This article investigates the magnetic properties and phase diagrams of a mixed spin‐1 and spin‐1/2 system on a 2D square‐octagon lattice within the quantum XXZ Heisenberg model using the Oguchi approximation. The sublattice magnetizations and total magnetization are analyzed as a function of temperature, providing insights into the thermal behavior of this mixed‐spin system. Additionally, the effect of the crystal field parameter on the magnetic properties is examined, revealing critical phenomena and phase transitions in the system. The influence of the anisotropy parameter is also explored, demonstrating its essential role in shaping the nature of the phase transitions and the emergence of tricritical behavior. The results contribute to a deeper understanding of the interplay between exchange interaction, crystal field, thermal fluctuations, and quantum effects in complex lattice systems.
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):
pc=1−2sin(π/18)≈0.6527 - For square-octagonal lattice:
pc≈0.6764 (exact value from 2D percolation theory) - Accuracy threshold ratio (hexagonal vs square-octagonal):
≈0.6527/0.6764≈0.965 - Mapping to spin model: critical exponents follow 2D Ising universality class, with
ν=1 (correlation length exponent) - Surface code threshold (toric code):
pth≈0.103 (for standard square lattice, from depolarizing noise) CONNECTION: - Hexagonal lattice: 6-fold rotational symmetry (crystallographic point group
D6), bond angles
120∘ — 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