Critical Phenomena on a 3D Fractal with Intermediate Dimensionality: Tensor-network study

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 \(d_{\rm H}=2.5\) 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.