  Recent studies have suggested a new phase in the extended Bose-Hubbard model
in one dimension at integer filling [1,2]. In this work, we show that this new
phase is phase-separated into a supersolid and superfluid part, generated by
mechanical instability. Numerical simulations are performed by means of the
density matrix renormalization group algorithm in terms of matrix product
states. In the phase-separated phase and the adjacent homogeneous superfluid
and supersolid phases, we find peculiar spatial patterns in the entanglement
spectrum and string-order correlation functions and show that they survive in
the thermodynamic limit. In particular, we demonstrate that the elementary
excitations of the homogeneous superfluid with enhanced periodic modulations
are phonons, find the central charge to be $c=1$, and show that the velocity of
sound, extracted from the intrinsic level splitting for finite systems, matches
with the propagation velocity of local excitations in dynamical simulations.
This suggests that the low-energy spectrum of the phase under investigation is
effectively captured by a spinless Luttinger liquid, for which we find
consistent results between the Luttinger parameter obtained from the linear
dependence of the structure factor and the algebraic decay of the one-body
density matrix.
