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 √ s_NN = 19.6 GeV in the centrality of 20−50%. (a) is the invariant mass distribution, solid blue circles are data and the red curve is fit to the background with the second order of polynomial. (b) presents extraction of the local polarization along the beam direction, solid blue circles are data and the red curve is fitting with the expression Eq.[].

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 and blue solid points are Λ and Λ¯ , respectively.  Solid and dashed curves are model predictions with AMPT initial conditions in the lambda scenario without SHE. (b) presents energy dependence of local polarization along out-of-plane, the meaning of the symbols and curves is the same as that in panel (a).

Fig. 3: (Color online) Comparison of net Λ local polarization P_{2,z,y}^{net} as a function of collision energy with the model predictions. Solid red points are data with statistics (vertical red line) and systematic (rectangle) errors; different curves are model predictions with and without SHE in the lambda or s-quark scenario. (a) and (b) are P_{2,z}^{net} and P_{2,y}^{net} as a function of collision energy, respectively.

Conclusion

1. Systematic measurements of Λ hyperons' local spin polarization P_{2,z(y)} and P_{2,z(y)}^{net} in Au+Au collisions at sqrt{s_{NN}}=7.7, 9.2, 11.5, 14.6, 17.3, 19.6 and 27 GeV (BES-II)

2. Monotonic energy dependence in P_{2,y} of Λ with decreasing collision energy

3. Positive magnitude of P_{2,z} no strong energy dependence with a hint of sign change at 7.7 GeV

4. First study of baryonic spin Hall effect (SHE) by measuring net local polarization P_{2,z(y)}^{net} in Au+Au collisions from 7.7 to 27 GeV --> Hints of sign change of P_{2,z(y)}^{net}