The Ising model exhibits qualitatively different properties in hyperbolic space in comparison to its flat space counterpart. Due to the negative curvature, a finite fraction of the total number of spins reside at the boundary of a volume in hyperbolic space. As a result, boundary conditions play an important role even when taking the thermodynamic limit. We investigate the bulk thermodynamic properties of the Ising model in two- and three-dimensional hyperbolic spaces using Monte Carlo and high- and low-temperature series expansion techniques. To extract the true bulk properties of the model in the Monte Carlo computations, we consider the Ising model in hyperbolic space with periodic boundary conditions. We compute the critical exponents and critical temperatures for different tilings of the hyperbolic plane and show that the results are of mean-field nature. We compare our results to the “asymptotic” limit of tilings of the hyperbolic plane: the Bethe lattice, explaining the relationship between the critical properties of the Ising model on Bethe and hyperbolic lattices. Finally, we analyze the Ising model on three-dimensional hyperbolic space using Monte Carlo and high-temperature series expansion. In contrast to recent field theory calculations, which predict a non-mean-field fixed point for the ferromagnetic-paramagnetic phase-transition of the Ising model on three-dimensional hyperbolic space, our computations reveal a mean-field behavior.

中文翻译：

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双曲空间中Ising模型的临界性质

与平面空间模型相比，伊辛模型在双曲线空间中显示出质的不同特性。由于负曲率，自旋总数的有限部分位于双曲空间中体积的边界。结果，即使在采取热力学极限时，边界条件也起着重要作用。我们使用蒙特卡洛以及高温和低温系列膨胀技术研究二维和三维双曲空间中伊辛模型的整体热力学性质。为了在蒙特卡洛计算中提取模型的真实体积特性，我们考虑了具有周期边界条件的双曲空间中的伊辛模型。我们计算了双曲平面的不同平铺的临界指数和临界温度，并表明结果具有均场性质。我们将结果与双曲平面的平铺“渐近”极限进行比较：贝特网格，解释了贝辛上的Ising模型的临界特性与双曲网格之间的关系。最后，我们使用蒙特卡洛和高温级数展开对三维双曲空间上的伊辛模型进行了分析。与最近的场论计算相反，后者预测了三维双曲空间上Ising模型的铁磁-顺磁相变的非平均场固定点，我们的计算揭示了平均场行为。Bethe晶格，解释了Bethe上Ising模型的临界特性与双曲晶格之间的关系。最后，我们使用蒙特卡洛和高温级数展开对三维双曲空间上的伊辛模型进行了分析。与最近的场论计算相反，后者预测了三维双曲空间上Ising模型的铁磁-顺磁相变的非平均场固定点，我们的计算揭示了平均场行为。Bethe晶格，解释了Bethe上Ising模型的临界特性与双曲晶格之间的关系。最后，我们使用蒙特卡罗和高温级数展开对三维双曲空间上的伊辛模型进行了分析。与最近的场论计算相反，后者预测了三维双曲空间上Ising模型的铁磁-顺磁相变的非平均场固定点，我们的计算揭示了平均场行为。