  The parent compounds of iron-based superconductors are magnetically-ordered
bad metals, with superconductivity appearing near a putative magnetic quantum
critical point. The presence of both Hubbard repulsion and Hund's coupling
leads to rich physics in these multiorbital systems, and motivated descriptions
of magnetism in terms of itinerant electrons or localized spins. The
NaFe$_{1-x}$Cu$_x$As series consists of magnetically-ordered bad metal ($x=0$),
superconducting ($x\approx0.02$) and magnetically-ordered
semiconducing/insulating ($x\approx0.5$) phases, providing a platform to
investigate the connection between superconductivity, magnetism and electronic
correlations. Here we use X-ray absorption spectroscopy and resonant inelastic
X-ray scattering to study the valence state of Fe and spin dynamics in two
NaFe$_{1-x}$Cu$_x$As compounds ($x=0$ and 0.47). We find that magnetism in both
compounds arises from Fe$^{2+}$ atoms, and exhibits underdamped dispersive spin
waves in their respective ordered states. The dispersion of spin excitations in
NaFe$_{0.53}$Cu$_{0.47}$As is consistent with being quasi-one-dimensional.
Compared to NaFeAs, the band top of spin waves in NaFe$_{0.53}$Cu$_{0.47}$As is
slightly softened with significantly more spectral weight of the spin
excitations. Our results indicate the spin dynamics in
NaFe$_{0.53}$Cu$_{0.47}$As arise from localized magnetic moments and suggest
the iron-based superconductors are proximate to a correlated insulating state
with localized iron moments.
