached, the system remains trapped for all subsequent times—there are no allowed dynamical moves or fluctuations that can restore activity (Chatterjee et al., 2011). The phase transition of interest separates a regime where activity persists indefinitely (active phase, nonzero order parameter) from one where fluctuations die out and the dynamics freezes (absorbing phase, vanishing order parameter). Canonical examples include the Domany-Kinzel automaton, contact processes, sandpile models, granular media under periodic drive, and certain classes of quantum circuits.

Generic order parameters for absorbing-state transitions are the steady-state activity density, such as the mean density of “active” sites, or defect density—the specific choice being model-dependent. The control parameter typically tunes the stochastic rates, particle/energy density, contact or driving amplitude, or feedback strength. Criticality is diagnosed via scaling collapse, diverging relaxation times and length scales, and nontrivial critical exponents (Sidoravicius et al., 2014, Harada et al., 2019, Sierant et al., 2023).

2. Prototypical Models and Microscopic Dynamics

2.1. Classical Lattice Models

Contact Process and Directed Percolation (DP). The order-disorder transition in the contact process and its lattice automaton generalizations (e.g., Domany-Kinzel model) is the standard paradigmatic example (Harada et al., 2019, Lesanovsky et al., 2018). Here, each site can be “active” or “inactive”; activity spreads to neighbors with a probability controlled by a percolation-like parameter and dies spontaneously. The unique absorbing state is the completely inactive configuration.

Conserved Lattice Gases: Stochastic Sandpiles and Activated Random Walks. Models with explicit conservation (stochastic sandpiles, ARW) extend this framework to include particle-number conservation, leading to unique scaling properties and new universality classes (Sidoravicius et al., 2014).

Predator-Prey, Multi-Species, and Traffic Models. Variants featuring multiple absorbing states (e.g., multi-species predator-prey lattices or traffic cellular automata) have more complex phase structures, including unusual absorbing configurations and rich universality phenomena (Chatterjee et al., 2011, Iannini et al., 2016).

2.2. Continuum and Granular Systems

Driven particulate models and granular suspensions exhibit absorbing-state transitions between regimes of hydrodynamically reversible dynamics (particle isolation), irreversible chaotic steady states (diffusive phase), and elasticity-dominated reversibility (caged/jammed regime) as control parameters such as density or shear amplitude are varied (Ness et al., 2020, Wang et al., 8 Oct 2025).

2.3. Quantum and Hybrid Quantum-Classical Systems

Quantum protocols with measurement and feedback (including stabilizer circuits, feedback-steered state preparation, and open quantum Lindblad systems) possess absorbing-state transitions where the order parameter may be the defect (non-target) density or coherence measures. In most such quantum circuit cases, a unique absorbing state (e.g., a product or "W" state) is robust under the local dynamics and certain feedback; otherwise, the system remains in a fluctuating, entangled active phase (Sierant et al., 2023, Wampler et al., 1 Oct 2024, Sierant et al., 2022).

3. Critical Exponents and Universality Classes

The universality class of an absorbing-state transition is set by the nature of the order parameter, number of absorbing states, existence of conservation laws, presence of symmetries, correlations, and microscopic dynamics:

3.1. Directed Percolation ([DP])

For generic systems with a unique absorbing state and no additional conserved quantities, symmetries, or quenched disorder, the critical behavior is governed by the DP universality class. Standard exponents in one spatial dimension include β ≈ 0.2765 (order parameter), ν⊥ ≈ 1.097 (spatial correlation length), ν∥ ≈ 1.734 (temporal correlation length), and z = ν∥ / ν⊥ ≈ 1.581 (Harada et al., 2019, Makki et al., 2023, Sierant et al., 2023). Critical scaling collapses have been confirmed both in classical and stabilizer quantum circuit contexts, quantum contact processes, and experimental Rydberg setups (Gutierrez et al., 2016).

3.2. Conserved Directed Percolation (Manna/CDP Class)

Absorbing-state transitions in models with strictly conserved quantities—such as energy, particles, or momentum—belong to the Manna or conserved directed percolation class. This class features larger exponents (e.g., β ≈ 0.64, ν_∥ ≈ 1.225) and stronger finite-size effects—a direct consequence of conservation-induced long-range temporal correlations and an uncountably infinite absorbing-state manifold (Basu et al., 2011, Ness et al., 2020, Wang et al., 8 Oct 2025, Bhowmik et al., 4 Mar 2024).

3.3. Multicritical and Symmetry-Driven Classes

Models with multiple symmetric absorbing states (e.g., Z₂, Z₃, S₃ or related cyclic symmetries) or parity conservation can exhibit different universality. For Z₂ (two symmetric absorbing states), the DP2 or parity-conserving (PC) class arises, characterized by distinct exponents (β ≈ 0.92, ν⊥ ≈ 1.83, ν∥ ≈ 3.22 in 1D) (Ha et al., 12 Feb 2025). For local Z₃/S₃ symmetric systems with branching, any nonzero branching destroys the absorbing phase, but controlled non-local feedback can resurrect a new universality class with unique exponents.

3.4. Crossover and Anomalous Criticality

Modified models with long-range annihilation, quenched disorder, or fractional dynamics exhibit critical exponents continuously interpolating between known classes (e.g., the family connecting DP and PC exponents as a function of annihilation kernel exponent α, or the fractional-time field theory across the Manna, glassy, and Griffiths phases) (O'Dea et al., 5 Sep 2024, Wang et al., 8 Oct 2025).

4. Scaling Laws, Finite-Size Effects, and Dynamical Signatures

Absorbing-state transitions are characterized by singularities in the order parameter and diverging correlation lengths and relaxation times at criticality. Time-dependent scaling at criticality generally has the form

$$\rho(t, e) \sim t^{-\theta} F\left[ (e - e_c) t^{1/\nu_\parallel} \right]$$

with steady-state scaling near the transition as

$$\rho_s(e) \sim (e - e_c)^\beta \quad \text{for } e \searrow e_c^+,$$

and finite-size scaling

$\rho(t, L)|_{e=e_c} = t^{-\theta} G( t / L^z ),$

where all exponents are model- and universality-class specific (Basu et al., 2011, Makki et al., 2023, Wang et al., 8 Oct 2025). Relaxation to the absorbing state may display slow, activated (logarithmic) coarsening in disordered systems, discontinuous jumps in first-order transitions, or Griffiths-phase scaling in the presence of strong disorder (Néel et al., 2014, Barghathi et al., 2015, Wang et al., 8 Oct 2025).

5. Role of Conservation Laws, Disorder, and Symmetries

The presence of conserved continuous local fields (e.g., energy) modifies the universality class of the absorbing-state transition, as shown in the CCLF model, where the order parameter exponent β ≈ 0.46 and ν_⊥ ≈ 1.9, both significantly larger than DP, reflecting the constraints and slow relaxation imposed by the conservation law (Basu et al., 2011). Similarly, strict particle-number conservation in sandpile and activated random walk models leads to unique fixation dynamics and perturbative bounds on escape probabilities and activity decay (Sidoravicius et al., 2014).

Quenched random-field disorder locally breaking symmetry between absorbing states (e.g., in the two-absorbing-state contact process) does not destroy the transition, but processes—such as coarsening—acquire ultraslow Sinai walk or random-field Ising criticality, with logarithmic rather than power-law decay (Barghathi et al., 2015).

Symmetry properties fundamentally shape accessible universality: in systems with multiple symmetric absorbing states, local rules can render the absorbing phase unstable (for q ≥ 3), unless supplemented with non-local feedback that properly suppresses branching events (Ha et al., 12 Feb 2025).

6. Extensions: Quantum Circuits, Measurement, and Entanglement

Recent work highlights the emergence of absorbing-state transitions in both classical feedback-controlled circuits and monitored quantum circuits (Sierant et al., 2022, Sierant et al., 2023, Wampler et al., 1 Oct 2024). Stabilizer circuits with projective measurements and flag-encoded feedback exhibit absorbing-state transitions in the order parameter (defect density) in the DP universality class across spatial dimensions, consistent with rigorous mapping to probabilistic cellular automata (Sierant et al., 2023, Sierant et al., 2022). Entanglement entropy transitions may coincide with or remain separate from the absorbing-state transition depending on the feedback range and circuit entanglement structure, with DP exponents governing the entanglement decay only in the strong feedback regime (Sierant et al., 2022).

In dissipative quantum state-preparation protocols with absorbing long-range coherent W-states, the critical behavior deviates from DP: the active-to-absorbing transition is controlled by the ratio of “error” and “preparation” rates, but the steady-state retains quantum coherence even above the critical error threshold, and critical exponents differ from DP in both static and dynamic scalings (Wampler et al., 1 Oct 2024).

7. Physical Consequences and Broader Implications

Absorbing-state transitions provide a unifying paradigm for nonequilibrium critical phenomena in systems ranging from sheared suspensions, granular materials, and population dynamics to open quantum systems and monitored circuits. The interplay between microscopic dynamical rules (conservation, feedback, disorder, symmetry), the structure of absorbing states (uniqueness, multiplicity, coherence), and the inclusion of quantum effects produces a rich taxonomy of universality classes, scaling behaviors, and dynamical signatures. Conservation laws, disorder, and feedback architecture are not perturbative modifications but can result in fundamentally new non-equilibrium critical points and crossovers (Basu et al., 2011, Ness et al., 2020, Ha et al., 12 Feb 2025, Wang et al., 8 Oct 2025).

Predictions regarding universality, scaling, and the fate of active or absorbing behavior in practical systems inform both numerical and experimental protocols, including granular experiment design, feedback control in quantum processors, and engineering of reversibility in driven matter (Makki et al., 2023, Gutierrez et al., 2016, Maire et al., 23 Jan 2024). The theoretical methodologies developed for absorbing-state transitions—such as multi-scale renormalization, stochastic PDEs with multiplicative noise, Doi–Peliti formalism, and tensor-network MPS evolution—form a standard analytical toolkit for nonequilibrium statistical mechanics.