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Physical Review A

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

Pierre Nataf

Published Feb 10, 2025• 3 citations
Publisher

Abstract

The authors find that the SU($N$) Fermi-Hubbard model on two sites can be mapped to the Lipkin-Meshkov-Glick model. It admits a Bethe ansatz solution and exhibits a quantum phase transition in the large-$N$ limit that could be experimentally accessible using current technologies involving SU($N$) ultracold atoms or molecules.

DOI
10.1103/physreva.111.l020201

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The authors find that the SU(NNN) Fermi-Hubbard model on two sites can be mapped to the Lipkin-Meshkov-Glick model. It admits a Bethe ansatz solution and exhibits a quantum phase transition in the large-NNN limit that could be experimentally accessible using current technologies involving SU(NNN) ultracold atoms or molecules.
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.
We prove that a special Lipkin–Meshkov–Glick model is quasi-exactly solvable with solutions that can be expressed in the SU (2) coherent state form. Ground-state properties of the model are studied analytically. We also show that the model reduces to the standard two-site Bose–Hubbard model in the large- N limit for finite U / t or large ( N − 1)| U |/ t cases with finite N , which proves that in these cases the ground state of the standard two-site Bose–Hubbard model is an SU (2) coherent state.
The one-dimensional Bose-Hubbard model in large-UUU limit has been studied via reducing and mapping the Hamiltonian to a simpler one. The eigenstates and eigenvalues have been obtained exactly in the subspaces with fixed numbers of single- and double-occupancies but without multiple-occupancies, and the thermodynamic properties of the system have been calculated further. These eigenstates and eigenvalues also enable us to develop a new perturbation treatment of the model, with which the ground-state energy has been calculated exactly to first order in 1/U1/U1/U.
It is shown that the two component Fermi or Bose many-body Hamiltonian, such as the two-orbit fermion pairing and two-site Bose-Hubbard model with arbitrary finite higher order terms can always be solved exactly by using Bethe ansatz vector construction based on the permutation of two components of bosons or fermions involved. As examples of the solution, the extended one-dimensional dimer Bose -Hubbard model with multi-body interactions and the mean-fifield plus orbit-dependent non-separable pairing model with two non-degenerate j-orbits are demonstrated with the eigenstates and the eigen-energy and the related Bethe ansatz equations. It is shown that the main feature of the solutions lies in the fact that the Bethe ansatz vectors can be expressed in terms of binomials of the boson or fermion operators times the related symmetric functions. As the consequence, two-component quantum many-body systems , such as the extended Lipkin-Meshkov-Glick model with higher-order interactions, can be solved in a similar way.
We show that the Lipkin-Meshkov-Glick 2N-fermion model is a particular case of one-spin Gaudin-type model in an external magnetic field corresponding to a limiting case of non-skew-symmetric elliptic r-matrix and to an external magnetic field directed along one axis. We propose an exactly-solvable generalization of the Lipkin-Meshkov-Glick fermion model based on the Gaudin-type model corresponding to the same r-matrix but arbitrary external magnetic field. This model coincides with the quantization of the classical Zhukovsky-Volterra gyrostat. We diagonalize the corresponding quantum Hamiltonian by means of the modified algebraic Bethe ansatz. We explicitly solve the corresponding Bethe-type equations for the case of small fermion number N=1,2.
Simple NNN-scaling above the superexchange energy of thermodynamic observables in the SU(NNN) Fermi Hubbard Model at 1/N1/N1/N filling in the two dimensional square lattice