  As a continuation of our previous work, we derive the optimal flux phase
which minimizes the ground state energy in the one-dimensional many particle
systems, when the number of particles is odd in the absence of on-site
interaction and external potential. Moreover, we study the relationship between
the flux on the ring and the spin of the ground state through which we derive
some information on the sum of the lowest eigenvalues of one-particle
Hamiltonians.
