Spin of the ground state and the flux phase problem on the ring

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.