Proximate Spin Liquid Ground State Arising from Competing Stripy and 120$^{\circ}$ Spin Correlations in the Triangular Quantum Antiferromagnet ErMgGaO$_4$

ErMgGaO$_4$ is a quantum antiferromagnet wherein the pseudospin-1/2 degrees of freedom of Er$^{3+}$ decorate two-dimensional triangular planes separated by disordered non-magnetic bilayers of Mg$^{2+}$ and Ga$^{3+}$. Unlike its sister compound, YbMgGaO$_4$, our powder ErMgGaO$_4$ sample shows a clear spin glass transition near $T_g \sim 2.5$~K, about 1/6 of its Curie-Weiss temperature. We have carried out new inelastic neutron scattering measurements on these powder ErMgGaO$_4$ samples. At high energies, we observed crystalline electric field (CEF) transitions within the $J=15/2$ multiplet of Er$^{3+}$, but with the first excited CEF level sufficiently low in energy ($\sim$ 3~meV) so as to allow the possibility that virtual CEF transitions influence the exchange couplings. At E=0, we observe diffuse elastic scattering which is analysed using Warren lineshapes appropriate for two dimensional correlations. This reveals dominant 2D stripy correlations below $T_g$, coexisting with 2D 120$^\circ$-type correlations that persist above $T_g$. At low temperatures, the low energy inelastic component of the scattering shows a continuum with bandwidth of $\sim$ 0.8~meV. This dynamic magnetic spectral weight can be modeled at all $Q$, energies, and temperatures as the sum of high energy and low energy damped harmonic oscillators (DHO), with the high energy DHO defining the bandwidth of $\sim$ 0.8~meV. We use linear spin wave theory to model this inelastic scattering and to estimate its spin Hamiltonian parameters in terms of a $J_1-J_2-\Delta$ model on the triangular lattice. This gives a good description of the low lying spectral weight for ErMgGaO4, and allows us to place it on the theoretical $J_1-J_2-\Delta$ phase diagram with $\frac{J_1}{J_2}=0.13 \pm 0.03$ and $\Delta=0.4 \pm 0.1$, which is close to the expected quantum phase boundary between the spin liquid and the stripy ordered phases.