The classic combinatorial construct of maximum matchings probes the random geometry of regions with local sublattice imbalance in a site-diluted bipartite lattice. We demonstrate that these regions, which host the monomers of any maximum matching of the lattice, control the localization properties of a zero-energy quantum particle hopping on this lattice. The structure theory of Dulmage and Mendelsohn provides us with a way of identifying a complete and nonoverlapping set of such regions. This motivates our large-scale computational study of the Dulmage-Mendelsohn decomposition of site-diluted bipartite lattices in two and three dimensions. Our computations uncover an interesting universality class of percolation associated with the end-to-end connectivity of such monomer-carrying regions with local sublattice imbalance, which we dub Dulmage-Mendelsohn percolation. Our results imply the existence of a monomer percolation transition in the classical statistical mechanics of the associated maximally packed dimer model and the existence of a phase with area-law entanglement entropy of arbitrary many-body eigenstates of the corresponding quantum dimer model. They also have striking implications for the nature of collective zero-energy Majorana fermion excitations of bipartite networks of Majorana modes localized on sites of diluted lattices, for the character of topologically protected zero-energy wavefunctions of the bipartite random hopping problem on such lattices, and thence for the corresponding quantum percolation problem, and for the nature of low-energy magnetic excitations in bipartite quantum antiferromagnets diluted by a small density of nonmagnetic impurities.

中文翻译：

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Dulmage-Mendelsohn 渗流：位点稀释二分晶格上最大堆积二聚体模型和拓扑保护零模的几何

最大匹配的经典组合构造探测了在位点稀释的二分晶格中具有局部亚晶格不平衡的区域的随机几何形状。我们证明了这些区域，它们承载晶格的任何最大匹配的单体，控制着零能量量子粒子在该晶格上跳跃的定位特性。Dulmage 和 Mendelsohn 的结构理论为我们提供了一种识别这些区域的完整且不重叠的集合的方法。这激发了我们对二维和三维位置稀释二分晶格的 Dulmage-Mendelsohn 分解的大规模计算研究。我们的计算揭示了一种有趣的普遍性渗透类，它与具有局部亚晶格不平衡的单体携带区域的端到端连通性相关，我们称之为 Dulmage-Mendelsohn 渗透。我们的结果意味着在相关的最大堆积二聚体模型的经典统计力学中存在单体渗透跃迁，并且在相应的量子二聚体模型的任意多体本征态中存在具有面积律纠缠熵的相。它们还对位于稀释晶格位置的马约拉纳模式二分网络的集体零能量马约拉纳费米子激发的性质，对于此类晶格上二分随机跳跃问题的拓扑保护零能量波函数的特征具有显着意义，以及因此对于相应的量子渗透问题，