The impact of lattice geometry on commensurate and incommensurate ultracold gases*

We consider the behaviour of the Bose-Hubbard model on a variety of quasi-1D ladder lattices. For hardcore bosons at half-filling a phase known as the Rung-Mott Insulator (RMI) has been theoretically predicted on regular 2-rung ladders. We show that the RMI persists for a range of softcore interaction strengths. A phase diagram in the thermodynamic limit is obtained via scaling analysis, finding the expected critical points. After studying the relevant properties, we observed phases analogous to the RMI for different quasi-1D ladder geometries, underlining a connection between these type of insulators and the underlying lattice geometry and its commensurability. Modifying the lattice geometry and commensurability accordingly, topological phases can be also obtained, leading to possible applications in information transport. We show how observables which can be easily measured in a quantum gas microscope with single-atom resolution can allow an experimental probe into how lattice geometry and incommensurability lead to exotic phases of matter.