Percolation Treatment of Frustrated Ising Lattices
In a previous paper , we proposed an exact transformation from frustrated Ising lattices into percolation systems. The percolation cluster is considered to express the spin cluster which is able to make up the spin-glass type order. As an example, we applied the above consideration to Bhatt and Young’s 3D spin glass simulation data and obtained the critical concentration of the corresponding frustrated percolation system . With the use of dimensional invariant relation for the critical concentration, we can obtain a value of effective dimension for this system, which is far lower than that of lattice dimension. This result can be explained by the coexistence of low dimensional infinite clusters. The avoiding tendency between clusters by frustration is consistent with the dimensional lowering. However, the structure of the coexisting state is not clear. To obtain more detailed information, we intend to perform the direct simulation of frustrated percolation system and to show the snap shot of the clustering state. Since the display of a 3D lattice is difficult, we treat the 2D frustrated lattice in this work. As a result, we observe at a sufficiently high concentration that the macroscopic cluster has a low dimensional structure although the magnitude of macroscopic cluster is the order of site number. In other words, the lower dimensional branches fill up nearly the whole lattice. It is probable that this tendency occurs similarly in 3D frustrated lattices. Our simulation has been done as follows.