  We investigate Lefschetz thimble structure of the complexified
path-integration in the one-dimensional lattice massive Thirring model with
finite chemical potential. The lattice model is formulated with staggered
fermions and a compact auxiliary vector boson (a link field), and the whole set
of the critical points (the complex saddle points) are sorted out, where each
critical point turns out to be in a one-to-one correspondence with a singular
point of the effective action (or a zero point of the fermion determinant). For
a subset of critical point solutions in the uniform-field subspace, we examine
the upward and downward cycles and the Stokes phenomenon with varying the
chemical potential, and we identify the intersection numbers to determine the
thimbles contributing to the path-integration of the partition function. We
show that the original integration path becomes equivalent to a single
Lefschetz thimble at small and large chemical potentials, while in the
crossover region multi thimbles must contribute to the path integration.
Finally, reducing the model to a uniform field space, we study the relative
importance of multiple thimble contributions and their behavior toward
continuum and low-temperature limits quantitatively, and see how the rapid
crossover behavior is recovered by adding the multi thimble contributions at
low temperatures. Those findings will be useful for performing Monte-Carlo
simulations on the Lefschetz thimbles.
