  In the absence of frustration, interacting bosons in the ground state exist
either in the superfluid or insulating phases. Superfluidity corresponds to
frictionless flow of the matter field, and in optical conductivity is revealed
through a distinct $\delta$-functional peak at zero frequency with the
amplitude known as the Drude weight. This characteristic low-frequency feature
is instead absent in insulating phases, defined by zero static optical
conductivity. Here we demonstrate that bosonic particles in disordered one
dimensional, $d=1$, systems can also exist in a conducting, non-superfluid,
phase when their hopping is of the dipolar type, often viewed as short-ranged
in $d=1$. This phase is characterized by finite static optical conductivity,
followed by a broad anti-Drude peak at finite frequencies. Off-diagonal
correlations are also unconventional: they feature an integrable algebraic
decay for arbitrarily large values of disorder. These results do not fit the
description of any known quantum phase and strongly suggest the existence of a
novel conducting state of bosonic matter in the ground state.
