Semiconductor Wannier equations: a real-time, real-space approach to the nonlinear optical response in crystals (ATATA)

TL;DR

The paper introduces the semiconductor Wannier equations (SWEs), a real-time, real-space formulation for ultrafast light–matter dynamics in crystals that eliminates structure-gauge ambiguities inherent to reciprocal-space SBEs. By propagating the one-electron reduced density matrix in a localized Wannier basis and incorporating time-dependent Hartree–Fock with static screened exchange (TD-HSEX) along with three decoherence channels, the SWEs capture nonlinear optical responses and attosecond phenomena with gauge cleanliness and numerical robustness. The authors demonstrate accurate linear response, excitonic effects, and faster convergence than SBEs, and show that real-space dephasing provides physically grounded damping for high-harmonic generation without the unphysical artifacts of pure dephasing. The framework bridges real-space semiclassical intuition with many-body solid-state optics, offering a scalable and ab-initio-compatible tool for HHG, nonlinear optics, and attosecond spectroscopy in solids, with future extensions to dynamical screening, phonons, and beyond-mean-field physics.

Abstract

We develop the semiconductor Wannier equations (SWEs), a real-time, real-space formulation of ultrafast light-matter dynamics in crystals, by deriving the equations of motion for the electronic reduced density matrix in a localized Wannier basis. Working in real space removes the structure-gauge ambiguities that hinder reciprocal-space semiconductor Bloch equations. Electron--electron interactions are included at the time-dependent Hartree plus static screened-exchange (TD-HSEX) level. Decoherence is modeled with three complementary channels: pure dephasing, population relaxation, and distance-dependent real-space dephasing; providing physically grounded damping for strong-field phenomena such as high-harmonic generation. Conceptually, the SWEs bridge real-space semiclassical intuition with many-body solid-state optics, offering a numerically robust and gauge-clean alternative to reciprocal-space approaches for nonlinear optical response and attosecond spectroscopy in solids.

Figures (5)

• Figure 1: Scatter plot of the coherences between localized orbitals for monolayer hBN (blue circles) and graphene, at zero (black circles) and finite temperature (orange circles).
• Figure 2: Band-structure and total density of states (DOS) of monolayer hBN (a) and of monolayer 1H-MoS$_{2}$ (b). In panels (c,d), we show the linear optical conductivity of monolayer hBN and 1H-MoS$_{2}$, respectively, in the IPA (orange lines) and TD-HSEX (black lines) approximations.
• Figure 3: Calculated current (a) and the relative error of the current (b) using the SWEs approach for different $N_{R}$. (c,d) The same for calculations using the SBEs approach.
• Figure 4: High harmonic spectrum of monolayer hBN with different dephasing mechanisms: (a) real-space dephasing and (b) pure dephasing.
• Figure 5: High harmonic spectrum at the IPA and TD-HSEX level in MoS$_{2}$. (a) HHG spectrum for a laser polarized along the $\Gamma-\mathrm{M}$ direction in the parallel direction. (b,c) Same as in (a) for a laser along the $\Gamma-\mathrm{K}$ in the parallel and perpendicular direction, respectively. The dashed vertical line corresponds to the bandgap of MoS$_{2}$.