  The Lieb-Liniger model describes one-dimensional bosons interacting through a
repulsive contact potential. In this work, we introduce an extended version of
this model by replacing the contact potential with a decaying exponential.
Using the recently developed continuous matrix product states techniques, we
explore the ground state phase diagram of this model by examining the
superfluid and density correlation functions. At weak coupling superfluidity
governs the ground state, in a similar way as in the Lieb-Liniger model.
However, at strong coupling quasi-crystal and super-Tonks-Girardeau regimes are
also found, which are not present in the original Lieb-Liniger case. Therefore
the presence of the exponentially-decaying potential leads to a
superfluid/super-Tonks-Girardeau/quasi-crystal crossover, when tuning the
coupling strength from weak to strong interactions. This corresponds to a
Luttinger liquid parameter in the range $K \in (0, \infty)$; in contrast with
the Lieb-Liniger model, where $K \in [1, \infty)$, and the screened long-range
potential, where $K \in (0, 1]$.
