Validate the fusion-spectrum diagnostic for many-body phase transitions

Establish whether redistribution of the bipartite fusion-matrix singular-value spectrum provides a reliable diagnostic of many-body phase transitions without requiring phase-specific physical observables, beyond the finite-size bosonic and fermionic benchmarks presented.

Background

The paper proposes the singular-value spectrum of a bipartite fusion matrix as an internal diagnostic of structural changes in optimized HyperDet fusion tensors. In the two-parton bosonic case, a dominant singular value is associated with a factorizable fusion tensor, while additional components emerge in neighboring superfluid phases. In the three-parton fermionic case, the spectrum changes across candidate anomalous Hall crystal–FCI and FCI–charge-density-wave boundaries, but its relation to tensor factorizability is less direct because the increased tensor order prevents factorizability from being inferred from singular-value strengths alone.

The authors therefore explicitly frame the broader interpretation of fusion-structure redistribution as a conjecture requiring validation beyond the studied examples and finite clusters.