  We consider the one-dimensional massive Thirring model formulated on the
lattice with staggered fermions and an auxiliary compact vector (link) field,
which is exactly solvable and shows a phase transition with increasing the
chemical potential of fermion number: the crossover at a finite temperature and
the first order transition at zero temperature. We complexify its
path-integration on Lefschetz thimbles and examine its phase transition by
hybrid Monte Carlo simulations on the single dominant thimble. We observe a
discrepancy between the numerical and exact results in the crossover region for
small inverse coupling $\beta$ and/or large lattice size $L$, while they are in
good agreement at the lower and higher density regions. We also observe that
the discrepancy persists in the continuum limit keeping the temperature finite
and it becomes more significant toward the low-temperature limit. This
numerical result is consistent with our analytical study of the model's thimble
structure. And these results imply that the contributions of subdominant
thimbles should be summed up in order to reproduce the first order transition
in the low-temperature limit.
