Xuanhua Wang, Fuyao University of Science and Technology, defined on a complete simplicial 2-complex with quenched random plaquette couplings exhibits compatible alternative paths. Parallel-tempering simulations exploring this behaviour reveal a continuous disorder-driven loss of global compatibility. In the high-disorder phase, the uniform compatibility measure (MP) decreases sharply with increasing system size while the integrated adjacent correlation weight (Iadj) maintains a finite value. The connected replica-overlap width approaches a numerical limit.

Networks where dynamical variables reside on links rather than nodes are under investigation. Most statistical models assign these variables to network vertices however, this framework explores assigning them directly to edges. Each node describes a multidimensional issue in a local frame and an oriented link carries a transformation Uij ∈G that maps information from one frame to another; physical observables must remain invariant under local frame redefinitions hi ∈G, enforcing the transformation law Uij −→hiUijh−1 j. Gauge covariance thus follows from representational freedom within the network.

Gauge-invariant combinations of link transformations, which compare accumulated maps along alternative paths, are the physically meaningful quantities. The choice of G is determined by how relational transformations compose; communication can be sequence dependent, requiring non-Abelian groups such as SU for accurate modelling. Loop consistency reduces to structural balance on signed networks when G = Z2. An ordered product of link maps gives the accumulated transformation and two alternative paths are compatible when holonomy around a closed loop joining them is trivial.

At Fuyao University of Science and Technology, researchers aimed to determine what collective phase replaces global compatibility under quenched frustration. In conventional disordered systems, dense quenched frustration produces glassy locking, but it remains unclear if this occurs with geometrically constrained non-Abelian holonomies. Understanding these dynamics requires exploring how frustrated interactions influence network behaviour at larger scales.

They addressed this question using fluctuating SU relations on a complete simplicial 2-complex with random plaquette couplings. A gauge-invariant decomposition separates uniform compatibility, disorder-pinned local structure, connected correlations between loops, and replica locking. Increasing disorder drives loss of global compatibility into a correlated gauge liquid where integrated correlation weight survives while individual pair correlations are diluted; there is no evidence for thermodynamic replica-symmetry breaking.

This finding establishes that sustained connectivity can exist even as global compatibility diminishes in the network. Under independent changes of local frames’ hi ∈SU, these variables transform as Uij −→hiUijh−1 j. Plaquette holonomies are generated by shared links and do not represent themselves as independent degrees of freedom.

Positive Jf favors relational closure while negative values favour nontrivial holonomy; these couplings represent heterogeneity in consistency pressures on triangles.

This scaling keeps both coherent and random contributions finite as N →∞.

A continuous loss of global compatibility was investigated using parallel tempering simulations on an SU model residing on a complete simplicial 2-complex with quenched random plaquette couplings. Results show that MP diminishes as disorder increases within a region where these values remain stable, indicating correlated gauge liquid behaviour rather than glassy locking.

Furthermore, 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 become geometrically diluted but a finite integrated correlation weight survives. Scientists have demonstrated that Iadj remains approximately constant even as MP diminishes; this represents a departure from conventional disordered systems which typically exhibit either glassy locking or replica symmetry breaking. They observed how individual connections contribute to overall stability by measuring the mean squared covariance between adjacent plaquettes finding it enhanced alongside increased disorder but scaled inversely with system size, meaning total integrated correlation weight remained stable despite weakening pairwise correlations. Supporting this observation was an increase in local thermal variance corresponding to actively fluctuating liquid-like structure rather than frozen one.

The research demonstrates a continuous loss of global compatibility within a simulated system experiencing increasing disorder. This means that as randomness is introduced into the network’s couplings, its ability to maintain consistent relationships across different pathways diminishes with larger systems.

Researchers found that while uniform compatibility decreased, the integrated adjacent correlation weight remained finite, indicating behaviour distinct from typical glassy materials or those exhibiting replica symmetry breaking. The study used simulations on an SU(2) model residing on a complete simplicial 2-complex and suggests this results in a non-glassy correlated gauge liquid where connections are diluted but overall stability persists.

👉 More information
🗞 Frustration without Glass in Non-Abelian Simplicial Networks
✍️ Xuanhua Wang
🧠 ArXiv: https://arxiv.org/abs/2608.17817

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