# 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

FIG. 7. (Color online) (a) ZF-\( \mu \) SR time spectra in the temper-ature range between 4.2 K and 10 K. (b) \( \mu ^ { \circ } \) SR time spectra measured at 30 mK in the presence of various longitudinal fields (LFs) up to 100 mT. The inset shows the earlier part of the spectrum at ZF. The solid lines are the fitted results by using Eq. (6)..

the base temperature. The initial fast depolarization ex-hibits a gradual suppression with increasing LF and is nearly quenched at \( ^ { \dag } \) F = 100 mT. Surprisingly, we also observed a finite slope due to the residual magnetic fluc-tuation in these spectra at a large time, which is almost independent of the LF, even at 30 mK. These features can be observed from the fast damped 2/3 amplitude followed by a slowly relaxing 1 \( / \) 3 component originating from the magnetic ordered state with slow persistent spin dynamics.

In a previous \( \mu \) SR study, Sugiyama et al. analyzed their ZF-\( \cdot \mu \) SR spectra by using a combination of an expo-nentially relaxing cosine oscillation for the damped 2 \( / \) 3 component and an exponential function for the 1 \( / \) 3 tail signal.11 They showed that the ratio of the relaxation rate of the 1 \( / \) 3 tail component \( \lambda _ { \mathrm { t a i l } } \) to the precession frequency \( \omega _ { \mathrm { A F } } \) is close to unity, \( \lambda _ { \mathrm { t a i l } } / \omega _ { \mathrm { A F } } \, \sim \, 1 \), where \( \lambda _ { \mathrm { t a i l } } / \omega _ { \mathrm { A F } } \) corresponds to the field distribution normalized to the mean value of \( B _ { \mathrm { l o c } } \). However, their fitting function is based on an unique precession signal at a spontaneous local field for the conventional magnetic ordered state, \( \lambda _ { \mathrm { t a i l } } / \omega _ { \mathrm { A F } } \ll 1 \) . The observed broad \( B _ { \mathrm { l o c } } \) distribution and residual magnetic fluctuation strongly suggest that a highly inhomogeneous magnetic state appears in the ground state of \( \mathrm { S r } _ { 2 } \)\( \ldots \) 4.

The inset of Fig. 7(b) shows the earlier part of the ZF-\( \mu \) SR spectrum at 30 mK. As mentioned above, the GKT and LKT functions are widely attributed to the \( \mu \) SR spectra for random magnetism in dense and diluted spin systems. However, the ZF spectrum at 30 mK lacks a decrease that usually precedes the asymptotic tail for the GKT function, although it exhibits a Gaussian-like initial depolarization. We attributed this to a broad dis-tribution of \( B _ { \mathrm { l o c } } \), which is not characterized by a sin-gle root-mean-square width, and adopted the Gaussian-broadened Gaussian (GbG) function as a recursive func-tion for the curve fitting.20–22

The GbG function is described as one of the general-ized forms of the GKT function, where \( \Delta \) in Eq. (2) has a distribution with the mean value \( \Delta _ { 0 } \) and root-mean-square width W. \( G _ { \mathrm { G b G } } ( t ) \) for ZF is written as

\[ \begin{array} { c c } { G _ { \mathrm { G b G } } ( t ) = \displaystyle \frac { 1 } { 3 } + \frac { 2 } { 3 } \bigg ( \displaystyle \frac { 1 } { 1 + R ^ { 2 } \Delta _ { 0 } ^ { 2 } t ^ { 2 } } \bigg ) ^ { 3 / 2 } \bigg ( 1 - \displaystyle \frac { \Delta _ { 0 } ^ { 2 } t ^ { 2 } } { 1 + R ^ { 2 } \Delta _ { 0 } ^ { 2 } t ^ { 2 } } \bigg ) } \\ { \times \exp \! \left( - \displaystyle \frac { \Delta _ { 0 } ^ { 2 } t ^ { 2 } } { 2 ( 1 + R ^ { 2 } \Delta _ { 0 } ^ { 2 } t ^ { 2 } ) } \right) \! , \! \! \! \! } \end{array} \]

where R is the ratio of the distribution width, and \( R = W / \Delta _ { 0 } \). For \( R \) = 0, the GbG function is identical to the GKT function with a deep minimum after the ini-tial depolarization. The depth of the minimum decreases with increasing R and becomes almost independent of R for \( R > \) 1. Thus, we imposed the condition 0 \( \le R \le \) 1 in the following analysis. We note that the GbG function does not approach the LKT function, even for \( R \gg \) 1.

For the curve fitting, we presume that G(t) is primar-ily described by the product of the GbG function and a slow exponential decay due to the persistent spin dynam-ics in addition to a supplementary term for exponential damping:

\[ G ( t ) = f G _ { \mathrm { G b G } } ( t ) e ^ { - \lambda t } + ( 1 - f ) e ^ { - \lambda _ { \mathrm { e x t } } t } , \qquad ( 6 ) \]

where f is the volume fraction exhibiting inhomoge-neous magnetism, and \( \lambda \) is the relaxation rate reflecting the spin dynamics. The second exponential term with the relaxation rate \( \lambda _ { \mathrm { e x t } } \) is for extrinsic contributions originat-ing from the magnetically ordered state region with rel-atively fast persistent dynamics, which makes the above muon spin depolarization model unreasonable. The T dependence of f is plotted in the inset of Fig. 8. The in-homogeneous magnetism develops over the almost entire volume of the sample below 6 K. We obtained \( R = 0 . 7 ( 1 ) \) below 4.2 K, whereas R is unity at higher temperatures. This discrepancy might be caused by the lack of data for \( t \sim 0 . 1 \)\( \cdot \ \mu \) s that is masked by the limited time resolu-tion of the pulsed muon beam (∼100 ns). We note that the dynamical LKT function and dynamical spin glass models19 could not reproduce the spectra over the early time range.

The absence of decrease in the \( \mu \) SR time spectra tells us important aspects of the magnetic ground state. Noakes et al. reproduced the shallow decrease in the GbG