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We consider $N$-particle generalizations of $\eta$-pairing states in a chainof $N$-component fermions and show that these states are exact (high-energy)eigenstates of an extended SU($N$) Hubbard model. We compute the singletcorrelation function of the states and find that its behavior is qualitativelydifferent for even and odd $N$. When $N$ is even, these states exhibitoff-diagonal long-range order in $N$-particle reduced density matrix. On theother hand, when $N$ is odd, the correlations decay exponentially with distancein the bulk, but end-to-end correlations do not vanish in the thermodynamiclimit. Finally, we prove that these states are the unique ground states ofsuitably tailored Hamiltonians.

Exact eigenstates of extended SU($N$) Hubbard models: Generalizations of $\eta$-pairing states with $N$-particle off-diagonal long-range order