Entanglement Theories: Packing vs. Percolation

Andrew Stewart Caldin

FINDING: Topological color codes on hexagonal and square-octagonal lattices exhibit accuracy thresholds determined by percolation theory, with mapping to spin models revealing critical ratios. MATH: - Percolation threshold for hexagonal lattice (bond percolation): \( p_c = 1 - 2\sin(\pi/18) \approx 0.6527 \) - For square-octagonal lattice: \( p_c \approx 0.6764 \) (exact value from 2D percolation theory) - Accuracy threshold ratio (hexagonal vs square-octagonal): \( \approx 0.6527 / 0.6764 \approx 0.965 \) - Mapping to spin model: critical exponents follow 2D Ising universality class, with \( \nu = 1 \) (correlation length exponent) - Surface code threshold (toric code): \( p_{\text{th}} \approx 0.103 \) (for standard square lattice, from depolarizing noise) CONNECTION: - Hexagonal lattice: 6-fold rotational symmetry (crystallographic point group \( D_6 \)), bond angles \( 120^\circ \) — related to hexagonal close-packing and graphene structure. - Square-octagonal lat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Shashaank Khanna, Ujjwal Sen, Sarika Jalan

Antonio Acín, J. I. Cirac, Maciej Lewenstein

Ritesh Bhola, Soham Biswas, Md Mursalin Islam +1 more

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.

Karen Pearlman, Richard J. Allen

Anonymous, Sounak Biswas, Md Mursalin Islam +1 more

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.