Molecular Hubbard Hamiltonian: Field regimes and molecular species

The molecular Hubbard Hamiltonian (MHH) naturally arises for ultracold ground-state polar alkali-metal dimer molecules in optical lattices. We show that, unlike ultracold atoms, different molecules display different many-body phases due to intrinsic variances in molecular structure even when the molecular symmetry is the same. We also demonstrate a wide variety of experimental controls on ${}^1\Sigma$ molecules via external fields, including applied static electric and magnetic fields, an ac microwave field, and the polarization and strength of optical lattice beams. We provide explicit numerical calculations of the parameters of the MHH, including tunneling and direct and exchange dipole-dipole interaction energies, for the molecules ${}^6$Li${}^{133}$Cs, ${}^{23}$Na${}^{40}$K, ${}^{87}$Rb${}^{133}$Cs, ${}^{40}$K${}^{87}$Rb, and ${}^6$Li${}^{23}$Na in weak and strong applied electric fields. As case studies of many-body physics, we use infinite-size matrix product state methods to explore the quantum phase transitions from the superfluid phase to half-filled and third-filled crystalline phases in one dimension.

Figure 1

Examples of experimental controls not available to atoms. An applied electric field $E_{\mathrm{dc}}$ gives rise to single-particle states with tunable expected dipole moments. Vertical dashed lines indicate the position of $E_{\mathrm{dc}}=5$ kV/cm for the various molecules. The expected transition dipole moments, giving rise to exchange interactions, are significantly different in the presence of an optical lattice ($d_{1,12}$, magenta short-dashed line) and in free space ($d_{1,12}^{\mathrm{RR}}$, dashed blue line).

Figure 2

Phase diagram of the MHH in the absence of an ac field. (a) Density phase diagram for LiCs, showing the $\rho =1/2$ crystalline phase (CP) with nonvanishing single-particle gap. The dashed line is a guide to the eye for the CP. (b),(c) Analogs of (a) for NaK and RbCs, respectively. (d) Density correlation function $\mathcal{N}\left(r\right)$ with period-2 oscillation removed for the points marked in (a). (e) Single-particle density matrix $\mathcal{A}\left(r\right)$ for the points marked in (a). (f) Density phase diagram for LiCs near the $\rho =1/3$ CP transition at fixed $E_{\mathrm{dc}}=3.5$ kV/cm showing small density fluctuations induced by our use of the grand canonical ensemble.

Figure 3

Scaling of single-particle density matrix with entanglement cutoff in iMPS. The single-particle density matrix $\mathcal{A}\left(r\right)$ computed by iMPS for $\chi =12$ (red solid line), 16 (green dashed line), 20 (blue dotted line), and 40 (purple fine-dashed line) in the crystalline and superfluid phases. In the crystalline phase, $\mathcal{A}\left(r\right)$ decays exponentially and so the results converge for finite $\chi$. In the superfluid phase $\mathcal{A}\left(r\right)$ has an algebraic decay, and the range over which this decay is well approximated increases with $\chi$.