Phase diagram of hard-core bosons on a frustrated zig-zag ladder

We study hard-core bosons with unfrustrated nearest-neighbor hopping $t$ and repulsive interaction $V$ on a zig-zag ladder. As a function of the boson density $\rho$ and $V/t$, the ground state displays different quantum phases. A standard one-component Tomonaga-Luttinger liquid is stable for $\rho<1/3$ (and $\rho>2/3$) at any value of $V/t$. At commensurate densities $\rho=1/3$, $1/2$, and $2/3$ insulating (crystalline) phases are stabilized for a sufficiently large interaction $V$. For intermediate densities $1/3<\rho<2/3$ and large $V/t$, the ground state shows a clear evidence of a bound state of two bosons, implying gapped single-particle excitations but gapless excitations of boson pairs. Here, the low-energy properties may be described by a two-component Tomonaga-Luttinger liquid with a finite gap in the antisymmetric sector. Finally, for the same range of boson densities and weak interactions, the system is again a one-component Tomonaga-Luttinger liquid with no evidence of any breaking of discrete symmetries, in contrast to the frustrated case, where a ${\cal Z}_2$ symmetry breaking has been predicted.