Non-equilibrium phase transition in sandpiles

We study the sandpile model on three-dimensional spanning Ising clusters with the temperature $T$ treated as the control parameter. By analyzing the three dimensional avalanches and their two-dimensional projections (which show scale-invariant behavior for all temperatures), we uncover two universality classes with different exponents (an ordinary BTW class, and SOC$_{T=\infty}$), along with a tricritical point (at $T_c$, the critical temperature of the host) between them. The SOC$_{T=\infty}$ universality class is characterized by the exponent of the avalanche size distribution $\tau^{T=\infty}=1.27\pm 0.03$, consistent with the exponent of the size distribution of the Barkhausen avalanches in amorphous ferromagnets (Phys. Rev. L 84, 4705 (2000)). The tricritical point is characterized by its own critical exponents and also some additional scaling behavior in its vicinity. In addition to the avalanche exponents, some other quantities like the average height, the spanning avalanche probability (SAP) and the average coordination number of the Ising clusters change significantly the behavior at this point, and also exhibit power-law behavior in terms of $\epsilon\equiv \frac{T-T_c}{T_c}$, defining further critical exponents. Importantly the finite size analysis for the activity (number of topplings) per site shows the scaling behavior with exponents $\beta=0.19\pm 0.02$ and $\nu=0.75\pm 0.05$. A similar behavior is also seen for the SAP and the average avalanche height. The fractal dimension of the external perimeter of the two-dimensional projections of avalanches is shown to be robust against $T$ with the numerical value $D_f=1.25\pm 0.01$.