a,b, Zero-field (ZF, panel a) and longitudinal-field (LF, parallel to the c-axes of co-aligned crystals and at 0.3 K, panel b) muon decay asymmetry spectra of ECA across a range of temperatures and magnetic fields. Solid lines show fitted results using the function: \(Asy(t)={A}_{1}\cdot {e}^{-{\lambda }_{1}\cdot t}+{A}_{2}\cdot {e}^{-{\lambda }_{2}\cdot t}+{A}_{bg}\), where λ 1 and λ 2 are the relaxation rates corresponding to the fast and slow depolarization components, and A bg represents a flat background term originating from the muons stopping in the silver sample holder. The two measurement paths crossing CLO*-MSY and UUD-MSY phase boundaries are indicated by the vertical and horizontal arrows, respectively, in the inset of a. Upon lowering the temperature from 1.7 K to 1.2 K, we find the relaxation rate increases significantly, and enhanced spin fluctuations are observed near the spin-supersolid thermal transition (at about 1.1 K). At lower temperature of 0.3 K and within the MSY phase, the missing initial asymmetry (“A 0 lost”) implies increasingly strong spin fluctuations. In panel (b), at μ 0 H = 0.35 T (in the UUD phase), we find entire decoupling; however, as the field decreases to 0.3 T (supersolid QCP) a well-defined, faster depolarization appears at early times (t < 2μs). This component vanishes again when entering the MSY phase. The field dependence of the muon-spin depolarization rules out that the loss of A 0 in the MSY phase arises from muon spin decoupling. Muon spin relaxation experiments are performed in ZF and LF configurations using the ARTEMIS spectrometer at the S1 area of the Materials and Life Science Experimental Facility (MLF), J-PARC under a user program (Proposal No. 2024B0263), Japan. A 3He cryostat provides a base temperature of 0.3 K, and co-aligned single crystals are mounted on a thin silver plate with their c-axes oriented parallel to both the initial muon-spin polarization and the applied longitudinal magnetic field.
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