  String bit models provide a possible method to formulate a string as a
discrete chain of pointlike string bits. When the bit number $M$ is large, a
chain behaves as a continuous string. We study the simplest case that has only
one bosonic bit and one fermionic bit. The creation and annihilation operators
are adjoint representations of the $U\left(N\right)$ color group. We show that
the supersymmetry reduces the parameter number of a Hamiltonian from 7 to 3
and, at $N=\infty$, ensures a continuous energy spectrum, which implies the
emergence of one spatial dimension. The Hamiltonian $H_{0}$ is constructed so
that in the large $N$ limit it produces a world sheet spectrum with one
Grassmann world sheet field. We concentrate on numerical study of the model in
finite $N$. For the Hamiltonian $H_{0}$, we find that the would-be ground
energy states disappear at $N=\left(M-1\right)/2$ for odd $M\leq11$. Such a
simple pattern is spoiled if $H$ has an additional term $\xi\Delta H$ which
does not affect the result of $N=\infty$. The disappearance point moves to
higher (lower) $N$ when $\xi$ increases (decreases). Particularly, the
$\pm\left(H_{0}-\Delta H\right)$ cases suggest a possibility that the ground
state could survive at large $M$ and $M\gg N$. Our study reveals that the model
has stringy behavior: when $N$ is fixed and large enough, the ground energy
decreases linearly with respect to $M$, and the excitation energy is roughly of
order $M^{-1}$. We also verify that a stable system of Hamiltonian $\pm
H_{0}+\xi\Delta H$ requires $\xi\geq\mp1$.
