We report ${}^{35}\mathrm{Cl}$ NMR, ESR, $\mu \mathrm{SR}$, and specific-heat measurements on the $S=\frac{1}{2}$ frustrated kagome magnet kapellasite $\alpha \text{−}{\mathrm{Cu}}_3{\text{Zn(OH)}}_6{\mathrm{Cl}}_2$, where a gapless spin-liquid phase is stabilized by a set of competing exchange interactions. Our measurements confirm the ferromagnetic character of the nearest-neighbor exchange interaction $J_1$ and give an energy scale for the competing interactions $\left|J\right|\sim 10$ K. The study of the temperature-dependent ESR line shift reveals a moderate symmetric exchange anisotropy term $D$, with $|D/J|\sim 3$%. These findings validate a posteriori the use of the $J_1\text{−}{J}_2\text{−}{J}_d$ Heisenberg model to describe the magnetic properties of kapellasite [Bernu et al., Phys. Rev. B 87, 155107 (2013)]. We further confirm that the main deviation from this model is the severe random depletion of the magnetic kagome lattice by 27%, due to Cu/Zn site mixing, and specifically address the effect of this disorder by ${}^{35}\mathrm{Cl}$ NMR, performed on an oriented polycrystalline sample. Surprisingly, while being very sensitive to local structural deformations, our NMR measurements demonstrate that the system remains homogeneous with a unique spin susceptibility at high temperature, despite a variety of magnetic environments. Unconventional spin dynamics is further revealed by NMR and $\mu \mathrm{SR}$ in the low-$T$, correlated, spin-liquid regime, where a broad distribution of spin-lattice relaxation times is observed. We ascribe this to the presence of local low-energy modes.

We report muon spin resonance experiments on Mg${}_x$Cu${}_{4-x}$(OH)${}_6$Cl${}_2$ with $x\sim 1$, a material isostructural to Herbertsmithite exhibiting regular kagome planes of $S=\frac{1}{2}$ ions (Cu${}^{2+}$), and therefore a candidate for a spin-liquid ground state. We evidence the absence of any magnetic ordering down to 20 mK ($\sim$ $J/{10}^4$). We present a detailed investigation of the spin dynamics on well-characterized samples in zero and applied longitudinal fields, and a defect-based interpretation is proposed to explain the unconventional dynamics observed in the quantum spin-liquid phase.

We present a method to build magnetic models of insulators based on high-temperature expansions by fitting both the magnetic susceptibility and the low-temperature specific heat data. It is applied to the frustrated magnet kapellasite [Cu${}_{3}$Zn(OH)${}_{6}$Cl${}_{2}$] with the ${J}_{1}-{J}_{2}-{J}_{d}$-Heisenberg model on the kagome lattice. Experimental data are reproduced with a set of competing exchange energies centered at ${J}_{1}=$-${}12,\phantom{\rule{0.28em}{0ex}}{J}_{2}=$-${}4$, and ${J}_{d}=15.6\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, where ${J}_{d}$ is the third-neighbor exchange energy across the hexagon. Strong constraints between these exchange energies are established. These values confirm the results of F\aa{}k et al. [Phys. Rev. Lett. 109, 037208 (2012)] regarding the location of kapellasite in the cuboc2 phase of the Heisenberg model. The quality and limits of this modeling are discussed.