Comparison of the canonical transformation and energy functional formalisms for ab initio calculations of self-localized polarons

In materials with strong electron-phonon ($e$-ph) interactions, charge carriers can distort the surrounding lattice and become trapped, forming self-localized (small) polarons. We recently developed an ab initio approach based on canonical transformations to efficiently compute the formation and energetics of small polarons [N.-E. Lee et al., Phys. Rev. Mater. 5, 063805 (2021)]. A different approach based on a Landau-Pekar energy functional has been proposed in the recent literature [W. H. Sio et al., Phys. Rev. Lett. 122, 246403 (2019); Phys. Rev. B 99, 235139 (2019)]. In this work, we analyze and compare these two methods in detail. We show that the small polaron energy is identical in the two formalisms when using the same polaron wave function. We also show that our canonical transformation formalism can predict polaron band structures and can properly treat zero- and finite-temperature lattice vibration effects, although at present using a fixed polaron wave function. Conversely, the energy functional approach can compute the polaron wave function, but as we show here, it neglects lattice vibrations and cannot address polaron self-localization and thermal band narrowing. Taken together, this work relates two different methods developed recently to study polarons from first-principles, highlighting their merits and shortcomings and discussing them both in a unified formalism.