Gallai-Edmonds percolation
Ritesh Bhola (Tata institute, Mumbai)
Maximally-packed dimer coverings (maximum matchings) of non-bipartite site-diluted lattices, such as the triangular and Shastry-Sutherland lattices in d = 2 dimensions and the stacked-triangular and corner-sharing octahedral lattices in d = 3, generically exhibit a nonzero density of monomers (unmatched vertices). Following the construction of Ref. [1], we use the the structure theory of Gallai and Edmonds to decompose the disordered lattice into “R-type” regions which host the monomers of any maximum matching, and perfectly matched “P-type” regions from which such monomers are excluded. When the density nv of quenched vacancies lies well within the low-nv geometrically percolated phase of the disordered lattice, we find that the random geometry of these regions exhibits rather unusual Gallai-Edmonds percolation phenomena. In d = 2, we find two
phases separated by a critical point, namely a phase in which all R-type and P-type regions are small and a percolated phase that displays a striking lack of self-averaging in the thermodynamic limit: Each sample has a single percolating region which is of type P or R with respective probability fP or 1 − fP, where fP ≈ 0.5 independent of nv (away from the critical region). In d = 3, apart from a phase with small R-type and P-type clusters, the thermodynamic limit exhibits four distinct percolated phases separated by critical points at successively lower nv. In the first such phase, each sample has one percolating R-type region and no such P-type region. In the second, each sample
simultaneously has one percolating region of each type (R and P). The third phase exhibits another interesting violation of self-averaging in the thermodynamics limit: R-type regions percolate with unit probability, while P-type regions percolate with probability fP ≈ 0.5. Finally, the lowest-nv phase is identical in character to the unusual percolated phase found in d = 2, again with fP ≈ 0.5 away from the transition region. These phenomena have interesting implications for the vacancy-induced local moment instabilities of short-range resonating valence bond spin liquid states of frustrated magnets and for the character of disorder-induced collective Majorana excitations of Kitaev magnets and SU(2) symmetric Majorana spin liquids.
[1] K. Damle, Theory of collective topologically protected Majorana fermion excitations of networks of localized Majorana modes, Phys. Rev. B 105, 235118 (2022).