Harmonic potential effect on the ground state of the “g-e” model

Alkaline-earth fermionic atoms confined in one-dimensional optical lattices can be described by the so-called "g-e" model [Nat. Phys. 6, 289 (10)], which at half filling shows four interesting insulating phases (CDW, Orbital density wave (ODW), Spin Haldane (SH) and Spin-Peierls) [PRB 91, 075121(15)]. However, the confining potential has not been considered so far, for this reason we explore an extended "g-e" model where a harmonic potential for both ¹S₀ (g) and ³P₀ (e) atoms was considered. Using the density-matrix renormalization group method, we found that diverse insulating phase coexist in the ground state and that the spin Haldane and the spin Peierls phases disappear with the confining potential, meaning that these phases will not be observed in the experiments. The evolution of the CDW and the ODW phases around the middle of the trap was studied for different parameters. This work is motivated by recent experimental results about ¹⁷¹Yb atoms confined in optical lattices [Arxiv:1810.00536v1].