Distinguishing Spontaneous Symmetry Breaking from Topological Order via Volume Scaling — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Spontaneous symmetry breaking (SSB) and topological order are distinct paradigms; SSB involves a reduced matrix element scaling with inverse volume, while topological order is robust to local perturbations and lacks local order parameters. | MATH: SSB order parameter ⟨ ϕ ⟩ ≠ 0 in thermodynamic limit; reduced matrix element ⟨ Ω | ϕ(x) | Ω ⟩ ∼ V−1/2 for finite volume V, vanishing as V → ∞. Topological ground state degeneracy D on genus-g surface scales as D = kg (e.g., ℤk gauge theory). | CONNECTION: Topological order connects to modular data and anyonic braiding — fusion rules and S,T matrices satisfying S2 = (ST)3 = C, with S-matrix elements containing e2π i / k (root-of-unity phases, base-60 compatible via cyclotomic fields). Crystallographic analog: fractional quantum Hall states have filling fractions ν = p/q (e.g., 1/3, 2/5) — ratios not golden, b
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026distinguishing,
title = {{Distinguishing Spontaneous Symmetry Breaking from Topological Order via Volume Scaling — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23587025},
url = {https://zenodo.org/records/23587025},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Distinguishing Spontaneous Symmetry Breaking from Topological Order via Volume Scaling — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23587025
UR - https://zenodo.org/records/23587025
AB - FINDING: Spontaneous symmetry breaking (SSB) and topological order are distinct paradigms; SSB involves a reduced matrix element scaling with inverse volume, while topological order is robust to local perturbations and lacks local order parameters. | MATH: SSB order parameter ⟨ ϕ ⟩ ≠ 0 in thermodynamic limit; reduced matrix element ⟨ Ω | ϕ(x) | Ω ⟩ ∼ V^−1/2 for finite volume V, vanishing as V → ∞. Topological ground state degeneracy D on genus-g surface scales as D = k^g (e.g., ℤ_k gauge theory). | CONNECTION: Topological order connects to modular data and anyonic braiding — fusion rules and S,T matrices satisfying S^2 = (ST)^3 = C, with S-matrix elements containing e^2π i / k (root-of-unity phases, base-60 compatible via cyclotomic fields). Crystallographic analog: fractional quantum Hall states have filling fractions ν = p/q (e.g., 1/3, 2/5) — ratios not golden, b Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -