Supersymmetry breakdown for an extended version of the Nicolai supersymmetric fermion lattice model

Sannomiya-Katsura-Nakayama have recently studied supersymmetry breakdown for an extension of the Nicolai supersymmetric fermion lattice model. This model called the extended Nicolai model is parameterized by an adjustable constant $g\in \mathbb{R}$ in its defining supercharge, and satisfies ${{\cal{N}}}=2$ supersymmetry. We show that for any non-zero $g$ the extended Nicolai model breaks supersymmetry dynamically, and the energy density of any homogeneous ground state (that breaks supersymmetry) is strictly positive.