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arXiv:2506.17053

Phase Transition of the Ising Model on a 3-Dimensional Fractal Lattice

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

Published Jun 20, 2025• 0 citations
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Abstract

The critical behavior of the classical Ising model on a three-dimensional fractal lattice with Hausdorff dimension $d_H = \ln32 / \ln4 = 2.5$ is investigated using the higher-order tensor renormalization group (HOTRG) method. We determine the critical temperature $T_c \approx 2.65231$ and the critical exponents for magnetization $β\approx 0.059$ and field response $δ\approx 35$. Unlike a previously studied 2D fractal with $d_H \approx 1.792$, the specific heat for this 3D fractal exhibits a divergent singularity at $T_c$. The results are compared with those for regular lattices and other fractal structures to elucidate the role of dimensionality in critical phenomena.

DOI
10.48550/arxiv.2506.17053

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The critical behavior of the classical Ising model on a three-dimensional fractal lattice with Hausdorff dimension dH=ln⁡32/ln⁡4=2.5d_H = \ln32 / \ln4 = 2.5dH​=ln32/ln4=2.5 is investigated using the higher-order tensor renormalization group (HOTRG) method. We determine the critical temperature Tc≈2.65231T_c \approx 2.65231Tc​≈2.65231 and the critical exponents for magnetization β≈0.059β\approx 0.059β≈0.059 and field response δ≈35δ\approx 35δ≈35. Unlike a previously studied 2D fractal with dH≈1.792d_H \approx 1.792dH​≈1.792, the specific heat for this 3D fractal exhibits a divergent singularity at TcT_cTc​. The results are compared with those for regular lattices and other fractal structures to elucidate the role of dimensionality in critical phenomena.
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=2.5d_{\rm H}=2.5dH​=2.5 and boundary dimension d=2d=2d=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≈2.65231T_c \approx 2.65231Tc​≈2.65231 with exotic critical exponents (α≈0.09\alpha\approx 0.09α≈0.09, α′≈0.06\alpha'\approx 0.06α′≈0.06, β≈0.059\beta \approx 0.059β≈0.059, δ≈35\delta \approx 35δ≈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.
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.
The phase transition of the classical Ising model on the Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpet, which has the fractal dimension ${log}_{3}^{\phantom{\rule{0.16em}{0ex}}}8\ensuremath{\approx}1.8927$, 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 ${T}_{\mathrm{c}}^{\phantom{\rule{0.16em}{0ex}}}\ensuremath{\approx}1.478$. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent $\ensuremath{\beta}$ associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas Tc{T}_{\mathrm{c}}^{\phantom{\rule{0.16em}{0ex}}}Tc​ 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 $\ensuremath{\beta}\ensuremath{\approx}0.135$.
We study the quantum Ising model on the Sierpiński triangle, whose Hausdorff dimension is log⁡3/log⁡2≈1.585\log 3/ \log 2 \approx 1.585log3/log2≈1.585, and demonstrate that it undergoes second-order phase transition with scaling relations satisfied precisely. We also study the quantum 333-state Potts model on the Sierpiński triangle and find first-order phase transition, which is consistent with a prediction from εεε-expansion that the transition becomes first-order for D>1.3D > 1.3D>1.3. We then compute critical exponents of the Ising model on higher-dimensional Sierpiński pyramids with various Hausdorff dimension via Monte-Carlo simulations and real-space RG analysis for D∈[1,3]D\in[1,3]D∈[1,3]. We find that only the correlation length exponent ννν interpolates the values of integer-dimensional models. This implies that, contrary to a generally held belief, the universality class of quantum phase transition may not be uniquely determined by symmetry and spatial dimension of the system. This work initiates studies on quantum critical phenomena on graphs and networks which may be of significant importance in the context of quantum networks and communication.
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.