{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T23:38:50Z","timestamp":1792625930208,"version":"4.5.0"},"reference-count":0,"publisher":"IOP Publishing","license":[{"start":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T00:00:00Z","timestamp":1792540800000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T00:00:00Z","timestamp":1792540800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Phys.: Condens. Matter"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>The length-gauge (LG) time-dependent Liouville equation is a major workhorse to simulate ultrafast dynamics. The equation features the commutator between the density operator \u03c1 and the position operator r, nk|[r, \u03c1]|mk \u2032 , where |nk and |mk \u2032 are Bloch states for bands n and m at crystal momenta k and k \u2032 , respectively. This commutator has two irreconcilable versions: Kohn and Luttinger (KL) (Phys. Rev. 108, 590 (1954)) have two partial derivatives with respect to k and k \u2032 , but Aversa and Sipe (AS) (Phys. Rev. B 52, 14636 (1995)) have one partial derivative and one Dirac delta. For several decades, this outstanding discrepancy has received little attention. Here, we show that the generalized LG time-dependent Liouville equation has two partial derivatives, but no Dirac delta, regardless of whether the electric field is homogeneous or not. Physically, the k-off-diagonal matrix elements of the position operator provide a channel by which the electric field generates k-off-diagonal matrix elements for \u03c1 in LG. The discrepancy emerges when the position matrix is assumed to be diagonal in k. Consequently the AS' equation is a limiting case of our generalized Liouville equation, where the relative momentum Q between two states is ignored.Our numerical test reveals that the k-diagonal formalism yields qualitatively different results and underestimates both energy absorption and polarization by a factor of 7. Our study establishes a general formalism for the length-gauge time-dependent Liouville equation and opens the door for future research on the ultrafast time scale, while prior results based on the k-diagonal only equation may need to be revisited.<\/jats:p>","DOI":"10.1088\/1361-648x\/aeb785","type":"journal-article","created":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T22:59:00Z","timestamp":1792623540000},"source":"Crossref","is-referenced-by-count":0,"title":["Generalized length-gauge time-dependent Liouville equation"],"prefix":"10.1088","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1792-2701","authenticated-orcid":false,"given":"Guoping","family":"Zhang","sequence":"first","affiliation":[{"name":"Indiana State University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mingsu","family":"Si","sequence":"additional","affiliation":[{"name":"Lanzhou University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"266","published-online":{"date-parts":[[2026,10,21]]},"container-title":["Journal of Physics: Condensed Matter"],"original-title":[],"link":[{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/aeb785","content-type":"text\/html","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/aeb785\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/aeb785\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/aeb785\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T22:59:01Z","timestamp":1792623541000},"score":1,"resource":{"primary":{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-648X\/aeb785"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,21]]},"references-count":0,"URL":"https:\/\/doi.org\/10.1088\/1361-648x\/aeb785","relation":{},"ISSN":["0953-8984","1361-648X"],"issn-type":[{"value":"0953-8984","type":"print"},{"value":"1361-648X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,21]]}}}