Title: Fundamental role of nonlocal orders in 1D Extended Bose-Hubbard Model

Abstract: Nonlocal order parameters capture the presence of correlated fluctuations between specific degrees of freedom, in otherwise disordered quantum matter. Here we provide a further example of their fundamental role, deriving the ground state phase diagram of the filling one extended Bose Hubbard model exclusively in terms of their ordering. By means of a density matrix renormalization group numerical analysis, we show that besides the (even) parity order characteristic of the Mott insulating phase, and the string order non vanishing in the Haldane insulator, the recently proposed odd parity order completes the picture, becoming nonvanishing at the transition from the normal superfluid to the paired superfluid phase. The above three nonlocal parameters capture all the distinct phases, including the density wave phase which local order is seen as the simultaneous presence of correlated fluctuations in different channels. They provide a unique tool for the experimental observation of the full phase diagram of strongly correlated quantum matter, by means of local density measurements.