Self-organized criticality as an absorbing-state phase transition

We explore the connection between self-organized criticality and phase transitions in models with absorbing states. Sandpile models are found to exhibit criticality only when a pair of relevant parameters — dissipation $\epsilon$ and driving field $h$ — are set to their critical values. The critical values of $\epsilon$ and $h$ are both equal to zero. The first result is due to the absence of saturation (no bound on energy) in the sandpile model, while the second result is common to other absorbing-state transitions. The original definition of the sandpile model places it at the point $\left(\epsilon {=0,h=0\mathrm{}}^{+}\right):$ it is critical by definition. We argue power-law avalanche distributions are a general feature of models with infinitely many absorbing configurations, when they are subject to slow driving at the critical point. Our assertions are supported by simulations of the sandpile at $\epsilon =h=0\mathrm{}$ and fixed energy density $\zeta$ (no drive, periodic boundaries), and of the slowly driven pair contact process. We formulate a field theory for the sandpile model, in which the order parameter is coupled to a conserved energy density, which plays the role of an effective creation rate.