{
  "abstract": "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 $\\hatρ$ and the position operator ${\\bf r}$, $\\langle n{\\bf k}|[{\\bf r},\\hatρ]|m{\\bf k}'\\rangle$, where $|n{\\bf k}\\rangle$ and $|m{\\bf k}'\\rangle$ are Bloch states for bands $n$ and $m$ at crystal momenta ${\\bf k}$ and ${\\bf k}'$, respectively. This commutator has two irreconcilable versions: Kohn and Luttinger (Phys. Rev. ${\\bf 108}$, 590 (1954)) have two partial derivatives with respect to ${\\bf k}$ and ${\\bf k}'$, but Aversa and Sipe (AS) (Phys. Rev. B ${\\bf 52}$, 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. Physically, the ${\\bf k}$-off-diagonal matrix elements of the position operator provide a channel by which the electric field generates ${\\bf k}$-off-diagonal matrix elements for $\\hatρ$ in LG. The discrepancy emerges when the position matrix is assumed to be diagonal in ${\\bf k}$. Consequently the AS' equation is a limiting case of our generalized Liouville equation, where the relative momentum ${\\bf Q}$ between two states is ignored. Our numerical test reveals that the ${\\bf k}$-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 ${\\bf k}$-diagonal only equation may need to be revisited.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22799",
  "author_total": 2,
  "authors": [
    {
      "name": "G. P. Zhang"
    },
    {
      "name": "M. S. Si"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/785m2z80gxrn0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22799",
  "fulltext_url": null,
  "id": "785m2z80gxrn0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22799",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.mtrl-sci"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.22799",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/785m2z80gxrn0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:32Z",
      "native_id": "2610.22799",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22799"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mtrl-sci",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Generalized length-gauge time-dependent Liouville equation",
  "title_text": "Generalized length-gauge time-dependent Liouville equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:28Z",
  "url": "https://research.draconitic.com/paper/785m2z80gxrn0",
  "venue": "arXiv",
  "versions": []
}