# Caminhada do Monopolo Magnético no Gelo de Spins

Monopolos magnéticos emergentes no gelo de spins pirocloro

> Página canônica: https://elysiatools.com/pt/visualizations/spin-ice-magnetic-monopole-walk

- **Categoria:** Physics

## Visão geral

Interactive pyrochlore spin-ice model with emergent magnetic monopoles — distinct from existing magnetism cases (magnetic-field, magnetic-hysteresis, magnetic-dipole-field are classical/continuous magnetism; meissner-effect-levitation is superconductivity; bragg-diffraction is X-ray crystallography, not spin configurations; none cover the condensed-matter spin-ice monopole, a quasiparticle between classical magnetism and quantum topology). The model builds the pyrochlore lattice via its geometric duality with the diamond lattice: each diamond-lattice edge carries one Ising spin (spin site), each diamond-lattice vertex is the centre of one tetrahedron. Built with periodic boundary conditions on an L×L×L conventional-cell block (L=1..4), every tetrahedron is exactly 4-coordinated so each carries an even magnetic charge Q=(#out−#in)∈{−4,−2,0,+2,+4} and every single-spin flip moves exactly ±2 units of charge between two tetrahedra — the clean monopole picture. The ground state obeys Pauling's ice rule (2-in/2-out, Q=0); a 3-in/1-out tetrahedron is a negative monopole (Q=−2), 1-in/3-out a positive monopole (Q=+2), all-in/all-out double charges (Q=±4) — the condensed-matter realisation of magnetic monopoles proposed by Castelnovo, Moessner & Sondhi (Nature 2008). Total charge ΣQ is conserved exactly under all dynamics (a Gauss-law invariant, tested). Energy E = (J/2)Σ_t Q(t)² + Σ_{t<t'} μ0Q²·q_t·q_{t'}/r − h·Σ spin·x̂ + (string-tension)·(new charges), and the Monte-Carlo engine uses exact flip-delta ΔE (sign-corrected by the current spin value) with the Metropolis acceptance min(1, exp(−ΔE/T)), so detailed balance holds and the system thermalises to the correct Boltzmann defect density (verified: hotter ⇒ more monopoles, low-T ⇒ ice ground state). Dirac strings — connected chains of flipped spins linking opposite monopoles, the locus of reversed magnetic flux — are detected as connected components of the charged-tetrahedron graph; finite strings have ± ends whose charge-sums are even. Two visualization panels: (1) a Three.js 3D pyrochlore lattice showing the tetrahedron-centre skeleton with spin cylinders (colour-coded by orientation), monopole spheres (red +, blue −, larger for double charges) with glow shells, and gold Dirac-string line segments; drag to orbit, scroll to zoom, and click a tetrahedron to inject a ±2 monopole pair across one of its bonds. (2) a control panel with 6 presets (Ice ground state, Warm monopole gas, Hot dense plasma, Coulomb-confined, Tense Dirac strings, Field-driven walk) and sliders for lattice size L, temperature T/J, Coulomb coupling μ0Q², Dirac-string tension, external [100] field, and MC-sweeps-per-frame, plus Run/Anneal/Reset buttons and a live observables strip (monopole count, defect density, conserved net charge ΣQ, Dirac-string count, energy per spin, accept rate) and an automatic phase detector (ice / monopole gas / dense plasma). Educational content covers the ice rule (Pauling, Bernal–Fowler water ice), emergent monopoles (Castelnovo–Moessner–Sondhi 2008), monopoles walking on the diamond lattice with charge conservation (Gauss's law), Dirac strings and deconfinement vs quark-confinement analogy, the Coulomb phase and Debye–Hückel screening, and real materials (Dy₂Ti₂O₇, Ho₂Ti₂O₇, the Wien effect and Pauling residual entropy). Multi-language support (zh, en, es, fr, de, ru, pt).

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