We present a new cluster-update Monte Carlo algorithm for a fluctuating interface in the 3D Ising model. Critical slowing down is drastically reduced and the dynamical critical exponent at the roughening temperature of the 3D Ising model is found to be ${\mathit{z}}_{\mathrm{cl}}$ =0.44\ifmmode\pm\else\textpm\fi{}0.03. For the Metropolis algorithm we obtain ${\mathit{z}}_{\mathrm{Met}}$ =2.0\ifmmode\pm\else\textpm\fi{}0.1 while the Swendsen-Wang algorithm performs even worse than local updating. A new method which suppresses fluctuations in the two bulk phases is introduced to determine moments of the interfacial thickness.We present a new cluster-update Monte Carlo algorithm for a fluctuating interface in the 3D Ising model. Critical slowing down is drastically reduced and the dynamical critical exponent at the roughening temperature of the 3D Ising model is found to be ${\mathit{z}}_{\mathrm{cl}}$=0.44\ifmmode\pm\else\textpm\fi{}0.03. For the Metropolis algorithm we obtain ${\mathit{z}}_{\mathrm{Met}}$=2.0\ifmmode\pm\else\textpm\fi{}0.1 while the Swendsen-Wang algorithm performs even worse than local updating. A new method which suppresses fluctuations in the two bulk phases is introduced to determine moments of the interfacial thickness.

Physical Review LettersFeb 4, 199136 citations

Possible new phase transition in the 3D Ising model associated with boundary percolation

Michael Grady

Abstract In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13% of spins are minority. Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships. Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here. A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation. This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition. Its critical exponent ν ∼ 1.3 is seen to match the finite-size shift exponent of the percolation transition further cementing their connection. This predicts a very weak specific heat singularity with exponent α ∼ − 1.9 . The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition. Unlike random boundary percolation, the Ising boundary percolation has two different ν exponents, one associated with largest-cluster scaling and the other with finite-size transition-point shift. This suggests there may be two different correlation lengths present.Abstract In the ordered phase of the 3D Ising model, minority spin clusters are surrounded by a boundary of dual plaquettes. As the temperature is raised, these spin clusters become more numerous, and it is found that eventually their boundaries undergo a percolation transition when about 13% of spins are minority. Boundary percolation differs from the more commonly studied site and link percolation, although it is related to an unusual type of site percolation that includes next to nearest neighbor relationships. Because the Ising model can be reformulated in terms of the domain boundaries alone, there is reason to believe boundary percolation should be relevant here. A symmetry-breaking order parameter is found in the dual theory, the 3D gauge Ising model. It is seen to undergo a phase transition at a coupling close to that predicted by duality from the boundary percolation. This transition lies in the disordered phase of the gauge theory and has the nature of a spin-glass transition. Its critical exponent ν ∼ 1.3 is seen to match the finite-size shift exponent of the percolation transition further cementing their connection. This predicts a very weak specific heat singularity with exponent α ∼ − 1.9 . The third energy cumulant fits well to the expected non-infinite critical behavior in a manner consistent with both the predicted exponent and critical point, indicating a true thermal phase transition. Unlike random boundary percolation, the Ising boundary percolation has two different ν exponents, one associated with largest-cluster scaling and the other with finite-size transition-point shift. This suggests there may be two different correlation lengths present.

arXiv:2301.08424Journal of Physics Condensed MatterApr 11, 20235 citations

Phase Diagram of Three Dimensional Site Random Ising Model

之康尾関, Yukiyasu Ozeki

Jan 1, 19970 citations

{\mathit{z}}_{\mathrm{cl}}

=0.44\ifmmode\pm\else\textpm\fi{}0.03. For the Metropolis algorithm we obtain

{\mathit{z}}_{\mathrm{Met}}

=2.0\ifmmode\pm\else\textpm\fi{}0.1 while the Swendsen-Wang algorithm performs even worse than local updating. A new method which suppresses fluctuations in the two bulk phases is introduced to determine moments of the interfacial thickness.