SIMBAD references

We select 1052 469 (754 635) thin disc stars from Gaia eDR3 and LAMOST DR7 in the range of Galactocentric radius R (guiding centre radius R_g) from 8 to 11 kpc to investigate the asymmetries between the North and South of the disc mid-plane. More specifically, we analyse the vertical velocity dispersion profiles ($\sigma _{v_{z}}(z$)) in different bins of R (R_g) and [Fe/H]. We find troughs in the profiles of $\sigma _{v_{z}}(z)$ located in both the North (z ∼ 0.7 kpc) and South (z ∼ -0.5 kpc) of the disc 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_g. A similar shift exists in these North-South (NS) asymmetry profiles further divided into different [Fe/H] ranges. The sample binned with R_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.