We present a Monte Carlo study in dimension
d=1 of the two-species reaction-diffusion process $A+\stackrel{\ensuremath{\rightarrow}}{B}2B$ and $\stackrel{\ensuremath{\rightarrow}}{B}A.$ Below a critical value ${\ensuremath{\rho}}_{c}$ of the conserved total density $\ensuremath{\rho}$ the system falls into an absorbing state without B particles. Above ${\ensuremath{\rho}}_{c}$ the steady state B particle density ${\ensuremath{\rho}}_{B}^{\mathrm{st}}$ is the order parameter. This system is related to directed percolation but in a different universality class identified by Kree et al. [Phys. Rev. A 39, 2214 (1989)]. We present an algorithm that enables us to simulate simultaneously the full range of densities $\ensuremath{\rho}$ between zero and some maximum density. From finite-size scaling we obtain the steady state exponents $\ensuremath{\beta}=0.435(10),\ensuremath{\nu}=2.21(5),$ and $\ensuremath{\eta}=\ensuremath{-}0.606(4)$ for the order parameter, the correlation length, and the critical correlation function, respectively. Independent simulation indicates that the critical initial increase exponent takes the value ${\ensuremath{\theta}}^{\ensuremath{'}}=0.30(2),$ in agreement with the theoretical relation ${\ensuremath{\theta}}^{\ensuremath{'}}=\ensuremath{-}\ensuremath{\eta}/2$ due to Van Wijland et al. [Physica A 251, 179 (1998)].
In the preceding article Carlon et al. investigate the critical behavior of the pair contact process with diffusion. Using density matrix renormalization group methods, they estimate the critical exponents, raising the possibility that the transition might belong to the same universality class as branching annihilating random walks with even numbers of offspring. This is surprising since the model does not have an explicit parity-conserving symmetry. In order to understand this contradiction, we estimate the critical exponents by Monte Carlo simulations. The results suggest that the transition might belong to a different universality class that has not been investigated before.
In this comment we address the preprint Z. Yao et al. (arXiv:1402.5417v1) concerning the conditions upon which the correct critical temperature exponent
ϕ can be observed (experimentally or numerically) close to the superfluid-Bose glass quantum phase transition in three dimensions. Yao et al. announce to have resolved the contradiction between the predictions of Fisher et al. [Phys. Rev. B 40, 546 (1989)] that
ϕ=νz≥2 (where
ν and
z are critical exponents at the quantum phase transition) and recent experiments and simulations, showing an exponent
ϕ∼1.1. Yao et al's resolution hinges on claiming that the
ϕ∼1.1 observations, which were conducted by varying the chemical potential, were not made to sufficiently close to the critical point to observe universal behavior. Here we critically examine their criteria for observing universal behavior. We show that past simulations were in fact conducted closer to the quantum critical point than the new results of Yao et al.; and that Yao et al.'s own results for varying chemical potential are consistent with previous results showing
ϕ∼1.1, and do not show a clear crossover to a different
ϕ value.
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.
Hove, Mo, and Sudbo [Phys. Rev. Lett. 85, 2368 (2000)] derived a simple connection,
η+DH=2, between the anomalous scaling dimension
η of the U(1) universality class order parameter and the Hausdorff dimension
DH of critical loops in loop representations of U(1) models. We show that the above relation is wrong and establish a correct relation that contains a new critical exponent.