First-principles electron-phonon interactions and polarons in the parent cuprate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>La</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>CuO</mml:mi><mml:mn>4</mml

Understanding electronic interactions in high-temperature superconductors is an outstanding challenge. In the widely studied cuprate materials, experimental evidence points to strong electron-phonon (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>e</a:mi></a:math>-ph) coupling and broad photoemission spectra. Yet, the microscopic origin of this behavior is not fully understood. Here, we study <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>e</b:mi></b:math>-ph interactions and polarons in a prototypical parent (undoped) cuprate, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msub><c:mi>La</c:mi><c:mn>2</c:mn></c:msub><c:msub><c:mi>CuO</c:mi><c:mn>4</c:mn></c:msub></c:mrow></c:math> (LCO), by means of first-principles calculations. Leveraging parameter-free Hubbard-corrected density functional theory, we obtain a ground state with the band gap and Cu magnetic moment in nearly exact agreement with experiments. This enables a quantitative characterization of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>e</d:mi></d:math>-ph interactions. Our calculations reveal two classes of longitudinal optical (LO) phonons with strong <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>e</e:mi></e:math>-ph coupling to hole states. These modes consist of bond stretching and bond bending in the Cu-O plane as well as vibrations of apical O atoms. The hole spectral functions, obtained with a cumulant method that can capture strong <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>e</f:mi></f:math>-ph coupling, exhibit broad quasiparticle peaks with a small spectral weight (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>Z</g:mi><g:mo>≈</g:mo><g:mn>0.25</g:mn></g:mrow></g:math>) and pronounced LO-phonon sidebands characteristic of polaron effects. Our calculations predict features observed in photoemission spectra, including a 40-meV peak in the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>e</h:mi></h:math>-ph coupling distribution function not explained by existing models. These results show that the universal strong <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>e</i:mi></i:math>-ph coupling found experimentally in doped lanthanum cuprates is also present in the parent compound, and elucidate its microscopic origin.