Exploring Unconventional Hubbard Models with Doubly Modulated Lattice Gases

Recent experiments show that periodic modulations of cold atoms in optical lattices may be used to engineer and explore interesting models. We show that double modulation combining lattice shaking and modulated interactions allows for the engineering of a much broader class of lattice with correlated hopping, which we study for the particular case of one-dimensional systems. We show, in particular, that by using this double modulation it is possible to study Hubbard models with asymmetric hopping, which, contrary to the standard Hubbard model, present insulating phases with both parity and string order. Moreover, double modulation allows for the simulation of lattice models in unconventional parameter regimes, as we illustrate for the case of the spin-$1/2$ Fermi-Hubbard model with correlated hopping, a relevant model for cuprate superconductors.

Figure 1

(top) Infinite DMRG [43] results for the effective model (2) (with four bosons per site, keeping up to $M=200$ states) for ${\mathcal{O}}_P^2$ (solid), $4{\mathcal{O}}_S^2$ (dashed), and $S_{vN}$ (dot dashed) as a function of $F_1/\hslash \omega$ for $U_0/J=1.2$, unit filling, and $U_1/\hslash \omega =3$. In the shaded SF regions, we find a logarithmic divergence of $S_{vN}$ with the number of kept matrix states $M$. (bottom) Relevant hopping rates.

Figure 2

Dynamics of ${\mathcal{O}}_P^2$ (solid) and ${\mathcal{O}}_S^2$ (dashed) after a sudden onset of DM with $F_1/\hslash \omega =2.4$, $U_1/\hslash \omega =3$ for a system initially prepared as a deep Mott insulator. The inset shows the dynamics for $U_0/J=1$. The curves are obtained using the effective model (2), whereas the crosses indicate the results directly obtained from Eq. (1). In the main figure we depict the time average of the orders for $2<Jt/\hslash <6$ as a function of $U_0$ for the same $F_1$ and $U_1$. The shaded regions indicate the variances of the orders associated to the dynamics after the quench. In the DMRG simulations of Eq. (1), we employ $L=60$ sites, $N=60$ particles, and $\hslash \omega =20\pi J$; the simulations of the effective model (2) are performed in an infinite scheme [43]. In both cases we keep up to $M=400$ matrix states.

Figure 3

Phase diagram for the Fermi-Hubbard model as a function of $\mu /t_{AA}$ and $t_{AB}/t_{AA}$ for $U_0=0$ (see text). The dashed line denotes values of $\mu$ for which the Luttinger-liquid parameter $K_{\rho }=1$ extracted from the long wavelength behavior of the static charge structure factor [53] for $t_{AA}\ne t_{AB}$. This is consistent with the perturbative result from Ref. [48] for $t_{AA}=t_{AB}$. The size for the BOW phase corresponds to the charge gap at unit filling. The shaded regions denote the vacuum or the fully occupied state. All results are extrapolated to the thermodynamic limit from open boundary DMRG calculations with up to $L=144$ sites.