Proposition: Restricted gaps can approach zero

On a two-qubit induced edge, let

These Hamiltonians have a unique ground vector and full gap at least $3/4$. The restricted Hamiltonian consisting of terms supported entirely in $\{a\}$ has gap $ϵ$.

Proof. The first operator has eigenvalues $0,1,1,1$, and the perturbation is positive with norm $ϵ$. By the min–max principle, the second eigenvalue of $H_{ϵ}$ is at least one, while the first is at most $ϵ$. They are separated by at least $1-ϵ\ge 3/4$, which also proves uniqueness. The restriction is $ϵ|1⟩⟨1{|}_a$.