Characterize hopping operators for general Hilbert spaces

Determine what conditions, beyond the standard cochain-form Hilbert space and operator ansatz, single out hopping operators for topological excitations in a general Hilbert space.

Background

The paper proves that, within the standard cochain ansatz, a family of hopping operators satisfies the topological-excitation axioms precisely when its elementary operators commute with the relevant global symmetry. This result depends on the specific form of the Hilbert space and operators assumed in Eq. (the ansatz).

The unresolved issue is to identify an intrinsic criterion that applies to general Hilbert spaces and operators, rather than only to the cochain-based constructions analyzed in the paper. Such a criterion would clarify how hopping operators should be distinguished from other local or symmetry-related operators in more general lattice systems.

References

Although this statement applies to arbitrary $H{d+2}(BqG,)$, it still assumes that the Hilbert space and operators have the standard form in Eq.~eq: ansatz. For a general Hilbert space, we do not know what conditions should single out the hopping operators.

eq: ansatz:

U(s)vX=exp⁡(2π∫XL(s,v))v+sX,s,v∈Cq−1(X,G),U(s) v_X =\exp\left(2\pi\int_XL(s,v)\right){v+s}_X, \qquad s,v\in C^{q-1}(X,G),

— Statistics, 't Hooft Anomaly, and the Else-Nayak Index: a careful comparison of concepts  (2609.20813 - Xue, 17 Sep 2026) in Section Conclusion, paragraph beginning “Several aspects of this work remain incomplete”