Topology-Constrained Lattice Monte Carlo --- First Soliton Energy Measurement
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2026
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| author | Novickis, Alexander |
| author_facet | Novickis, Alexander |
| contents | <p><strong>Title:</strong> Topology-Constrained Lattice Monte Carlo --- First Soliton Energy Measurement</p> <p><strong>Author:</strong> Alexander Novickis (alex.novickis@gmail.com)</p> <p>The energy of the fundamental Hopf soliton ($Q = 1$) is the single most important numerical prediction of the topological EM structures programme, yet no stable lattice measurement with proper thermal fluctuations exists. We report the first systematic investigation of topology preservation under lattice Monte Carlo evolution of the Faddeev-Niemi CP$^1$ field theory. Three regimes are studied on a $16^3$ lattice: (i) unconstrained Metropolis MC at $\beta = \kappa = 5$ and $\beta = \kappa = 20$, where topology is destroyed in $< 20$ sweeps via gradual phase decoherence; (ii) whole-sweep topology-constrained MC at $\beta = \kappa = 10$ with tolerance $\delta = 0.1$, where the Hopf charge is preserved ($H_W = 1.038$) but the rejection rate is 100%; and (iii) gradient-flow cooling with topology enforcement, which yields a local energy minimum at $E = 70.49$ (lattice units) with $H_W = 1.04$ and virial ratio $E_2/E_4 = 0.506$. We present a detailed analysis of the topology loss mechanism, establish the entropy argument for the dominance of the trivial sector, and identify per-site topology constraints as the necessary algorithmic advance. The cooled configuration represents the best current lattice approximation to the Hopf soliton but lacks thermal fluctuations required for a proper partition function measurement. We outline three concrete algorithms --- per-site constrained Metropolis, projected Hybrid Monte Carlo, and topology-penalized action --- for achieving the first genuine lattice soliton energy measurement.</p> <p><i>(195 words)</i></p> <p><strong>Keywords:</strong> physics, soliton, topology, lattice, Monte Carlo, numerical, Hopf charge, energy, CP1, Faddeev Niemi</p> <p><strong>DOI:</strong> <a href="https://doi.org/10.5281/zenodo.19111685">10.5281/zenodo.19111685</a></p> <p><strong>Series:</strong> Paper XLV in the Hopf Soliton Programme</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19590457 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Topology-Constrained Lattice Monte Carlo --- First Soliton Energy Measurement Novickis, Alexander physics soliton topology lattice Monte-Carlo numerical Hopf-charge energy CP1 Faddeev-Niemi <p><strong>Title:</strong> Topology-Constrained Lattice Monte Carlo --- First Soliton Energy Measurement</p> <p><strong>Author:</strong> Alexander Novickis (alex.novickis@gmail.com)</p> <p>The energy of the fundamental Hopf soliton ($Q = 1$) is the single most important numerical prediction of the topological EM structures programme, yet no stable lattice measurement with proper thermal fluctuations exists. We report the first systematic investigation of topology preservation under lattice Monte Carlo evolution of the Faddeev-Niemi CP$^1$ field theory. Three regimes are studied on a $16^3$ lattice: (i) unconstrained Metropolis MC at $\beta = \kappa = 5$ and $\beta = \kappa = 20$, where topology is destroyed in $< 20$ sweeps via gradual phase decoherence; (ii) whole-sweep topology-constrained MC at $\beta = \kappa = 10$ with tolerance $\delta = 0.1$, where the Hopf charge is preserved ($H_W = 1.038$) but the rejection rate is 100%; and (iii) gradient-flow cooling with topology enforcement, which yields a local energy minimum at $E = 70.49$ (lattice units) with $H_W = 1.04$ and virial ratio $E_2/E_4 = 0.506$. We present a detailed analysis of the topology loss mechanism, establish the entropy argument for the dominance of the trivial sector, and identify per-site topology constraints as the necessary algorithmic advance. The cooled configuration represents the best current lattice approximation to the Hopf soliton but lacks thermal fluctuations required for a proper partition function measurement. We outline three concrete algorithms --- per-site constrained Metropolis, projected Hybrid Monte Carlo, and topology-penalized action --- for achieving the first genuine lattice soliton energy measurement.</p> <p><i>(195 words)</i></p> <p><strong>Keywords:</strong> physics, soliton, topology, lattice, Monte Carlo, numerical, Hopf charge, energy, CP1, Faddeev Niemi</p> <p><strong>DOI:</strong> <a href="https://doi.org/10.5281/zenodo.19111685">10.5281/zenodo.19111685</a></p> <p><strong>Series:</strong> Paper XLV in the Hopf Soliton Programme</p> |
| title | Topology-Constrained Lattice Monte Carlo --- First Soliton Energy Measurement |
| topic | physics soliton topology lattice Monte-Carlo numerical Hopf-charge energy CP1 Faddeev-Niemi |
| url | https://doi.org/10.5281/zenodo.19590457 |