We report the second harmonic sine and cosine modulations of the local spin polarization of $\Lambda$ and $\overline{\Lambda}$ hyperons along the in-plane ($z$) and out-of-plane ($y$) directions, denoted as $P_{2z}^{\text{local}}$ and $P_{2y}^{\text{local}}$, respectively, at $\sqrt{s_{NN}} = 7.7, 11.5, 14.6, 17.3, 19.6$, and $27$ GeV in Au+Au collisions from the RHIC Beam Energy Scan-II (BES-II) program. We observe a monotonic increase in $P_{2y}^{\text{local}}$ for $\Lambda$ with decreasing beam energy, and a smaller positive $P_{2z}^{\text{local}}$ with hints of a sign change at $7.7$ GeV. In contrast, the uncertainties for $\overline{\Lambda}$ are large and show no significant dependence on collision energy. The local polarization components are influenced by shear-induced effects, and gradients in thermal vorticity and baryon chemical potential. Our data can provide strong constraints on the magnitude and sign of these contributions across BES energy ranges. The observed differences between $\Lambda$ and $\overline{\Lambda}$ polarization, including the sign change in $P_{2y,2z}^{\text{net-local}}$ ($=P_{2y,2z}^{\text{local,}\Lambda} - P_{2y,2z}^{\text{local,}\overline{\Lambda}}$), offer a promising avenue for probing the spin Hall effect driven by baryon chemical potential gradients. Above measurements provide valuable insights into the spin dynamics of QCD matter in high-density baryonic environments.

FIG. 1: (Color online) Invariant mass distribution and local polarization extraction in Au+Au collisions at √ sNN = 19.6 GeV in the centrality of 20 − 50%. (a) is the invariant mass distribution, solid blue circles are data and red curve is background fitting with the second order of polynomial. (b) presents extraction of the local polarization along the beam direction, solid blue circles are data and red curve is fitting with the expression [].

FIG. 2: (Color online) Comparisons of Λ hyperons local polarization as a function of energy in Au+Au collisions. (a) is energy dependence of local polarization along the beam direction, red solid squares are Λ and blue empty circles are Λ, different curves ¯ are model predictions. (b) presents energy dependence of local polarization along out-of-plane, the meaning of the symbols and curves are the same as that in (a).

FIG. 3:  (Color online) Comparisons of net Λ local polarization as a function of energy in Au+Au collisions. (a) and (b) are the net Λ polarization in-plane and out-of-plane, respectively, as a function of energy in Au+Au collisions.