Data-Driven Compression of Electron-Phonon Interactions

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Data-Driven Compression of Electron-Phonon Interactions

Yao Luo, Dhruv Desai, Benjamin K. Chang, Jinsoo Park, and Marco Bernardi

Phys. Rev. X 14, 021023 – Published 1 May 2024

Abstract

First-principles calculations of electron interactions in materials have seen rapid progress in recent years, with electron-phonon ( $e − ph$ ) interactions being a prime example. However, these techniques use large matrices encoding the interactions on dense momentum grids, which reduces computational efficiency and obscures interpretability. For $e − ph$ interactions, existing interpolation techniques leverage locality in real space, but the high dimensionality of the data remains a bottleneck to balance cost and accuracy. Here we show an efficient way to compress $e − ph$ interactions based on singular value decomposition (SVD), a widely used matrix and image compression technique. Leveraging (un)constrained SVD methods, we accurately predict material properties related to $e − ph$ interactions—including charge mobility, spin relaxation times, band renormalization, and superconducting critical temperature—while using only a small fraction (1%–2%) of the interaction data. These findings unveil the hidden low-dimensional nature of $e − ph$ interactions. Furthermore, they accelerate state-of-the-art first-principles $e − ph$ calculations by about 2 orders of magnitude without sacrificing accuracy. Our Pareto-optimal parametrization of $e − ph$ interactions can be readily generalized to electron-electron and electron-defect interactions, as well as to other couplings, advancing quantitative studies of condensed matter.

Received 16 July 2023
Revised 17 January 2024
Accepted 29 March 2024

DOI:https://doi.org/10.1103/PhysRevX.14.021023

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Electron-phonon coupling Phonons Spin-phonon coupling

Data analysis Data mining First-principles calculations

Condensed Matter, Materials & Applied PhysicsInterdisciplinary Physics

Authors & Affiliations

Yao Luo ¹, Dhruv Desai¹, Benjamin K. Chang¹, Jinsoo Park¹, and Marco Bernardi ^1,2,*

¹Department of Applied Physics and Materials Science, California Institute of Technology, Pasadena, California 91125, USA
²Department of Physics, California Institute of Technology, Pasadena, California 91125, USA

^*bmarco@caltech.edu

Popular Summary

Understanding how electrons interact in materials is key to advancing condensed-matter physics and for engineering novel electronic and quantum devices. State-of-the-art numerical techniques can accurately describe these interactions on dense momentum grids but require large matrices with billions of entries. This constitutes a bottleneck for predicting materials properties, both in terms of computational cost and model interpretability. We demonstrate a data-driven method to compress the matrices for electron-phonon interactions, accelerating their calculation by about 2 orders of magnitude while preserving the accuracy. The method enables quantitative calculations in real materials of behaviors such as transport, nonequilibrium dynamics, spin relaxation, and superconductivity, all at a significantly reduced computational cost.

The approach is based on a clever application of singular value decomposition in a localized basis, which separates the electronic and vibrational degrees of freedom, thereby splitting a costly double Fourier transform into two single transforms. In addition, the singular value decomposition reduces the number of parameters to a small fraction (1%–2%) of the original interaction data, achieving an optimal formulation that simultaneously minimizes error and the number of parameters.

Conceptually, this work is an example of Occam’s razor—the idea that one should favor simple physical models—because it provides minimal models of electron-phonon interactions. We show that using only 1%–2% of the interaction parameters suffices to obtain accurate results, revealing the hidden low-dimensional nature of the problem. This work provides a blueprint to treat other interactions quantitatively with low computational effort, opening new directions at the intersection of condensed-matter physics, materials, and data science.

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Vol. 14, Iss. 2 — April - June 2024

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Images

Figure 1
(a) Error on the compressed $e − ph$ matrix, computed using Eq. (8), as a function of the fraction of SVs used in the low-rank approximation. The Pareto-optimal region is shown with a shaded rectangle. (b) Mode-resolved $e − ph$ coupling strength computed using the full $e − ph$ matrix (blue) and the low-rank approximate matrix (orange) for silicon. The full $e − ph$ matrix elements are computed on a real-space grid with size $N_{R_{e}} = 1325$ and $N_{R_{p}} = 1325$ , the smallest values to achieve convergence, setting the electron momentum to $k = Γ$ and using the $N_{b} = 3$ highest valence bands. In the SVD calculation, we keep 20 out of 1325 SVs, corresponding to a $\sim 1.5 %$ fraction of SVs, as noted in the legend.
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Figure 2
(a) Mobility of electrons in silicon and (b) mobility of hole carriers in silicon, computed with the full $e − ph$ matrix ( $N_{R_{e}} = N_{R_{p}} = 1325$ ) and compared with standard and constrained SVD, in both cases using 1.5% of the SVs. (c) Spin relaxation times of electrons in silicon, computed with the full $e − ph$ matrix ( $N_{R_{e}} = N_{R_{p}} = 1325$ ) and using SVD with 1.5% of the SVs. Experimental data from Refs. [61, 62] are shown for comparison. (d) Eliashberg spectral function $α^{2} F (ω)$ for Pb, comparing full $e − ph$ matrix ( $N_{R_{e}} = N_{R_{p}} = 279$ ) with SVD results using 1.8% of the SVs. (e) Superconducting gap $Δ_{0}$ as a function of temperature, comparing full $e − ph$ matrix results with SVD using 1.8% of the SVs. The inset shows the convergence of the critical temperature $T_{c}$ with number of SVs; the darker (lighter) colored regions indicate 5% (10%) error relative to the full calculation. (f) Band renormalization for electronic states near the Dirac cone in graphene at 20 K, comparing the full calculation with SVD using 1.5% of the SVs.
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Figure 3
(a) $e − ph$ coupling constant $λ$ in doped monolayer ${MoS}_{2}$ computed with our compression method using different fractions of SV. The doping concentration is 0.22 electrons per formula unit. The shaded region corresponds to an accuracy greater than 95% relative to the fully converged calculation. (b) Comparison of the wall time for computing the $e − ph$ coupling constant $λ$ with the full $e − ph$ matrices and with our SVD compression technique using 1% of the SVs. The $38 \times$ speedup achieved by the SVD compression is indicated in red font.
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Figure 4
(a) Decay of the SVs in graphene and (b) decay of the SVs in silicon, shown by plotting the SVs referenced to the largest SV. Here, ${\tilde{s}}_{i}$ refers to the SVs averaged over Wannier orbitals and vibrational modes, ${\tilde{s}}_{i} = \sqrt{\sum_{F} (s_{i}^{F})^{2}}$ . The respective insets show the real part of the second versus tenth principal components, obtained from the PCA of the $e − ph$ matrices; in parentheses we give the fraction of explained variance [67], ${\tilde{λ}}_{i} = σ_{i}^{2} / \sum_{i} σ_{i}^{2}$ , where $σ_{i}^{2}$ is the variance of the $i$ th principal component. The red oval shows the standard deviation of the corresponding principal components, obtained by dropping feature vectors with norm smaller than $10^{- 3} \times {\tilde{s}}_{0}$ . (c) Atomic vibrations associated with the dominant $e − ph$ interactions in graphene, and (d) the same quantity in silicon, obtained by analyzing the phonon singular vectors, $\tilde{v} (κ α R_{p})$ in Eq. (10), for the two largest SVs.
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