TD-HSEX: Real-Time Electron Dynamics in Solids

• TD-HSEX is a hybrid approximation that combines the Hartree mean field with static screened exchange to model electron interactions and nonlinear optical responses in solids.
• It leverages a real-space Wannier basis to simplify many-body interactions, resulting in rapid convergence and enhanced physical interpretability of carrier dynamics.
• The framework eliminates gauge ambiguities and supports scalable simulations of nonlinear phenomena, including excitonic effects and high-harmonic generation in semiconductors.

The time-dependent Hartree plus static screened-exchange (TD-HSEX) approximation is a widely used framework for modeling electron-electron interactions in time-dependent quantum many-body simulations, particularly in the context of nonlinear optical response in solids. TD-HSEX enables the incorporation of both the classical mean field (Hartree) and quantum exchange interactions with static (instantaneous) dielectric screening, offering a balanced trade-off between accuracy and computational tractability. This approach is central to recent real-space, real-time methodologies such as the semiconductor Wannier equations (SWEs), which circumvent limitations of reciprocal-space techniques, especially for ultrafast and strong-field phenomena in crystals.

1. Fundamental TD-HSEX Formalism

The TD-HSEX scheme operates at the mean-field level, propagating the one-electron reduced density matrix (1RDM) under combined single-particle, Hartree, and static screened-exchange dynamics. In a real-space Wannier basis, the central dynamical variable is

$$\rho_{ij}(t) = \langle c^\dagger_j(t) c_i(t) \rangle$$

where $i, j$ label localized Wannier orbitals incorporating both spatial and orbital degrees of freedom.

The evolution equation is

$$i \frac{d \rho_{ij}}{dt} = [h(t) + \Sigma^{\mathrm{HSEX}}[\rho] - \Sigma_0, \rho]_{ij}$$

with

• $h(t)$: time-dependent single-particle Hamiltonian (including light-matter interaction, typically in length gauge),
• $$\Sigma^{\mathrm{HSEX}}[\rho] = \Sigma^{\mathrm{SEX}}[\rho] + \Sigma^{\mathrm{H}}[\rho]$$: sum of screened-exchange and Hartree self-energies,
• $\Sigma_0 = \Sigma^{\mathrm{HSEX}}[\rho^0]$: equilibrium self-energy, subtracted to avoid double counting from the reference electronic structure (e.g., DFT or GW Hamiltonian) (Molinero et al., 24 Oct 2025).

The static screened-exchange (SEX) and Hartree contributions are

$$\Sigma^{\mathrm{SEX}}[\rho]_{ij} = -\sum_{kl} W^{il}_{kj} \rho_{kl}, \qquad \Sigma^{\mathrm{H}}[\rho]_{ij} = \sum_{kl} V^{il}_{jk} \rho_{kl}$$

where $V$ is the bare Coulomb interaction and $W$ is the statically screened Coulomb interaction, typically obtained in the RPA and taken at $\omega \to 0$.

2. Real-Space Construction and Orbital Localization

The use of localized Wannier functions allows for a substantial simplification of the TD-HSEX terms, exploiting electronic nearsightedness. Under the ultra-localized orbital approximation (ULOA), matrix elements become

$$V_{kl}^{ij} \approx \delta_{ik} \delta_{jl} V(\vec{\tau}_i - \vec{\tau}_j), \qquad W_{kl}^{ij} \approx \delta_{ik} \delta_{jl} W(\vec{\tau}_i - \vec{\tau}_j)$$

with Wannier center $\vec{\tau}_i$. The self-energies thus reduce to

$$\Sigma^{\mathrm{SEX}}[\rho]_{ij} \approx -W(\vec{\tau}_i - \vec{\tau}_j) \rho_{ij}$$

$$\Sigma^{\mathrm{H}}[\rho]_{ij} \approx \delta_{ij} \sum_k s_k \rho_{kk} V(\vec{\tau}_i - \vec{\tau}_k)$$

where $s_k$ is the spin degeneracy. The Hartree term acts strictly on the density, while the SEX term is inherently nonlocal in the density matrix, encoding quantum exchange and screening.

For periodic crystals, the expressions generalize to supercell notation:

$$\Sigma^{\mathrm{SEX}}_{\alpha\beta}[\rho](\mathbf{R}) = -W(\vec{\tau}_\alpha - (\vec{\tau}_\beta + \mathbf{R})) \rho_{\alpha\beta}(\mathbf{R})$$

$$\Sigma^H_{\alpha\beta}[\rho](\mathbf{R}) = \delta_{\alpha\beta}\delta_{\mathbf{R},0} \sum_\gamma \rho_{\gamma\gamma}(0) X_{\gamma\alpha}$$

$$X_{\gamma\alpha} = \sum_{\mathbf{R}'}\, s_\gamma V(\vec{\tau}_\alpha-(\vec{\tau}_\gamma+\mathbf{R}'))$$

3. Physical Implications in Optical and Electronic Response

The TD-HSEX approximation captures essential Coulombic effects pivotal for the correct description of both linear and nonlinear response, as well as ultrafast solid-state phenomena:

• The Hartree term accounts for the mean electrostatic interaction, essential for maintaining charge neutrality and enforcing collective plasma oscillations.
• The screened exchange term incorporates quantum exchange with static dielectric screening, thereby accessing:
  • Excitonic effects: Binding of electron-hole pairs below the band gap, crucial in optical absorption and high-harmonic generation (HHG).
  • Local-field effects: Nonlocal polarization response due to crystalline structure.
  • Dielectric screening: Modulation of Coulomb interaction by the electronic environment.
• The form of the screened interaction ensures that ultrafast electron-electron correlation dynamics are modeled beyond the independent-particle approximation, with computational efficiency not possible in fully dynamical MBPT.

In nonlinear strong-field regimes (e.g., HHG), the inclusion of TD-HSEX leads to resonance shifts and quantitative changes in harmonic spectra—a consequence of correct excitonic and screening physics.

4. Advantages over Reciprocal-Space and Band Approaches

Implementing TD-HSEX within real-space SWEs yields several critical advantages:

• Elimination of structure-gauge ambiguities: SWEs circumvent the arbitrary phase and gauge issues afflicting reciprocal-space semiconductor Bloch equations (SBEs), especially for position-related operators and Berry connections.
• Numerical robustness: Locality and sparsity of the real-space representation enable:
  • Rapid convergence with system size due to truncation of the density matrix at finite distances,
  • Supercells orders of magnitude smaller than the $k$-point meshes required in SBEs for comparably accurate results.
• Physical interpretability: Real-space propagation provides direct insight into semiclassical carrier trajectories, spatial localization of excitons, and visualization of real-space dephasing.
• Advanced decoherence models: Enables the use of physically-grounded, distance-dependent dephasing mechanisms that cannot be directly formulated in the Bloch basis, improving the realism of simulations under strong drives.

A summary comparison is provided in the table below:

Feature	Reciprocal-space (SBEs)	Real-space (SWEs, TD-HSEX)
Gauge issues	Structure-gauge ambiguities	Gauge-clean, no structure-gauge ambiguities
Numerical convergence	Slow ($k$-mesh)	Fast (localized, sparse density matrix)
Excitonic & correlation effects	Not natural	Natural and efficient via TD-HSEX
Dephasing/dissipation	Pure dephasing, unphysical	Physically-motivated, spatially dependent
Physical interpretability	Less direct	Direct, semiclassical trajectories
Complex system scaling	Challenging	Favors linear scaling

5. Theoretical Relations and Extensions

The TD-HSEX paradigm is connected to other mean-field and time-dependent approaches:

• In finite systems or atomic cases, time-dependent exact-exchange (TDEXX) and TD-HF calculations are formally similar, with TDEXX replacing the four-point kernel of TDHF with a two-point, frequency-dependent kernel (0807.0091).
• Screening within TD-HSEX is most commonly treated in the static limit but can in principle be generalized to include frequency-dependence; however, this sacrifices the locality in time and practical numerical advantages central to the real-time SWEs.
• For extended Fermi systems with pairing, the TD-HSEX concept underlies screened-gap equations and connects with screened-interaction approaches in time-dependent Hartree-Fock-Bogoliubov theory (Tohyama et al., 2014).

6. Implementation Considerations

The practical formulation of TD-HSEX in the real-space SWEs is conducive to numerical implementation:

• Storage requirements scale with the number of localized orbitals and the sparsity of the density matrix.
• At each time step, evaluation of local Hartree self-energies and nonlocal screened-exchange is computationally efficient due to truncation in the spatial domain.
• Decoherence and dissipation channels—identifiable with pure dephasing, population relaxation, and spatially-dependent real-space dephasing—are straightforwardly included by acting locally or nonlocally on the 1RDM.
• Subtraction of the equilibrium self-energy $\Sigma_0$ ensures consistency with the underlying DFT or GW reference, eliminating artificial double-counting.

7. Application Scope and Impact

TD-HSEX provides a foundational tool for the simulation of ultrafast and nonlinear optical phenomena in correlated crystals and nanostructures:

• It enables physically robust and numerically stable calculation of strong-field, attosecond, and high-harmonic generation processes in solids, essential for advancing real-time materials spectroscopy (Molinero et al., 24 Oct 2025).
• The real-space, gauge-clean approach can be scaled to multi-band and multi-orbital models, supporting predictive simulations of complex materials where standard band approaches are numerically or conceptually inadequate.
• The formalism facilitates a direct bridge between semiclassical models of carrier dynamics and ab initio many-body treatments, underpinning a wide scope of modern solid-state optics.

Conclusion:

TD-HSEX within the SWEs framework constitutes a gauge-clean, computationally robust method for real-time simulations of electron dynamics in solids, capturing both local electrostatic and nonlocal, statically screened exchange interactions. Its advantages in stability, interpretability, and scalability mark it as a preferred method for modeling nonlinear and ultrafast phenomena in condensed matter systems, especially where excitonic and strong-field effects are significant (Molinero et al., 24 Oct 2025).