Stability of the spectral gap under a small half‑plane chemical potential step

Establish that for any short-range interacting lattice fermion Hamiltonian H0 on Z^2 possessing a locally unique gapped ground state (in the sense of inequality (1) defining a gap g>0), the spectral gap persists under the perturbation by a small half‑plane number operator Λ1; that is, prove that there exists ε0>0 such that for all |ε|<ε0 the perturbed Hamiltonian Hε := H0 + ε Λ1 has a gapped ground state. This addresses the general interacting case beyond non‑interacting and weakly interacting regimes, for which gap stability follows from known results.

Background

The charge pump approach in prior work assumes that along the adiabatic path H_{ε,η}(t) = H0 + f(η t)Λ1 the Hamiltonian remains gapped for all times. The authors argue that one expects the gap to remain open when adding a small chemical potential step Λ1 to a gapped H0, but they note that a general proof is lacking.

For non‑interacting systems, the statement follows directly from perturbation theory at the one‑body level, and for weakly interacting systems it follows from established spectral gap stability results. Extending this to general interacting gapped H0 would remove an assumption used in the charge pump proofs and would align the rigor of that approach with the NEASS framework, which does not rely on the gap‑persistence assumption.