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Journal of Physics A Mathematical and General

Scaling properties of the damage cloud in the 3D Ising model

L. de Arcangelis, Hans Jürgen Herrmann, Antonio Coniglio A

Published Mar 21, 1990• 12 citations
Publisher

Abstract

The authors show that in the three-dimensional Ising model at the critical temperature the fractal dimension of the touching damage, determined by the box counting method, asymptotically converges to the expected value d f =d- beta /v. In contrast, the finite-size scaling analysis indicates for the touching damage an effective fractal dimension whose value is 20% smaller. The expected d f is, however, recovered again when instead of the system size L the quantity L/ln L is used as a scaling variable. This behaviour could be explained by the recently discovered dynamical multiscaling.

DOI
10.1088/0305-4470/23/6/006

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arXiv:2506.17053•Jun 20, 2025•0 citations
The authors show that in the three-dimensional Ising model at the critical temperature the fractal dimension of the touching damage, determined by the box counting method, asymptotically converges to the expected value d f =d- beta /v. In contrast, the finite-size scaling analysis indicates for the touching damage an effective fractal dimension whose value is 20% smaller. The expected d f is, however, recovered again when instead of the system size L the quantity L/ln L is used as a scaling variable. This behaviour could be explained by the recently discovered dynamical multiscaling.
The authors use the method of 'damage spreading' to measure the time taken for the equilibrium damage (the magnetization) to disappear in finite systems (10<or=L<or=103) at the critical temperature of the infinite system. This time is found to scale as L z with an exponent z of 2.16+or-0.02. This value is compatible with the expected value of the dynamic exponent for the 2D Ising model, but they argue that this method is unable to measure fluctuations properly which they show are responsible for the higher values recently reported in the literature. They suggest that the slowest mode of relaxation has an exponent of z>or=2.27.
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.