Physics Division, National Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan.
Phys Rev E. 2023 Mar;107(3-1):034131. doi: 10.1103/PhysRevE.107.034131.
The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension log_{4}12≈1.792, is investigated using a modified higher-order tensor renormalization group algorithm supplemented with automatic differentiation to compute relevant derivatives efficiently and accurately. The complete set of critical exponents characteristic of a second-order phase transition was obtained. Correlations near the critical temperature were analyzed through two impurity tensors inserted into the system, which allowed us to obtain the correlation lengths and calculate the critical exponent ν. The critical exponent α was found to be negative, consistent with the observation that the specific heat does not diverge at the critical temperature. The extracted exponents satisfy the known relations given by various scaling assumptions within reasonable accuracy. Perhaps most interestingly, the hyperscaling relation, which contains the spatial dimension, is satisfied very well, assuming the Hausdorff dimension takes the place of the spatial dimension. Moreover, using automatic differentiation, we have extracted four critical exponents (α, β, γ, and δ) globally by differentiating the free energy. Surprisingly, the global exponents differ from those obtained locally by the technique of the impurity tensors; however, the scaling relations remain satisfied even in the case of the global exponents.
使用修正的高阶张量重整化群算法并结合自动微分技术来高效准确地计算相关导数，研究了分形晶格上伊辛模型的临界行为，该分形晶格的豪斯多夫维数 log_{4}12≈1.792。得到了具有二阶相变特征的完整临界指数集。通过向系统中插入两个杂质张量来分析接近临界温度的相关性，这使我们能够获得相关长度并计算临界指数 ν。发现临界指数 α 为负，与观察到的比热在临界温度处不发散的情况一致。提取的指数在合理的精度范围内满足各种标度假设给出的已知关系。也许最有趣的是，假设豪斯多夫维数代替空间维数，超标度关系（包含空间维数）得到了很好的满足。此外，我们通过对自由能进行微分，使用自动微分全局提取了四个临界指数（α、β、γ 和 δ）。令人惊讶的是，全局指数与通过杂质张量技术局部获得的指数不同；然而，即使在全局指数的情况下，标度关系仍然成立。
