Department of Theoretical Physics, Royal Institute of Technology, Stockholm 106 91, Sweden.
Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA.
Phys Rev E. 2018 Mar;97(3-1):032104. doi: 10.1103/PhysRevE.97.032104.
The fractal dimension of domain walls produced by changing the boundary conditions from periodic to antiperiodic in one spatial direction is studied using both the strong-disorder renormalization group algorithm and the greedy algorithm for the Edwards-Anderson Ising spin-glass model for up to six space dimensions. We find that for five or fewer space dimensions, the fractal dimension is lower than the space dimension. This means that interfaces are not space filling, thus implying that replica symmetry breaking is absent in space dimensions fewer than six. However, the fractal dimension approaches the space dimension in six dimensions, indicating that replica symmetry breaking occurs above six dimensions. In two space dimensions, the strong-disorder renormalization group results for the fractal dimension are in good agreement with essentially exact numerical results, but the small difference is significant. We discuss the origin of this close agreement. For the greedy algorithm there is analytical expectation that the fractal dimension is equal to the space dimension in six dimensions and our numerical results are consistent with this expectation.
使用强无序重整化群算法和贪婪算法研究了在一个空间方向上从周期性边界条件变为反周期性边界条件时产生的畴壁的分形维数，对爱德华兹-安德森伊辛自旋玻璃模型进行了高达六个空间维度的研究。我们发现，对于五个或更少的空间维度，分形维数低于空间维度。这意味着界面不是空间填充的，因此意味着在小于六个空间维度的情况下不存在副本对称性破坏。然而，分形维数在六个维度上接近空间维度，表明副本对称性在六个以上维度上发生。在两个空间维度上，分形维数的强无序重整化群结果与基本上精确的数值结果非常吻合，但微小的差异非常显著。我们讨论了这种紧密一致性的起源。对于贪婪算法，我们有分析性的期望，即在六个维度上分形维数等于空间维度，我们的数值结果与这一期望一致。
