  We study the decay of superflow of a one-dimensional (1D) superfluid in the
presence of a periodic potential. In 1D, superflow at zero temperature can
decay via quantum nucleation of phase slips even when the flow velocity is much
smaller than the critical velocity predicted by mean-field theories. Applying
the instanton method to the O(2) quantum rotor model, we calculate the
nucleation rate of quantum phase slips $\Gamma$. When the flow momentum $p$ is
small, we find that the nucleation rate per unit length increases algebraically
with $p$ as $\Gamma/L \propto p^{2K-2}$, where $L$ is the system size and $K$
is the Tomonaga-Luttinger parameter. Based on the relation between the
nucleation rate and the quantum superfluid-insulator transition, we present a
unified explanation on the scaling formulae of the nucleation rate for
periodic, disorder, and single-barrier potentials. Using the time-evolving
block decimation method, we compute the exact quantum dynamics of the superflow
decay in the 1D Bose-Hubbard model at unit filling. From the numerical
analyses, we show that the scaling formula is valid for the case of the
Bose-Hubbard model, which can quantitatively describe Bose gases in optical
lattices.
