Ohzeki Masayuki, Fujii Keisuke
Department of Systems Science, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Nov;86(5 Pt 1):051121. doi: 10.1103/PhysRevE.86.051121. Epub 2012 Nov 19.
The conventional duality analysis is employed to identify a location of a critical point on a uniform lattice without any disorder in its structure. In the present study, we deal with the random planar lattice, which consists of the randomized structure based on the square lattice. We introduce the uniformly random modification by the bond dilution and contraction on a part of the unit square. The random planar lattice includes the triangular and hexagonal lattices in extreme cases of a parameter to control the structure. A modern duality analysis fashion with real-space renormalization is found to be available for estimating the location of the critical points with a wide range of the randomness parameter. As a simple test bed, we demonstrate that our method indeed gives several critical points for the cases of the Ising and Potts models and the bond-percolation thresholds on the random planar lattice. Our method leads to not only such an extension of the duality analyses on the classical statistical mechanics but also a fascinating result associated with optimal error thresholds for a class of quantum error correction code, the surface code on the random planar lattice, which is known as a skillful technique to protect the quantum state.
传统的对偶分析用于确定结构无任何无序的均匀晶格上临界点的位置。在本研究中，我们处理的是随机平面晶格，它由基于正方形晶格的随机结构组成。我们通过对单位正方形的一部分进行键稀释和收缩来引入均匀随机修改。随机平面晶格在控制结构的参数的极端情况下包括三角形和六边形晶格。发现一种具有实空间重整化的现代对偶分析方式可用于估计具有广泛随机性参数的临界点的位置。作为一个简单的测试平台，我们证明我们的方法确实为伊辛模型和波茨模型以及随机平面晶格上的键渗流阈值的情况给出了几个临界点。我们的方法不仅导致了对经典统计力学对偶分析的这种扩展，还带来了与一类量子纠错码（随机平面晶格上的表面码）的最优误差阈值相关的迷人结果，表面码是一种保护量子态的巧妙技术。
