Mott-insulating phases of the Bose-Hubbard model on quasi-1D ladder lattices

Abstract

We calculate the phase diagram of the Bose-Hubbard model on a half-filled ladder lattice including the effect of finite on-site interactions. This shows that the rung-Mott insulator (RMI) phase persists to finite interaction strength, and we calculate the RMI-superfluid phase boundary in the thermodynamic limit. We show that the phases can still be distinguished using the number and parity variances, which are observables accessible in a quantum-gas microscope. Phases analogous to the RMI were found to exist in other quasi-1D lattice structures, with the lattice connectivity modifying the phase boundaries. This shows that the the presence of these phases is the result of states with one-dimensional structures being mapped onto higher dimensional systems, driven by the reduction of hopping rates along different directions.

Figures (9)

• Figure 1: (a) Geometry of the two-leg ladder. Hopping along each chain occurs at rate $J$ (blue links), while the inter-chain hopping across the rungs at rate $J_\perp$ (red links). In the HCB limit ($U\rightarrow\infty$) doubly occupied sites are prohibited. (b) In the RMI phase each boson is delocalized across the two sites of a rung but localized along the chains, leading to insulating behavior. This configuration is analogous to the MI phase of a fully filled single chain with an effective hopping amplitude $2J$.
• Figure 2: Phase diagrams showing (a) the energy gap, $\Delta E$, and (b) the normalized correlation length, $\xi_\text{chain}$, along the chains for a ladder with $\rho=0.5$ and $L=40$. The hopping correlation function, $\Gamma_\text{chain}$, for (c) $U/J=1$ and $J_\perp/J=15$, (d) $U/J=15$ and $J_\perp/J=0.5$, (e) $U/J=J_\perp/J=15$. The colored markers show the location of these parameters in the phase diargam of panel (b).
• Figure 3: (a) Phase diagram in the thermodynamic limit for the half-filled ladder lattice. The RMI (white) and SF (checked) phases are highlighted. The phase boundary (solid, black line) includes errorbars showing thedeviation from results obtained with a more lenient cutoff [App. \ref{['app:Scaling']}]. The approximate boundary from Eq. \ref{['eq:boundary_intext']} is shown as a red dashed line. (b) Example scaling analysis for $J_\perp/J=15$. The critical point was derived by comparing the data for different values of $L$. The extracted value is marked with a black vertical line while the black dashed borders display the error in this estimate.
• Figure 4: (a) Population density $\langle n_\text{rung}\rangle$ and (b) corresponding population variances $\kappa$ and $\kappa_\text{rung}$ for $U/J=J_\perp/J=15$. The value of $\kappa$ is calculated using only the first chain. (c) Population variances for $U/J=1$ and $J_\perp/J=15$.
• Figure 5: Average (a) on-site and (b) rung number variances, $\kappa$ and $\kappa_\text{rung}$, along with the corresponding parity variances, $\sigma$ and $\sigma_\text{rung}$, calculated for a half-filled ladder ($L=40$) at $J_\perp/J=10$. The dashed black line is located at the critical point, calculated via scaling analysis. (c) Same quantities in the HCB limit, where the on-site variances become identical, while the rung variances are distinguishable for $J_\perp\lesssim 4J$.