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.
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.
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, J1 and J2, chosen to transform a regular square lattice (J1=J2) onto a fractal lattice if decreasing J2 to zero (the fractal fully reconstructs when J2=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 ${J}_{1}^{}\ensuremath{-}{J}_{2}^{}$ planar fractal whose Hausdorff dimension at J2=0 is ${d}^{(H)}=ln12/ln4\ensuremath{\approx}1.792$. The generalized tensor-network algorithm is applicable to a wide range of fractal patterns and is suitable for models without translational invariance.
The Ising model on fractal lattices has attracted renewed interest, yet a quantitative understanding of how the critical temperature depends on the Hausdorff dimension (D_{H}) and connectivity (Q) remains limited. We study the ferromagnetic Ising model on the one-parameter "legs-extension" family of fractal lattices originally introduced by Genzor et al. and apply the higher-order tensor renormalization group (HOTRG) method, following the tensor-network construction developed for these geometries in earlier work. By tuning a single structural parameter, we access lattices with D_{H} decreasing from 1.792 to 1.260 and connectivity Q from 0.50 to 0.131, and obtain high-precision critical temperatures T_{c} and magnetization exponents β with carefully quantified uncertainties from systematic bond-dimension and fitting-window analyses, finding that β is strongly and almost exponentially suppressed as D_{H} is reduced. Along this one-dimensional trajectory in the (D_{H},Q) plane, T_{c} decreases monotonically with D_{H} and, when plotted against Q, exhibits a robust near-linear behavior within this lattice family over almost an order of magnitude in connectivity. A finite-temperature phase transition persists throughout the explored range down to our lowest Hausdorff dimension, D_{H}≈1.260, implying that the lower critical dimension of this family, if it exists, lies below this value. Comparison with zero-order and first-order mean-field theory shows that HOTRG systematically incorporates fluctuation effects beyond mean-field approximations. The tabulated T_{c} and β values thus provide a stringent benchmark for theories of criticality on fractal lattices and a quantitative reference for future numerical and analytical studies on other complex geometries.
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.
Abstract The critical behavior of spin systems is fundamentally governed by dimensionality and connectivity. Moving beyond translationally invariant lattices, we explore a new paradigm where fractality itself becomes a tunable parameter, engineering magnetic order and critical behavior. By implementing the classical Ising model on a three-dimensional fractal lattice---a static realization of a spin cluster with Hausdorff dimension dH=2.5 and boundary dimension d=2---we demonstrate how fractal geometry dictates unique critical phenomena. Using the higher-order tensor renormalization group (HOTRG) method, we identify a finite-temperature phase transition at Tc≈2.65231 with exotic critical exponents (α≈0.09, α′≈0.06, β≈0.059, δ≈35) and a diverging specific heat---a hallmark absent in lower-dimensional fractals. This work establishes that fractal geometry serves as a powerful and untapped degree of freedom for spintronics. It provides a blueprint for designing materials with programmable magnetic phase transitions, paving the way for next-generation, geometry-driven devices in magnonics and neuromorphic computing.
Phase transitions of the J_{1}-J_{2} Ising model on a square lattice are studied using the higher-order tensor renormalization group (HOTRG) method. This system involves a competition between the ferromagnetic interaction J_{1} and antiferromagnetic interaction J_{2}, and in previous studies, weak first-order and second-order transitions were observed near the ratio g=J_{2}/|J_{1}|=1/2. It has also been suggested that the universality class of the second-order phase transition connected to the first-order transition line for g>1/2 belongs to the Ashkin-Teller class, which is characterized by a continuously varying critical exponent with g, as predicted by field-theoretical and other studies. Our results, based on the HOTRG calculations for significantly larger sizes, indicate that the region of the first-order transition is marginally narrower than that in previous studies. Furthermore, it is suggested that the region where the critical exponent changes does not necessarily coincide with the Ashkin-Teller region.
We investigate the phase transition of the four-dimensional Ising model with two types of tensor network scheme, one is the higher-order tensor renormalization group and the other is the anisotropic tensor renormalization group. The results for the internal energy and magnetization obtained by the former algorithm with the impure tensor method, enlarging the lattice volume up to 10244, are consistent with the weak first-order phase transition. For the later algorithm, our implementation successfully reduces the execution time thanks to the parallel computation and the results provided by ATRG seems comparable to those with HOTRG.