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<p>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 <i>d</i><sub>H</sub> = 2.5 and boundary dimension <i>d</i> = 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 <i>T</i><sub><i>c</i></sub> ≈ 2.65231 with exotic critical exponents (<i>β</i> ≈ 0.059, <i>δ</i> ≈ 35) and a diverging specific heat consistent with a logarithmic singularity (i.e., <i>α</i> = 0) within the present numerical accuracy—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.</p>

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Critical phenomena on a 3D fractal with intermediate dimensionality: tensor-network study

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月22日

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  • Jozef Genzor,
  • Roman Krčmár,
  • Hiroshi Ueda,
  • Tomotoshi Nishino,
  • Denis Kochan,
  • Andrej Gendiar

摘要

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 (β ≈ 0.059, δ ≈ 35) and a diverging specific heat consistent with a logarithmic singularity (i.e., α = 0) within the present numerical accuracy—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.