Ground state of hard-core bosons in one-dimensional periodic potentials

With Girardeau's Fermi-Bose mapping, we find the exact ground states of hard-core bosons residing in a one-dimensional periodic potential. The analysis of these ground states shows that when the number of bosons N is commensurate with the number of wells M in the periodic potential, the boson system is a Mott insulator whose energy gap, however, is given by the single-particle band gap of the periodic potential; when N is not commensurate with M, the system is a metal (not a superfluid). In fact, we argue that there may be no superfluid phase for any one-dimensional boson system in terms of Landau's criterion of superfluidity. The Kronig-Penney potential is used to illustrate our results.