# 1. Anomalous paramagnetism \( T > T _ { c 0 } \))

In the paramagnetic state of ferromagnetic and anti-ferromagnetic systems, the muon spin depolarization at zero external field has contributions from the static nu-clear dipolar fields and the fluctuating fields from electron spins. The nuclear spin contribution is well reproduced by the Gaussian Kubo-Toyabe (GKT) function derived from the Gaussian distribution of the local field (\( B _ { \mathrm { l o c } } \)),which is a good approximation for a dense ensemble of randomly orientated nuclear magnetic moments.18 The GKT function is written as

\[ G _ { \mathrm { G K T } } ( t ) = \frac { 1 } { 3 } + \frac { 2 } { 3 } \left[ 1 - ( \Delta t ) ^ { 2 } \right] \exp \left[ - \frac { ( \Delta t ) ^ { 2 } } { 2 } \right] , \qquad ( 2 ) \]

where \( \Delta / \gamma _ { \mu } = \langle B _ { \mathrm { l o c } } ^ { 2 } \rangle ^ { 1 / 2 } \) corresponds to the root-mean-square for the distribution of the local magnetic field (where \( \gamma _ { \mu } = 2 \pi \times 1 3 5 . 5 3 \) MHz/T is the muon gyromag-netic ratio). On the other hand, the dynamical electron spin contribution is given as \( \exp ( - \lambda t ) \) with the relax-ation rate \( \lambda \). Then, the depolarization function G(t) is described by the product of the two contributions:

\[ G ( t ) = G _ { \mathrm { G K T } } ( t ) \exp ( - \lambda t ) . \eqno ( 3 ) \]

Figure 6(a) shows the zero-field (ZF)-\( \mu ! \) SR spectra at 300 K and 80 K (i.e., above and below \( T _ { \mathrm { c 1 } } \)), where no sign of the emergence of the spontaneous internal mag-netic field, such as oscillation and/or fast damping, is observed below \( T _ { \mathrm { c 1 } } \). This immediately leads us to the conclusion that no long-range magnetic order is present below \( T _ { c 1 } \), which is in qualitative agreement with the pre-vious result.11 The temperature dependence of \( \lambda \) deduced by a curve fitting is plotted in the inset of Fig. 6(a).

The spectra in Fig. 6(a) including the one at 18 K show that the exponential damping gradually overcomes the GKT depolarization below 40 K. Figure 6(b) shows \( \mu \) SR spectra at 18 K measured with ZF and longitudinal fields (LFs) of 10 mT and 100 mT. Generally, the expo-nential damping due to the electron spin fluctuation does not depend much on the LF because the fluctuation rate of the dynamical field from the ordinary paramagnetic moments is much larger than the Larmor frequency of the muon spin precession. However, the depolarization is quenched by the application of a relatively weak LF, which indicates that \( B _ { \mathrm { l o c } } \) is much weaker than the LF and that it is quasistatic within the time window of \( \mu \) SR (\( > 1 0 ^ { - 5 } \) s). These features allow us to estimate an up-per limit for the mean hyperfine field at the muon site, \( B _ { \mathrm { l o c } } \, \simeq \, \Delta / \gamma _ { \mu } \, \lesssim \, 2 \) mT, and that of the fluctuation rate \( \nu \lesssim \) 0.5 MHz. We also performed curve fitting for the LF dependence of the \( \mu \) SR spectra at 18 K by replac-ing the exponential term in Eq. (3) with the Lorentzian Kubo-Toyabe (LKT) function for a finite LF:

FIG. 6. (Color online) (a) Zero-field (ZF)-\( \mu ^ { 6 } \) SR time spectra at 300 K, 80 K, and 18 K. The solid curves are the fitted results of Eq. (3). The inset shows the temperature dependence of the relaxation rate. (b) \( \mu ^ { \star } \) SR spectra at 18 K measured with ZF and longitudinal fields (LF = 10 mT and 100 mT) with the results of the curve fitting with Eq.(4).

\[ \begin{array} { l } { \displaystyle G _ { \mathrm { L K T } } ( t ) = \! 1 - \frac { \Delta } { \omega _ { 0 } } j _ { 1 } ( \omega _ { 0 } t ) e ^ { - \Delta t } } \\ { \displaystyle \quad \, - \big ( \frac { \Delta } { \omega _ { 0 } t } \big ) ^ { 2 } [ j _ { 0 } ( \omega _ { 0 } t ) e ^ { - \Delta t } - 1 ] } \\ { \displaystyle \quad \, - \big [ 1 + \big ( \frac { \Delta } { \omega _ { 0 } } \big ) ^ { 2 } \big ] \Delta \int _ { 0 } ^ { t } j _ { 0 } ( \omega _ { 0 } t ^ { \prime } ) e ^ { - \Delta t ^ { \prime } } d t ^ { \prime } , \, \, \, \, ( 4 ) } \end{array} \]

where \( \omega _ { 0 } = \gamma _ { \mu } B _ { 0 } \) is the Larmor frequency of the muon spin precession in the presence of an external field \( B _ { 0 } \), and \( j _ { 0 } \) and \( j _ { 1 } \) denote spherical Bessel functions.19 We note that the results of the curve fitting indicate that the entire volume of the sample exhibits a Lorentzian-like field distribution. Thus, we are led to the conclusion that the dilute magnetic moments of d electrons having an unusually quasistatic characteristic make a significant contribution to the muon depolarization for \( T _ { c 0 } \leq T \leq \)\( T _ { c 1 } \), although their origin is not clear at this stage.

# 2. Inhomogeneous magnetism \( ( T < T _ { c 0 } \))

As is observed for the ZF-\( \mu \) SR spectra in Fig. 7(a), a rapid depolarization develops with decreasing tempera-ture below \( T _ { \mathrm { c 0 } } \). We show the ZF- and LF-\( \mu \) SR spectra at 30 mK measured in the presence of various LFs up to 100 mT in Fig. 7(b). The ZF spectra clearly demonstrate the absence of conventional long-range magnetic order at