Nonperturbative many-body calculations of polarons in real materials from first principles: the strong-coupling small-polaron regime

Sio, Verdi, Poncé, and Giustino (Phys. Rev. Lett. 122, 246403 (2019); companion paper Phys. Rev. B 99, 235139 (2019)) reformulated the first-principles polaron problem as a nonlinear eigenvalue equation built entirely from unit-cell quantities — Kohn–Sham band structures, phonons, and electron–phonon matrix elements — describing both large (Fröhlich-type) and small (Holstein-type) polarons in LiF and Li2O2, but within an adiabatic, static-distortion picture. They posed as an open program the coupling of such first-principles data (optionally reduced to model Hamiltonians via the computed spectral amplitudes) to nonperturbative many-body techniques such as path-integral or diagrammatic Monte Carlo, so as to enable predictive ab initio many-body calculations of polarons in real materials. That objective has since been demonstrated for weak-to-intermediate coupling and large/delocalized polarons: first-principles diagrammatic Monte Carlo has yielded numerically exact polaron properties in LiF, SrTiO3, and TiO2, with mobility predictions for anatase TiO2 validated against experiment. A precise open core survives in the strong-coupling, small-polaron regime: independent 2026 work reports significant bias in first-principles diagrammatic Monte Carlo for the strong-coupling LiF hole polaron, new scalable methods are still under development, and the source's own small-polaron test material Li2O2 has no nonperturbative ab initio treatment at all. The problem posed is to establish and validate a systematically improvable, nonperturbative ab initio many-body treatment of strong-coupling small polarons in real materials — formation energies, structures, dynamics or transport.

Background

A polaron is the composite quasiparticle formed when an excess electron or hole in a crystal distorts the surrounding lattice and becomes dressed by, and partially trapped in, the potential well of its own making. Model Hamiltonians bracket the physics: in the Fröhlich model the carrier couples to the long-range polar field of longitudinal optical phonons and stays comparatively delocalized (a "large" polaron), while in the Holstein model the carrier couples to short-range on-site lattice distortions and self-traps onto one or a few unit cells (a "small" polaron). Nonperturbative solutions of these models are classical results of many-body theory: Feynman's path-integral variational treatment of the Fröhlich polaron, and more recently diagrammatic Monte Carlo (DMC) — stochastic summation of the Feynman diagrams of many-body perturbation theory, numerically exact where it converges — which yields benchmark energies, dispersions, and mobilities for the model Hamiltonians (see, e.g., Mishchenko et al., Phys. Rev. Lett. 123, 076601 (2019)). For real materials, the workhorse has instead been density-functional theory (DFT) with large periodic supercells containing the localized carrier. Supercell calculations become prohibitive for extended polarons, and semilocal exchange-correlation functionals suffer from self-interaction error (a spurious interaction of the carrier with itself and its periodic images) that artificially delocalizes the charge. The electron–phonon matrix elements $g_{mn\nu}(\mathbf{k},\mathbf{q})$, which couple Kohn–Sham electronic states $n,m$ at wavevector $\mathbf{k}$ to phonon branch $\nu$ at $\mathbf{q}$, can nonetheless be computed in the primitive unit cell by density-functional perturbation theory (DFPT). Sio, Verdi, Poncé, and Giustino exploited this to recast the polaron problem as a nonlinear eigenvalue problem in the basis of Kohn–Sham states and phonon eigenmodes, in close analogy with the Bethe–Salpeter equation, the standard two-particle equation used to compute excitons (bound electron–hole pairs). The polaron wavefunction is a coherent superposition of Kohn–Sham states with Fourier amplitudes $A_{n\mathbf{k}}$, and the accompanying lattice distortion a superposition of phonons with amplitudes $B_{\mathbf{q}\nu}$; the formation energy is the electronic energy cost of localizing the carrier minus the lattice relaxation energy. The method treats large polarons (electrons in LiF) and small polarons (holes in LiF, electrons in Li2O2) seamlessly, without supercells. Its limitation, however, is that — like the classic Landau–Pekar variational polaron, an electron in a self-consistent static polarization well — it is adiabatic and classical: the lattice distortion is static, the DFT energy is expanded only to lowest order in the atomic displacements, a single electronic eigenstate is retained, and nuclear motion is treated classically. Excited states, nonadiabatic dynamics, quantum nuclear fluctuations, and finite-temperature transport lie outside it. The authors therefore posed as an explicit open program (companion paper, Sec. VII "Future developments"): write down the ab initio electron–phonon Hamiltonian — the counterpart of the Fröhlich Hamiltonian built from unit-cell bands, phonons, and $g$-matrix elements — optionally simplified by retaining the dominant bands, phonons, and couplings identified through the $A_{n\mathbf{k}}$, $B_{\mathbf{q}\nu}$ spectral decomposition, and solve it with nonperturbative many-body techniques such as path-integral or diagrammatic Monte Carlo, thereby realizing "predictive ab initio many-body calculations of polarons in real materials". The subsequent literature shows this flagship goal now achieved for weak-to-intermediate coupling and large/delocalized polarons: Luo, Park, and Bernardi (Nat. Phys. 21, 1275 (2025)) developed first-principles diagrammatic Monte Carlo combining DFT-grade electron–phonon interactions with DMC, obtaining numerically exact ground-state and dynamical polaron properties in LiF, SrTiO3, and rutile and anatase TiO2, with polaron mobility predictions for anatase TiO2 subsequently validated against experiment on high-quality films. What survives as a precise open core is the strong-coupling, small-polaron regime — precisely the Holstein-like self-trapping that motivated the unit-cell methodology: 2026 work reports a significant bias in first-principles DMC for the strong-coupling hole polaron of LiF, several groups are still constructing new scalable methods, and Li2O2 has no nonperturbative ab initio treatment.

Problem Statement

Establish and validate a nonperturbative many-body treatment of polarons in real materials, parameterized entirely by first-principles quantities — Kohn–Sham band structures, phonon dispersions, and electron–phonon matrix elements computed in the crystal unit cell, either used directly or reduced to a model polaron Hamiltonian via the spectral amplitudes $A_{n\mathbf{k}}$ and $B_{\mathbf{q}\nu}$ as envisioned by the source — and demonstrate it with quantified accuracy in the regime that remains unrealized: the strong-coupling, small-polaron (Holstein-like self-trapping) regime, exemplified by the hole polaron of LiF and the electron polaron of Li2O2. Concretely, a complete answer must provide, for at least one representative strongly coupled small-polaron system of this kind (the hole polaron of LiF and the electron polaron of Li2O2 are the canonical examples): (i) polaron formation energies and structural characterization (wavefunction, spatial extent, spectral make-up over electronic bands and phonon branches) computed beyond the adiabatic, static-distortion, lowest-order treatment, with controlled numerical and methodological errors; (ii) at least one dynamical or transport observable in the strong-coupling regime — excited-state spectrum, effective mass or dispersion, optical response, or mobility — where nonadiabaticity and quantum nuclear fluctuations matter; and (iii) cross-validation against independent references (supercell DFT with controlled self-interaction corrections, established weak- and strong-coupling model limits, other independent solvers, or experiment) sufficient to establish the quantified accuracy of the treatment in this regime. Any systematically improvable nonperturbative solver — diagrammatic Monte Carlo, path-integral Monte Carlo, translationally invariant variational, coupled-cluster, or equivalent — is admissible. The demonstrated capability for weak-to-intermediate coupling and large/delocalized polarons (LiF electrons, SrTiO3, TiO2) is context, not a deliverable: the completion boundary is a treatment of strongly coupled small polarons in real materials whose accuracy is independently corroborated.

References

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