Kryzhanovsky Boris, Litinskii Leonid, Egorov Vladislav
Center of Optical Neural Technologies, Scientific Research Institute for System Analysis RAS, Nakhimov Ave, 36-1, 117218 Moscow, Russia.
Entropy (Basel). 2021 Dec 10;23(12):1665. doi: 10.3390/e23121665.
We use an -vicinity method to examine Ising models on hypercube lattices of high dimensions d≥3. This method is applicable for both short-range and long-range interactions. We introduce a small parameter, which determines whether the method can be used when calculating the free energy. When we account for interaction with the nearest neighbors only, the value of this parameter depends on the dimension of the lattice d. We obtain an expression for the critical temperature in terms of the interaction constants that is in a good agreement with the results of computer simulations. For d=5,6,7, our theoretical estimates match the numerical results both qualitatively and quantitatively. For d=3,4, our method is sufficiently accurate for the calculation of the critical temperatures; however, it predicts a finite jump of the heat capacity at the critical point. In the case of the three-dimensional lattice (d=3), this contradicts the commonly accepted ideas of the type of the singularity at the critical point. For the four-dimensional lattice (d=4), the character of the singularity is under current discussion. For the dimensions d=1, 2 the -vicinity method is not applicable.
我们使用一种邻域方法来研究维度(d\geq3)的超立方晶格上的伊辛模型。该方法适用于短程和长程相互作用。我们引入一个小参数，它决定了在计算自由能时该方法是否可以使用。当我们仅考虑与最近邻的相互作用时，这个参数的值取决于晶格的维度(d)。我们得到了一个用相互作用常数表示的临界温度表达式，它与计算机模拟结果吻合得很好。对于(d = 5,6,7)，我们的理论估计在定性和定量上都与数值结果相符。对于(d = 3,4)，我们的方法对于临界温度的计算足够准确；然而，它预测在临界点热容量有有限的跳跃。在三维晶格（(d = 3)）的情况下，这与关于临界点奇异性类型的普遍接受的观点相矛盾。对于四维晶格（(d = 4)），奇异性的特征目前正在讨论中。对于维度(d = 1,2)，邻域方法不适用。
