Najafi M N, Cheraghalizadeh J, Luković M, Herrmann H J
Department of Physics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran.
Computational Physics, IfB, ETH Zurich, Stefano-Franscini-Platz 3, CH-8093 Zurich, Switzerland.
Phys Rev E. 2020 Mar;101(3-1):032116. doi: 10.1103/PhysRevE.101.032116.
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=∞}), along with a tricritical point (at T_{c}, the critical temperature of the host) between them. The transition between these two criticalities is induced by the transition in the support. The SOC_{T=∞} universality class is characterized by the exponent of the avalanche size distribution τ^{T=∞}=1.27±0.03, consistent with the exponent of the size distribution of the Barkhausen avalanches in amorphous ferromagnets Durin and Zapperi [Phys. Rev. Lett. 84, 4705 (2000)PRLTAO0031-900710.1103/PhysRevLett.84.4705]. The tricritical point is characterized by its own critical exponents. 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 ε≡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 β=0.19±0.02 and ν=0.75±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±0.01.
我们研究了以温度(T)作为控制参数的三维生成伊辛团簇上的沙堆模型。通过分析三维雪崩及其二维投影（在所有温度下均表现出尺度不变行为），我们发现了具有不同指数的两个普适类（一个普通的 BTW 类和(SOC_{T = ∞})），以及它们之间的一个三临界点（在(T_{c})，即主体的临界温度处）。这两种临界状态之间的转变是由支撑的转变引起的。(SOC_{T = ∞})普适类的特征是雪崩规模分布的指数(\tau^{T = ∞} = 1.27 ± 0.03)，这与非晶铁磁体中巴克豪森雪崩规模分布的指数一致[《物理评论快报》84, 4705 (2000)PRLTAO0031 - 900710.1103/PhysRevLett.84.4705]。三临界点由其自身的临界指数表征。除了雪崩指数外，一些其他量，如平均高度、生成雪崩概率（SAP）以及伊辛团簇的平均配位数，在这一点上会显著改变行为，并且在(\varepsilon≡(T - T_{c})/T_{c})方面也表现出幂律行为，从而定义了更多的临界指数。重要的是，对每个位点的活动（ toppling 数）进行的有限尺寸分析显示出指数为(\beta = 0.19 ± 0.02)和(\nu = 0.75 ± 0.05)的标度行为。对于 SAP 和平均雪崩高度也观察到了类似的行为。雪崩二维投影的外周分形维数显示出对(T)具有鲁棒性，数值为(D_{f} = 1.25 ± 0.01)。
