Atacamite Cu$_2$Cl(OH)$_3$ in High Magnetic Fields: Quantum Criticality and Dimensional Reduction of a Sawtooth-Chain Compound

We report an extensive high-field study of atacamite Cu$_2$Cl(OH)$_3$, a material realization of quantum sawtooth chains with weak interchain couplings, in continuous and pulsed magnetic fields up to 58 T. In particular, we have mapped the entropy landscape for fields as high as 35 T and have identified a field-induced quantum critical point at 21.9(1) T for $\mathbf{H} \parallel c$ axis. The quantum critical point separates field regions with and without magnetic order, evidenced by our thermodynamic study and $^1$H nuclear magnetic resonance spectroscopy, but lies far below full saturation of the magnetization. Corroborated by numerical results using density-matrix renormalization group (DMRG) calculations, we find this behavior associated with a dimensional reduction of the spin system: the sawtooth chain effectively decouples into an antiferromagnetic spin-$1/2$ chain (backbone of the sawtooth chain) in the presence of an exchange field produced by the remaining field-polarized spins.