Hardcore bosons on the dual of the bowtie lattice

We study the zero temperature phase diagram of hardcore bosons on the dual of the bowtie lattice. Two types of striped diagonal long-range order (striped order I and striped order II) are discussed. A state with type-II striped order and superfluidity is found, even without nearest-neighbor repulsion. The emergence of such a state is due to the inhomogeneity and the anisotropy of the lattice structure. However, neither the translational symmetry nor the symmetry between sublattices of the original lattice is broken. In this paper, we restrict a 'solid state' of lattice bosons as a diagonal long-range ordered state breaking either the translational symmetry of the original lattice or the symmetry of different sublattices. We thus name such a phase a striped superfluid phase (SSF). In the presence of a nearest-neighbor repulsion, we find two striped charge density wave phases(SCDW I and II) with boson density $ρ=1/2$ (with striped order I) and $ρ=2/3$ (with striped order II) respectively, when the hopping amplitude is small compared with the repulsion. The SCDW I state is a solid, in which the translational symmetry of the original lattice is broken. We observe a rather special first-order phase transition showing an interesting multi-loop hysteresis phenomenon between the two SCDW phases when the hopping term is small enough. This can be accounted for by the special degeneracy of the ground states near the classical limit. The SSF re-appears outside the two SCDW phases. The transition between the SCDW I and SSF phases is first order, while the transition between SCDW II and SSF phases is continuous. We find that the superfluid stiffness is anisotropic in the SSF states with and without repulsion. In the SSF with repulsion, the superfluid stiffness is subject to different types of anisotropy in the region near half filling and above 2/3-filling.