Optimally embedded tight binding for reproducing geometry dependent observables

Technical Summary: Optimally Embedded Tight Binding for Reproducing Geometry Dependent Observables

Problem Statement
Tight-binding (TB) models are the standard technique for computing electronic properties of solid-state systems, typically defined by on-site energies ($\epsilon_\alpha$) and hopping terms ($t_{\alpha\beta}$). While these models accurately reproduce geometry-independent quantities like band structures, they often fail to capture geometry-dependent observables (e.g., optical responses, polarization, magnetization) which rely explicitly on the position operator $\hat{r}$.

In standard TB implementations, the position operator is approximated by assigning diagonal elements $\tau_\alpha$ (orbital embeddings) to each basis orbital, neglecting off-diagonal position matrix elements. This "Tight-Binding Assumption" (TBA) is often justified by constructing Maximally Localized Wannier Functions (MLWFs), where the orbital centers are chosen as Wannier Charge Centers (WCCs) to minimize the spread of the position operator. However, the authors argue that WCCs minimize the error in the position operator itself, not necessarily the error in physical observables. Consequently, TB models using WCCs can yield significant quantitative and even qualitative errors in geometry-dependent responses, such as shift currents and optical conductivities, despite reproducing the band structure perfectly.

Methodology
The paper introduces a framework of Optimally Embedded Tight Binding, treating orbital embeddings $\tau_\alpha$ not as fixed physical centers (like WCCs) but as tunable geometric parameters. The methodology proceeds as follows:

1. Reference Generation: An ab-initio electronic structure calculation (e.g., DFT) is performed and Wannierized to fix the hopping and on-site energy parameters of the TB model. This ensures the geometry-independent band structure is exact.
2. Geometry Decomposition: The authors derive a general decomposition of TB observables into geometry-independent and geometry-dependent parts. By expressing the Berry connection and Quantum Geometric Tensor (QGT) in terms of orbital displacements, they obtain analytical expressions for the first and second derivatives of observables with respect to $\tau_\alpha$.
3. Optimization: Using these derivatives, the authors perform a least-squares minimization to find the set of orbital embeddings $\tau_\alpha$ that best reproduce a specific reference observable (obtained from Wannier interpolation or experiment). This is done efficiently without recomputing the full Hamiltonian for every iteration.
4. Target Quantities: The framework is applied to fit:
   • The interspace Berry connection.
   • The Abelian Quantum Geometric Tensor (QGT).
   • Specific optical responses (linear optical conductivity, injection current, and shift current).

Key Contributions

• Theoretical Framework: The paper establishes that orbital embeddings should be treated as genuine model parameters to be fixed against physical data, rather than being rigidly tied to WCCs. It provides explicit analytical derivatives for key geometric quantities, enabling efficient optimization.
• Decomposition of Geometric Effects: The authors demonstrate how to isolate the geometric contribution to observables, revealing that geometry can dramatically alter the local metric trace even when the band structure remains unchanged.
• Efficient Fitting Procedure: The method allows for the generation of TB models that reproduce specific geometric quantities (like non-linear optical responses) with high precision while maintaining the simplicity and band-structure accuracy of standard TB models.

Results
The authors validate their approach on two material systems and several toy models:

• GaAs (Zincblende): Previous studies using MLWF WCCs showed poor quantitative agreement for the shift current spectrum compared to Wannier interpolation. The optimally embedded models, fitted to the Abelian QGT or the shift current tensor directly, reproduce the reference response with negligible error. Furthermore, these tailored embeddings successfully suppress symmetry-disallowed components of the response, a feat not achieved by standard WCCs.
• CdS (Wurtzite): While MLWFs provided reasonable results for linear optical conductivity, they exhibited notable qualitative errors in the shift current component $\sigma_{xxz}$ above 6 eV. Optimally embedded models fitted to the optical conductivity tensor or injection current significantly improved the agreement with reference data. However, the authors note that fitting one component (e.g., $\sigma_{xxz}$) did not guarantee a good fit for all components simultaneously, suggesting limits to the TBA for certain observables.
• Toy Models and V$_2$O$_3$: In the Chern insulator V$_2$O$_3$ and Haldane/QWZ toy models, the authors show that varying orbital embeddings (while keeping hoppings fixed) can drastically alter the qualitative shape of the quantum metric trace. For instance, the peak of the metric trace can be shifted between high-symmetry points or redistributed, even when the average metric trace (quantum weight) is held constant. This demonstrates that geometry can qualitatively change local geometric responses without affecting the band gap or topology.

Significance and Claims
The paper claims that the choice of orbital embeddings is a critical, often overlooked, factor in the accuracy of tight-binding models for geometry-dependent phenomena.

• Accuracy vs. Simplicity: The authors demonstrate that it is possible to retain the computational simplicity of tight-binding (avoiding full real-space Wannier function evaluations for every calculation) while achieving DFT-level accuracy for non-linear optical responses by simply tuning the orbital positions.
• Physical Interpretation: The work challenges the notion that MLWFs are the "correct" physical centers for all applications. Instead, it posits that for specific observables (like optical responses), the "correct" embedding is the one that minimizes the error for that specific observable.
• Geometric Engineering: The results suggest that the "shape" of geometric objects like the quantum metric is highly sensitive to orbital placement. This implies that in systems with flat bands or specific symmetries, geometric effects can be dominant, and standard approximations may fail to capture the physics of correlated phenomena or quantum sensing.
• Limitations: The authors remain modest regarding the physical realizability of "engineering" orbital embeddings in real materials (independently of hopping terms). They acknowledge that while the mathematical framework allows for this separation, physical perturbations (like strain) typically affect both positions and hoppings. However, they assert that the method remains highly valuable for constructing accurate minimal models for theoretical study and prediction, regardless of whether the specific embedding can be physically realized.

In conclusion, the paper provides a practical recipe for improving tight-binding models: use ab-initio data to fix the Hamiltonian, then use experimental or high-level theoretical data to optimize the orbital embeddings, thereby resolving the long-standing discrepancy between TB predictions and geometry-dependent observables.