Data-Driven Compression of Electron-Phonon Interactions

Yao Luo ,

Dhruv Desai,

Benjamin K. Chang,

Jinsoo Park,

and Marco Bernardi

1,2

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

(Received 16 July 2023; revised 17 January 2024; accepted 29 March 2024; published 1 May 2024)

First-principles calculations of electron interactions in materials have seen rapid progress in recent years,

with electron-phonon (

-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

-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

-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

-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

-ph interactions. Furthermore, they accelerate state-of-the-art first-principles

-ph calculations by about 2

orders of magnitude without sacrificing accuracy. Our Pareto-optimal parametrization of

-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.

DOI:

10.1103/PhysRevX.14.021023

Subject Areas: Computational Physics,

Condensed Matter Physics,

Quantum Physics

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 quan-

titative 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 eco-

nomical, accurate, and interpretable. Examples of efficient

models exist across domains: in quantum chemistry,

low-rank approximations

–

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 electron-electron (

)

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 (

-ph) interactions, ana-

lytic 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

-ph interactions, are still widely utilized

[16,17]

. In recent

years, first-principles calculations of

-ph interactions

using density functional theory (DFT)

[18]

and its linear-

response variant, density functional perturbation theory

(DFPT)

[19]

, have enabled quantitative studies of proper-

ties ranging from transport to excited state dynamics to

superconductivity

[20

–

33]

. Unlike the analytic models, in a

typical first-principles calculation one represents the

-ph

interactions using a multidimensional 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

-ph interactions. Yet this complexity is also a barrier

toward obtaining minimal models and tackling new phys-

ics. For example, materials with strong or correlated

-ph

bmarco@caltech.edu

Published by the American Physical Society under the terms of

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Creative Commons Attribution 4.0 International

license.

Further distribution of this work must maintain attribution to

the author(s) and the published article

’

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PHYSICAL REVIEW X

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14(2)

021023(12)

021023-1

Published by the American Physical Society

interactions need specialized treatments to capture polaron

effects

[28,30,34,35]

and electron correlations

[36,37]

Reducing the high dimensionality of first-principles

-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

–

43]

, serves as inspiration.

Here we show a low-rank approximation of first-princi-

ples

-ph interactions which significantly accelerates

-ph

calculations while using only a small fraction (1%

–

2%) of

the data and preserving quantitative accuracy. This is

achieved by developing singular value decomposition

(SVD) calculations of

-ph matrices in Wannier basis to

achieve a minimal representation of

-ph coupling. We use

our compressed

-ph matrices to compute a range of

properties, including charge transport, spin relaxation, band

renormalization, and superconductivity, in both metals and

semiconductors. Across all benchmarks, the highly com-

pressed

-ph representation achieves a quantitative accuracy

comparable to the standard workflow, while also providing a

deeper understanding of the dominant patterns governing

-ph interactions. Principal-component analysis (PCA)

sheds light on the inherent compressibility of

-ph coupling

matrices. Recent interesting work on improving the effi-

ciency of

-ph calculations

[44,45]

is distinct in method and

scope from our data-driven approach.

II. RESULTS

A. Compression of

-ph

interactions

The key quantities in first-principles

-ph calculations

are the

-ph matrix elements

;

, which represent the

probability amplitude for an electron in a band state

with band index

and crystal momentum

, to scatter into a

final state

by emitting or absorbing a phonon with

mode index

, wave vector

, energy

ℏ

, and polarization

vector

[46]

;

Þ¼

ffiffiffiffiffiffiffiffiffiffi

ℏ

κα

ffiffiffiffiffiffiffi

∂

κα

;

where

∂

κα

≡

∂

κα

is the lattice-periodic

-ph

perturbation potential, given by the change in the DFT

Kohn-Sham potential with respect to the position of atom

(with mass

and located in unit cell at

) in the

Cartesian direction

. The inset in Fig.

1(a)

shows

schematically such an

-ph scattering process. We separate

the

-ph interactions into short- and long-ranged

[47

–

53]

;

Þ¼

;

Þþ

;

The long-range part

;

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

;

cannot be written in closed

form and needs numerical quantum mechanics to be com-

puted, a consequence of the nearsightedness of electronic

matter

[54]

. Because

;

is a smooth function of

electron and phonon momenta, it is short-ranged in a real-

space representation using a localized basis set suchas atomic

orbitals

[55]

or Wannier functions

[46,56]

The short-range

-ph coupling matrix in Wannier basis

κα

;

is obtained by transforming DFPT results

computed on a coarse momentum grid

;

[46]

κα

;

Þ¼

−

†

mn;

κα

;

where

is a unitary transformation from Bloch to Wannier

basis, and

mn;

κα

;

Þ¼h

∂

κα

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:

μα

;

Þ¼

κα

;

where

exp

κμ

adds a relative phase to

different atoms in the unit cell, and

∈

;

...

−

labels phonon modes (

is the number of atoms in the unit

cell). This way,

corresponds to the acoustic sub-

space, where all the atoms in the unit cell move in phase,

and

≠

labels the optical modes. Here and below, we use

a collective index

¼ð

ij;

μα

to label Wannier orbital

pairs

and phonon mode and direction

μα

, simplifying the

notation of the Wannier-basis

-ph matrices to

;

When viewed as a matrix for each mode and orbital pair,

;

decays rapidly with lattice vectors

and

and has a typical size

ranging between

and

. After carrying out SVD on

;

,we

obtain

;

Þ¼

;

where

is the singular value (SV) with index

, and

and

are the left and right singular vectors, respec-

tively. One can interpret

as the coupling strength between

the generalized electron cloud

†

and

phonon mode

(

†

μα

Þþ

μα

)

, where

†

are creation and annihilation operators for electrons

and

†

for phonons.Foreachchannel

,thereisa total of

LUO, DESAI, CHANG, PARK, and BERNARDI

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min

SVs; we keep only the

largest ones,

resulting in a truncated, low-rank

-ph matrix

̃

̃

;

Þ¼

This matrix can be conveniently transformed to momentum

space using

̃

;

Þ¼

;

̃

;

where

Þ¼

and

Þ¼

are singular vectors in momentum space. (Note that

the long-range part of the

-ph matrix is added after

interpolation ofthisshort-rangedpart.) Equation

(7)

provides

a generic parametrization of

-ph interactions, where by

increasing the number ofSVs one can systematically tune the

accuracy and computational cost. According to the Eckart-

Young-Mirskytheorem,thetruncatedmatrix

̃

obtainedfrom

SVD is an optimal low-rank approximation of

-ph inter-

actions, in the sense that it minimizes the Frobenius-norm

distance between the original and low-rank

-ph matrices

[57]

. From a computational viewpoint, Eq.

(7)

can greatly

accelerate the calculation of

-ph interactions and the

associated material properties, with a speedup by the inverse

fraction of SVs kept in the truncated

-ph matrix. In most

cases, we will keep only 1%

–

2% of SVs, resulting in a

–

100 times speedup for the key step in

-ph calculations

(see Appendix

for details).

B. Error and Pareto-optimal interactions

To test the accuracy of the truncated

-ph matrix and its

convergence with respect to the number of SVs (

), we

define a relative error for the

-ph matrix averaged over

electron bands and momenta, and phonon modes and wave

vectors:

Þ¼

;

−

̃

;

Þj

;

Þj

;

where

̃

;

is the low-rank (approximate) and

;

is the full first-principles

-ph matrix.

Figure

1(a)

shows this error as a function of the fraction

of SVs,

, 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

-ph interactions. We find that the error

decreases rapidly with the number of SVs

—

for example,

is as low as 1% when using only 2% of SVs, which

achieves a

× compression of the original

-ph matrix.

This error curve defines a Pareto-optimal region, high-

lighted in Fig.

1(a)

, where

-ph calculations are both

accurate and parsimonious

[57]

. This region spans

–

4% of SVs in most of our calculations

—

which

corresponds to keeping

≈

–

SVs

—

and suggests

that many materials may possess only

∼

dominant

elementary

-ph interaction patterns. Accordingly, the

-ph coupling strength for each phonon mode

[46,58]

Þ¼

ℏ

−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

;

Þj

(where

is the mass of the unit cell and the band indices

and

run over

bands), shown in Fig.

1(b)

, can be

computed accurately using just the largest 1.5% of SVs,

matching closely results using the full

-ph matrix.

FIG. 1. (a) Error on the compressed

-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

-ph coupling strength

computed using the full

-ph matrix (blue) and the low-rank

approximate matrix (orange) for silicon. The full

-ph matrix

elements are computed on a real-space grid with size

1325

and

1325

, the smallest values to achieve convergence,

setting the electron momentum to

and using the

highest valence bands. In the SVD calculation, we keep 20 out of

1325 SVs, corresponding to a

∼

fraction of SVs, as noted in

the legend.

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C. Application to transport, spin,

and superconductivity

We showcase the accuracy of the low-rank approximate

-ph interactions by computing a wide range of material

properties, including charge mobility, spin relaxation,

phonon-assisted superconductivity, and phonon-induced

band renormalization. Figures

2(a)

and

2(b)

show the

electron and hole mobility in silicon for temperatures

between 100 and 400 K, obtained using the full

-ph

matrix and compared with SVD using 1.5% of the SVs (see

Appendix

). The mobility is overestimated for electrons,

and underestimated for hole carriers, despite the accuracy

of the low-rank

-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

-ph matrix

—

but carry a considerable contribu-

tion to the mobility due to their large thermal occupation.

To improve the treatment of acoustic phonons, we develop

a constrained SVD (cSVD) which preserves the deforma-

tion potential

[59]

for long-wavelength acoustic phonons in

the compressed

-ph matrix (see Appendix

). When using

cSVD, 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

-ph matrices governing spin-flip

-ph interactions;

these matrices enable first-principles calculations of spin

relaxation times (SRTs) in centrosymmetric materials via

the Elliot-Yafet mechanism

[60]

(see Appendix

). The

SRTs for electrons in silicon between 150 and 400 K are

shown in Fig.

2(c)

. Our results from SVD with

(corresponding to

∼

of the SVs) match closely the full

-ph matrix calculations and agree with experimental

results

[61,62]

. 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

cSVD approach can be readily extended to the spin-

dependent case.

Calculations of phonon-mediated superconductivity, pre-

sented here using lead (Pb) as an example, can also leverage

our low-rank approximation (see Appendix

). Figure

2(d)

compares the Eliashberg spectral function

from full

-ph matrix calculations with SVD results; using only the

first

SVs (here equal to 1.8% of the SVs) suffices to

reproduce the full calculation. We solve the isotropic

Eliashberg equation self-consistently (see Appendix

for details) and compute the superconducting gap

a function of temperature [Fig.

2(e)

] as well as the critical

temperature

versus number of SVs [inset of Fig.

2(e)

The low-rank

-ph matrix with

SVs provides a gap

Electron

Hole

abc

def

(a)

(b)

(c)

(e)

(d)

(f)

FIG. 2. (a) Mobility of electrons in silicon and (b) mobility of hole carriers in silicon, computed with the full

-ph matrix

(

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

-ph matrix (

1325

) and using SVD with 1.5% of the SVs. Experimental

data from Refs.

[61,62]

are shown for comparison. (d) Eliashberg spectral function

for Pb, comparing full

-ph matrix

(

279

) with SVD results using 1.8% of the SVs. (e) Superconducting gap

as a function of temperature, comparing full

-ph matrix results with SVD using 1.8% of the SVs. The inset shows the convergence of the critical temperature

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.

LUO, DESAI, CHANG, PARK, and BERNARDI

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function in good agreement with the full-matrix calcula-

tion, which can be further improved by using a larger

fraction of SVs; the critical temperature for

SVs is

very accurate;

K which is within 5% of the

K value obtained using the full

-ph matrix. This result

implies that as few as five elementary

-ph interactions

determine

in Pb. Finally, we compute band renormal-

ization from

-ph interactions

[63]

] focusing on the

contribution from the Fan-Migdal

-ph self-energy (see

Appendix

). 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

-ph matrix correctly predicts the kinks near the Dirac cone

and matches full

-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

-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

(Fröhlich) dipole contribution is dominant and is well

modeled by an analytic formula using Born effective

charges

[47

–

49]

. To illustrate this point, we demonstrate

accurate mobility calculations in GaAs and PbTiO

using

only 1% of the SVs (see the Supplemental Material

[65]

D. Computational speedup from compression

We illustrate the computational speedup achieved by our

compression method using the

-ph coupling constant

doped monolayer MoS

as a case study

[66]

. Following

Ref.

[66]

, we employ a grid size of

288

and

points for

numerical integration and a Gaussain smearing of

0.002 Ry. We also leverage the improved Brillouin-zone

sampling technique where only electronic states in a small

energy window (0.006 Ry) near the Fermi surface are

included in the calculation. In Fig.

3(a)

, we show the

convergence of

with respect to the fraction of SVs. The

shaded region corresponds to an accuracy greater than 95%

compared to the fully converged result. Similar to other

quantities computed in this work,

converges rapidly with

the fraction of SVs; in particular, using only 1% of the SVs

gives

within 2% of the converged result. Figure

3(b)

compares the computational wall time for the calculation

employing the full

-ph matrices and for our SVD

compression method with 1% SVs (note that both calcu-

lations use the same improved Brillouin-zone sampling

scheme). Our approach achieves a speedup by 38 times

relative to using the full

-ph matrices; if we count solely

the time for

-ph interpolation, the speedup is 83 times.

This example illustrates the 1

–

2 orders of magnitude

speedup deriving from compressing the

-ph matrices with

SVD. For a more detailed analysis of computational

complexity, see Appendix

E. Dominant modes and principal-component analysis

To understand the inherent compressibility of the

-ph

matrices, we analyze the SV spectrum in graphene and

silicon [Figs.

4(a)

and

4(b)

]. In both materials, the SVs

decay rapidly, dropping by 1

–

2 orders of magnitude from

the largest to the tenth largest SV. In principal-component

analysis

[57,67]

, this decay can be understood as a

consequence of high-variance generalized directions in

the

-ph matrix,

;

, which capture the vast

majority of the physics, while other principal components

can be viewed as noise and neglected

[67]

. We carry out

PCA by treating each row of the matrix

;

as a

feature vector, and find that the variance of the two leading

principal components is one order of magnitude greater

than for the tenth or following principal components,

(a)

(b)

4.6 h

0.12 h

(38 )

FIG. 3. (a)

-ph coupling constant

in doped monolayer MoS

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

-ph coupling constant

with

the full

-ph matrices and with our SVD compression technique

using 1% of the SVs. The

× speedup achieved by the SVD

compression is indicated in red font.

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indicating that most of the physical information is already

captured by the first few SVs [see the insets of Figs.

4(a)

and

4(b)

]. This analysis reveals that only a few atomic

vibrational patterns dominate

-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

-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:

κα

;

Þ¼

̃

̃

κα

In this global SVD, the singular vectors

̃

depend only on

electron variables and

̃

only on phonon variables. This

way, the phonon singular vectors

̃

κα

can be inter-

preted as local vibrational modes (in the Wigner-Seitz cell

associated with the coarse grid

[46]

) and visualized to study

the dominant

-ph couplings. We show these singular

vectors for the two modes with largest SVs in graphene

and silicon in Figs.

4(c)

and

4(d)

, using arrows on each

atom, with length proportional to the singular vector

̃

κα

, to indicate the atomic displacements in the

modes obtained from SVD.

In graphene, where the electronic states consist of

orbitals centered on each carbon atom, the dominant mode

resembles a longitudinal optical phonon that brings the

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.

4(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

-like Wannier orbitals

oriented along the chemical bond directions, the two modes

with dominant

-ph coupling are associated with compres-

sion and stretching of the bonds [Fig.

4(d)

]. The intuition

gained from this mode analysis can aid the formulation of

model Hamiltonians in chemically and structurally com-

plex materials, where keeping only the dominant

-ph

interactions can provide effective models of transport and

(a)

(b)

(c)

(d)

Graphene

Silicon

FIG. 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,

̃

refers to the SVs averaged over Wannier orbitals and vibrational modes,

̃

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

. The respective insets show the real

part of the second versus tenth principal components, obtained from the PCA of the

-ph matrices; in parentheses we give the fraction of

explained variance

[67]

̃

, where

is the variance of the

th principal component. The red oval shows the standard

deviation of the corresponding principal components, obtained by dropping feature vectors with norm smaller than

−

̃

-ph interactions in graphene, and (d) the same quantity in silicon, obtained by

analyzing the phonon singular vectors,

̃

κα

in Eq.

(10)

, for the two largest SVs.

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polaron physics informed by first-principles calculations

[34,68

–

71]

III. DISCUSSION AND CONCLUSION

The accuracy of the low-rank

-ph matrices implies that

current brute-force first-principles calculations overpara-

metrize

-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

-ph couplings, may fall short of

achieving quantitative accuracy by using too few param-

eters. 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

-ph physics in just a few

parameters

—

in our case, the leading SVs and singular

vectors, which represent dominant

-ph interactions.

In summary, our results unveil the hidden low-

dimensional nature of

-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

-ph cou-

pling. 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

-ph matrices. Compressing

-ph interactions

significantly accelerates calculations of material properties

ranging from transport to spin relaxation to superconduc-

tivity. Future work will extend these ideas to other electronic

interactions, with the goal of advancing

“

precise but parsi-

monious

”

quantummany-body calculations inreal 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

-ph calculations that are currently inaccessible with

standard Wannier interpolation, including the development

of first-principles diagrammatic Monte Carlo simulations,

which sum

-ph diagrams to all orders from first principles,

thus providing a gold standard for quantitative

-ph studies.

ACKNOWLEDGMENTS

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

-ph interactions and the asso-

ciated material properties. The key bottleneck in these

calculations is obtaining the

-ph matrix elements on fine-

momentum grids,

;

, starting from the Wannier

representation, with a cost scaling as

for an

optimal implementation

[46]

(for a fixed number of Wannier

functions and atoms in the unit cell), where

and

are

the number of points in the fine-momentum electron and

phonon grids, with typical values of order

≈

In contrast, when using SVD, this interpolationstep costs only

, with a speedup by a factor

,the

inverse fraction of SVs kept in the truncated

-ph matrix. In

most cases, we will keep only 1%

–

2% of SVs, resulting in a

–

100 times speedup for the key step in

-ph calculations.

We show specific timing comparisons for all the materials

studied in this work in Fig. S1 of the Supplemental Material

[65]

. Inall cases,our algorithm achieves a speedup close to the

ideal value of

. The memory improvement is also

dramatic. A convergedtransportcalculationin siliconrequires

grid of

100

and

grid of

points

[46]

; on these fine

grids, the memory required to store the entire

-ph matrix

;

is 700 TB, while the memory needed to store the

singular vectors

and

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

-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

;

αβγ

¼ð

−

αβγ

;

where

αβγ

is the Levi-Civita tensor and the value of

is taken from Refs.

[72,73]

We compute the phonon-limited mobility at temperature

using the Boltzmann transport equation (BTE) in the

relaxation time approximation

[46]

. We first obtain the

-ph

scattering rate

using Fermi

’

s golden rule, which is

equivalent to using the imaginary part of the lowest-order

-ph self-energy

[74]

ℏ

;

Þj

½ð

−

ℏ

þð

−

ℏ

;

DATA-DRIVEN COMPRESSION OF ELECTRON-PHONON

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