  Due to the different environments in the Milky Way's disk and halo, comparing
wide binaries in the disk and halo is key to understanding wide binary
formation and evolution. By using Gaia Early Data Release 3, we search for
resolved wide binary companions in the H3 survey, a spectroscopic survey that
has compiled $\sim$150,000 spectra for thick-disk and halo stars to date. We
identify 800 high-confidence (a contamination rate of 4%) wide binaries and two
resolved triples, with binary separations mostly between $10^3$-$10^5$ AU and a
lowest [Fe/H] of $-2.7$. Based on their Galactic kinematics, 33 of them are
halo wide binaries, and most of those are associated with the accreted
Gaia-Sausage-Enceladus galaxy. The wide binary fraction in the thick disk
decreases toward the low metallicity end, consistent with the previous findings
for the thin disk. Our key finding is that the halo wide binary fraction is
consistent with the thick-disk stars at a fixed [Fe/H]. There is no significant
dependence of the wide binary fraction on the $\alpha$-captured abundance.
Therefore, the wide binary fraction is mainly determined by the iron abundance,
not their disk or halo origin nor the $\alpha$-captured abundance. Our results
suggest that the formation environments play a major role for the wide binary
fraction, instead of other processes like radial migration that only apply to
disk stars.
