  We select 1,052,469 (754,635) thin disk stars from {\it Gaia} eDR3 and LAMOST
DR7 in the range of Galactocentric radius $R$ (guiding center radius
$R_\mathrm{g}$) from 8 to 11\,kpc to investigate the asymmetries between the
North and South of the disk midplane. More specifically we analyze the vertical
velocity dispersion profiles ($\sigma_{v_{z}}(z$)) in different bins of $R$
($R_\mathrm{g}$) and $[\mathrm{Fe/H}]$. We find troughs in the profiles of
$\sigma_{v_{z}}(z)$ located in both the North ($z \sim 0.7$\,kpc) and South ($z
\sim -0.5$\,kpc) of the disk at all radial and chemical bins studied. The
difference between the Northern and Southern vertical velocity dispersion
profiles ($\Delta\sigma_{v_{z}}(|z|)$) shows a shift between curves of
different $R$ and $R_\mathrm{g}$. A similar shift exists in these NS asymmetry
profiles further divided into different $[\mathrm{Fe/H}]$ ranges. The sample
binned with $R_\mathrm{g}$ more clearly displays the features in the velocity
dispersion profiles. The shift in the peaks of the $\Delta\sigma_{v_{z}}$
profiles and the variation in the phase spiral shape binned by metallicity
indicate the variation of the vertical potential profiles and the radial
metallicity gradient. The wave-like signal in NS asymmetry of
$\sigma_{v_{z}}(z)$ largely originates from phase spiral; while the NS
asymmetry profiles of [Fe/H] only display a weak wave-like feature near solar
radius. We perform a test particle simulation to qualitatively reproduce the
observed results. A quantitative explanation of the NS asymmetry in the
metallicity profile needs careful consideration of the spiral shape and the
perturbation model, and we leave this for future work.
