  We study a model of interacting spinless fermions in a one dimensional
lattice with supersymmetry (SUSY). The Hamiltonian is given by the
anti-commutator of two supercharges $Q$ and $Q^\dagger$, each of which is
comprised solely of fermion operators and possesses one adjustable parameter
$g$. When the parameter $g$ vanishes, the model is identical to the one studied
by Nicolai [H. Nicolai, J. Phys. A: Math. Gen. \textbf{9}, 1497 (1976)], where
the zero-energy ground state is exponentially degenerate. On the other hand, in
the large-$g$ limit the model reduces to the free-fermion chain with a
four-fold degenerate ground state. We show that for finite chains SUSY is
spontaneously broken when $g > 0$. We also rigorously prove that for
sufficiently large $g$ the ground-state energy density is nonvanishing in the
infinite-volume limit. We further analyze the nature of the low-energy
excitations by employing various techniques such as rigorous inequalities,
exact numerical diagonalization, and renormalization group method with
bosonization. The analysis reveals that the low-energy excitations are
described by massless Dirac fermions (or Thirring fermions more generally),
which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY
breaking.
