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

Establish that for any interacting lattice-fermion Hamiltonian H₀ on ℤ² with a locally unique gapped ground state, the perturbed Hamiltonian H_ε = H₀ + εΛ₁ (where Λ₁ is the number operator of the right half-plane) also has a gapped ground state for all sufficiently small ε > 0; equivalently, prove that the spectral gap remains open under this potential-step perturbation for general gapped H₀.

Background

The paper compares two approaches to defining Hall conductance in interacting fermion systems: the NEASS approach on infinite lattices and the charge pump approach on finite tori. The charge pump proofs assume that a spectral gap remains open during the adiabatic cycle for the time-dependent Hamiltonian H_{ε,η}(t) = H₀ + f(ηt)Λ₁. The authors argue that, for the specific perturbation Λ₁ (a half-plane potential step), one expects the gap to remain open for sufficiently small ε and show that this is straightforward in the non-interacting case and follows for weakly interacting systems from gap stability results. However, a general proof for arbitrary gapped H₀ is not available.

Clarifying gap stability under the half-plane step perturbation would remove an assumption in charge pump-based proofs and further align the charge pump and NEASS frameworks without relying on additional conditions. It would also strengthen the theoretical foundations for linear response in general gapped interacting systems beyond special cases.