Power-law dynamics in the spin-liquid kagome lattices ${\mathrm{SrCr}}_8{\mathrm{Ga}}_4{\mathrm{O}}_{19}$ and ${\mathrm{ZnCu}}_3{\mathrm{(OH)}}_6{\mathrm{Cl}}_2$

We consider the polarization function $P_Z^{\exp }\left(t\right)$ measured by the muon-spin relaxation ($\mu \mathrm{SR}$) technique for the ${\mathrm{SrCr}}_8{\mathrm{Ga}}_4{\mathrm{O}}_{19}$ and ${\mathrm{ZnCu}}_3{\left(\mathrm{OH}\right)}_6{\mathrm{Cl}}_2$ spin-liquid systems. We show the functional form of $P_Z^{\exp }\left(t\right)$ to imply that, in the temperature range of order 0.1 K, the spectral-weight function $F\left(\omega \right)$ of the magnetic correlations scales with $1/{\left|\omega \right|}^{1-x}$ ($0<x<1$) in the energy range of $\hslash \omega \approx 1\mu \mathrm{eV}$. We derive the parameters involved in $F\left(\omega \right)$ from fits to available experimental data. Inelastic neutron scattering data probing $F\left(\omega \right)$ in the millielectronvolt energy range are consistent with a more conventional behavior. These differences could be due to a variety of spin-dynamics mechanisms, i.e., intrinsic to the kagome layer or related to the magnetic defects that have been evidenced in these compounds, acting at different energies. New $\mu \mathrm{SR}$ measurements are proposed and theoretical developments are suggested to pinpoint the mechanisms at play.

Figure 1

Variation of $K_x\left(z\right)$ as a function of $z$ for different values of the exponent $x$ as indicated in the plot. Note the analytical expression $K_x(z=0)=1/(2-x)(1-x)$ [38].

Figure 2

Polarization functions measured for two kagome lattice systems for different applied fields as indicated in the graphs: (a) SCGO at 100 mK and (b) HS at 50 mK. The data are respectively taken from Refs. [17] and [20]. The lines are the result of a fit of Eq. (2) together with Eq. (11) to the respective combined sets of data with parameters given in Table 1. A small time-independent background is allowed in addition to the $P_Z\left(t\right)$ function, in particular for a better match of the low-field data [20]. For the HS zero-field data, the effect of the coupling of the muon spin to nearby H, Cu, and Cl nuclei is accounted for as in Ref. [20] with similar parameters. The dotted line shows the model prediction at zero field, in the absence of this latter coupling. It implies the absence of resonance effect around 8 mT as found, e.g., in metallic copper [42].