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}^{}\ensuremath{-}{J}_{2}^{}

planar fractal whose Hausdorff dimension at

{J}_{2}^{}=0

{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.

DOI: 10.1103/physreve.105.024124

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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

{J}_{1}^{}\ensuremath{-}{J}_{2}^{}

planar fractal whose Hausdorff dimension at

{J}_{2}^{}=0

{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.

d_{\rm H}=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

T_c \approx 2.65231

with exotic critical exponents (

\alpha\approx 0.09

\alpha'\approx 0.06

\beta \approx 0.059

\delta \approx 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.