Revealing Short- and Long-range Li-ion diffusion in Li$_2$MnO$_3$ from finite-temperature dynamical mean field theory

Li$_2$MnO$_3$ is a key component of Li-excess layered cathodes of the form $(1-x),\mathrm{LiMO_2} + x,\mathrm{Li_2MnO_3}$ ($M$ = Mn, Ni, Co, \dots), yet its role in setting Li-ion transport limitations remains under debate. Here we combine DFT+$U$, finite-temperature DFT+DMFT with a continuous-time quantum Monte Carlo impurity solver, and nudged-elastic-band (NEB) calculations to study Li$^{+}$ migration in paramagnetic Li$_2$MnO$_3$ in the presence of a single Li vacancy. Evaluating DMFT total energies along the DFT+$U$ NEB geometries reveals that dynamical correlations strongly renormalize the lowest-barrier processes, reducing the activation energies to $E_a = 0.18$ eV for the shortest-range hop and $E_a = 0.50$ eV for the next-lowest (transport-controlling) step. The 0.18 eV barrier quantitatively reproduces the short-range activation energy from $\mu^{+}$SR, while the 0.50 eV barrier is consistent with the long-range transport scale extracted from ac-impedance measurements. This single-vacancy, paramagnetic DMFT description thus provides a unified interpretation of local and macroscopic probes without invoking clustered vacancy configurations or strong extrinsic disorder, consistent with nearly stoichiometric Li$_2$MnO$_3$ powders. More broadly, our results highlight finite-temperature dynamical correlations as an essential ingredient for predicting ionic migration energetics in correlated oxide electrodes.