Decoupling lattice and magnetic instabilities in frustrated CuMnO$_2$

The $A$MnO$_{2}$ delafossites ($A$=Na, Cu), are model frustrated antiferromagnets, with triangular layers of Mn$^{3+}$~spins. At low temperatures ($T_{N}$=65 K), a $C2/m \rightarrow P\overline{1}$ transition is found in CuMnO$_2$, which breaks frustration and establishes magnetic order. In contrast to this clean transition, $A$=Na only shows short-range distortions at $T_N$. Here we report a systematic crystallographic, spectroscopic, and theoretical investigation of CuMnO$_2$. We show that, even in stoichiometric samples, non-zero anisotropic Cu displacements co-exist with magnetic order. Using X-ray/neutron diffraction and Raman scattering, we show that high pressures acts to decouple these degrees of freedom. This manifests as an isostuctural phase transition at $\sim$10 GPa, with a reversible collapse of the $c$-axis. This is shown to be the high pressure analog of the $c$-axis negative thermal expansion seen at ambient pressure. DFT simulations confirm that dynamical instabilities of the Cu$^{+}$ cations and edge-shared MnO$_{6}$ layers are intertwined at ambient pressure. However, high pressure selectively activates the former, before an eventual predicted re-emergence of magnetism at the highest pressures. Our results show that the lattice dynamics and local structure of CuMnO$_2$ are quantitatively different to non-magnetic Cu delafossites, and raise questions about the role of intrinsic inhomogeniety in frustrated antiferromagnets.