The Ising model in a Bak-Tang-Wiesenfeld sandpile
  We study the spin-1 Ising model with non-local constraints imposed by the Bak-Tang-Wiesenfeld sandpile model of self-organized criticality (SOC). The model is constructed as if the sandpile is being built on a (honeycomb) lattice with Ising interactions. In this way we combine two models that exhibit power-law decay of correlation functions characterized by different exponents. We discuss the model properties through an order parameter and the mean energy per node, as well as the temperature dependence of their fourth-order Binder cumulants. We find (i) a thermodynamic phase transition at a finite T_c between paramagnetic and antiferromagnetic phases, and (ii) that above T_c the correlation functions decay in a way typical of SOC. The usual thermodynamic criticality of the two-dimensional Ising model is not affected by SOC constraints (the specific heat critical exponent \alpha \approx 0), nor are SOC-induced correlations affected by the interactions of the Ising model. Even though the constraints imposed by the SOC model induce long-range correlations, as if at standard (thermodynamic) criticality, these SOC-induced correlations have no impact on the thermodynamic functions. 
Zbigniew Koza, Marcel Ausloos