Calculation of critical exponents on fractal lattice Ising model by higher-order tensor renormalization group method

Abstract

The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension $\log_{4} 12 \approx 1.792$, 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.