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 $\hat{\rho }$ and the position operator $\mathbf{r}$, $⟨n\mathbf{k}|[\mathbf{r},\hat{\rho }]|m{\mathbf{k}}^{\prime }⟩$, where $|n\mathbf{k}⟩$ and $|m{\mathbf{k}}^{\prime }⟩$ are Bloch states for bands $n$ and $m$ at crystal momenta $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$, respectively. This commutator has two irreconcilable versions: Kohn and Luttinger (Phys. Rev. $108$, 590 (1954)) have two partial derivatives with respect to $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$, 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. Physically, the $\mathbf{k}$-off-diagonal matrix elements of the position operator provide a channel by which the electric field generates $\mathbf{k}$-off-diagonal matrix elements for $\hat{\rho }$ in LG. The discrepancy emerges when the position matrix is assumed to be diagonal in $\mathbf{k}$. Consequently the AS' equation is a limiting case of our generalized Liouville equation, where the relative momentum $\mathbf{Q}$ between two states is ignored. Our numerical test reveals that the $\mathbf{k}$-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 $\mathbf{k}$-diagonal only equation may need to be revisited.

23 pages, 5 figures, 46 references (submitted)