Quantum simulation of the Hubbard model: The attractive route

We study the conditions under which, using a canonical transformation, the phases sought after for the repulsive Hubbard model, namely, a Mott insulator in the paramagnetic and antiferromagnetic phases, and a putative $d$-wave superfluid can be deduced from observations in an optical lattice loaded with a spin-imbalanced ultracold Fermi gas with attractive interactions, thus realizing the attractive Hubbard model. We argue that the Mott insulator and antiferromagnetic phase of the repulsive Hubbard model are easier to observe in the attractive Hubbard mode as a band insulator of Cooper pairs and superfluid phase, respectively. The putative $d$-wave superfluid phase of the repulsive Hubbard model doped away from half filling is related to a $d$-wave antiferromagnetic phase for the attractive Hubbard model. We discuss the advantages of this approach to “quantum simulate” the Hubbard model in an optical lattice over the simulation of the doped Hubbard model in the repulsive regime. We also point out a number of technical difficulties of the proposed approach and, in some cases, suggest possible solutions.

Figure 1

(Color online) Schematic of how the transformation of Eq. (4) works. In (a) we sketch how it acts on a single site (e.g., red stands for spin up and blue for spin down) and in (b) the way it transforms different types of states on a uniform optical lattice. In addition to the phases depicted above, an antiferromagnetic state at $U>0$ ordered along the $x$ or $y$ direction corresponds to a superfluid phase of fermion pairs. The bottom diagram illustrates one of the main points of this paper, namely, that doping the attractive Hubbard model corresponds to introducing spin imbalance in the repulsive Hubbard model.

Figure 2

Density profile for the repulsive Hubbard model in a harmonic trap (model R2) as a function of the radial distance from the trap center $r$ in units of $a$, the optical lattice spacing. Dashed line: $T/U=0.05$, solid line: $T/U=0.1$, and dotted-dashed line: $T/U=0.2$. The total number of particles is $N=6599$. The trapping energy ${ϵ}_0/2U=0.0003$.

Figure 3

Density profile for the attractive Hubbard model in a harmonic trap (model A1 at $h=0$) as a function of the radial distance from the trap center. Dashed line: $T/U=0.05$, solid line: $T/U=0.1$, and dotted-dashed line: $T/U=0.2$. The total number of particles is $N=11728$. The trapping energy ${ϵ}_0/2U=0.0003$.

Figure 4

(Color online) Schematic of the shell structure of (a) the half-filled repulsive Hubbard model in a site-dependent Zeeman field (model R1) and (b) the corresponding attractive model A1 state configuration in a trap.

Figure 5

Density (grayscale) plot of the noise correlation for the 2D $d_{x^2-y^2}$-wave AFM against $k_x$ and $k_y$ when $k^{\prime }=k+Q$. Parameters used: $g_Q{M}_Q=0.10$, $\mu =0$, and $h=0.23$.