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arXiv:2007.04974 · Physical Review X

Dulmage-Mendelsohn Percolation: Geometry of Maximally Packed Dimer Models and Topologically Protected Zero Modes on Site-Diluted Bipartite Lattices

Anonymous, Sounak Biswas, Md Mursalin Islam, Kedar Damle

Published Jun 15, 2022• 13 citations
arXivHTMLPDF

Abstract

A new kind of quantum percolation phenomenon---of particles hopping on a lattice with missing sites---has implications for the effect of nonmagnetic impurities in quantum antiferromagnets and could aid the search for Majorana excitations in solid-state devices.
DOI
10.1103/physrevx.12.021058

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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.
We introduce a new class of site-percolation models and study them by Monte Carlo methods. These percolation models are defined as the zero-temperature limit of a quench-diluted spin model whose symmetries are those of the antiferromagnetic three-state Potts model on a triangular lattice. Many of these percolation models exhibit a sequence of two percolation transitions. A new phase, excluded in the absence of dilution, is found at intermediate spin concentrations. This phase, which we call the chiral phase, exhibits the symmetries of only a subgroup of the full symmetry group of the Potts model. The critical properties of the transitions found in these percolation models have been studied. The results indicate that, for this entire class of percolation models, the observed transitions are all contained in the universality class of the geometric percolation (q=1 Potts) model. From this result we conclude that the breaking of symmetry, which is relevant for thermally driven transitions, is not relevant for percolation transitions.
The field-induced antiferromagnetic ordering in systems of weakly coupled S = 1/2 dimers at zero temperature can be described as a Bose-Einstein condensation of triplet quasiparticles (singlet quasiholes) in the ground state. For the case of a Heisenberg bilayer, it is here shown how the above picture is altered in the presence of site dilution of the magnetic lattice. Geometric randomness leads to quantum localization of the quasiparticles or quasiholes and to an extended Bose-glass phase in a realistic disordered model. This localization phenomenon drives the system towards a quantum-disordered phase well before the classical geometric percolation threshold is reached.
The interplay of geometric randomness and strong quantum fluctuations is an exciting topic in quantum many-body physics, leading to the emergence of novel quantum phases in strongly correlated electron systems. Recent investigations have focused on the case of homogeneous site and bond dilution in the quantum antiferromagnet on the square lattice, reporting a classical geometric percolation transition between magnetic order and disorder. In this study we show how inhomogeneous bond dilution leads to percolative quantum phase transitions, which we have studied extensively by quantum Monte Carlo simulations. Quantum percolation introduces a new class of two-dimensional spin liquids, characterized by an infinite percolating network with vanishing antiferromagnetic order parameter.
We theoretically investigate the quantum percolation problem on Lieb lattices in two and three dimensions. We study the statistics of the energy levels through random matrix theory and determine the level spacing distributions, which, with the aid of finite-size scaling theory, allows us to obtain accurate estimates for site- and bond-percolation thresholds and critical exponents. Our numerical investigation supports a localized-delocalized transition at finite threshold, which decreases as the average coordination number increases. The precise determination of the localization length exponent enables us to claim that quantum site- and bond-percolation problems on Lieb lattices belong to the same universality class, with ν decreasing with lattice dimensionality d, similarly to the classical percolation problem. In addition, we verify that, in three dimensions, quantum percolation on Lieb lattices belongs to the same universality class as the Anderson impurity model.
Quantum phase transition of site-diluted and bond-diluted Heisenberg antiferromagnets on square lattices is studied. By using the novel continuous-time loop algorithm, we perform quantum Monte Carlo simulations on quite larger lattices at extremely lower temperatures than the previous numerical studies. It is found that the antiferromagnetic long-range order at T = 0 persists so long as a cluster of magnetic sites percolates, that is, the critical concentration is equal to the classical percolation threshold, in both of the site-diluted and bond-diluted cases. Furthermore, we find that some critical exponents, such as the magnetization exponent β, are non-classical and strongly depend on the spin size S. On the other hand, we show that the correlation-length exponent ν is universal and is equal to the classical value (ν=4/3).