Self-consistent evaluation of the Berry connection for Wannier functions

This paper proposes a self-consistent interpolation scheme based on the matrix logarithm that significantly improves the accuracy of Berry connection evaluations and optical conductivity calculations by explicitly accounting for the matrix structure of overlap matrices and quantifying the impact of basis set incompleteness.

The Gist

The Big Picture: Mapping a Mountain Range

Imagine you are a cartographer trying to draw a detailed map of a massive, complex mountain range (the electrons in a solid material).

In the world of quantum physics, calculating exactly how electrons move and interact is incredibly expensive and slow. It's like trying to survey every single rock and tree on the mountain using a drone. To make this manageable, scientists use a shortcut called Wannier functions.

Think of Wannier functions as local guideposts. Instead of mapping the whole mountain at once, you place a few highly accurate guideposts in key valleys. Once you have these guideposts, you can use a mathematical "zoom" (Fourier interpolation) to guess what the terrain looks like between them. This allows you to create a super-detailed map without needing to fly a drone over every single inch.

The Problem: The "Berry Connection" is a Tricky Compass

To understand how light interacts with these materials (like why silicon is shiny or why MoS2 glows), you need to know the Berry connection.

Think of the Berry connection as a magnetic compass that tells the electrons which way to turn as they move across the mountain.

• The Issue: In the past, the methods used to calculate this compass were a bit "loose." They treated every part of the compass independently, like trying to fix a broken clock by just adjusting the hour hand without looking at the gears.
• The Consequence: Sometimes, this led to the compass pointing in the wrong direction or spinning wildly. This resulted in inaccurate predictions about how the material would react to light (optical conductivity). It was like predicting the weather based on a broken thermometer.

The Old Solutions: "Sym" and "Lihm"

Scientists tried to fix this with two main methods:

1. The "Sym" Method: They tried to force the compass to be symmetrical (if it points left, it shouldn't point right). It was better, but still treated the gears of the clock as separate pieces.
2. The "Lihm" Method: This was a clever tweak that adjusted for where the guideposts were placed. It was good, but it still didn't fully understand how the gears (the matrix elements) were connected to each other.

The New Solution: The "Self-Consistent Logarithmic" Scheme

The authors of this paper (Thümmler, Croy, et al.) realized that the compass and the guideposts aren't separate; they are part of a single, complex machine.

They proposed a new method called the Self-Consistent Logarithmic (sclog) scheme. Here is how it works, using an analogy:

The Analogy of the "Recursive Refinement":
Imagine you are trying to guess the exact shape of a curved road by looking at two points far apart.

• Old Way: You draw a straight line between the two points. It's okay, but it misses the curve.
• The New Way (sclog):
  1. Step 1: You take a "logarithmic guess" (a mathematical tool that understands curves better) to draw a rough line.
  2. Step 2: You check your guess against the actual road. You see where you were wrong.
  3. Step 3: You adjust your guess and check again.
  4. Step 4: You repeat this process (recursively) until your guess matches the road perfectly.

This "self-consistent" loop means the method keeps refining itself until it stops making mistakes. It treats the compass as a whole, interconnected system rather than a collection of loose parts.

The "Leaky Bucket" Problem (Basis Incompleteness)

There is one catch. Even with the best map, if your guideposts (Wannier functions) are too few or in the wrong spots, you can't map the whole mountain perfectly. Some information "leaks" out.

The authors call this basis incompleteness.

• The Metaphor: Imagine trying to fill a bucket with water using a sieve. No matter how hard you try, some water will always leak through the holes.
• The Insight: The authors developed a way to measure exactly how much water is leaking (using something called "singular values"). They found that while their new method is amazing, it can't fix the leak if the sieve itself is too coarse. However, their method is much better at handling the water that does stay in the bucket.

The Results: Why Should We Care?

The team tested their new method on two materials: Monolayer MoS2 (a thin, 2D material used in future electronics) and Bulk Silicon (the stuff computer chips are made of).

• The Result: Their new "self-consistent" compass was incredibly accurate.
• The Impact: When they calculated how these materials absorb light (optical conductivity), the old methods were off by as much as 26%. The new method was off by less than 0.3%.

In plain English:
If you were designing a new solar panel or a faster computer chip, using the old methods might make you think the material works 26% better (or worse) than it actually does. That's a huge mistake. The new method gives you a nearly perfect prediction, saving time and money in the lab.

Summary

The paper introduces a smarter, self-correcting mathematical tool to calculate how electrons move in solids. By treating the problem as a connected whole and refining the answer repeatedly, they created a method that is far more accurate than previous techniques, especially for predicting how materials interact with light. It's like upgrading from a hand-drawn sketch to a high-definition GPS for the quantum world.

Technical Summary

1. Problem Statement

The Berry connection ($A_k$) is a gauge-dependent quantity essential for describing optical responses, polarization, and topological properties in solids. In computational materials science, it is typically evaluated using Wannier functions to interpolate matrix elements from a coarse ab-initio $k$-grid to a dense grid.

Key Challenges Identified:

• Matrix Structure Neglect: Existing interpolation schemes (e.g., Marzari-Vanderbilt, Lihm) treat the overlap matrices ($M^{kb}$) between neighboring $k$-points element-wise or distinguish only between diagonal and off-diagonal entries. They fail to account for the intrinsic matrix structure where $M^{kb}$ acts as a path-ordered exponential of the Berry connection.
• Hermiticity and Gauge Issues: Traditional schemes often yield non-Hermitian Berry connections, leading to non-physical results (e.g., in semiconductor Bloch equation propagation). While recent schemes (like Lihm's) enforce translational invariance, they do not fully resolve the transformation behavior or the matrix coupling.
• Basis Set Incompleteness: The finite number of Wannier functions used in ab-initio calculations creates a "basis spill" (coupling to un-described bands). This imposes a fundamental limit on the accuracy of any interpolation scheme, yet this limit is rarely quantified in relation to the interpolation error.
• Convergence Sensitivity: Derived properties like optical conductivity are highly sensitive to the choice of Wannierization (e.g., Maximally Localized Wannier Functions vs. separate Conduction/Valence band stitching) and the interpolation scheme, often requiring extremely fine grids for convergence.

2. Methodology

The authors propose a new Self-Consistent Logarithmic (sclog) interpolation scheme. The methodology proceeds in three main theoretical steps:

1. Matrix Logarithm Formulation:
   Instead of treating matrix elements independently, the authors recognize that the overlap matrix $M^{kb}$ is the path-ordered matrix exponential of the Berry connection:
   $$M^{kb} = \mathcal{T} \exp\left( \int_0^{|b|} A_{k+t\frac{b}{|b|}} dt \right)$$
   They invert this relationship using the matrix logarithm to obtain an initial estimate of the Berry connection:
   $$\log M^{kb} \approx -ib A_{k + b/2}$$
   This approach respects the unitary nature of the transformation and the mixing of matrix elements.

2. Magnus Expansion and Path-Ordered Integrals:
   To improve accuracy beyond a local approximation, the authors employ a Magnus expansion (4th order) to evaluate the path-ordered integral explicitly. This allows for a global Fourier interpolation of the estimated Berry connection.

3. Self-Consistent Iteration:
   The scheme iteratively refines the Berry connection:

   • Compute an initial $A_k$ via the matrix logarithm.
   • Reconstruct the overlap matrices using the global Fourier-interpolated $A_k$ and the Magnus expansion.
   • Compare the reconstructed overlaps with the original ab-initio overlaps.
   • Update $A_k$ to minimize the deviation.
   • Repeat until self-consistency is reached (typically ~20 iterations).

4. Quantification of Basis Incompleteness:
   The authors derive a metric to quantify the "basis spill" (error due to incomplete basis sets) using the singular values of the overlap matrices. They relate this to the invariant part of the Wannier spread functional ($\Omega_I$), providing a theoretical bound on the achievable accuracy.

3. Key Contributions

• Novel Interpolation Scheme: Introduction of the sclog scheme, which unifies the treatment of diagonal and off-diagonal elements by utilizing the matrix logarithm and self-consistent refinement.
• Theoretical Rigor: Proof that the overlap matrices are generated by the Berry connection, correcting the element-wise assumptions of previous methods.
• Error Quantification: Development of a rigorous framework to quantify basis set incompleteness via singular values of overlap matrices and its relation to $\Omega_I$.
• Robust Velocity Operator: Introduction of a BZ-averaged mismatch measure between the interpolated velocity operator and the ab-initio velocity operator as a systematic benchmark for interpolation quality, independent of specific physical property convergence.

4. Numerical Results

The authors tested their schemes on monolayer MoS$_2$ and bulk Silicon (Si) using two different Wannierizations: Maximally Localized Wannier Functions (MLWF) and stitched Conduction/Valence Band (CB-VB) Wannierizations.

• Velocity Operator Accuracy:

  • The sclog scheme consistently showed the lowest mismatch with the ab-initio velocity operator.
  • For MoS$_2$ (low basis spill), the sclog scheme achieved error scaling better than quadratic ($O(|b|^4)$) for CB-VB Wannierizations, whereas standard schemes (MV, Lihm) remained at $O(|b|^2)$.
  • For Si (high basis spill), all schemes struggled, but sclog remained the most accurate, significantly outperforming Marzari-Vanderbilt (MV) and Lihm schemes.

• Optical Conductivity:

  • Optical conductivity is highly sensitive to the interpolation scheme.
  • MoS$_2$: For a coarse $8 \times 8 \times 1$ grid with CB-VB Wannierization, the relative deviation of the maximal optical conductivity peak dropped from 26% (using MV scheme) to < 0.3% (using sclog).
  • Si: The sclog scheme reduced errors significantly compared to MV, though the high basis spill in Si limited the absolute accuracy.

• Basis Spill Analysis:

  • The study confirmed that the "basis spill" (singular values of the spill matrix) is the fundamental limit to accuracy.
  • The invariant spread $\Omega_I$ converges slowly with grid size, particularly when band disentanglement is required, indicating that disentanglement introduces significant basis incompleteness.

5. Significance and Conclusion

• Improved Accuracy: The self-consistent logarithmic scheme provides a substantial improvement in the accuracy of interpolated optical properties, particularly for coarse ab-initio grids.
• Reduced Sensitivity: The sclog scheme makes calculated optical conductivities much less sensitive to the specific details of the Wannierization (e.g., MLWF vs. CB-VB), making it a more robust tool for high-throughput material screening.
• Practical Implication: The authors suggest that in practical applications, one should evaluate multiple interpolation schemes (sym, Lihm, log, sclog) and automatically select the one yielding the lowest mismatch to the ab-initio velocity operator, as higher-order schemes cannot overcome the fundamental limit imposed by basis set incompleteness.
• Future Outlook: The work highlights the need to quantify basis incompleteness ($\Omega_I$) as a standard metric in Wannier interpolation studies and suggests that the proposed scheme preserves crystal symmetries, though this requires further investigation.

In summary, this paper resolves long-standing issues in the interpolation of the Berry connection by treating the overlap matrices as a coherent unitary object rather than a collection of independent elements, leading to significantly more accurate predictions of optical and transport properties in solids.