Remark: Conventions are part of the theorem

The one-excitation hopping amplitude is $J$, because of the factor $1/2$ in (eq:H). With ${L_k}^{\mathrm{d}}=\sqrt{{\gamma }_{\mathrm{d}}/2}{Z}_k$, an isolated qubit's dephasing rate is ${\gamma }_{\mathrm{d}}$, while an intersite one-excitation coherence receives dephasing rate $2{\gamma }_{\mathrm{d}}$. Replacing this collapse operator by $\sqrt{{\gamma }_{\mathrm{d}}}{Z}_k$ doubles these dephasing contributions.

The relaxation vacuum is $|0⟩$, since $a=|0⟩⟨1|$. Software operator names do not substitute for checking this action.