Dulmage-Mendelsohn percolation: Geometry of maximally-packed dimer models and topologically-protected zero modes on site-diluted bipartite lattices

Abstract

The classic combinatorial construct of {\em 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 a way of identifying a complete and non-overlapping 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 {\em 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.

version to appear in PRX; minor changes to text, slightly expanded discussion of computational methods and implications for diluted antiferromagnets, no new results

References in corpus (17)

• Anderson Transitions
• Hyperbolic Geometry of Complex Networks
• Multichannel Generalization of Kitaev's Majorana End States and a Practical Route to Realize Them in Thin Films
• Classical dimers with aligning interactions on the square lattice
• Some formal results for the valence bond basis
• Antiferromagnetic order in the Hubbard Model on the Penrose Lattice
• Quantum criticality, lines of fixed points, and phase separation in doped two-dimensional quantum dimer models
• Critical Correlations for Short-Range Valence-Bond Wave Functions on the Square Lattice
• Majorana-Hubbard model on the square lattice
• Unusual localisation effects in quantum percolation
• Dynamical Aspects of 2D Quantum Percolation
• Nature of Protected Zero Energy States in Penrose Quasicrystals
• Coexistence of long-range and algebraic correlations for short-range valence-bond wave functions in three dimensions
• Generalization of the singlet sector valence bond loop algorithm to antiferromagnetic ground states with total spin $S_{tot}=1/2$
• Numerical results for crossing, spanning and wrapping in two-dimensional percolation
• Hardcore dimer aspects of the SU(2) Singlet wavefunction
• Bilayer Coulomb phase of two dimensional dimer models: Absence of power-law columnar order

Cited by in corpus (7)

• Physical properties of an Aperiodic monotile: Graphene-like features, chirality and zero-modes
• Nematic Superconductivity and Its Critical Vestigial Phases in the Quasi-crystal
• Statistical mechanics of dimers on quasiperiodic Ammann-Beenker tilings
• Superconductivity and charge-density-wave in the Holstein model on the Penrose Lattice
• Magnetic effects of non-magnetic impurities in gapped short-range resonating valence bond spin liquids
• Hamiltonian Cycles on Ammann-Beenker Tilings
• Theory of collective topologically-protected Majorana fermion excitations of networks of localized Majorana modes