Exact Eigenvalues of the Constant-Coupled One-Band Model with a Finite Number of Sites

The exact eigenvalues (for ground and excited states) of the constant-coupled one-band model for $N_f$ fermions on $N$ sites were obtained. Both electrons ($T<\mathrm{}0\mathrm{}$, where $T$ is the coefficient of the term representing the "hopping energy") and holes ($T>\mathrm{}0\mathrm{}$) were considered, and results were obtained for $2\le N_f\le N$ and $N=2, 3, \mathrm{and} 4$. Only the $N_f=2\mathrm{}$, $N=3\mathrm{}$ and $N_f=3\mathrm{}$, $N=4\mathrm{}$ with $T>\mathrm{}0\mathrm{}$ cases gave a purely ferromagnetic ground state. These cases also showed a ferromagnetic state at higher energy. The eigenvalues were calculated as a function of $\frac{U}{T}$, where $U$ represents the Coulomb energy of repulsion for electrons on the same site. The total spins of the states were obtained from their splitting in a magnetic field. The hopping term was found to be dominant for $\left|\frac{U}{T}\right|$ as large as one. The calculations support the suggestion that the ground state is usually antiferromagnetic.