  We show that 1D lattice models exhibit superdiffusive transport in the
presence of random "nodal impurities" in the absence of interaction. Here a
nodal impurity is defined as a localized state, the wave function of which has
zeros (nodes) in momentum space. The dynamics exponent $z$, a defining quantity
for transport behaviors, is computed to establish this result. To be specific,
in a disordered system having only nodal impurities, the dynamical exponent
$z=4n/(4n-1)$ where $n$ is the order of the node. If the system has time
reversal, the nodes appear in pairs and the dynamical exponent can be enhanced
to $z=8n/(8n-1)$. As $1<z<2$, both cases indicate superdiffusive transport.
