Generalized length-gauge time-dependent Liouville equation
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 ρ̂ and the position operator bf r, ⟨ nbf k|[bf r,ρ̂]|mbf k'⟩, where |nbf k⟩ and |mbf k'⟩ 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 ρ̂ 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.
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BibTeX
@misc{zhang2026generalized,
title = {{Generalized length-gauge time-dependent Liouville equation}},
author = {G. P. Zhang and M. S. Si},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22799},
eprint = {2610.22799},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22799},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generalized length-gauge time-dependent Liouville equation
AU - G. P. Zhang
AU - M. S. Si
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22799
UR - https://arxiv.org/abs/2610.22799
AB - 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 ρ̂ and the position operator bf r, ⟨ nbf k|[bf r,ρ̂]|mbf k'⟩, where |nbf k⟩ and |mbf k'⟩ 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 ρ̂ 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.
ER -