We show that the $\mathrm{SU}\left(N\right)$ Fermi-Hubbard model (FHM) on two sites, where $N$ is the number of flavors of each fermion, corresponds to an exactly solvable two-level many-boson model that Richardson [J. Math. Phys. 9, 1327 (1968)] analytically solved long ago. We express the Bethe ansatz solutions as a function of the physical parameters of the $\mathrm{SU}\left(N\right)$ FHM, and recast its eigenvalues and eigenstates in terms of the Richardson pair energies and creation operators. In this context, the connection with the well-studied Lipkin-Meshkov-Glick (LMG) model, known as equivalent to the Richardson model, is established and serves as a guideline to the prediction of some $N$-body physics phenomena in the two-site $\mathrm{SU}\left(N\right)$ FHM with $N$ particles. In particular, the LMG second-order quantum phase transition (QPT) is shown to occur in the $\mathrm{SU}\left(N\right)$ FHM for an attractive density-density interaction $U$ equal to $U_c=-1/\left(2N\right)$, in units of the (absolute value of the) tunneling amplitude between the two sites. We show the finite-size energies, the gap, and the kinetic energy, which all reveal the transition, as a function of $U$ for values of $N$ from $N=3$ to $N=36$, suggesting that the QPT could be experimentally achieved with current technologies involving $\mathrm{SU}\left(N\right)$ ultracold atoms or molecules. Finally, we show the entanglement entropy of the first site with respect to the second, and it scales like $N$ at the transition, in contrast with several two-mode models.

Close

PrevNext

View figure in article

• Cold and ultracold molecules
• Nuclear many-body theory
• Quantum phase transitions
• Quantum many-body systems
• Bethe ansatz
• Continuous symmetries in condensed matter
• Exact solutions for many-body systems
• Holstein-Primakoff method
• Hubbard model
• Large-N expansion in field theory
• Models based on symmetries