I study a nonequilibrium lattice model, the pair contact process, in which pairs of particles annihilate with probability p or else create a particle at a vacant nearest neighbor. The model exhibits a continuous phase transition from an active state, with an ongoing production of particles, to an absorbing state without pairs. The model has infinitely many absorbing states. Computer simulations in 1D yielded critical exponents consistent with directed percolation, for the first time placing a model with infinitely many absorbing states firmly in the directed-percolation universality class.
Systems with absorbing (trapped) states may exhibit a nonequilibrium phase transition from a noise-free inactive phase into an ever-lasting active phase. We briefly review the absorbing critical phenomena and universality classes, and discuss over the controversial issues on the pair contact process with diffusion (PCPD). Two different approaches are proposed to clarify its universality issue, which unveil strong evidences that the PCPD belongs to a new universality class other than the directed percolation class.
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.
Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension
dc=4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes $\stackrel{\ensuremath{\rightarrow}}{A}A+A,$ $A+\stackrel{\ensuremath{\rightarrow}}{A}A$, and $\stackrel{\ensuremath{\rightarrow}}{A}0$. We study a hierarchy of such DP processes for particle species
A,B,…, unidirectionally coupled via the reactions $\stackrel{\ensuremath{\rightarrow}}{A}B,\dots{}$ (with rates ${\ensuremath{\mu}}_{\mathrm{AB}},\dots{})$. When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents ${\ensuremath{\beta}}_{i}$ which are markedly reduced at each hierarchy level
i> 2. This scenario can be understood on the basis of the mean-field rate equations, which yield ${\ensuremath{\beta}}_{i}{=1/2}^{i\ensuremath{-}1}$ at the multicritical point. Using field-theoretic renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, we identify a new crossover exponent $\ensuremath{\varphi}$, and compute $\ensuremath{\varphi}=1+O({\ensuremath{\epsilon}}^{2})$ in the multicritical regime (for small ${\ensuremath{\mu}}_{\mathrm{AB}})$ of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, ${\ensuremath{\beta}}_{2}=1/2\ensuremath{-}\ensuremath{\epsilon}/8+O({\ensuremath{\epsilon}}^{2})$. Outside the multicritial region, we discuss the crossover to ordinary DP behavior, with the density exponent ${\ensuremath{\beta}}_{1}=1\ensuremath{-}\ensuremath{\epsilon}/6+O({\ensuremath{\epsilon}}^{2})$. Monte Carlo simulations are then employed to confirm the crossover scenario, and to determine the values for the new scaling exponents in dimensions
d< 3, including the critical initial slip exponent. Our theory is connected to specific classes of growth processes and to certain cellular automata, and the above ideas are also applied to unidirectionally coupled pair annihilation processes. We also discuss some technical as well as conceptual problems of the loop expansion, and suggest some possible interpretations of these difficulties.
The contact process is a paradigmatic classical stochastic system displaying critical behavior even in one dimension. It features a nonequilibrium phase transition into an absorbing state that has been widely investigated and shown to belong to the directed percolation universality class. When the same process is considered in a quantum setting, much less is known. So far, mainly semiclassical studies have been conducted and the nature of the transition in low dimensions is still a matter of debate. Also, from a numerical point of view, from which the system may look fairly simple-especially in one dimension-results are lacking. In particular, the presence of the absorbing state poses a substantial challenge, which appears to affect the reliability of algorithms targeting directly the steady state. Here we perform real-time numerical simulations of the open dynamics of the quantum contact process and shed light on the existence and on the nature of an absorbing state phase transition in one dimension. We find evidence for the transition being continuous and provide first estimates for the critical exponents. Beyond the conceptual interest, the simplicity of the quantum contact process makes it an ideal benchmark problem for scrutinizing numerical methods for open quantum nonequilibrium systems.