Recent advances in experimental techniques have made it possible to simulate multicomponent fermions with ultracold atoms. In particular, N-component fermions with SU(N) symmetry in optical lattices have attracted much attention because they are predicted to exhibit various exotic phases that do not appear in the SU(2) counterpart. These systems are well described by the SU(N) Hubbard model, but this model is notoriously difficult to analyze mathematically and new theoretical tools must be developed.
In this talk, I would like to present rigorous results for (i) the ground states of the attractive SU(N) Hubbard model [1] and (ii) N-body clustering eigenstates of the extended SU(N) Hubbard chain [2]. For (i), I present the results in the form of theorems on the degeneracy, the SU(N) quantum number, and the long-range order of the ground states. To prove the theorems, I make full use of a newly developed method of Majorana reflection positivity, inspired by recent progress in the sign-problem-free quantum Monte Carlo simulations. For (ii), we generalize η-pairing states, which are exact eigenstates of the SU(2) Hubbard model. When N is even, these states exhibit off-diagonal long-range order in N-particle reduced density matrix. On the other hand, when N is odd, the correlations decay exponentially with distance in the bulk, but end-to-end correlations do not vanish in the thermodynamic limit.
[1] Hironobu Yoshida and Hosho Katsura, Phys. Rev. Lett. 126, 100201 (2021)
[2] Hironobu Yoshida and Hosho Katsura, Phys. Rev. B 105, 024520 (2022)