Extended operator algebra for abelian sandpile models

We generalize the definition of the operators corresponding to particle addition in the abelian sandpile models to include a phase factor eiεn, where n is the number of topplings in the avalanche, and ε is a real parameter. The new operators so defined are still abelian, and their derivatives with respect to ε satisfy an equation similar to the Heisenberg equation for the time-derivative of operators in standard quantum mechanics. The role of the Hamiltonian is played by the toppling function, an operator linear in the height variables of the sandpile, which does not commute with the particle addition operators. We show that moments of the distribution of the number of topplings in avalanches in the steady state are expressed in a simple way in terms of these operators.