Frequency-dependent electron-phonon coupling and vibrational responses in tight-binding and continuous Dirac models with nuclear velocity correction

Technical Summary: Frequency-Dependent Electron-Phonon Coupling and Vibrational Responses in Tight-Binding and Continuous Dirac Models with Nuclear Velocity Correction

Problem Statement
Effective Hamiltonians, such as tight-binding (TB) and low-energy continuous models (e.g., Dirac models), are widely used to describe electronic and vibrational properties of materials. However, standard formulations often neglect the effect of nuclear motion on the electronic atomic orbitals. Within the Born-Oppenheimer approximation, effective Hamiltonians are typically derived for fixed nuclear configurations, with dynamic treatments often involving only rigid shifts of orbitals. This approach introduces discrepancies between the non-adiabatic, frequency-dependent vibrational responses of effective models and the all-electron Hamiltonian. Specifically, standard TB models fail to satisfy the all-electron sum rules relating Born effective charges and the force-constant matrix to frequency-dependent electromagnetic susceptibilities. In TB methods, these sum rules vanish entirely, leading to qualitative failures in describing phenomena such as vibrational circular dichroism and non-adiabatic electron-phonon coupling, particularly in metallic systems or when lattice excitations resonate with electronic transitions. The root cause is the neglect of nuclear-velocity-dependent phases (electron-translation factors) in the non-local part of the interactions.

Methodology
The authors employ a semiclassical Ehrenfest Lagrangian approach, treating nuclei as classical particles and electrons as quantum mechanical particles. The methodology proceeds through the following steps:

1. Velocity-Including Atomic Orbitals (LCAO): The authors define a basis of localized atomic orbitals that includes a nuclear-velocity-dependent phase factor, $e^{i\alpha_s(\hat{r})}$, where $\alpha_s(\hat{r}) = \frac{m}{\hbar}\dot{R}_s(t) \cdot (\hat{r} - R_s(t))$. This modification accounts for the electron-translation factor arising from moving nuclei.
2. Derivation of Effective Hamiltonians: Using a Lagrangian formalism, they derive the equations of motion for both electrons and nuclei. This yields an effective Hamiltonian that includes terms dependent on nuclear position, velocity, and acceleration.
3. Tight-Binding and Dirac Formulations:
   • Tight-Binding: The LCAO Lagrangian is projected onto an orthogonal tight-binding basis. The authors derive explicit expressions for the hopping terms and on-site energies. Crucially, the hopping terms acquire Peierls-like phases dependent on nuclear velocities, and the on-site energy is corrected by the kinetic energy of the electron in the co-moving frame.
   • Dirac Models: These corrections are mapped to low-energy Dirac Hamiltonians (e.g., for gapped graphene and the Haldane model). The nuclear velocity correction to the electron-phonon coupling is shown to be proportional to the band velocity.
4. Linear Response Theory: The authors apply linear response theory to calculate frequency-dependent vibrational responses, specifically the Born effective charges ($Z^*$) and the force-constant matrix ($C$). They derive explicit expressions for the derivatives of the Hamiltonians with respect to nuclear displacements and velocities.
5. Sum Rule Analysis: The derived expressions are tested against all-electron sum rules. The authors demonstrate that the inclusion of nuclear velocity corrections restores these sum rules, which are otherwise zero in standard TB models.
6. Numerical Validation: The theory is applied to metallic gapped graphene and the topological Haldane model. Results are benchmarked against ab initio calculations (using Quantum Espresso) and analytical limits.

Key Contributions and Results

• Restoration of Sum Rules: The primary theoretical contribution is the demonstration that including nuclear-velocity-dependent phases in TB and Dirac models restores the all-electron sum rules for frequency-dependent Born effective charges and force-constant matrices. Without these corrections, the sum rules vanish in TB models; with them, they correctly relate to optical conductivity and electromagnetic susceptibilities.
• Modified Electron-Phonon Coupling: The work establishes that the electron-phonon coupling vertex acquires an additional term proportional to the nuclear velocity (and thus the phonon frequency $\omega$). In Dirac models, this appears as a "velocity vertex" added to the standard displacement vertex.
• Quantitative and Qualitative Corrections:
  • Metallic Gapped Graphene: In doped gapped graphene, the nuclear velocity correction to the Born effective charges is significant, reaching up to 50% for high doping levels. The corrected TB results show excellent agreement with ab initio calculations regarding the sum rule for Born effective charges.
  • Resonance Effects: In both gapped graphene and the Haldane model, the corrections are enhanced when phonon frequencies resonate with electronic interband transitions.
  • Topological Systems: For the Haldane model (a Chern insulator), the corrections to the off-diagonal components of the Born effective charges depend on the Hall conductivity and the topological state, distinguishing them from the trivial case.
• Phonon Lifetime and Frequency: The corrections modify the force-constant matrix, affecting phonon frequencies and linewidths. The correction to the phonon lifetime is found to be enhanced by mass asymmetry between sublattices (e.g., in h-BN analogues) and scales inversely with the electron-phonon coupling strength.
• Vibrational Circular Dichroism (VCD): The authors note that the inclusion of these velocity-dependent phases is a prerequisite for capturing phenomena like VCD in solids, which are absent in standard adiabatic TB approximations.

Significance and Claims
The paper claims that the inclusion of nuclear velocity effects is not merely a quantitative refinement but a fundamental qualitative correction required for the validity of tight-binding and low-energy models in non-adiabatic regimes.

• Qualitative Necessity: The authors assert that without these corrections, TB models fail to describe the correct physics of frequency-dependent vibrational responses, leading to zero sum rules where non-zero values are physically required.
• Physical Interpretation: The corrections introduce electronic inertia into the nuclear dynamics problem, accounting for the fact that electrons do not instantaneously follow nuclear motion. This is quantified by the sum rules linking vibrational responses to electronic susceptibilities.
• Broad Applicability: While demonstrated on graphene and Haldane models, the framework is presented as general for any effective Hamiltonian constructed from localized orbitals or continuous low-energy models.
• Future Implications: The authors suggest these corrections are essential for studying chiral phonons and their associated magnetic moments, as well as for accurately modeling systems with weak electron-phonon coupling or strong electronic topology.

The work concludes that the proposed procedure provides a rigorous pathway to recover all-electron non-adiabatic vibrational responses within computationally efficient effective models, bridging the gap between simplified TB descriptions and first-principles accuracy.