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arXiv:2409.03280 · Physical review. E

Absorbing state transitions with long-range annihilation

Nicholas O’Dea, Sayak Bhattacharjee, Sarang Gopalakrishnan, Vedika Khemani

Published Jun 15, 2026• 0 citations

Abstract

We introduce a family of classical stochastic processes describing diffusive particles undergoing branching and long-range annihilation in the presence of a parity constraint. The probability of a pair-annihilation event decays as a power-law in the distance between particles, with a tunable exponent. Such long-range processes arise naturally in various classical settings, and increasingly play a role in the study of quantum dynamics: for example, quantum state preparation or error correction processes aim to prepare ordered ground states, which requires removing pointlike excitations in pairs via nonlocal feedback operations conditioned on a global set of measurement outcomes. We analytically and numerically describe features of absorbing phases and phase transitions in this family of classical models as pairwise annihilation is performed at larger and larger distances. Notably, we find that the two canonical absorbing-state universality classes-directed percolation and parity conserving-are endpoints of a new line of universality classes with continuously varying critical exponents. These new classes follow from an interplay of long-range interactions and symmetries: decreasing the long-range exponent effectively weakens the local effects of global symmetry constraints. This competition significantly broadens known nonequilibrium phases and phase transitions.

DOI: 10.1103/njzm-t262

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We study absorbing phase transitions in systems of branching annihilating random walkers and pair contact process with diffusion on a one-dimensional ring, where the walkers hop to their nearest neighbor with a bias ε. For ε=0, three universality classes-directed percolation (DP), parity-conserving (PC), and pair contact process with diffusion (PCPD)-are typically observed in such systems. We find that the introduction of ε does not change the DP universality class but alters the other two universality classes. For nonzero ε, the PCPD class crosses over to DP, and the PC class changes to a new universality class.

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Many non-equilibrium systems display dynamic phase transitions from active to absorbing states, where fluctuations cease entirely. Based on a field theory representation of the master equation, the critical behavior can be analyzed by means of the renormalization group. The resulting universality classes for single-species systems are reviewed here. Generically, the critical exponents are those of directed percolation (Reggeon field theory), with critical dimension d_c = 4. Yet local particle number parity conservation in even-offspring branching and annihilating random walks implies an inactive phase (emerging below d_c' = 4/3) that is characterized by the power laws of the pair annihilation reaction, and leads to different critical exponents at the transition. For local processes without memory, the pair contact process with diffusion represents the only other non-trivial universality class. The consistent treatment of restricted site occupations and quenched random reaction rates are important open issues.

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