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First-principles diagrammatic Monte Carlo for electron–phonon interactions and polaron

Nature Physics volume 21, pages 1275–1282 (2025)Cite this article

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Abstract

In condensed matter, phonons—quanta of the lattice vibration field—couple with electrons, leading to the formation of entangled electron–phonon states called polarons. In the intermediate- and strong-coupling regimes common to many conventional and quantum materials, a many-body treatment of polarons requires adding up a large number of electron–phonon Feynman diagrams. In this regard, diagrammatic Monte Carlo is an efficient method for diagram summation and has been used to study polarons within simplified electron–phonon models. Here we develop diagrammatic Monte Carlo calculations based on accurate first-principles electron–phonon interactions, enabling numerically exact results for the ground-state and dynamical properties of polarons in real materials. We implement these calculations in LiF, SrTiO3, and rutile and anatase TiO2, and describe both localized and delocalized polarons. Our work enables the precise modeling of electron–phonon interactions and polarons in coupling regimes ranging from weak to strong. The results will provide deeper insights into transport phenomena, linear response and superconductivity within the strong electron–phonon coupling regime.

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Fig. 1: Sign problem in e–ph DMC.
Fig. 2: Polaron formation energy.
Fig. 3: Electronic make-up and phonon cloud of polarons.
Fig. 4: Polaron energy vs. momentum curves.
Fig. 5: Polaronic charge transport in rutile and anatase TiO2.

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Data availability

The datasets generated and analysed in this study are available in the Materials Cloud repository75. Additional data and information are available from the corresponding author upon reasonable request. Source data are provided with this paper.

Code availability

The Perturbo code used to generate the datasets is an open-source software, which can be downloaded at https://perturbo-code.github.io/. The first-principle e–ph DMC routines are available via GitHub at https://github.com/yaoluo/FEP-DMC.

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Acknowledgements

Y.L. thanks I. Maliyov, J. Yang and A. Lee for fruitful discussions. Y.L. acknowledges partial support from the Eddleman Graduate Fellowship. M.B. is grateful to the Scuola Normale Superiore in Pisa, Italy, for hosting him during the writing of this manuscript. J.P. acknowledges support from the Chicago Prize Postdoctoral Fellowship in Theoretical Quantum Science. Methods development was supported by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, and Office of Basic Energy Sciences, Scientific Discovery through Advanced Computing (SciDAC), program under award no. DESC0022088. Code development was supported by the National Science Foundation under grant no. OAC-2209262. Calculations of transport and polarons in oxides were supported by the AFOSR and Clarkson Aerospace Corp under award no. FA9550-24-1-0004. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a US Department of Energy Office of Science User Facility using NERSC award no. DDR-ERCAP0026831.

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Authors and Affiliations

  1. Department of Applied Physics and Materials Science, and Department of Physics, California Institute of Technology, Pasadena, CA, USA

    Yao Luo, Jinsoo Park & Marco Bernardi

  2. Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL, USA

    Jinsoo Park

  3. Department of Physics, Pohang University of Science and Technology, Pohang, Korea

    Jinsoo Park

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  1. Yao Luo
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  2. Jinsoo Park
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Contributions

Y.L. and M.B. conceived and designed the research. Y.L. performed the calculations and analysis with support from all authors. Y.L. and M.B. wrote the manuscript with input from all authors.

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Correspondence to Marco Bernardi.

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Nature Physics thanks Cesare Franchini and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Information (download PDF )

Supplementary Sections I and II, Figs. 1–13, Equations (1)–(56), Table 1, and theory and implementation details.

Source data

Source Data Fig. 1 (download XLSX )

Source data for Green’s function and the averaged sign from band sampling and matrix product.

Source Data Fig. 2 (download XLSX )

Polaron formation energy for LiF, SrTiO3 and anatase TiO2.

Source Data Fig. 3 (download XLSX )

Polaron wavefunction amplitude in phonon and electron sectors.

Source Data Fig. 4 (download XLSX )

Polaron dispersion along high-symmetry paths for LiF, SrTiO3 and anatase TiO2.

Source Data Fig. 5 (download XLSX )

Polaron mobility for rutile and anatase TiO2, and polaron spectral function.

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Luo, Y., Park, J. & Bernardi, M. First-principles diagrammatic Monte Carlo for electron–phonon interactions and polaron. Nat. Phys. 21, 1275–1282 (2025). https://doi.org/10.1038/s41567-025-02954-1

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