$d$-wave correlated Bose liquid phases on multi-leg ladders with ring exchange

We discuss recent progress on the study of ladder descendants of a novel two-dimensional quantum phase of bosons moving on the square lattice which is characterized by singular surfaces in momentum space, namely the $d$-wave correlated Bose liquid (DBL). Using a combination of numerics (e.g., density matrix renormalization group, variational Monte Carlo, and exact diagonalization) and analytics (e.g., bosonization of a compact U(1) lattice gauge theory) we explore the existence and stability of ladder analogs of the DBL on $N$-leg ladders, with $N\ge3$, in the context of a model of itinerant hard-core bosons with frustrating four-site ring exchange. As in the case of $N=2$, see [1], we find numerical evidence for various strong-coupling DBL phases which can rather remarkably be understood within a slave-fermion picture in which the boson wave function is written as a product of two Slater determinants. The additional features and difficulties associated with taking $N>2$ will be addressed. The boson ring model we consider has potential physical realizations in the contexts of low-dimensional frustrated quantum magnets and in ultracold quantum gases. [1] D. N. Sheng \emph{et al.}, Phys. Rev. B {\bf 78}, 054520 (2008).