The SU(N) Fermi-Hubbard Model on two sites: Bethe Ansatz solution and Quantum Phase Transition of the Lipkin-Meshkov-Glick Model in the large-N limit

TL;DR

This paper provides an exact solution for the two-site SU(N) Fermi-Hubbard model using Bethe ansatz, reveals a quantum phase transition akin to the Lipkin-Meshkov-Glick model, and discusses potential experimental realizations with ultracold atoms.

Contribution

It establishes a connection between the SU(N) Fermi-Hubbard model and the Richardson and LMG models, deriving exact solutions and identifying a quantum phase transition at finite N.

Findings

Quantum phase transition at U_c=-1/(2N)

Finite-size energies and gap reveal the transition

Entanglement entropy scales with N at the transition

Abstract

We show that the SU(N) 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 SU(N) 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 SU(N) FHM with N particles. In particular, the LMG second-order quantum phase transition (QPT) is shown to occur in the SU(N) FHM for an attractive density-density interaction U equal to U_c=-1/(2N), 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 SU(N) 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.