Neural Canonical Transformations for Quantum Anharmonic Solids of Lithium

Lithium is a typical quantum solid, characterized by cubic structures at ambient pressure. As the pressure increases, it forms more complex structures and undergoes a metal-to-semiconductor transformation, complicating theoretical and experimental analyses. We employ the neural canonical transformation approach, an \textit{ab initio} variational method based on probabilistic generative models, to investigate the quantum anharmonic effects in lithium solids at finite temperatures. This approach combines a normalizing flow for phonon excited-state wave functions with a probabilistic model for the occupation of energy levels, optimized jointly to minimize the free energy. Our results indicate that quantum anharmonicity lowers the \textit{bcc}-\textit{fcc} transition temperature compared to classical molecular dynamics predictions. At high pressures, the predicted fractional coordinates of lithium atoms in the \textit{cI16} structure show good quantitative agreement with experimental observations. Finally, contrary to previous beliefs, we find that the poor metallic \textit{oC88} structure is stabilized by the potential energy surface obtained via high-accuracy electronic structure calculations, rather than thermal or quantum nuclear effects.