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This preprint proposes a non-Abelian theoretical framework for structural consistency in simplicial networks, utilizing parallel-tempering simulations to model disorder-driven transitions. The study investigates whether dense frustration leads to glassy behavior or instead results in a correlated gauge liquid where global compatibility is lost without replica-symmetry breaking. While rooted in condensed matter and high-energy physics theory, the work addresses fundamental questions about disordered systems that are directly relevant to advanced machine learning models involving complex energy landscapes and neural network dynamics.
Frustration without Glass in Non-Abelian Simplicial Networks
arXiv:2608.17817v1 Announce Type: cross Abstract: We formulate a non-Abelian theory of structural consistency for systems in which dynamical transformations reside on the links of a simplicial network. Gauge covariance follows from freedom to choose local representation frames, while plaquette holonomies quantify the incompatibility of alternative paths. For an SU(2) model on the complete simplicial $2$-complex with quenched random plaquette couplings, parallel-tempering simulations reveal a continuous disorder-driven loss of global compatibility. In the high-disorder phase, the uniform compatibility $M_P$ decreases with system size, the integrated adjacent correlation weight $\mathcal I_{\mathrm{adj}}$ remains finite value. Moreover, the connected replica-overlap width approaches the numerical noise floor and its distribution narrows, providing no evidence for thermodynamic replica-symmetry breaking. Dense frustration therefore produces a non-glassy correlated gauge liquid in which individual pair correlations are geometrically diluted while a finite integrated correlation weight survives.
Abstract:We formulate a non-Abelian theory of structural consistency for systems in which dynamical transformations reside on the links of a simplicial network. Gauge covariance follows from freedom to choose local representation frames, while plaquette holonomies quantify the incompatibility of alternative paths. For an SU(2) model on the complete simplicial $2$-complex with quenched random plaquette couplings, parallel-tempering simulations reveal a continuous disorder-driven loss of global compatibility. In the high-disorder phase, the uniform compatibility $M_P$ decreases with system size, the integrated adjacent correlation weight $\mathcal I_{\mathrm{adj}}$ remains finite value. Moreover, the connected replica-overlap width approaches the numerical noise floor and its distribution narrows, providing no evidence for thermodynamic replica-symmetry breaking. Dense frustration therefore produces a non-glassy correlated gauge liquid in which individual pair correlations are geometrically diluted while a finite integrated correlation weight survives.
4 pages, 2 figures. Feedback welcome!
Statistical Mechanics (cond-mat.stat-mech); Disordered Systems and Neural Networks (cond-mat.dis-nn); Other Condensed Matter (cond-mat.other); High Energy Physics - Theory (hep-th); Quantum Physics (quant-ph)
arXiv:2608.17817 [cond-mat.stat-mech]
(or arXiv:2608.17817v1 [cond-mat.stat-mech] for this version)
https://doi.org/10.48550/arXiv.2608.17817 arXiv-issued DOI via DataCite (pending registration)
Full text under the publisher's open licence · 175 words
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