Local and global magnetization on the Sierpinski carpet.

The phase transition of the classical Ising model on the Sierpinski carpet, which has the fractal dimension log_{3}^{}8≈1.8927, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature T_{c}^{}≈1.478. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent ß associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas T_{c}^{} is not affected. Furthermore, we employ automatic differentiation to accurately and efficiently compute the average spontaneous magnetization per site as a first derivative of free energy with respect to the external field, yielding the global critical exponent of ß≈0.135.

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

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, ß, γ, and Î´) 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.

J_{1}-J_{2} fractal studied by multirecursion tensor-network method.

We generalize a tensor-network algorithm to study the thermodynamic properties of self-similar spin lattices constructed on a square-lattice frame with two types of couplings, J_{1}^{} and J_{2}^{}, chosen to transform a regular square lattice (J_{1}^{}=J_{2}^{}) onto a fractal lattice if decreasing J_{2}^{} to zero (the fractal fully reconstructs when J_{2}^{}=0). We modified the higher-order tensor renormalization group (HOTRG) algorithm for this purpose. Single-site measurements are performed by means of so-called impurity tensors. So far, only a single local tensor and uniform extension-contraction relations have been considered in HOTRG. We introduce 10 independent local tensors, each being extended and contracted by 15 different recursion relations. We applied the Ising model to the J_{1}^{}-J_{2}^{} planar fractal whose Hausdorff dimension at J_{2}^{}=0 is d^{(H)}=ln12/ln4≈1.792. The generalized tensor-network algorithm is applicable to a wide range of fractal patterns and is suitable for models without translational invariance.

Free-energy analysis of spin models on hyperbolic lattice geometries.

We investigate relations between spatial properties of the free energy and the radius of Gaussian curvature of the underlying curved lattice geometries. For this purpose we derive recurrence relations for the analysis of the free energy normalized per lattice site of various multistate spin models in the thermal equilibrium on distinct non-Euclidean surface lattices of the infinite sizes. Whereas the free energy is calculated numerically by means of the corner transfer matrix renormalization group algorithm, the radius of curvature has an analytic expression. Two tasks are considered in this work. First, we search for such a lattice geometry, which minimizes the free energy per site. We conjecture that the only Euclidean flat geometry results in the minimal free energy per site regardless of the spin model. Second, the relations among the free energy, the radius of curvature, and the phase transition temperatures are analyzed. We found out that both the free energy and the phase transition temperature inherit the structure of the lattice geometry and asymptotically approach the profile of the Gaussian radius of curvature. This achievement opens new perspectives in the AdS-CFT correspondence theories.

Phase transition of the Ising model on a fractal lattice.

The phase transition of the Ising model is investigated on a planar lattice that has a fractal structure. On the lattice, the number of bonds that cross the border of a finite area is doubled when the linear size of the area is extended by a factor of 4. The free energy and the spontaneous magnetization of the system are obtained by means of the higher-order tensor renormalization group method. The system exhibits the order-disorder phase transition, where the critical indices are different from those of the square-lattice Ising model. An exponential decay is observed in the density-matrix spectrum even at the critical point. It is possible to interpret that the system is less entangled because of the fractal geometry.