# Data-driven compression of electron-phonon interactions

Yao Luo  Department of Applied Physics and Materials Science, California
Institute of Technology, Pasadena, California 91125, USA  Dhruv Desai
Department of Applied Physics and Materials Science, California Institute of
Technology, Pasadena, California 91125, USA  Jinsoo Park  Department of
Applied Physics and Materials Science, California Institute of Technology,
Pasadena, California 91125, USA  Marco Bernardi  Department of Applied Physics
and Materials Science, California Institute of Technology, Pasadena,
California 91125, USA  Department of Physics, California Institute of
Technology, Pasadena, California 91125, USA<EMAIL_ADDRESS>
###### 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 / 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 two orders of magnitudes 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.

##  I Introduction

Electrons in materials are subject to various interactions, including those
with phonons, other electrons, and defects. Modeling of these interactions
follows two main approaches $-$ analytic treatments that qualitatively capture
the main physics with minimal models using only a few parameters, and first-
principles calculations aiming at quantitative accuracy but often requiring
specialized workflows, high computational cost, and large amounts of data. A
middle ground between these extremes would require formulating models of
electron interactions that are economical, accurate, and interpretable.
Examples of efficient models exist across domains $-$ in quantum chemistry,
low-rank approximations [1, 2, 3] can compress two-electron integrals to
reduce the computational cost of post-Hartree Fock calculations [4, 5] and
extract the critical vibrational modes in a chemical reaction [6, 7]; in
correlated-electron physics, efficient parametrization of $e$-$e$ interactions
[8] enables the solution of functional renormalization-group flow [9, 10] and
the Bethe-Salpeter equation [11, 12]. However, despite these isolated
examples, it remains challenging to formulate widely applicable approaches to
represent electron interactions both efficiently and accurately.
Focusing on electron-phonon ($e$-ph) interactions, analytic treatments such as
deformation potential for acoustic phonons [13, 14] and the Fröhlich model for
optical phonons [15], which use only a few parameters to describe $e$-ph
interactions, are still widely utilized [16, 17]. In recent years, first-
principles calculations of $e$-ph interactions using density functional theory
(DFT) [18] and its linear-response variant, density functional perturbation
theory (DFPT) [19], have enabled quantitative studies of properties ranging
from transport to excited state dynamics to superconductivity [20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. Unlike the analytic models, in a
typical first-principles calculation one represents the $e$-ph interactions
using a multi-dimensional matrix with millions or billions of entries. This
enormous number of parameters, which are computed rather than assumed,
guarantees a faithful description of microscopic details such as the
dependence on electronic states and phonon modes of $e$-ph interactions. Yet
this complexity is also a barrier toward obtaining minimal models and tackling
new physics. For example, materials with strong or correlated $e$-ph
interactions need specialized treatments to capture polaron effects [28, 30,
34, 35] and electron correlations [36, 37]. Reducing the high dimensionality
of first-principles $e$-ph interactions would allow one to more efficiently
describe this physics while retaining quantitative accuracy. The development
of data-driven methods to tackle the high-dimensional Hilbert space in the
many-electron problem, including neural network states [38, 39] and tensor
network methods [40, 41, 42, 43], serve as inspiration.
Here we show a low-rank approximation of first-principles $e$-ph interactions
which significantly accelerates $e$-ph calculations while using only a small
fraction (1$-$2%) of the data and preserving quantitative accuracy. This is
achieved by developing SVD calculations of $e$-ph matrices in Wannier basis to
achieve a minimal representation of $e$-ph coupling. We use our compressed
$e$-ph matrices to compute a range of properties, including charge transport,
spin relaxation, band renormalization, and superconductivity, both in metals
and semiconductors. Across all benchmarks, the highly compressed $e$-ph
representation achieves a quantitative accuracy comparable to the standard
workflow, while also providing a deeper understanding of the dominant patterns
governing $e$-ph interactions. Principal component analysis sheds light on the
inherent compressibility of $e$-ph coupling matrices. Recent interesting work
on improving the efficiency of e-ph calculations [44, 45] is distinct in
method and scope from our data-driven approach.

##  II results

###  II.1 compression of e-ph interactions

The key quantities in first-principles $e$-ph calculations are the $e$-ph
matrix elements $g_{mn\nu}(\bm{k},\bm{q})$, which represent the probability
amplitude for an electron in a band state $\ket{n\bm{k}}$, with band index $n$
and crystal momentum $\bm{k}$, to scatter into a final state
$\ket{m\bm{k}+\bm{q}}$ by emitting or absorbing a phonon with mode index
$\nu$, wave-vector $\bm{q}$, energy $\hbar\omega_{\nu\bm{q}}$, and
polarization vector $\textbf{e}_{\nu\bm{q}}$ [46]:

$g_{mn\nu}(\bm{k},\bm{q})=\sqrt{\frac{\hbar}{2\omega_{\nu\bm{q}}}}\sum_{\kappa\alpha}\frac{\bf{e}^{\kappa\alpha}_{\nu\bm{q}}}{\sqrt{M_{\kappa}}}\braket{m\bm{k}+\bm{q}}{\partial_{\bm{q}\kappa\alpha}V}{n\bm{k}},$
(1)

where
$\partial_{\bm{q}\kappa\alpha}V\equiv\sum_{p}e^{i\bm{q}\textbf{R}_{p}}\partial_{p\kappa\alpha}V$
is the lattice-periodic $e$-ph perturbation potential, given by the change in
the DFT Kohn-Sham potential with respect to the position of atom $\kappa$
(with mass $M_{\kappa}$ and located in unit cell at $\textbf{R}_{p}$) in the
Cartesian direction $\alpha$. The inset in Fig. 1(a) shows schematically such
an $e$-ph scattering process. We separate the $e$-ph interactions into short-
and long-ranged [47, 48, 49, 50, 51, 52, 53],

$g_{mn\nu}(\bm{k},\bm{q})=g^{L}_{mn\nu}(\bm{k},\bm{q})+g^{S}_{mn\nu}(\bm{k},\bm{q}).$
(2)

The long-range part $g^{L}_{mn\nu}(\bm{k},\bm{q})$ includes dipole (Fröhlich)
and quadrupole contributions, which can be written analytically, using
classical electromagnetism, in terms of Born effective charges and dynamical
quadrupoles obtained from DFPT. The short-ranged part
$g^{S}_{mn\nu}(\bm{k},\bm{q})$ cannot be written in closed form and needs
numerical quantum mechanics to be computed, a consequence of the
nearsightedness of electronic matter [54]. Because
$g^{S}_{mn\nu}(\bm{k},\bm{q})$ is a smooth function of electron and phonon
momenta, it is short-ranged in a real-space representation using a localized
basis set such as atomic orbitals [55] or Wannier functions [56, 46].
The short-range $e$-ph coupling matrix in Wannier basis,
$g_{ij}^{\kappa\alpha}(\bm{R}_{e},\bm{R}_{p})$, is obtained by transforming
DFPT results computed on a coarse momentum grid $(\bm{k}_{c},\bm{q}_{c})$
[46]:

$\displaystyle
g_{ij}^{\kappa\alpha}(\bm{R}_{e},\bm{R}_{p})=\frac{1}{N_{\bm{k}_{c}}N_{\bm{q}_{c}}}\sum_{mn\bm{k}_{c}}\sum_{\bm{q}_{c}}e^{-i(\bm{k}_{c}\bm{R}_{e}+\bm{q}_{c}\bm{R}_{p})}\,$
(3) $\displaystyle\times\
\mathcal{U}^{\dagger}_{im}(\bm{k}_{c}+\bm{q}_{c})\Delta
V^{S}_{mn,\kappa\alpha}(\bm{k}_{c},\bm{q}_{c})\,\mathcal{U}_{nj}(\bm{k}_{c}),$

where $\mathcal{U}$ is a unitary transformation from Bloch to Wannier basis,
and $\Delta
V^{S}_{mn,\kappa\alpha}(\bm{k}_{c},\bm{q}_{c})=\braket{m\bm{k}+\bm{q}}{\partial_{\bm{q}\kappa\alpha}V^{S}}{n\bm{k}}$
is the short-ranged part of the perturbation potential in Bloch basis. To
separate acoustic and optical modes, we carry out a rotation in atomic basis:

$g^{\mu\alpha}_{ij}(\bm{R}_{e},\bm{R}_{p})=\sum_{\kappa}A^{\mu}_{\kappa}\,g_{ij}^{\kappa\alpha}(\bm{R}_{e},\bm{R}_{p}),$
(4)

where $A^{\kappa}_{\mu}=\exp(i\frac{2\pi}{N_{\rm at}}\kappa\mu)$ adds a
relative phase to different atoms in the unit cell, and $\mu\in(0,...,N_{\rm
at}-1)$ labels phonon modes ($N_{\rm at}$ is the number of atoms in the unit
cell). This way, $\mu=0$ corresponds to the acoustic subspace, where all the
atoms in the unit cell move in phase, and $\mu\neq 0$ labels the optical
modes. Here and below, we use a collective index $F=(ij,\mu\alpha)$ to label
Wannier orbital pairs $ij$ and phonon mode and direction $\mu\alpha$,
simplifying the notation of the Wannier-basis $e$-ph matrices to
$g^{F}(\bm{R}_{e},\bm{R}_{p})$.
When viewed as a matrix for each mode and orbital pair,
$g^{F}(\bm{R}_{e},\bm{R}_{p})$ decays rapidly with lattice vectors
$\bm{R}_{e}$ and $\bm{R}_{p}$ and has a typical size $N_{\bm{R}_{e}}\times
N_{\bm{R}_{p}}$ ranging between $10^{2}\times 10^{2}$ and $10^{3}\times
10^{3}$. After carrying out SVD on $g^{F}(\bm{R}_{e},\bm{R}_{p})$, we obtain

$g^{F}(\bm{R}_{e},\bm{R}_{p})=\sum_{\gamma}s^{F}_{\gamma}u^{F}_{\gamma}(\bm{R}_{e})\,{v_{\gamma}^{F}}^{*}(\bm{R}_{p}),$
(5)

where $s^{F}_{\gamma}$ is the singular value (SV) with index $\gamma$, and
$u^{F}_{\gamma}(\bm{R}_{e})$ and ${v^{F}}^{*}_{\gamma}(\bm{R}_{p})$ are the
left and right singular vectors, respectively. One can interpret
$s^{F}_{\gamma}$ as the coupling strength between the generalized electron
cloud
$\sum_{\bm{R}_{e}}\\!u^{F}_{\gamma}(\bm{R}_{e})c^{\dagger}_{i}(\bm{R}_{e})c_{j}(\textbf{0})$
and phonon mode
$\sum_{\bm{R}_{p}}\\!{v_{\gamma}^{F}}^{*}(\bm{R}_{p})\\!\left(b^{\dagger}_{\mu\alpha}(\bm{R}_{p})+b_{\mu\alpha}(\bm{R}_{p})\right)$,
where $(c^{\dagger},c)$ are creation and annihilation operators for electrons
and $(b^{\dagger},b)$ for phonons. For each channel $F$, there is a total of
$\min(N_{\bm{R}_{e}},N_{\bm{R}_{p}})$ SVs; we keep only the $N_{c}$ largest
ones, resulting in a truncated, low-rank $e$-ph matrix $\tilde{g}$:

$\tilde{g}_{N_{c}}^{F}(\bm{R}_{e},\bm{R}_{p})=\sum_{\gamma=1}^{N_{c}}s^{F}_{\gamma}\,u^{F}_{\gamma}(\bm{R}_{e})\,{v_{\gamma}^{F}}^{*}(\bm{R}_{p}).$
(6)

This matrix can be conveniently transformed to momentum space using

$\displaystyle\tilde{g}^{F}_{N_{c}}(\bm{k},\bm{q})$
$\displaystyle=\sum_{\bm{R}_{p},\bm{R}_{e}}e^{i\bm{k}\bm{R}_{e}+i\bm{q}\bm{R}_{p}}\,\tilde{g}_{N_{c}}^{F}(\bm{R}_{e},\bm{R}_{p})$
$\displaystyle\approx\sum_{\gamma=1}^{N_{c}}s^{F}_{\gamma}\,u^{F}_{\gamma}(\bm{k})\,{v_{\gamma}^{F}}^{*}(\bm{q}),$
(7)

where
$u^{F}_{\gamma}(\bm{k})\\!=\\!\sum_{\bm{R}_{e}}e^{i\bm{k}\bm{R}_{e}}u^{F}_{\gamma}(\bm{R}_{e})$
and
${v_{\gamma}^{F}}^{*}(\bm{q})\\!=\\!\sum_{\bm{R}_{p}}e^{i\bm{q}\bm{R}_{p}}v^{F}_{\gamma}(\bm{R}_{p})$
are singular vectors in momentum space. (Note that the long-range part of the
$e$-ph matrix is added after interpolation of this short-ranged part.) Eq. (7)
provides a generic parametrization of $e$-ph interactions, where by increasing
the number of SVs one can systematically tune the accuracy and computational
cost. According to the Eckart–Young-Mirsky theorem, the truncated matrix
$\tilde{g}$ obtained from SVD is an optimal low-rank approximation of $e$-ph
interactions, in the sense that it minimizes the Frobenius-norm distance
between the original and low-rank $e$-ph matrices [57]. From a computational
viewpoint, Eq. (7) can greatly accelerate the calculation of e-ph interactions
and the associated material properties, with a speed up by the inverse
fraction of SVs kept in the truncated e-ph matrix. In most cases, we will keep
only 1-2$\%$ of SVs, resulting in a $\sim$100 times speed-up for the key step
in e-ph calculations (see Appendix A for details).

###  II.2 Error and Pareto-optimal interactions

To test the accuracy of the truncated $e$-ph matrix and its convergence with
respect to the number of SVs ($N_{c}$), we define a relative error for the
$e$-ph matrix averaged over electron bands and momenta, and phonon modes and
wave-vectors:

$\epsilon_{g}(N_{c})=\frac{\sum_{mn\nu,\bm{k}\bm{q}}|g_{mn\nu}(\bm{k},\bm{q})-\tilde{g}^{N_{c}}_{mn\nu}(\bm{k},\bm{q})|^{2}}{\sum_{mn\nu,\bm{k}\bm{q}}|g_{mn\nu}(\bm{k},\bm{q})|^{2}},$
(8)

where $\tilde{g}^{N_{c}}_{mn\nu}(\bm{k},\bm{q})$ is the low-rank (approximate)
and $g_{mn\nu}(\bm{k},\bm{q})$ is the full first-principles $e$-ph matrix.

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
$N_{\bm{R}_{e}}\\!=\\!1325$ and $N_{\bm{R}_{p}}\\!=\\!1325$, the smallest
values to achieve convergence, setting the electron momentum to
$\bm{k}\\!=\\!\Gamma$ 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.

Fig. 1(a) shows this error as a function of the fraction of SVs,
$N_{c}/N_{\bm{R}_{p}}$, kept in the approximate matrix. In the language of
model selection [57], the resulting curve of error versus number of parameters
is the Pareto frontier for modeling $e$-ph interactions. We find that the
error decreases rapidly with the number of SVs $-$ for example, $\epsilon_{g}$
is as low as $1\%$ when using only $2\%$ of SVs, which achieves a $98\%$
compression of the original $e$-ph matrix. This error curve defines a Pareto-
optimal region, highlighted in Fig. 1(a), where $e$-ph calculations are both
accurate and parsimonious [57]. This region spans 1$-$4% of SVs in most of our
calculations $-$ which corresponds to keeping $N_{c}\approx$ 10$-$50 SVs $-$
and suggests that many materials may possess only $\sim$10 dominant elementary
$e$-ph interaction patterns. Accordingly, the $e$-ph coupling strength for
each phonon mode [58, 46],

$D_{\nu}(\bm{q})=\hbar^{-1}\sqrt{2\omega_{\nu\bm{q}}M_{\rm
uc}\sum_{mn}|g_{mn\nu}(\bm{k}=\Gamma,\bm{q})|^{2}/N_{b}}\,,$ (9)

(where $M_{\rm uc}$ is the mass of the unit cell and the band indices $m$ and
$n$ run over $N_{b}$ bands), shown in Fig. 1(b), can be computed accurately
using just the largest 1.5% of SVs, matching closely results using the full
$e$-ph matrix.

Figure 2:  (a) Mobility of electrons in silicon, and (b) mobility of hole
carriers in silicon, computed with the full $e$-ph matrix
($N_{\bm{R}_{e}}\\!=\\!N_{\bm{R}_{p}}\\!=\\!1325$) 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
($N_{\bm{R}_{e}}\\!=\\!N_{\bm{R}_{p}}\\!=\\!1325$) and using SVD with 1.5% of
the SVs. Experimental data from Refs. [59, 60] are shown for comparison. (d)
Eliashberg spectral function $\alpha^{2}F(\omega)$ for Pb, comparing full
$e$-ph matrix ($N_{\bm{R}_{e}}\\!=\\!N_{\bm{R}_{p}}\\!=\\!279$) with SVD
results using 1.8% of the SVs. (e) Superconducting gap $\Delta_{0}$ 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.

###  II.3 Application to transport, spin and superconductivity

We showcase the accuracy of the low-rank approximate $e$-ph interactions by
computing a wide range of material properties, including charge mobility, spin
relaxation, phonon-assisted superconductivity, and phonon-induced band
renormalization. Figs. 2(a) and 2(b) show the electron and hole mobility in
silicon for temperatures between 100$-$400 K, obtained using the full $e$-ph
matrix and compared with SVD using 1.5% of the SVs (see Appendix B). The
mobility is overestimated for electrons, and underestimated for hole carriers,
despite the accuracy of the low-rank $e$-ph interactions in silicon (Fig.
1(a)). The error comes from the acoustic phonons, which interact weakly with
electrons $-$ and therefore are ignored in the low-rank $e$-ph matrix $-$ but
carry a considerable contribution to the mobility due to their large thermal
occupation. To improve the treatment of acoustic phonons, we develop a
constrained SVD (c-SVD) which preserves the deformation potential [61] for
long-wavelength acoustic phonons in the compressed $e$-ph matrix (see Appendix
C). When using c-SVD, the mobility computed using only 1.5% of the SVs is
nearly identical to the full-matrix result for both carriers.
We also apply the low-rank approximation to spin-dependent $e$-ph matrices
governing spin-flip $e$-ph interactions; these matrices enable first-
principles calculations of spin relaxation times (SRTs) in centrosymmetric
materials via the Elliot-Yafet mechanism [62] (see Appendix D). The SRTs for
electrons in silicon between 150$-$400 K are shown in Fig. 2(c). Our results
from SVD with $N_{c}\\!=\\!20$ (corresponding to $\sim$1.5% of the SVs) match
closely the full $e$-ph matrix calculations and agree with experimental
results [59, 60]. Different from charge transport, standard SVD gives accurate
SRTs in silicon because the optical phonons govern spin-flip processes. For
materials where acoustic phonons contribute to spin relaxation, our c-SVD
approach can be readily extended to the spin-dependent case.

Figure 3:  (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, $\tilde{s}_{i}=\sqrt{\sum_{F}(s^{F}_{i})^{2}}$. The
respective insets show the real part of the 2nd versus 10th principal
components, obtained from the PCA of the $e$-ph matrices; in parentheses we
give the fraction of explained variance [66],
$\tilde{\lambda}_{i}=\sigma^{2}_{i}/{\sum_{i}\sigma_{i}^{2}}$, where
$\sigma_{i}^{2}$ 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,
$\tilde{v}(\kappa\alpha\mathbf{R}_{p})$ in Eq. (10), for the two largest SVs.

Calculations of phonon-mediated superconductivity, presented here using lead
(Pb) as an example, can also leverage our low-rank approximation (see Appendix
E). Fig. 2(d) compares the Eliashberg spectral function $\alpha^{2}F(\omega)$
from full $e$-ph matrix calculations with SVD results; using only the first
$N_{c}\\!=\\!5$ SVs (here equal to 1.8% of the SVs) suffices to reproduce the
full calculation. We solve the isotropic Eliashberg equation self-consistently
(see Methods for details) and compute the superconducting gap $\Delta_{0}$ as
a function of temperature (Fig. 2(e)) as well as the critical temperature
$T_{c}$ versus number of SVs (inset of Fig. 2(e)). The low-rank $e$-ph matrix
with $N_{c}=5$ SVs provides a gap function in good agreement with the full-
matrix calculation, which can be further improved by using a larger fraction
of SVs; the critical temperature for $N_{c}=5$ SVs is very accurate,
$T_{c}=6.9$ K which is within $5\%$ of the $T_{c}=6.6$ K value obtained using
the full $e$-ph matrix. This result implies that as few as five elementary
$e$-ph interactions determine $T_{c}$ in Pb. Finally, we compute band
renormalization from e-ph interactions [63] ] focusing on the contribution
from the Fan-Migdal e-ph self-energy (see Appendix F). The Debye-Waller term
can also be added following Ref. [64]. Results for graphene show that band
renormalization near the Dirac cone can be computed keeping only the five
largest SVs; this highly compressed $e$-ph matrix correctly predicts the kinks
near the Dirac cone and matches full $e$-ph matrix results over a 2 eV energy
range. We also carry out convergence tests with respect to the fraction of SVs
included in the calculation for all the materials studied here (see the
Supplemental Material [65]). The rapid convergence with respect to the
fraction of SVs guarantees that the full e-ph matrix calculation is not
needed, and one can obtain accurate results by converging the desired property
with respect to the fraction of SVs with minimal computational overhead.
While all the examples discussed above are for materials with nonpolar bonds,
polar materials are even simpler to study with our compression method because
the long-range (Frohlich) dipole contribution is dominant and is well modeled
by an analytic formula using Born effective charges [47, 48, 49]. To
illustrate this point, we demonstrate accurate mobility calculations in GaAs
and PbTiO3 using only 1% of the SVs (see the Supplemental Material [65]).

###  II.4 Dominant modes and
principal-component analysis

To understand the inherent compressibility of the $e$-ph matrices, we analyze
the SV spectrum in graphene and silicon (Figs. 3(a) and 3(b)). In both
materials, the SVs decay rapidly, dropping by 1$-$2 orders of magnitude from
the largest to the 10th largest SV. In principal component analysis (PCA) [57,
66], this decay can be understood as a consequence of high-variance
generalized directions in the $e$-ph matrix,
$g^{F}(\mathbf{R}_{e},\mathbf{R}_{p})$, which capture the vast majority of the
physics, while other principal components can be viewed as noise and neglected
[66]. We carry out PCA by treating each row of the matrix
$g^{F}(\bm{R}_{e},\bm{R}_{p})$ as a feature vector, and find that the variance
of the two leading principal components is one order of magnitude greater than
for the 10th or following principal components, indicating that most of the
physical information is already captured by the first few SVs (see the insets
of Figs. 3(a) and 3(b)). This analysis reveals that only a few atomic
vibrational patterns dominate $e$-ph coupling. Although these dominant modes
are not known a priori, they can be learned efficiently with SVD. We also
apply the PCA to lead (see Fig. S4 in Supplemental Material [65]) ]) and
observe a similar rapid decay of the SVs. We remark that the dimensionality
reduction is general – it occurs in all the materials studied here, and it is
associated to the rapid decay of the SVs, which we view as a consequence of
the nearsightedness of electronic interactions.
We visualize the atomic vibrations with dominant $e$-ph interactions by
analyzing the vibrational singular vectors. To that end, we introduce a
modified SVD that includes Wannier orbitals and phonon modes in the
decomposition:

$g_{ij}^{\kappa\alpha}(\bm{R}_{e},\bm{R}_{p})=\sum_{\gamma}s_{\gamma}\,\tilde{u}_{\gamma}(ij\bm{R}_{e})\,\tilde{v}_{\gamma}^{*}(\kappa\alpha\bm{R}_{p}).$
(10)

In this global SVD, the singular vectors $\tilde{u}$ depend only on electron
variables and $\tilde{v}$ only on phonon variables. This way, the phonon
singular vectors $\tilde{v}^{*}_{\gamma}(\kappa\alpha\bm{R}_{p})$ can be
interpreted as local vibrational modes (in the Wigner-Seitz cell associated
with the coarse grid [46]) and visualized to study the dominant $e$-ph
couplings. We show these singular vectors for the two modes with largest SVs
in graphene and silicon in Figs. 3(c) and 3(d), using arrows on each atom,
with length proportional to the singular vector
$\tilde{v}^{*}_{\gamma}(\kappa\alpha\bm{R}_{p})$, to indicate the atomic
displacements in the modes obtained from SVD.
In graphene, where the electronic states consist of $p_{z}$ orbitals centered
on each carbon atom, the dominant mode resembles a longitudinal optical phonon
that brings the $p_{z}$ orbitals closer together in the unit cell. The second-
strongest mode is a shear vibration resembling a transverse optical phonon,
which spreads over multiple unit cells (Fig. 3(c)). For the other modes, we
observe that the vibrational pattern progressively delocalizes over multiple
unit cells for decreasing values of the SVs. In silicon, where the electronic
states consist of $sp^{3}$-like Wannier orbitals oriented along the chemical
bond directions, the two modes with dominant $e$-ph coupling are associated
with compression and stretching of the bonds (Fig. 3(d)). The intuition gained
from this mode analysis can aid the formulation of model Hamiltonians in
chemically and structurally complex materials, where keeping only the dominant
$e$-ph interactions can provide effective models of transport and polaron
physics informed by first-principles calculations [67, 68, 69, 70, 71].

##  III Discussion and conclusion

The accuracy of the low-rank $e$-ph matrices implies that current brute-force
first-principles calculations overparametrize $e$-ph interactions, falling too
far on the right side of the Pareto-optimal region in Fig. 1(a). Conversely,
textbook approaches such as Holstein and Fröhlich models, which use only a
handful of $e$-ph couplings, may fall short of achieving quantitative accuracy
by using too few parameters. Our SVD compression in Wannier basis (followed by
interpolation) provides a systematic route to achieve Pareto-optimal
calculations. These optimal models enhance interpretability and enable a
deeper understanding because they concentrate all the relevant $e$-ph physics
in just a few parameters $-$ in our case, the leading SVs and singular
vectors, which represent dominant $e$-ph interactions.
In summary, our results unveil the hidden low-dimensional nature of $e$-ph
interactions. While accurate, current first-principles calculations
overparametrize these interactions due to a lack of a priori knowledge of the
dominant atomic vibrational patterns governing $e$-ph coupling. We have shown
that when this optimal representation is achieved via SVD, using only 10$-$20
parameters (for each orbital pair and vibrational mode) is sufficient to
obtain results with state-of-the-art accuracy. Surprisingly, this is only a
small fraction (1$-$2%) of the typical size of first-principles $e$-ph
matrices. Compressing $e$-ph interactions significantly accelerates
calculations of material properties ranging from transport to spin relaxation
to superconductivity. Future work will extend these ideas to other electronic
interactions, with the goal of advancing “precise but parsimonious” quantum
many-body calculations in real materials. Our approach works equally as well
for small systems with a few atoms and for large systems with tens of atoms in
the unit cell (see Fig. S5 in Supplemental Material [65]). In addition, as we
plan to show elsewhere, our method enables many-body e-ph calculations that
are currently inaccessible with standard Wannier interpolation, including the
development of first-principles diagrammatic Monte Carlo, which sums e-ph
diagrams to all orders from first principles, thus providing a gold standard
for quantitative e-ph studies.

###### Acknowledgements.

Y.L. thanks Ivan Maliyov and Junjie Yang for fruitful discussions. This work
was supported by the National Science Foundation under Grant No. OAC-2209262.
This research used resources of the National Energy Research Scientific
Computing Center (NERSC), a U.S. Department of Energy Office of Science User
Facility located at Lawrence Berkeley National Laboratory, operated under
Contract No. DE-AC02-05CH11231.

##  Appendix A Computational complexity analysis

From a computational viewpoint, Eq. (7) can greatly accelerate the calculation
of $e$-ph interactions and the associated material properties. The key
bottleneck in these calculations is obtaining the $e$-ph matrix elements on
fine momentum grids, $g_{mn\nu}(\bm{k}_{f},\bm{q}_{f})$, starting from the
Wannier representation, with a cost scaling as
$\mathcal{O}(N_{\bm{R}_{p}}N_{\bm{k}_{f}}N_{\bm{q}_{f}})$ for an optimal
implementation [46] (for a fixed number of Wannier functions and atoms in the
unit cell), where $N_{\bm{k}_{f}}$ and $N_{\bm{q}_{f}}$ are the number of
points in the fine-momentum electron and phonon grids, with typical values of
order $N_{\bm{k}_{f}}\approx N_{\bm{q}_{f}}\approx 10^{6}$. In contrast, when
using SVD, this interpolation step costs only
$\mathcal{O}(N_{c}N_{\bm{k}_{f}}N_{\bm{q}_{f}})$, with a speed up by a factor
$N_{\bm{R}_{p}}/N_{c}$, the inverse fraction of SVs kept in the truncated
$e$-ph matrix. In most cases, we will keep only 1$-$2% of SVs, resulting in a
$\sim$100 times speed-up for the key step in $e$-ph calculations. We show
specific timing comparisons for all the materials studied in this work in Fig.
S1 of the Supplemental Material [65]. In all cases, our algorithm achieves a
speed up close to the ideal value of $N_{\bm{R}_{p}}/N_{c}$. The memory
improvement is also dramatic. A converged transport calculation in silicon
requires a $\mathbf{k}$ grid of $100^{3}$ and $\mathbf{q}$ grid of $50^{3}$
points [46]; on these fine grids, the memory required to store the entire
$e$-ph matrix $g_{mn\nu}(\bm{k}_{f},\bm{q}_{f})$ is 700 TB, while the memory
needed to store the singular vectors $u_{\gamma}^{F}(\bm{k}_{f})$ and
$v_{\gamma}^{F}(\bm{q}_{f})$ is only 128 GB when we retain $1.5\%$ of SVs,
which guarantees accurate results as we show in Figs. 2(a) and 2(b). This
efficiency removes the key bottleneck in first-principles $e$-ph calculations.

##  Appendix B Mobility calculations

The first-principles mobility calculations in silicon follows our previous
work [51]. We include the quadrupole contribution analytically for silicon.
The quadrupole tensor can be written as

$Q_{\mathrm{si},\alpha\beta\gamma}=(-1)^{\kappa+1}Q_{\mathrm{si}}|\epsilon_{\alpha\beta\gamma}|,$
(11)

where $\epsilon_{\alpha\beta\gamma}$ is the Levi-Civita tensor and the value
of $Q_{\mathrm{si}}=13.67$ is taken from Refs. [72, 73].

We compute the phonon-limited mobility at temperature $T$ using the BTE in the
relaxation time approximation (RTA) [46]. We first obtain the $e$-ph
scattering rate $\Gamma_{n\bm{k}}$ using Fermi’s golden rule, which is
equivalent to using the imaginary part of the lowest-order $e$-ph self-energy
[74]:

$\displaystyle\Gamma_{n\bm{k}}$
$\displaystyle=\frac{2\pi}{\hbar}\frac{1}{\mathcal{N}_{\bm{q}}}\sum_{m\nu\bm{q}}|g_{mn\nu}(\bm{k},\bm{q})|^{2}$
$\displaystyle\times[(N_{\nu\bm{q}}+1-f_{m\bm{k}+\bm{q}})\delta(\epsilon_{n\bm{k}}-\epsilon_{m\bm{k}+\bm{q}}-\hbar\omega_{\nu\bm{q}})$
$\displaystyle+(N_{\nu\bm{q}}+f_{m\bm{k}+\bm{q}})\,\delta(\epsilon_{n\bm{k}}-\epsilon_{m\bm{k}+\bm{q}}+\hbar\omega_{\nu\bm{q}})],$
(12)

where $\mathcal{N}_{\bm{q}}$ is the number of $\bm{q}$ points and $\delta$ is
the Dirac delta function. Then we obtain the mobility from the BTE by summing
over contributions from different electronic states and scattering processes
[46]:

$\mu_{\alpha\beta}(T)=\frac{e}{n_{c}\Omega\mathcal{N}_{\bm{k}}}\int
dE\,\left(-\frac{\partial f}{\partial
E}\right)\sum_{n\bm{k}}\tau_{n\bm{k}}\,\bm{v}_{n\bm{k}}^{\alpha}\bm{v}_{n\bm{k}}^{\beta}\delta(E-\epsilon_{n\bm{k}}),$
(13)

where $\Omega$ is the volume of the unit cell,
$\tau_{n\bm{k}}=(\Gamma_{n\bm{k}})^{-1}$ are relaxation times, $n_{c}$ is the
carrier concentration, $f$ is the Fermi-Dirac distribution, and
$\mathcal{N}_{\bm{k}}$ is the number of $\bm{k}$ points; $\epsilon_{n\bm{k}}$
and $\bm{v}_{n\bm{k}}$ are electron energies and band velocities,
respectively. Our calculations in silicon use a uniform grid with $200^{3}$
$\bm{k}$ points and a uniform random grid with $10^{5}$ $\bm{q}$ points, where
$\bm{k}$ and $\bm{q}$ are electron and phonon momenta respectively. The delta
function is approximated as a Gaussian with a 10 meV smearing [46].

##  Appendix C c-SVD algorithm

Let us briefly describe our c-SVD algorithm. Similar to the acoustic sum rule
(ASR) for the dynamical matrix [19], we formulate an ASR for the $e$-ph matrix
elements:

$g_{\nu_{A}}(\bm{k},\bm{q}=0)=0,$ (14)

where $\nu_{A}$ labels the acoustic modes, and we omit band indices for
simplicity. The rationale for this $e$-ph ASR is that a rigid translation of
the lattice will not change the electronic band structure. The real-space
version of this $e$-ph ASR reads

$\sum_{\kappa,\bm{R}_{p}}g^{\kappa\alpha}_{ij}(\bm{R}_{e},\bm{R}_{p})=\sum_{\bm{R}_{p}}g^{\mu=0,\alpha}_{ij}(\bm{R}_{e},\bm{R}_{p})=0,$
(15)

where $g^{\mu=0,\alpha}_{ij}(\bm{R}_{e},\bm{R}_{p})$ accounts for the acoustic
subspace of the $e$-ph matrix defined in Eq. (4). With this ASR, the $e$-ph
matrix for long-wavelength acoustic phonons can be approximated to first order
in $\bm{q}$ as

$\displaystyle\lim_{\bm{q}\rightarrow 0}\,g^{\mu=0,\alpha}(\bm{R}_{e},\bm{q})$
$\displaystyle=\lim_{\bm{q}\rightarrow
0}\,\sum_{\bm{R}_{p}}g^{\mu=0,\alpha}(\bm{R}_{e},\bm{R}_{p})e^{i\bm{q}\bm{R}_{p}}$
$\displaystyle\approx i\bm{q}\cdot\textbf{A}^{\alpha}(\bm{R}_{e})\,,$ (16)

where we defined a real-space deformation potential, which in general can be
anisotropic, as

$\textbf{A}^{\alpha}(\bm{R}_{e})\equiv\sum_{\bm{R}_{p}}\bm{R}_{p}g^{\mu=0,\alpha}(\bm{R}_{e},\bm{R}_{p}).$
(17)

In the limit of $|\bm{q}|\to 0$, $\tilde{g}^{\mu=0,\alpha}(\bm{R}_{e},\bm{q})$
vanishes linearly in $|\bm{q}|$, but the phonon occupation number diverges as
$\frac{1}{|\bm{q}|}$; therefore in the long-wavelength limit acoustic phonon
scattering is often important. This acoustic phonon contribution is
challenging to preserve when using the compressed $e$-ph matrices because
standard SVD primarily captures large entries in the $e$-ph matrix. To address
this point, we compress the $e$-ph matrix while conserving
$\textbf{A}^{\alpha}(\bm{R}_{e})$ by imposing the following constraint:

$\sum_{\bm{R}_{p}}\bm{R}_{p}\,\tilde{g}^{\mu=0,\alpha}_{N_{c}}(\bm{R}_{e},\bm{R}_{p})=\textbf{A}^{\alpha}(\bm{R}_{e}).$
(18)

Satisfying this set of linear equations leads to a constrained low-rank
approximation [75], a more general optimization problem.
The c-SVD is applied only to the acoustic subspace, which corresponds to
$F=(ij,\mu=0\alpha)$, resulting in a compressed $e$-ph matrix of rank $N_{c}$:

$\displaystyle\tilde{g}^{F}_{N_{c}}(\bm{R}_{e},\bm{R}_{p})$
$\displaystyle=\tilde{g}^{F}_{N_{c}-3}(\bm{R}_{e},\bm{R}_{p})$ (19)
$\displaystyle+\sum_{\beta\beta^{\prime}\in(x,y,z)}\left(\delta\textbf{A}^{\alpha}(\bm{R}_{e})\right)_{\beta}\lambda_{\beta\beta^{\prime}}\left(\bm{R}_{p}\right)_{\beta^{\prime}},$

where $\tilde{g}^{F}_{N_{c}-3}(\bm{R}_{e},\bm{R}_{p})$ is the truncated SVD of
the $e$-ph matrix with $N_{c}-3$ singular values (see Eq. (6));
$\delta\textbf{A}^{\alpha}$ is the residual term for
$\tilde{g}^{F}_{N_{c}-3}(\bm{R}_{e},\bm{R}_{p})$, defined as

$\delta\textbf{A}^{\alpha}(\bm{R}_{e})=\textbf{A}^{\alpha}(\bm{R}_{e})-\sum_{\bm{R}_{p}}\bm{R}_{p}\,\tilde{g}^{\mu=0,\alpha}_{N_{c}-3}(\bm{R}_{e},\bm{R}_{p}),$
(20)

and $\lambda_{\beta\beta^{\prime}}$ is the inverse of the overlap matrix
between $\bm{R}_{p}$ vectors:

$\sum_{\beta^{\prime}}\lambda_{\beta\beta^{\prime}}\sum_{\bm{R}_{p}}\left(\bm{R}_{p}\right)_{\beta^{\prime}}\left(\bm{R}_{p}\right)_{\beta^{\prime\prime}}=\delta_{\beta\beta^{\prime\prime}}.$
(21)

Using this approach, the compressed $e$-ph matrix
$\tilde{g}^{F}_{N_{c}}(\bm{R}_{e},\bm{R}_{p})$ gives the same deformation
potential as the full $e$-ph matrix in Eq. (17), and its rank is smaller than
or equal to $N_{c}$. This c-SVD workflow requires only a minimal computational
overhead relative to standard SVD.

##  Appendix D Spin relaxation times

The first-principles calculation of SRTs in silicon follows our recent work
[62]. The spin-flip relaxation time $\tau^{\text{flip}}_{n\bm{k}}$, for a band
electron in state $\ket{n\bm{k}}$, accounts for the Elliott-Yafet spin
relaxation mechanism and is computed using [76, 77, 62]

$\displaystyle\frac{1}{\tau^{\text{flip}}_{n\bm{k}}}$
$\displaystyle=\frac{4\pi}{\hbar}\sum_{m\nu\bm{q}}|g^{\text{flip}}_{mn\nu}(\bm{k},\bm{q})|^{2}$
$\displaystyle\times[(N_{\nu\bm{q}}+1-f_{m\bm{k}+\bm{q}})\,\delta(\epsilon_{n\bm{k}}-\epsilon_{m\bm{k}+\bm{q}}-\hbar\omega_{\nu\bm{q}})$
$\displaystyle+(N_{\nu\bm{q}}+f_{m\bm{k}+\bm{q}})\,\delta(\epsilon_{n\bm{k}}-\epsilon_{m\bm{k}+\bm{q}}+\hbar\omega_{\nu\bm{q}})].$
(22)

The key ingredients in this equation are the spin-flip $e$-ph matrix elements,

$g^{\text{flip}}_{mn\nu}(\bm{k},\bm{q})=\braket{m\bm{k}+\bm{q}\Downarrow}{\Delta\hat{V}_{\nu\bm{q}}}{n\bm{k}\Uparrow},$
(23)

with $\Downarrow$ and $\Uparrow$ denoting nearly spin-down and nearly spin-up
states, respectively. These matrix elements describe the probability amplitude
to flip the spin of a band electron due to a particular phonon mode
$\nu\bm{q}$. The macroscopic spin relaxation time, $\tau_{s}(T)$, is a thermal
average over electronic states of the spin-flip scattering rates [62]:

$\tau_{s}(T)=\left\langle\frac{1}{\tau^{\text{flip}}_{n\bm{k}}}\right\rangle^{-1}_{T}=\left(\frac{\sum_{n\bm{k}}\frac{1}{\tau^{\text{flip}}_{n\bm{k}}}\left(-\frac{df_{n\bm{k}}}{dE}\right)}{\sum_{n\bm{k}}\left(-\frac{df_{n\bm{k}}}{dE}\right)}\right)^{-1}.$
(24)

The SRT calculations employ a uniform grid with up to $140^{3}$ $\bm{k}$
points and a 5 meV Gaussian smearing for the delta functions.

##  Appendix E Eliashberg spectral function and superconducting gap

We carry out DFT calculations on lead (Pb) using the generalized gradient
approximation [78] in the Quantum ESPRESSO code [79]. The ground state and
electron wave functions are computed on a $14\times 14\times 14$
$\bm{k}$-point grid with a kinetic energy cutoff of 90 Ry, and the lattice
constant is set to 4.88 Å. We use DFPT to calculate the phonon frequencies and
eigenvectors, and the $e$-ph matrix elements $g_{mn\nu}(\bm{k},\bm{q})$, on
coarse $6\times 6\times 6$ $\bm{k}$\- and $\bm{q}$-point grids. We wannierize
the 4 bands near the Fermi surface using the Wannier90 code [80], and obtain
the $e$-ph matrices in Wannier basis using Perturbo [46]. The Eliashberg
spectral function is computed as

$\displaystyle\alpha^{2}F(\omega)$
$\displaystyle=\frac{1}{2}\sum_{\nu\bm{q}}\omega_{\nu\bm{q}}\,\lambda_{\nu\bm{q}}\,\delta(\omega-\omega_{\nu\bm{q}}),~{}~{}~{}~{}\text{with}$
$\displaystyle\lambda_{\nu\bm{q}}$
$\displaystyle=\frac{1}{N(\epsilon_{F})\omega_{\nu\bm{q}}}\sum_{mn\bm{k}}|g_{mn\nu}(\bm{k},\bm{q})|^{2}$
(25)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\times\delta(\epsilon_{n\bm{k}}-\epsilon_{F})\delta(\epsilon_{m\bm{k}+\bm{q}}-\epsilon_{F}),$

where $N(\epsilon_{F})$ is the density of states at the Fermi energy
($\epsilon_{F}$). The Eliashberg function $\alpha^{2}F(\omega)$ encodes the
isotropic and retarded effective attraction between electronic states on the
Fermi surface. Using $\alpha^{2}F(\omega)$, we obtain the gap function by
solving the isotropic Migdal-Eliashberg Eq. [81]:

$\displaystyle Z(i\omega_{j})$ $\displaystyle=1+\frac{\pi
k_{B}T}{\omega_{j}}\sum_{j^{\prime}}\frac{\omega_{j^{\prime}}}{\sqrt{\omega_{j^{\prime}}^{2}+\Delta^{2}(i\omega_{j})}}\lambda(\omega_{j}-\omega_{j^{\prime}}),$
$\displaystyle Z(i\omega_{j})$ $\displaystyle\Delta(i\omega_{j})=\pi
k_{B}T\sum_{j^{\prime}}\frac{\Delta(i\omega_{j^{\prime}})}{\sqrt{\omega_{j^{\prime}}^{2}+\Delta^{2}(i\omega_{j^{\prime}})}}$
(26)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left[\lambda(\omega_{j}-\omega_{j^{\prime}})-\mu_{c}^{*}\right],$

where $T$ is the temperature, $\omega_{j}=(2j+1)\pi k_{B}T$ is the Matsubara
frequency, $\mu_{c}^{*}$ is the screened Coulomb potential, $Z(i\omega_{j})$
is the mass renormalization function, $\Delta(i\omega_{j})$ is the
superconducting gap function, and
$\lambda(\omega_{j})=\int_{0}^{\infty}d\omega\alpha^{2}F(\omega)\frac{2\omega}{\omega_{j}^{2}+\omega^{2}}$
is the isotropic $e$-ph coupling strength.

For the numerical integrations in Eq. (E), we employ a $\bm{k}$-point grid
consisting of 400,000 quasi-random Sobol points (generated using SciPy[82])
and a $\bm{q}$-point grid with 30,000 uniformly distributed random points; the
delta functions are approximated as Gaussians with a $30$ meV smearing for
electrons and $0.1$ meV smearing for phonons. Using the converged
$\alpha^{2}F(\omega)$ function, we set $\mu_{c}^{*}=0.1$ and solve Eq. (E)
iteratively for a range of temperatures. The critical temperature $T_{c}$ is
obtained as the temperature where $\Delta_{o}=\Delta(i\omega_{j}=i\pi k_{B}T)$
extrapolates to zero.

##  Appendix F Band structure
renormalization

The DFT ground state calculation in graphene uses the local density
approximation with a norm-conserving pseudopotential from Pseudo DOJO[83]. We
employ a 90 Ry plane-wave kinetic energy cutoff, a $60\times 60\times 1$
$\bm{k}$-point grid, and a 2.46 Å lattice constant. For the DFPT calculation,
we use coarse grids with $36\times 36\times 1$ $\bm{k}$-points for electrons
and $18\times 18\times 1$ $\bm{q}$-points for phonons. The band structure
renormalized by $e$-ph interactions, $\tilde{\epsilon}_{n\bm{k}}$, is obtained
as the DFT band structure plus the real part of the $e$-ph self-energy
evaluated on-shell,

$\tilde{\epsilon}_{n\bm{k}}=\epsilon_{n\bm{k}}+\real\Sigma_{n\bm{k}}(E=\epsilon_{n\bm{k}},T).$
(27)

We use the lowest-order (Fan-Migdal) $e$-ph self-energy,

$\displaystyle\Sigma_{n\bm{k}}(E,T)=\frac{1}{\mathcal{N}_{\bm{q}}}\sum_{\nu\bm{q}m}|g_{mn\nu}(\bm{k},\bm{q})|^{2}$
(28)
$\displaystyle\times\left[\frac{N_{\nu\bm{q}}+1-f_{m\bm{k}+\bm{q}}}{E-\epsilon_{m\bm{k}+\bm{q}}-\hbar\omega_{\nu\bm{q}}-i\eta}+\frac{N_{\nu\bm{q}}+f_{m\bm{k}+\bm{q}}}{E-\epsilon_{m\bm{k}+\bm{q}}+\hbar\omega_{\nu\bm{q}}-i\eta}\right],$

where $\eta$ is a Lorentzian smearing [84]. We employ $10^{7}$ uniform random
$\bm{q}$-points for the numerical integration in Eq. (28) and set the
Lorentzian smearing to 15 meV.

For more accurate band renormalization calculations, one could use Wannier
function perturbation theory (WFPT) to overcome errors resulting from finite
number of Wannier functions [85].

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