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Description
The second harmonic cosine and sine modulations of the local spin polarization of $\Lambda$($\bar{\Lambda}$) hyperons out-of-plane ($P_{y}$) and in-plane ($P_{z}$), denoted as $P_{2y}$ and $P_{2z}$, respectively, are newly proposed observables for probing spin Hall effect (SHE) driven by the gradient of baryonic chemical potential \cite{liu21, fu2021_she}. $P_{2y}$ and $P_{2z}$ are measured in Au+Au collisions at $\sqrt{s_{NN}} = $ 7.7, 9.2, 11.5, 14.6, 17.3, 19.6 and 27 GeV from the RHIC Beam Energy Scan-II. A monotonic decrease of $P_{2y}$ for $\Lambda$ with increasing collision energy, and smaller positive $P_{2z}$ with hints of sign change at $\sqrt{s_{NN}} = 7.7$ GeV are observed. However, $P_{2y}$ and $P_{2z}$ for $\bar{\Lambda}$ show no significant energy dependence within the large statistical uncertainties. The local polarization components are influenced by thermal vorticity, shear-induced effects, and baryon chemical potential. These measurements provide valuable insights into the spin dynamics of QCD matter in high baryon density environments.