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arXiv:1904.10645 · Physical review. E

Local and global magnetization on the Sierpiński carpet

Jozef Genzor, Andrej Gendiar, Tomotoshi Nishino

Published Apr 17, 2023• 5 citations
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Abstract

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 ${T}_{\mathrm{c}}^{\phantom{\rule{0.16em}{0ex}}}$ 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 $\ensuremath{\beta}\ensuremath{\approx}0.135$.

DOI
10.1103/physreve.107.044108

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The phase transition of the classical Ising model on the Sierpi\ifmmode \acute{n}\else '{n}\fi{}ski carpet, which has the fractal dimension log38\ensuremath≈1.8927{log}_{3}^{\phantom{\rule{0.16em}{0ex}}}8\ensuremath{\approx}1.8927log3​8\ensuremath≈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 Tc\ensuremath≈1.478{T}_{\mathrm{c}}^{\phantom{\rule{0.16em}{0ex}}}\ensuremath{\approx}1.478Tc​\ensuremath≈1.478. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent \ensuremathβ\ensuremath{\beta}\ensuremathβ 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 \ensuremathβ\ensuremath≈0.135\ensuremath{\beta}\ensuremath{\approx}0.135\ensuremathβ\ensuremath≈0.135.
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 transverse-field Ising model on the Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski fractal, which is characterized by the fractal dimension ${log}_{2}^{\phantom{\rule{3.33333pt}{0ex}}}3\ensuremath{\approx}1.585$, is studied by a tensor-network method known as the higher-order tensor renormalization group. We analyze the ground-state energy and the spontaneous magnetization in the thermodynamic limit. The system exhibits the second-order phase transition at the critical transverse field hc=1.865{h}_{\mathrm{c}}^{\phantom{\rule{3.33333pt}{0ex}}}=1.865hc​=1.865. The critical exponents $\ensuremath{\beta}\ensuremath{\approx}0.198$ and $\ensuremath{\delta}\ensuremath{\approx}8.7$ are obtained. Complementary to the tensor-network method, we make use of the real-space renormalization group and improved mean-field approximations for comparison.
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.
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 nonequilibrium critical dynamics of the Ising magnet on a fractal substrate, namely the Sierpinski carpet with Hausdorff dimension d(H)=1.7925, has been studied within the short-time regime by means of Monte Carlo simulations. The evolution of the physical observables was followed at criticality, after both annealing ordered spin configurations (ground state) and quenching disordered initial configurations (high temperature state), for three segmentation steps of the fractal. We have obtained evidence showing that during these relaxation processes both the growth and the fragmentation of magnetic domains become influenced by the hierarchical structure of the substrate. In fact, the interplay between the dynamic behavior of the magnet and the underlying fractal leads to the emergence of a logarithmic-periodic oscillation, superimposed to a power law, which has been observed in the time dependence of both the decay of the magnetization and its logarithmic derivative. These oscillations have been carefully characterized in order to determine the critical temperature of the second-order phase transition and the critical exponents corresponding to the short-time regime. The effects of the substrate can also be observed from the dependence of the effective critical exponents on the segmentation step. The exponent theta of the initial increase of the magnetization has also been obtained and the results suggest that it would be almost independent of the fractal dimension of the substrate, provided that d(H) is close enough to d=2. The oscillations have been discussed within the framework of the discrete scale invariance of the substrate.
We study the critical behavior of the transverse-field quantum Ising model on a fractal structure, namely the Sierpinski carpet. When a magnetic field Δ is applied perpendicular to the Ising spin direction, quantum fluctuations affect the transition between the ferromagnetic and the paramagnetic phases. Employing the continuous-time quantum Monte Carlo simulation method and the finite-size scaling analysis, we investigate the interplay between the quantum fluctuations and the exotic dimensionality of the fractal structure and its effect on the critical behavior. As the transverse magnetic field increases, the critical temperature monotonically decreases until it apparently vanishes at a critical field Δ(c), beyond which the system becomes paramagnetic at all temperatures. However, the critical exponents are independent of Δ and remain the same as in the purely classical(Δ=0) case.