- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Beyond-Quasiparticle Transport with Vertex Correction: Self-Consistent Ladder Formalism for Electron-Phonon Interactions

Jae-Mo Lihm1,* and Samuel Poncé1,2,†

  • *Contact author: jaemo.lihm@gmail.com
  • †Contact author: samuel.ponce@uclouvain.be
PDF

Phys. Rev. X 16, 011008 – Published 13 January, 2026

DOI: https://doi.org/10.1103/mtjf-ylf7

Abstract

We present a self-consistent many-body framework for computing phonon-limited electronic transport from first principles, incorporating both beyond-quasiparticle effects and vertex corrections. Using the recently developed first-principles scGD0 method, we calculate spectral functions with nonperturbative effects such as broadening, satellites, and energy-dependent renormalization. We show that the scGD0 spectral functions outperform one-shot G0D0 and cumulant approximations in model Hamiltonians and real materials, eliminating unphysical spectral kinks and correctly predicting the phonon emission continuum. Building on this, we introduce the self-consistent ladder formalism for transport, which captures vertex corrections due to electron-phonon interactions. This approach unifies and improves upon the two state-of-the-art approaches for first-principles phonon-limited transport: the bubble approximation and the Boltzmann transport equation. Moreover, as a charge-conserving approximation, it enables consistent calculations of the optical conductivity and dielectric function. We validate the developed method against numerically exact results for model Hamiltonians in the dilute polaronic limit and apply it to real materials. Our results show quantitative agreement with the experimental dc conductivities in intrinsic semiconductors Si and ZnO and the SrVO3 metal, as well as excellent agreement with the experimental THz optical and dielectric properties of Si and ZnO. This work unifies first-principles and many-body approaches for studying transport, opening new directions for applying many-body theory to materials with strong electron-phonon interactions.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Understanding how electrons move through materials is key to designing better electronics, solar cells, and thermoelectrics, but current predictive methods fail in many important materials where interactions between electrons and atomic vibrations (phonons) are strong. We addressed this challenge by creating a theoretical and computational framework that simultaneously includes the breakdown of the electron quasiparticle description and the essential conservation laws inherent in electron-phonon scattering, which leads to vertex corrections. Applying our method to silicon, zinc oxide, and strontium vanadate, we achieved more accurate predictions of their electrical and optical behavior; for instance, our results quantitatively reproduce light absorption and refraction in zinc oxide. Our work marks a step forward by bridging many-body theory and first-principles calculations, and it opens paths for studying complex materials with strong electron-phonon interactions, including the future possibility of incorporating electron-electron scattering effects.

Article Text

Supplemental Material

References (251)

  1. F. Giustino, Electron-phonon interactions from first principles, Rev. Mod. Phys. 89, 015003 (2017).
  2. S. Poncé, W. Li, S. Reichardt, and F. Giustino, First-principles calculations of charge carrier mobility and conductivity in bulk semiconductors and two-dimensional materials, Rep. Prog. Phys. 83, 036501 (2020).
  3. O. D. Restrepo, K. Varga, and S. T. Pantelides, First-principles calculations of electron mobilities in silicon: Phonon and Coulomb scattering, Appl. Phys. Lett. 94, 212103 (2009).
  4. J.-J. Zhou and M. Bernardi, Ab initio electron mobility and polar phonon scattering in GaAs, Phys. Rev. B 94, 201201(R) (2016).
  5. S. Poncé, E. R. Margine, and F. Giustino, Towards predictive many-body calculations of phonon-limited carrier mobilities in semiconductors, Phys. Rev. B 97, 121201(R) (2018).
  6. F. Macheda and N. Bonini, Magnetotransport phenomena in p-doped diamond from first principles, Phys. Rev. B 98, 201201(R) (2018).
  7. J.-J. Zhou, O. Hellman, and M. Bernardi, Electron-phonon scattering in the presence of soft modes and electron mobility in SrTiO3 perovskite from first principles, Phys. Rev. Lett. 121, 226603 (2018).
  8. M. Schlipf, S. Poncé, and F. Giustino, Carrier lifetimes and polaronic mass enhancement in the hybrid halide perovskite CH3NH3PbI3 from multiphonon Fröhlich coupling, Phys. Rev. Lett. 121, 086402 (2018).
  9. G. Brunin, H. P. C. Miranda, M. Giantomassi, M. Royo, M. Stengel, M. J. Verstraete, X. Gonze, G.-M. Rignanese, and G. Hautier, Phonon-limited electron mobility in Si, GaAs, and GaP with exact treatment of dynamical quadrupoles, Phys. Rev. B 102, 094308 (2020).
  10. F. Macheda, S. Poncé, F. Giustino, and N. Bonini, Theory and computation of Hall scattering factor in graphene, Nano Lett. 20, 8861 (2020).
  11. J. Park, J.-J. Zhou, V. A. Jhalani, C. E. Dreyer, and M. Bernardi, Long-range quadrupole electron-phonon interaction from first principles, Phys. Rev. B 102, 125203 (2020).
  12. S. Poncé, F. Macheda, E. R. Margine, N. Marzari, N. Bonini, and F. Giustino, First-principles predictions of Hall and drift mobilities in semiconductors, Phys. Rev. Res. 3, 043022 (2021).
  13. D. C. Desai, B. Zviazhynski, J.-J. Zhou, and M. Bernardi, Magnetotransport in semiconductors and two-dimensional materials from first principles, Phys. Rev. B 103, L161103 (2021).
  14. R. Claes, G. Brunin, M. Giantomassi, G.-M. Rignanese, and G. Hautier, Assessing the quality of relaxation-time approximations with fully automated computations of phonon-limited mobilities, Phys. Rev. B 106, 094302 (2022).
  15. S. Poncé, M. Royo, M. Gibertini, N. Marzari, and M. Stengel, Accurate prediction of Hall mobilities in two-dimensional materials through gauge-covariant quadrupolar contributions, Phys. Rev. Lett. 130, 166301 (2023).
  16. S. Poncé, M. Royo, M. Stengel, N. Marzari, and M. Gibertini, Long-range electrostatic contribution to electron-phonon couplings and mobilities of two-dimensional and bulk materials, Phys. Rev. B 107, 155424 (2023).
  17. J.-M. Lihm, S. Poncé, and C.-H. Park, Self-consistent electron lifetimes for electron-phonon scattering, Phys. Rev. B 110, L121106 (2024).
  18. N. Roisin, G. Brunin, G.-M. Rignanese, D. Flandre, J.-P. Raskin, and S. Poncé, Phonon-limited mobility for electrons and holes in highly-strained silicon, npj Comput. Mater. 10, 242 (2024).
  19. J. Zhou, S. Poncé, and J.-C. Charlier, Guidelines for accurate and efficient calculations of mobilities in two-dimensional semiconductors, Phys. Rev. B 110, 125304 (2024).
  20. J. I. Mustafa, M. Bernardi, J. B. Neaton, and S. G. Louie, Ab initio electronic relaxation times and transport in noble metals, Phys. Rev. B 94, 155105 (2016).
  21. D. C. Desai, J. Park, J.-J. Zhou, and M. Bernardi, Dominant two-dimensional electron-phonon interactions in the bulk Dirac semimetal Na3Bi, Nano Lett. 23, 3947 (2023).
  22. F. A. Goudreault, S. Poncé, F. Giustino, and M. Côté, Effects of spin-orbit coupling and thermal expansion on the phonon-limited resistivity of Pb from first principles, arXiv:2410.20157.
  23. S. Poncé, E. Margine, C. Verdi, and F. Giustino, EPW: Electron-phonon coupling, transport and superconducting properties using maximally localized Wannier functions, Comput. Phys. Commun. 209, 116 (2016).
  24. H. Lee, S. Poncé, K. Bushick, S. Hajinazar, J. Lafuente-Bartolome, J. Leveillee, C. Lian, J.-M. Lihm, F. Macheda, H. Mori, H. Paudyal, W. H. Sio, S. Tiwari, M. Zacharias, X. Zhang, N. Bonini, E. Kioupakis, E. R. Margine, and F. Giustino, Electron–phonon physics from first principles using the EPW code, npj Comput. Mater. 9, 156 (2023).
  25. J.-J. Zhou, J. Park, I.-T. Lu, I. Maliyov, X. Tong, and M. Bernardi, perturbo: A software package for ab initio electron-phonon interactions, charge transport and ultrafast dynamics, Comput. Phys. Commun. 264, 107970 (2021).
  26. X. Gonze et al., The Abinit project: Impact, environment and recent developments, Comput. Phys. Commun. 248, 107042 (2020).
  27. A. Cepellotti, J. Coulter, A. Johansson, N. S. Fedorova, and B. Kozinsky, Phoebe: A high-performance framework for solving phonon and electron Boltzmann transport equations, J. Nonlinear Opt. Phys. Mater. 5, 035003 (2022).
  28. N. H. Protik, C. Li, M. Pruneda, D. Broido, and P. Ordejón, The elphbolt ab initio solver for the coupled electron-phonon Boltzmann transport equations, npj Comput. Mater. 8, 28 (2022).
  29. R. Claes, S. Poncé, G.-M. Rignanese, and G. Hautier, Phonon-limited electronic transport through first principles, Nat. Rev. Phys. 7, 73 (2025).
  30. J. Park, A. M. Ganose, and Y. Xia, Advances in theory and computational methods for next-generation thermoelectric materials, Appl. Phys. Rev. 12, 011339 (2025).
  31. T. Valla, A. V. Fedorov, P. D. Johnson, and S. L. Hulbert, Many-body effects in angle-resolved photoemission: Quasiparticle energy and lifetime of a Mo(110) surface state, Phys. Rev. Lett. 83, 2085 (1999).
  32. A. Eiguren, S. de Gironcoli, E. V. Chulkov, P. M. Echenique, and E. Tosatti, Electron-phonon interaction at the Be(0001) surface, Phys. Rev. Lett. 91, 166803 (2003).
  33. S. Moser, L. Moreschini, J. Jaćimović, O. S. Barišić, H. Berger, A. Magrez, Y. J. Chang, K. S. Kim, A. Bostwick, E. Rotenberg, L. Forró, and M. Grioni, Tunable polaronic conduction in anatase TiO2, Phys. Rev. Lett. 110, 196403 (2013).
  34. C. Chen, J. Avila, E. Frantzeskakis, A. Levy, and M. C. Asensio, Observation of a two-dimensional liquid of Fröhlich polarons at the bare SrTiO3 surface, Nat. Commun. 6, 8585 (2015).
  35. C. Cancellieri, A. S. Mishchenko, U. Aschauer, A. Filippetti, C. Faber, O. Barišić, V. Rogalev, T. Schmitt, N. Nagaosa, and V. N. Strocov, Polaronic metal state at the LaAlO3/SrTiO3 interface, Nat. Commun. 7, 10386 (2016).
  36. Z. Wang et al., Tailoring the nature and strength of electron-phonon interactions in the SrTiO3(001) 2D electron liquid, Nat. Mater. 15, 835 (2016).
  37. J. M. Riley, F. Caruso, C. Verdi, L. B. Duffy, M. D. Watson, L. Bawden, K. Volckaert, G. Van Der Laan, T. Hesjedal, M. Hoesch, F. Giustino, and P. D. C. King, Crossover from lattice to plasmonic polarons of a spin-polarised electron gas in ferromagnetic EuO, Nat. Commun. 9, 2305 (2018).
  38. C. Franchini, M. Reticcioli, M. Setvin, and U. Diebold, Polarons in materials, Nat. Rev. Mater. 6, 560 (2021).
  39. J.-J. Zhou and M. Bernardi, Predicting charge transport in the presence of polarons: The beyond-quasiparticle regime in SrTiO3, Phys. Rev. Res. 1, 033138 (2019).
  40. M. Pickem, E. Maggio, and J. M. Tomczak, Prototypical many-body signatures in transport properties of semiconductors, Phys. Rev. B 105, 085139 (2022).
  41. C. Verdi, F. Caruso, and F. Giustino, Origin of the crossover from polarons to Fermi liquids in transition metal oxides, Nat. Commun. 8, 15769 (2017).
  42. J. P. Nery, P. B. Allen, G. Antonius, L. Reining, A. Miglio, and X. Gonze, Quasiparticles and phonon satellites in spectral functions of semiconductors and insulators: Cumulants applied to the full first-principles theory and the Fröhlich polaron, Phys. Rev. B 97, 115145 (2018).
  43. G. Antonius, Y.-H. Chan, and S. G. Louie, Polaron spectral properties in doped ZnO and SrTiO3 from first principles, Phys. Rev. Res. 2, 043296 (2020).
  44. F. Caruso, C. Verdi, and F. Giustino, Many-body calculations of plasmon and phonon satellites in angle-resolved photoelectron spectra using the cumulant expansion approach, in Handbook of Materials Modeling: Methods: Theory and Modeling, edited by W. Andreoni and S. Yip (Springer International Publishing, Cham, 2020), pp. 341–365.
  45. J. C. de Abreu, J. P. Nery, M. Giantomassi, X. Gonze, and M. J. Verstraete, Spectroscopic signatures of nonpolarons: the case of diamond, Phys. Chem. Chem. Phys. 24, 12580 (2022).
  46. J.-M. Lihm and S. Poncé, Nonperturbative self-consistent electron-phonon spectral functions and transport, Phys. Rev. Lett. 134, 186401 (2025).
  47. J. P. Nery and F. Mauri, Nonperturbative Green’s function method to determine the electronic spectral function due to electron-phonon interactions: Application to a graphene model from weak to strong coupling, Phys. Rev. B 105, 245120 (2022).
  48. M. Houtput, L. Ranalli, C. Verdi, S. Klimin, S. Ragni, C. Franchini, and J. Tempere, First-principles theory of nonlinear long-range electron-phonon interaction, Phys. Rev. B 111, 184320 (2025).
  49. I. Zappacosta, M. Houtput, and J. Tempere, Nonlinear electron-phonon interactions in the Migdal-Eliashberg theory, Phys. Rev. B 111, 214525 (2025).
  50. B. K. Chang, J.-J. Zhou, N.-E. Lee, and M. Bernardi, Intermediate polaronic charge transport in organic crystals from a many-body first-principles approach, npj Comput. Mater. 8, 63 (2022).
  51. D. J. Abramovitch, J.-J. Zhou, J. Mravlje, A. Georges, and M. Bernardi, Combining electron-phonon and dynamical mean-field theory calculations of correlated materials: Transport in the correlated metal Sr2RuO4, Phys. Rev. Mater. 7, 093801 (2023).
  52. D. J. Abramovitch, J. Mravlje, J.-J. Zhou, A. Georges, and M. Bernardi, Respective roles of electron-phonon and electron-electron interactions in the transport and quasiparticle properties of SrVO3, Phys. Rev. Lett. 133, 186501 (2024).
  53. A. Troisi and G. Orlandi, Charge-transport regime of crystalline organic semiconductors: Diffusion limited by thermal off-diagonal electronic disorder, Phys. Rev. Lett. 96, 086601 (2006).
  54. S. Fratini, D. Mayou, and S. Ciuchi, The transient localization scenario for charge transport in crystalline organic materials, Adv. Funct. Mater. 26, 2292 (2016).
  55. J. M. Frost, Calculating polaron mobility in halide perovskites, Phys. Rev. B 96, 195202 (2017).
  56. J. H. Fetherolf, D. Golež, and T. C. Berkelbach, A unification of the Holstein polaron and dynamic disorder pictures of charge transport in organic crystals, Phys. Rev. X 10, 021062 (2020).
  57. F. Marsiglio, The spectral function of a one-dimensional Holstein polaron, Phys. Lett. A 180, 280 (1993).
  58. H. Fehske, J. Loos, and G. Wellein, Lattice polaron formation: Effects of nonscreened electron-phonon interaction, Phys. Rev. B 61, 8016 (2000).
  59. M. Hohenadler, H. G. Evertz, and W. von der Linden, Quantum Monte Carlo and variational approaches to the Holstein model, Phys. Rev. B 69, 024301 (2004).
  60. G. Schubert, G. Wellein, A. Weisse, A. Alvermann, and H. Fehske, Optical absorption and activated transport in polaronic systems, Phys. Rev. B 72, 104304 (2005).
  61. H. Fehske and S. A. Trugman, Numerical solution of the Holstein polaron problem, in Polarons in Advanced Materials, edited by A. S. Alexandrov (Springer Netherlands, Dordrecht, 2007), pp. 393–461.
  62. A. S. Mishchenko, N. Nagaosa, G. De Filippis, A. de Candia, and V. Cataudella, Mobility of Holstein polaron at finite temperature: An unbiased approach, Phys. Rev. Lett. 114, 146401 (2015).
  63. A. S. Mishchenko, L. Pollet, N. V. Prokof’ev, A. Kumar, D. L. Maslov, and N. Nagaosa, Polaron mobility in the “beyond quasiparticles” regime, Phys. Rev. Lett. 123, 076601 (2019).
  64. Y.-C. Wang and Y. Zhao, Diagrammatic quantum Monte Carlo toward the calculation of transport properties in disordered semiconductors, J. Chem. Phys. 156, 204116 (2022).
  65. V. Janković, Holstein polaron transport from numerically “exact” real-time quantum dynamics simulations, J. Chem. Phys. 159, 094113 (2023).
  66. S. Miladić and N. Vukmirović, Method for obtaining polaron mobility using real and imaginary time path-integral quantum Monte Carlo, Phys. Rev. B 107, 184315 (2023).
  67. V. Janković, P. Mitrić, D. Tanasković, and N. Vukmirović, Vertex corrections to conductivity in the Holstein model: A numerical-analytical study, Phys. Rev. B 109, 214312 (2024).
  68. H. Rammal, A. Ralko, S. Ciuchi, and S. Fratini, Transient localization from the interaction with quantum bosons, Phys. Rev. Lett. 132, 266502 (2024).
  69. V. Janković, Charge transport limited by nonlocal electron-phonon interaction. I. Hierarchical equations of motion approach, Phys. Rev. B 112, 035111 (2025).
  70. V. Janković, Charge transport limited by nonlocal electron-phonon interaction. II. Numerically exact quantum dynamics in the slow-phonon regime, Phys. Rev. B 112, 035112 (2025).
  71. Y. Luo, J. Park, and M. Bernardi, First-principles diagrammatic monte carlo for electron–phonon interactions and polaron, Nat. Phys. 21, 1275 (2025).
  72. S. F. Edwards, A new method for the evaluation of electric conductivity in metals, Philos. Mag. J. Theor. Exp. Appl. Phys. 3, 1020 (1958).
  73. S. Engelsberg and J. R. Schrieffer, Coupled electron-phonon system, Phys. Rev. 131, 993 (1963).
  74. T. Holstein, Theory of transport phenomena in an electron-phonon gas, Ann. Phys. (N.Y.) 29, 410 (1964).
  75. R. E. Prange and L. P. Kadanoff, Transport theory for electron-phonon interactions in metals, Phys. Rev. 134, A566 (1964).
  76. G. D. Mahan, Many-Particle Physics, 3rd ed. (Kluwer Academic/Plenum Publishers, New York, 2000).
  77. W. Hänsch and G. D. Mahan, Transport equations for many-particle systems, Phys. Rev. B 28, 1902 (1983).
  78. S. Kim, S. Woo, and H. Min, Vertex corrections to the dc conductivity in anisotropic multiband systems, Phys. Rev. B 99, 165107 (2019).
  79. D. N. Basov, R. D. Averitt, D. van der Marel, M. Dressel, and K. Haule, Electrodynamics of correlated electron materials, Rev. Mod. Phys. 83, 471 (2011).
  80. J. B. Baxter and C. A. Schmuttenmaer, Carrier dynamics in bulk ZnO. I. Intrinsic conductivity measured by terahertz time-domain spectroscopy, Phys. Rev. B 80, 235205 (2009).
  81. H. Scher, Far-infrared absorptivity of normal lead, Phys. Rev. Lett. 25, 759 (1970).
  82. P. B. Allen, Electron-phonon effects in the infrared properties of metals, Phys. Rev. B 3, 305 (1971).
  83. I. Kupčić, Damping effects in doped graphene: The relaxation-time approximation, Phys. Rev. B 90, 205426 (2014).
  84. I. Kupčić, General theory of intraband relaxation processes in heavily doped graphene, Phys. Rev. B 91, 205428 (2015).
  85. P. Gosar and S.-I. Choi, Linear-response theory of the electron mobility in molecular crystals, Phys. Rev. 150, 529 (1966).
  86. P. Gosar and I. Vilfan, Phonon-assisted current in organic molecular crystals, Mol. Phys. 18, 49 (1970).
  87. K. Hannewald and P. A. Bobbert, Anisotropy effects in phonon-assisted charge-carrier transport in organic molecular crystals, Phys. Rev. B 69, 075212 (2004).
  88. J.-M. Lihm, electronphonon.jl (2025), https://github.com/jaemolihm/ElectronPhonon.jl.
  89. J.-M. Lihm and S. Poncé, Beyond-quasiparticle transport with vertex correction: Self-consistent ladder formalism for electron-phonon interactions, Mater. Cloud Arch. 2025, 95 (2025).
  90. M. Capone and S. Ciuchi, Polaron crossover and bipolaronic metal-insulator transition in the half-filled Holstein model, Phys. Rev. Lett. 91, 186405 (2003).
  91. J. Bauer, J. E. Han, and O. Gunnarsson, Quantitative reliability study of the Migdal-Eliashberg theory for strong electron-phonon coupling in superconductors, Phys. Rev. B 84, 184531 (2011).
  92. N. Säkkinen, Y. Peng, H. Appel, and R. Van Leeuwen, Many-body Green’s function theory for electron-phonon interactions: Ground state properties of the Holstein dimer, J. Chem. Phys. 143, 234101 (2015).
  93. N. Säkkinen, Y. Peng, H. Appel, and R. Van Leeuwen, Many-body Green’s function theory for electron-phonon interactions: The Kadanoff-Baym approach to spectral properties of the Holstein dimer, J. Chem. Phys. 143, 234102 (2015).
  94. L. Rademaker, Y. Wang, T. Berlijn, and S. Johnston, Enhanced superconductivity due to forward scattering in FeSe thin films on SrTiO3 substrates, New J. Phys. 18, 022001 (2016).
  95. H.-T. Chen, G. Cohen, A. J. Millis, and D. R. Reichman, Anderson-Holstein model in two flavors of the noncrossing approximation, Phys. Rev. B 93, 174309 (2016).
  96. I. Esterlis, B. Nosarzewski, E. W. Huang, B. Moritz, T. P. Devereaux, D. J. Scalapino, and S. A. Kivelson, Breakdown of the Migdal-Eliashberg theory: A determinant quantum Monte Carlo study, Phys. Rev. B 97, 140501(R) (2018).
  97. P. M. Dee, K. Nakatsukasa, Y. Wang, and S. Johnston, Temperature-filling phase diagram of the two-dimensional Holstein model in the thermodynamic limit by self-consistent Migdal approximation, Phys. Rev. B 99, 024514 (2019).
  98. P. M. Dee, J. Coulter, K. G. Kleiner, and S. Johnston, Relative importance of nonlinear electron-phonon coupling and vertex corrections in the Holstein model, Commun. Phys. 3, 145 (2020).
  99. P. Mitrić, V. Janković, N. Vukmirović, and D. Tanasković, Spectral functions of the Holstein polaron: Exact and approximate solutions, Phys. Rev. Lett. 129, 096401 (2022).
  100. P. Mitrić, V. Janković, N. Vukmirović, and D. Tanasković, Cumulant expansion in the Holstein model: Spectral functions and mobility, Phys. Rev. B 107, 125165 (2023).
  101. G. Stefanucci, R. van Leeuwen, and E. Perfetto, In and out-of-equilibrium ab initio theory of electrons and phonons, Phys. Rev. X 13, 031026 (2023).
  102. O. Gunnarsson, M. W. Haverkort, and G. Sangiovanni, Analytical continuation of imaginary axis data for optical conductivity, Phys. Rev. B 82, 165125 (2010).
  103. O. Goulko, A. S. Mishchenko, L. Pollet, N. Prokof’ev, and B. Svistunov, Numerical analytic continuation: Answers to well-posed questions, Phys. Rev. B 95, 014102 (2017).
  104. J. S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. (N.Y.) 2, 407 (1961).
  105. L. V. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz. 47, 1515 (1964), https://inspirehep.net/literature/3567.
  106. F. B. Kugler, Seung-Sup B. Lee, and J. von Delft, Multipoint correlation functions: Spectral representation and numerical evaluation, Phys. Rev. X 11, 041006 (2021).
  107. H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics: An Introduction (Oxford University Press, Oxford, 2004).
  108. G. Stefanucci and R. Van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction (Cambridge University Press, Cambridge, England, 2013).
  109. J.-M. Lihm and C.-H. Park, Phonon-induced renormalization of electron wave functions, Phys. Rev. B 101, 121102(R) (2020).
  110. V. Brousseau-Couture, G. Antonius, and M. Côté, Temperature dependence of the topological phase transition of BiTeI from first principles, Phys. Rev. Res. 2, 023185 (2020).
  111. F. Macheda, F. Mauri, and T. Sohier, Ab initio Van der Waals electrodynamics: Polaritons and electron scattering from plasmons and phonons in BN-capped graphene, Phys. Rev. B 110, 115407 (2024).
  112. J.-M. Lihm and C.-H. Park, Plasmon-phonon hybridization in doped semiconductors from first principles, Phys. Rev. Lett. 133, 116402 (2024).
  113. J. Krsnik, D. Novko, and O. S. Barišić, Superconductivity in two-dimensional systems enhanced by nonadiabatic phonon-production effects, Phys. Rev. B 110, L180505 (2024).
  114. J. Berges, N. Girotto, T. Wehling, N. Marzari, and S. Poncé, Phonon self-energy corrections: To screen, or not to screen, Phys. Rev. X 13, 041009 (2023).
  115. N.-E. Lee, J.-J. Zhou, H.-Y. Chen, and M. Bernardi, Ab initio electron-two-phonon scattering in GaAs from next-to-leading order perturbation theory, Nat. Commun. 11, 1607 (2020).
  116. B. Hatanpää, A. Y. Choi, P. S. Cheng, and A. J. Minnich, Two-phonon scattering in nonpolar semiconductors: A first-principles study of warm electron transport in Si, Phys. Rev. B 107, L041110 (2023).
  117. I. Esho and A. J. Minnich, Charge transport in BAs and the role of two-phonon scattering, Phys. Rev. B 108, 165202 (2023).
  118. J. Sun and A. J. Minnich, Transport and noise of hot electrons in GaAs using a semianalytical model of two-phonon polar optical phonon scattering, Phys. Rev. B 107, 205201 (2023).
  119. P. B. Allen and V. Heine, Theory of the temperature dependence of electronic band structures, J. Phys. C 9, 2305 (1976).
  120. P. B. Allen and M. Cardona, Theory of the temperature dependence of the direct gap of germanium, Phys. Rev. B 23, 1495 (1981).
  121. P. B. Allen and M. Cardona, Temperature dependence of the direct gap of Si and Ge, Phys. Rev. B 27, 4760 (1983).
  122. X. Gonze, P. Boulanger, and M. Côté, Theoretical approaches to the temperature and zero-point motion effects on the electronic band structure, Ann. Phys. (Amsterdam) 523, 168 (2011).
  123. S. Poncé, G. Antonius, P. Boulanger, E. Cannuccia, A. Marini, M. Côté, and X. Gonze, Verification of first-principles codes: Comparison of total energies, phonon frequencies, electron-phonon coupling and zero-point motion correction to the gap between ABINIT and QE/Yambo, Comput. Mater. Sci. 83, 341 (2014).
  124. J.-M. Lihm and C.-H. Park, Wannier function perturbation theory: Localized representation and interpolation of wave function perturbation, Phys. Rev. X 11, 041053 (2021).
  125. J.-M. Lihm and C.-H. Park, Nonlinear Hall effect from long-lived valley-polarizing relaxons, Phys. Rev. Lett. 132, 106402 (2024).
  126. J. J. Kas, J. J. Rehr, and L. Reining, Cumulant expansion of the retarded one-electron Green function, Phys. Rev. B 90, 085112 (2014).
  127. J. J. Kas and J. J. Rehr, Finite temperature Green’s function approach for excited state and thermodynamic properties of cool to warm dense matter, Phys. Rev. Lett. 119, 176403 (2017).
  128. S. M. Story, J. J. Kas, F. D. Vila, M. J. Verstraete, and J. J. Rehr, Cumulant expansion for phonon contributions to the electron spectral function, Phys. Rev. B 90, 195135 (2014).
  129. T. Holstein, Studies of polaron motion: Part I. The molecular-crystal model, Ann. Phys. (N.Y.) 8, 325 (1959).
  130. S. Barišić, J. Labbé, and J. Friedel, Tight binding and transition-metal superconductivity, Phys. Rev. Lett. 25, 919 (1970).
  131. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42, 1698 (1979).
  132. M. Capone, W. Stephan, and M. Grilli, Small-polaron formation and optical absorption in Su-Schrieffer-Heeger and Holstein models, Phys. Rev. B 56, 4484 (1997).
  133. D. J. J. Marchand, G. De Filippis, V. Cataudella, M. Berciu, N. Nagaosa, N. V. Prokof’ev, A. S. Mishchenko, and P. C. E. Stamp, Sharp transition for single polarons in the one-dimensional Su-Schrieffer-Heeger model, Phys. Rev. Lett. 105, 266605 (2010).
  134. H. P. H. Fröhlich and S. Zienau, XX. properties of slow electrons in polar materials, Philos. Mag. 41, 221 (1950).
  135. T. Hahn, S. Klimin, J. Tempere, J. T. Devreese, and C. Franchini, Diagrammatic Monte Carlo study of Fröhlich polaron dispersion in two and three dimensions, Phys. Rev. B 97, 134305 (2018).
  136. A. S. Mishchenko, N. V. Prokof’ev, A. Sakamoto, and B. V. Svistunov, Diagrammatic quantum Monte Carlo study of the Fröhlich polaron, Phys. Rev. B 62, 6317 (2000).
  137. C. Verdi and F. Giustino, Fröhlich electron-phonon vertex from first principles, Phys. Rev. Lett. 115, 176401 (2015).
  138. J. Sjakste, N. Vast, M. Calandra, and F. Mauri, Wannier interpolation of the electron-phonon matrix elements in polar semiconductors: Polar-optical coupling in GaAs, Phys. Rev. B 92, 054307 (2015).
  139. D. M. Larsen, Polaron energy spectrum, Phys. Rev. 144, 697 (1966).
  140. N. Kandolf, C. Verdi, and F. Giustino, Many-body Green’s function approaches to the doped Fröhlich solid: Exact solutions and anomalous mass enhancement, Phys. Rev. B 105, 085148 (2022).
  141. M. Berciu, Green’s function of a dressed particle, Phys. Rev. Lett. 97, 036402 (2006).
  142. F. J. Morin and J. P. Maita, Electrical properties of silicon containing arsenic and boron, Phys. Rev. 96, 28 (1954).
  143. G. W. Ludwig and R. L. Watters, Drift and conductivity mobility in silicon, Phys. Rev. 101, 1699 (1956).
  144. R. A. Logan and A. J. Peters, Impurity effects upon mobility in silicon, J. Appl. Phys. 31, 122 (1960).
  145. P. Norton, T. Braggins, and H. Levinstein, Impurity and lattice scattering parameters as determined from Hall and mobility analysis in n-type silicon, Phys. Rev. B 8, 5632 (1973).
  146. C. Jacoboni, C. Canali, G. Ottaviani, and A. Alberigi Quaranta, A review of some charge transport properties of silicon, Solid-State Electron. 20, 77 (1977).
  147. A. Hutson, Electronic properties of ZnO, J. Phys. Chem. Solids 8, 467 (1959).
  148. P. Wagner and R. Helbig, Halleffekt und anisotropie der beweglichkeit der elektronen in ZnO, J. Phys. Chem. Solids 35, 327 (1974).
  149. D. Look, D. Reynolds, J. Sizelove, R. Jones, C. Litton, G. Cantwell, and W. Harsch, Electrical properties of bulk ZnO, Solid State Commun. 105, 399 (1998).
  150. T. Makino, Y. Segawa, A. Tsukazaki, A. Ohtomo, and M. Kawasaki, Electron transport in ZnO thin films, Appl. Phys. Lett. 87, 022101 (2005).
  151. M. van Exter and D. Grischkowsky, Carrier dynamics of electrons and holes in moderately doped silicon, Phys. Rev. B 41, 12140 (1990).
  152. T.-I. Jeon and D. Grischkowsky, Nature of conduction in doped silicon, Phys. Rev. Lett. 78, 1106 (1997).
  153. A. Nozaki, H. Yoshikawa, T. Wada, H. Yamauchi, and S. Tanaka, Layered perovskite compounds Srn+1VnO3n+1 (n=1, 2, 3, and ∞), Phys. Rev. B 43, 181 (1991).
  154. Y. Lan, X. Chen, and M. He, Structure, magnetic susceptibility and resistivity properties of SrVO3, J. Alloys Compd. 354, 95 (2003).
  155. M. Itoh, M. Shikano, H. Kawaji, and T. Nakamura, Structural aspects on the variations of electric and magnetic properties of the layered compound system Srn+1VnO3n+1−δ(n=1,2,3∞), Solid State Commun. 80, 545 (1991).
  156. I. H. Inoue, O. Goto, H. Makino, N. E. Hussey, and M. Ishikawa, Bandwidth control in a perovskite-type 3 d1-correlated metal Ca1−xSrxVO3. I. Evolution of the electronic properties and effective mass, Phys. Rev. B 58, 4372 (1998).
  157. D. Reyes Ardila, J. Andreeta, and H. Basso, Preparation, microstructural and electrical characterization of SrVO3 single crystal fiber, J. Cryst. Growth 211, 313 (2000).
  158. M. Brahlek, L. Zhang, C. Eaton, H.-T. Zhang, and R. Engel-Herbert, Accessing a growth window for SrVO3 thin films, Appl. Phys. Lett. 107, 143108 (2015).
  159. A. Fouchet, M. Allain, B. Bérini, E. Popova, P.-E. Janolin, N. Guiblin, E. Chikoidze, J. Scola, D. Hrabovsky, Y. Dumont, and N. Keller, Study of the electronic phase transition with low dimensionality in SrVO3 thin films, Mater. Sci. Eng. B 212, 7 (2016).
  160. L. Zhang, Y. Zhou, L. Guo, W. Zhao, A. Barnes, H.-T. Zhang, C. Eaton, Y. Zheng, M. Brahlek, H. F. Haneef, N. J. Podraza, M. H. Chan, V. Gopalan, K. M. Rabe, and R. Engel-Herbert, Correlated metals as transparent conductors, Nat. Mater. 15, 204 (2016).
  161. R. Xu, Y. Ji, R. Bouchilaoun, F. Qian, M. Li, X. Zhang, R. Tang, R. Zhao, S. Misra, H. Wang, W. Li, C. Kan, D. Shi, J. Fan, and H. Yang, Optical and electrical properties of (111)-oriented epitaxial SrVO3 thin films, Ceram. Int. 45, 11304 (2019).
  162. L. Shoham, M. Baskin, M.-G. Han, Y. Zhu, and L. Kornblum, Scalable synthesis of the transparent conductive oxide SrVO3, Adv. Electron. Mater. 6, 1900584 (2020).
  163. M. Mirjolet, F. Rivadulla, P. Marsik, V. Borisov, R. Valentí, and J. Fontcuberta, Electron-phonon coupling and electron-phonon scattering in SrVO3, Adv. Sci 8, 2004207 (2021).
  164. J. Roth, T. Kuznetsova, L. Miao, A. Pogrebnyakov, N. Alem, and R. Engel-Herbert, Self-regulated growth of [111]-oriented perovskite oxide films using hybrid molecular beam epitaxy, APL Mater. 9, 021114 (2021).
  165. G. Ahn, M. Zingl, S. J. Noh, M. Brahlek, J. D. Roth, R. Engel-Herbert, A. J. Millis, and S. J. Moon, Low-energy interband transition in the infrared response of the correlated metal SrVO3 in the ultraclean limit, Phys. Rev. B 106, 085133 (2022).
  166. M. Brahlek, J. D. Roth, L. Zhang, M. Briggeman, P. Irvin, J. Lapano, J. Levy, T. Birol, and R. Engel-Herbert, Hidden transport phenomena in an ultraclean correlated metal, Nat. Commun. 15, 5304 (2024).
  167. J. Lee-Hand, H. LaBollita, F. B. Kugler, L. V. Muñoz, J. Kaye, S. Beck, A. Hampel, A. Georges, and C. E. Dreyer, Fermi-liquid T2 resistivity: Dynamical mean-field theory meets experiment, arXiv:2412.16363.
  168. H. LaBollita, J. Lee-Hand, F. B. Kugler, L. V. Muñoz, S. Beck, A. Hampel, J. Kaye, A. Georges, and C. E. Dreyer, Low-temperature transport in high-conductivity correlated metals: A density-functional plus dynamical mean-field study of cubic perovskites, arXiv:2505.04508.
  169. J. Coulter, F. B. Kugler, H. LaBollita, A. Georges, and C. E. Dreyer, Mechanisms for the ultralow room-temperature resistivity of SrMoO3, arXiv:2506.10143.
  170. S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Verstraete, and X. Gonze, Temperature dependence of the electronic structure of semiconductors and insulators, J. Chem. Phys. 143, 102813 (2015).
  171. W. Bludau, A. Onton, and W. Heinke, Temperature dependence of the band gap of silicon, J. Appl. Phys. 45, 1846 (1974).
  172. P. M. M. C. de Melo, J. C. de Abreu, B. Guster, M. Giantomassi, Z. Zanolli, X. Gonze, and M. J. Verstraete, High-throughput analysis of Fröhlich-type polaron models, npj Comput. Mater. 9, 147 (2023).
  173. I. A. Nekrasov, K. Held, G. Keller, D. E. Kondakov, T. Pruschke, M. Kollar, O. K. Andersen, V. I. Anisimov, and D. Vollhardt, Momentum-resolved spectral functions of SrVO3 calculated by LDA+DMFT, Phys. Rev. B 73, 155112 (2006).
  174. R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12, 570 (1957).
  175. P. C. Martin and J. Schwinger, Theory of many-particle systems. I, Phys. Rev. 115, 1342 (1959).
  176. L. G. Aslamazov and A. I. Larkin, Effect of fluctuations on the priperties of a superconductor above the critical temperature, Sov. Phys.-Solid State (Engl. Transl.) 10, 875 (1968), https://www.osti.gov/biblio/7298727.
  177. E. Cappelluti, L. Benfatto, M. Manzardo, and A. B. Kuzmenko, Charged-phonon theory, Fano effect in the optical spectroscopy of bilayer graphene, Phys. Rev. B 86, 115439 (2012).
  178. O. Bistoni, P. Barone, E. Cappelluti, L. Benfatto, and F. Mauri, Giant effective charges and piezoelectricity in gapped graphene, 2D Mater. 6, 045015 (2019).
  179. L. Binci, P. Barone, and F. Mauri, First-principles theory of infrared vibrational spectroscopy of metals and semimetals: Application to graphite, Phys. Rev. B 103, 134304 (2021).
  180. G. Marchese, F. Macheda, L. Binci, M. Calandra, P. Barone, and F. Mauri, Born effective charges and vibrational spectra in superconducting and bad conducting metals, Nat. Phys. 20, 88 (2024).
  181. J. Park, Y. Luo, J.-J. Zhou, and M. Bernardi, Many-body theory of phonon-induced spin relaxation and decoherence, Phys. Rev. B 106, 174404 (2022).
  182. J. Park, J.-J. Zhou, Y. Luo, and M. Bernardi, Predicting phonon-induced spin decoherence from first principles: Colossal spin renormalization in condensed matter, Phys. Rev. Lett. 129, 197201 (2022).
  183. J. Hu, W. Li, H. Wang, and K. Chang, Phonon-mediated nonlinear optical responses and quantum geometry, arXiv:2410.09677 [Phys. Rev. Lett. (to be published)].
  184. E. Blount, Formalisms of Band Theory (Academic Press, New York, 1962), pp. 305–373.
  185. M. Graf and P. Vogl, Electromagnetic fields and dielectric response in empirical tight-binding theory, Phys. Rev. B 51, 4940 (1995).
  186. See Supplemental Material at http://link.aps.org/supplemental/10.1103/mtjf-ylf7 for theoretical details, computational details, and additional results, which include Refs. [187–196].
  187. David Abramovitch (private communication).
  188. N. Marzari, D. Vanderbilt, A. De Vita, and M. C. Payne, Thermal contraction and disordering of the Al(110) surface, Phys. Rev. Lett. 82, 3296 (1999).
  189. P. Giannozzi et al., Advanced capabilities for materials modelling with quantum espresso, J. Phys. Condens. Matter 29, 465901 (2017).
  190. D. R. Hamann, Optimized norm-conserving Vanderbilt pseudopotentials, Phys. Rev. B 88, 085117 (2013).
  191. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  192. M. van Setten, M. Giantomassi, E. Bousquet, M. Verstraete, D. Hamann, X. Gonze, and G.-M. Rignanese, The PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table, Comput. Phys. Commun. 226, 39 (2018).
  193. G. Pizzi et al., wannier90 as a community code: New features and applications, J. Phys. Condens. Matter 32, 165902 (2020).
  194. J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, julia: A fresh approach to numerical computing, SIAM Rev. 59, 65 (2017).
  195. G. Brunin, H. P. C. Miranda, M. Giantomassi, M. Royo, M. Stengel, M. J. Verstraete, X. Gonze, G.-M. Rignanese, and G. Hautier, Electron-phonon beyond Fröhlich: Dynamical quadrupoles in polar and covalent solids, Phys. Rev. Lett. 125, 136601 (2020).
  196. V. A. Jhalani, J.-J. Zhou, J. Park, C. E. Dreyer, and M. Bernardi, Piezoelectric electron-phonon interaction from ab initio dynamical quadrupoles: Impact on charge transport in wurtzite GaN, Phys. Rev. Lett. 125, 136602 (2020).
  197. N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous Hall effect, Rev. Mod. Phys. 82, 1539 (2010).
  198. X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt, Ab initio calculation of the anomalous Hall conductivity by wannier interpolation, Phys. Rev. B 74, 195118 (2006).
  199. I. C. Cheeseman, The structure of the long wave absorption edge of insulating crystals, Proc. Phys. Soc. London Sect. A 65, 25 (1952).
  200. L. H. Hall, J. Bardeen, and F. J. Blatt, Infrared absorption spectrum of germanium, Phys. Rev. 95, 559 (1954).
  201. F. Bassani and G. P. Parravicini, Electronic States and Optical Transitions in Solids (Pergamon Press, New York, 1975).
  202. E. Kioupakis, P. Rinke, A. Schleife, F. Bechstedt, and C. G. Van de Walle, Free-carrier absorption in nitrides from first principles, Phys. Rev. B 81, 241201(R) (2010).
  203. J. Noffsinger, E. Kioupakis, C. G. Van de Walle, S. G. Louie, and M. L. Cohen, Phonon-assisted optical absorption in silicon from first principles, Phys. Rev. Lett. 108, 167402 (2012).
  204. S. Tiwari, E. Kioupakis, J. Menendez, and F. Giustino, Unified theory of optical absorption and luminescence including both direct and phonon-assisted processes, Phys. Rev. B 109, 195127 (2024).
  205. W. H. Butler, Theory of electronic transport in random alloys: Korringa-Kohn-Rostoker coherent-potential approximation, Phys. Rev. B 31, 3260 (1985).
  206. P. Mitrić, Spectral functions and mobility of the Holstein polaron, Ph.D. thesis, University of Belgrade, Belgrade, 2023.
  207. J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford University Press, Oxford, 2001).
  208. J. Kovačević, M. Ferrero, and J. c. v. Vučičević, Toward numerically exact computation of conductivity in the thermodynamic limit of interacting lattice models, Phys. Rev. Lett. 135, 016502 (2025).
  209. W. Yang, S. Tiwari, F. Giustino, and Y.-W. Son, First-principles electron-phonon interactions with self-consistent Hubbard interaction: Application to transparent conducting oxides, Phys. Rev. B 112, 075203 (2025).
  210. D. Rode, Chapter 1 Low-Field Electron Transport (Elsevier, New York, 1975), pp. 1–89.
  211. D. J. Abramovitch, J. Coulter, S. Beck, and A. Millis, Electron-phonon coupling in correlated metals: A dynamical mean-field theory study, Phys. Rev. B 112, 075113 (2025).
  212. J.-J. Zhou, J. Park, I. Timrov, A. Floris, M. Cococcioni, N. Marzari, and M. Bernardi, Ab initio electron-phonon interactions in correlated electron systems, Phys. Rev. Lett. 127, 126404 (2021).
  213. A. Poliukhin, N. Colonna, F. Libbi, S. Poncé, and N. Marzari, Carrier mobilities and electron-phonon interactions beyond DFT, arXiv:2508.14852.
  214. M. Fox, Optical Properties of Solids, 2nd ed. (Cambridge University Press, Cambridge, England, 2010), p. 183.
  215. J. Vučičević, S. Predin, and M. Ferrero, Charge fluctuations, hydrodynamics, and transport in the square-lattice Hubbard model, Phys. Rev. B 107, 155140 (2023).
  216. G. Baym, Self-consistent approximations in many-body systems, Phys. Rev. 127, 1391 (1962).
  217. J. C. Ward, An identity in quantum electrodynamics, Phys. Rev. 78, 182 (1950).
  218. Y. Takahashi, On the generalized Ward identity, Nuovo Cimento (1955–1965) 6, 371 (1957).
  219. F. Krien, E. G. C. P. van Loon, H. Hafermann, J. Otsuki, M. I. Katsnelson, and A. I. Lichtenstein, Conservation in two-particle self-consistent extensions of dynamical mean-field theory, Phys. Rev. B 96, 075155 (2017).
  220. F. Krien, Conserving dynamical mean-field approaches to strongly correlated systems, Ph.D. thesis, Staats-und Universitätsbibliothek Hamburg Carl von Ossietzky, 2018.
  221. H. Haug, A.-P. Jauho et al., Quantum Kinetics in Transport and Optics of Semiconductors (Springer, New York, 2008), Vol. 2.
  222. M. Luisier and G. Klimeck, Atomistic full-band simulations of silicon nanowire transistors: Effects of electron-phonon scattering, Phys. Rev. B 80, 155430 (2009).
  223. A. Szabó, R. Rhyner, and M. Luisier, Ab initio simulation of single- and few-layer MoS2 transistors: Effect of electron-phonon scattering, Phys. Rev. B 92, 035435 (2015).
  224. J. Backman, Y. Lee, and M. Luisier, Phonon-limited transport in two-dimensional materials: A unified approach for ab initio mobility and current calculations, Phys. Rev. Appl. 21, 054017 (2024).
  225. G. Stefanucci and E. Perfetto, Semiconductor electron-phonon equations: A rung above Boltzmann in the many-body ladder, SciPost Phys. 16, 073 (2024).
  226. P. Lipavský, V. Špička, and B. Velický, Generalized Kadanoff-Baym ansatz for deriving quantum transport equations, Phys. Rev. B 34, 6933 (1986).
  227. D. Karlsson, R. van Leeuwen, Y. Pavlyukh, E. Perfetto, and G. Stefanucci, Fast Green’s function method for ultrafast electron-boson dynamics, Phys. Rev. Lett. 127, 036402 (2021).
  228. Y. Pavlyukh, E. Perfetto, D. Karlsson, R. van Leeuwen, and G. Stefanucci, Time-linear scaling nonequilibrium Green’s function methods for real-time simulations of interacting electrons and bosons. I. formalism, Phys. Rev. B 105, 125134 (2022).
  229. W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phenomena (Springer Science & Business Media, New York, 2013).
  230. M. Calandra, G. Profeta, and F. Mauri, Adiabatic and nonadiabatic phonon dispersion in a Wannier function approach, Phys. Rev. B 82, 165111 (2010).
  231. W. H. Sio, C. Verdi, S. Poncé, and F. Giustino, Ab initio theory of polarons: Formalism and applications, Phys. Rev. B 99, 235139 (2019).
  232. W. H. Sio, C. Verdi, S. Poncé, and F. Giustino, Polarons from first principles, without supercells, Phys. Rev. Lett. 122, 246403 (2019).
  233. J. Lafuente-Bartolome, C. Lian, W. H. Sio, I. G. Gurtubay, A. Eiguren, and F. Giustino, Ab initio self-consistent many-body theory of polarons at all couplings, Phys. Rev. B 106, 075119 (2022).
  234. J. Lafuente-Bartolome, C. Lian, W. H. Sio, I. G. Gurtubay, A. Eiguren, and F. Giustino, Unified approach to polarons and phonon-induced band structure renormalization, Phys. Rev. Lett. 129, 076402 (2022).
  235. L. D. Landau, J. S. Bell, M. Kearsley, L. Pitaevskii, E. Lifshitz, and J. Sykes, Electrodynamics of Continuous Media (Elsevier, Amsterdam, 1984), Vol. 8.
  236. D. B. Melrose and R. C. McPhedran, Electromagnetic Processes in Dispersive Media (Cambridge University Press, Cambridge, England, 1991).
  237. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68, 13 (1996).
  238. G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Electronic structure calculations with dynamical mean-field theory, Rev. Mod. Phys. 78, 865 (2006).
  239. G. Rohringer, H. Hafermann, A. Toschi, A. A. Katanin, A. E. Antipov, M. I. Katsnelson, A. I. Lichtenstein, A. N. Rubtsov, and K. Held, Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory, Rev. Mod. Phys. 90, 025003 (2018).
  240. A. Kauch, P. Pudleiner, K. Astleithner, P. Thunström, T. Ribic, and K. Held, Generic optical excitations of correlated systems: π-tons, Phys. Rev. Lett. 124, 047401 (2020).
  241. J. Krsnik, O. Simard, P. Werner, A. Kauch, and K. Held, Displaced Drude peak from π-ton vertex corrections, Phys. Rev. B 110, 075118 (2024).
  242. A. Gleis, Seung-Sup B. Lee, G. Kotliar, and J. von Delft, Dynamical scaling, Planckian dissipation due to heavy-fermion quantum criticality, Phys. Rev. Lett. 134, 106501 (2025).
  243. C. Aversa and J. E. Sipe, Nonlinear optical susceptibilities of semiconductors: Results with a length-gauge analysis, Phys. Rev. B 52, 14636 (1995).
  244. G. B. Ventura, D. J. Passos, J. M. B. Lopes dos Santos, J. M. Viana Parente Lopes, and N. M. R. Peres, Gauge covariances and nonlinear optical responses, Phys. Rev. B 96, 035431 (2017).
  245. D. J. Passos, G. B. Ventura, J. M. Viana Parente Lopes, J. M. B. Lopesdos Santos, and N. M. R. Peres, Nonlinear optical responses of crystalline systems: Results from a velocity gauge analysis, Phys. Rev. B 97, 235446 (2018).
  246. B. Adolph, V. I. Gavrilenko, K. Tenelsen, F. Bechstedt, and R. Del Sole, Nonlocality and many-body effects in the optical properties of semiconductors, Phys. Rev. B 53, 9797 (1996).
  247. F. Giustino, M. L. Cohen, and S. G. Louie, Electron-phonon interaction using Wannier functions, Phys. Rev. B 76, 165108 (2007).
  248. J. Bennetto and D. Vanderbilt, Semiconductor effective charges from tight-binding theory, Phys. Rev. B 53, 15417 (1996).
  249. J. Ibañez-Azpiroz, F. de Juan, and I. Souza, Assessing the role of interatomic position matrix elements in tight-binding calculations of optical properties, SciPost Phys. 12, 070 (2022).
  250. M. Brambilla and R. Bilato, Advances in numerical simulations of ion cyclotron heating of non-Maxwellian plasmas, Nucl. Fusion 49, 085004 (2009).
  251. R. Bilato, O. Maj, and M. Brambilla, An algorithm for fast Hilbert transform of real functions, Adv. Comput. Math. 40, 1159 (2014).

Outline

Information

Phys. Rev. X 16, 011008– Published 13 January, 2026
  • Received 22 June 2025
  • Revised 13 October 2025
  • Accepted 4 December 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation