Combining electron-phonon and dynamical mean-field theory calculations of correlated materials: Transport in the correlated metal ${\rm Sr}_2{\rm RuO}_4$

PHYSICAL REVIEW MATERIALS

, 093801 (2023)

Editors’ Suggestion

Combining electron-phonon and dynamical mean-field theory calculations of correlated materials:

Transport in the correlated metal Sr

RuO

David J. Abramovitch,

Jin-Jian Zhou

Jernej Mravlje,

Antoine Georges,

and Marco Bernardi

†

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

School of Physics, Beijing Institute of Technology, Beijing 100081, China

Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia

Collège de France, Paris, France

CCQ-Flatiron Institute, New York, New York 10010, USA

Department of Physics, California Institute of Technology, Pasadena, California 91125, USA

(Received 14 April 2023; accepted 1 August 2023; published 1 September 2023)

Electron-electron (

) and electron-phonon (

-ph) interactions are challenging to describe in correlated

materials, where their joint effects govern unconventional transport, phase transitions, and superconductivity.

Here we combine first-principles

-ph calculations with dynamical mean-field theory (DMFT) as a step toward

a unified description of

and

-ph interactions in correlated materials. We compute the

-ph self-energy

using the DMFT electron Green’s function and combine it with the

self-energy from DMFT to obtain

a Green’s function including both interactions. This approach captures the renormalization of quasiparticle

dispersion and spectral weight on equal footing. Using our method, we study the

-ph and

contributions

to the resistivity and spectral functions in the correlated metal Sr

RuO

. In this material, our results show that

interactions dominate transport and spectral broadening in the temperature range we study (50–310 K),

while

-ph interactions are relatively weak and account for only

∼

10% of the experimental resistivity. We also

compute effective scattering rates and find that the

interactions result in scattering several times greater

than the Planckian value

, whereas

-ph interactions are associated with scattering rates lower than

.Our

work demonstrates a first-principles approach to combine electron dynamical correlations from DMFT with

-ph

interactions in a consistent way, advancing quantitative studies of correlated materials.

DOI:

10.1103/PhysRevMaterials.7.093801

I. INTRODUCTION

In strongly correlated materials, characteristic behaviors

such as high-temperature superconductivity [

], phase tran-

sitions [

], multiferroicity [

], and unconventional transport

[

] involve a subtle interplay of charge and lattice de-

grees of freedom. The atomic vibrations (phonons) couple

with the strongly interacting

electrons, resulting in

electron-electron (

) and electron-phonon (

-ph) interac-

tions with nontrivial dependence on orbital, spin, and crystal

symmetry [

]. Although heuristic models can address this

rich phenomenology [

–

], deriving generic quantitative

frameworks to combine

and

-ph interactions remains

challenging [

], especially in correlated materials.

First-principles calculations based on density functional

theory (DFT) and its linear-response extension, density func-

tional perturbation theory (DFPT), can address the electronic

structure [

], lattice dynamics [

], and

-ph interactions

[

] in many materials. Yet these methods often fail to

describe important features of correlated systems [

], pre-

dicting incorrect ground states, lattice vibrations, and

-ph

coupling [

]. Recent work has focused on two directions

These authors contributed equally to this work.

†

bmarco@caltech.edu

to improve the description of

-ph interactions in correlated

materials: Hubbard-corrected DFT (DFT

), which has en-

abled calculations of

-ph interactions in a Mott insulator [

and DFPT with improved electronic correlations (from

hybrid functionals) which has revealed correlation-enhanced

-ph interactions in metallic systems [

–

However, while

and DFT

can renormalize the

band structure and

-ph coupling, both methods describe

the electronic states in the quasiparticle (QP) picture, map-

ping them to noninteracting bands. The QP approximation

is typically used as a starting point in studies of equilibrium

properties and nonequilibrium dynamics [

]. Developing

first-principles calculations of

-ph interactions beyond the

QP picture remains challenging [

]. Ideally, one would

use the full electron Green’s function renormalized by

interactions, which includes the renormalization of both QP

weight and band dispersion, as a starting point to study

-ph

interactions in correlated materials [

Dynamical mean-field theory (DMFT) and its

ab initio

variant, DFT

DMFT, can capture dynamical electronic cor-

relations by mapping the solid to an embedded atomic site

with a self-energy local in the atomic orbital basis [

–

These methods have been successful in describing the ground

state and transport properties in materials with strongly in-

teracting

electrons, including both correlated metals

and insulators [

–

]. A key question addressed in this

2475-9953/2023/7(9)/093801(10)

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DAVID J. ABRAMOVITCH

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PHYSICAL REVIEW MATERIALS

, 093801 (2023)

work is how one can combine first-principles DMFT and

-ph calculations to explicitly treat

and

-ph interactions

together in correlated materials.

Here we show first-principles calculations combining

-ph

and DMFT

interactions using a Green’s function approach.

We compare two treatments of the

-ph interactions—the

Fan-Migdal self-energy with standard approximations [

]

(using DFT or DMFT-renormalized band structures) and the

same self-energy computed as a convolution integral of the

DMFT Green’s function. As a proof of principle, we ap-

ply this method to Sr

RuO

(SRO) in the normal state, a

prototypical correlated metal [

–

] (and unconventional

superconductor below 1 K [

–

]) requiring treatment of

strong

interactions between

orbitals [

–

]. In SRO,

we find that the

-ph interactions are relatively weak and

momentum dependent, in contrast with the strong local

interactions. From Green-Kubo calculations, we find that

interactions account for

∼

50%, and

-ph interactions

∼

10%,

of the experimental resistivity. In the

-ph calculation using

the DMFT Green’s function the resistivity equals the sum of

the

-ph and

contributions; in contrast, the standard

-ph

calculation with renormalized QP bands leads to an artificial

enhancement of

-ph interactions and nonadditive resistivities.

The origin of these trends is discussed in detail, together with

possible improvements for resistivity calculations in corre-

lated materials and future extensions of our method.

II. NUMERICAL METHODS

We compute the electron Green’s function by combining

the DMFT and

-ph self-energies:

(

ω,

)

[

(

)

−

(

ω,

)

−

-ph

(

ω,

)

]

−

(1)

where

is temperature,

are electronic band energies,

the electron energy,

is a band index,

is crystal momentum,

and

is a temperature dependent chemical potential obtained

from DMFT. This Green’s function includes the DMFT

self-energy in the band basis,

(

ω,

), which is computed

starting from DFT, and the

-ph self-energy

-ph

(

ω,

which we compute using different approximations as dis-

cussed below.

Our calculations in SRO use a DMFT self-energy taken

from recent work [

] and transformed from the Wannier

orbital to the band basis using [

]

†

(2)

where

are unitary matrices made up by eigenvectors of the

Wannier Hamiltonian, while

and

are respectively

the self-energies in the Wannier and band basis; the former

is calculated directly from DMFT and is therefore

indepen-

dent. As in the DMFT calculation, both the Wannier and band

basis self-energies are taken to be diagonal [

These DMFT calculations use a three-orbital correlated

subspace with

symmetry resulting from the hybridization

of Ru 4

(

) and O 2

orbitals; the orbitals in-

teract with Hubbard and Hund type Coulomb repulsion and

exchange [

]. The DMFT impurity problem is solved on

the imaginary-time axis with the TRIQS

CTHYB quantum

Monte Carlo solver [

] and analytically continued to

the real-frequency axis using Padé approximants [

]. These

methods are discussed and validated in greater detail in recent

work [

]. Previous studies have clarified the role of spin-orbit

coupling (SOC) in SRO [

–

], which is relevant

primarily near band crossings. Because such crossings occur

away from the Fermi surface in the majority of the Brillouin

zone, we neglect SOC in our calculations.

Our DFT and DFPT calculations are carried out using

UANTUM

SPRESSO

[

] with 10

-point and

-point grids, and then projected onto Wannier or-

bitals using the W

ANNIER

90 code [

]. We use the P

ERTURBO

code for Wannier interpolation of the electronic structure,

phonon modes, and

-ph coupling and for computing the

-ph

self-energy, spectral functions, and transport properties [

The transport calculations use a 60

60 fine

-point

grid and 10

points randomly sampled in the Brillouin zone.

Transport results were converged with respect to the

-point

grid density, number of

points sampled, the energy window

in which the

points were chosen, and the frequency grid of

the self-energy and spectral functions.

III. RESULTS

A. Electronic structure and

e

-ph coupling

The electronic structure of SRO is strongly renormalized

by electron correlations. The DFT band structure near the

Fermi energy consists of relatively narrow

bands [Fig.

1(a)

The

interactions significantly decrease the

-band Fermi

velocity relative to DFT, as captured by the DMFT electronic

spectral functions [Fig.

1(b)

]. In the 50–310 K temperature

range studied here, the

interactions also reduce the spectral

weight

of the electronic states on the Fermi surface—to

≈

2forthe

and

≈

3forthe

and

bands [

]—and

cause a large broadening of the associated spectral functions.

These effects are a signature of strong electron correlations in

SRO and highlight the need to treat the electronic structure

beyond the band picture.

The phonon dispersion with a color map of the

-ph cou-

pling strength is shown in Fig.

1(c)

. The dispersions are

calculated from real space interpolation of DFPT forces [

]

and are in good agreement with inelastic neutron scattering

measurements [

]. We find that the

-ph interactions are

overall relatively weak in SRO, with coupling strength

100 meV for all phonon modes. These values are signifi-

cantly smaller than in insulating transition metal oxides—for

example, CoO and SrTiO

[

]—where the long-range

Fröhlich interaction with longitudinal optical (LO) modes

[

] reaches coupling strengths of order

|≈

1eV.Dueto

its metallic character, such polar LO phonons are screened out

in SRO, resulting in short-range

-ph interactions with weaker

coupling strengths.

B. Electron-phonon self-energy calculations

We describe the

and

-ph interactions using the

corresponding self-energies. When

-ph coupling is rela-

tively weak, the

-ph interactions are well described by the

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FIG. 1. Electronic structure and

-phcouplinginSRO.(a)DFT

band structure for the

bands included in the Wannierization.

(b) DMFT spectral function at 77 K, plotted in an enlargement of the

black rectangle in (a), showing significant renormalization relative to

the DFT bands near the Fermi energy (shown with a horizonal line).

-ph coupling strength,

which is computed as

(

)

[

∑

(

)

]

[

]

by averaging over

3 bands.

lowest-order (so-called Fan-Migdal) self-energy [

]

-ph

(

)

∑

(

)

(

−

)

(

)

(3)

written as a convolution of electron and phonon propaga-

tors,

(

) and

(

), respectively, using Matsubara

frequencies

for electrons and

for phonons;

(

)

are

-ph coupling matrix elements [

In current first-principles calculations, this expression is

evaluated using noninteracting electron (and phonon) Green’s

functions, tacitly assuming that the electron spectral func-

tions consist of a sharp QP peak. In this approximation, the

-ph self-energy on the real-frequency axis

, computed as

in Fig.

2(a)

with noninteracting electron Green’s functions

(

)

(

−

)

−

, becomes [

]

-ph

(

ω,

)

∑

(

)

[

(

)

(

)

−

(

)

−

(

)

−

]

(4)

FIG. 2. Feynman diagrams for the

-ph self-energy

-ph

, com-

puted with (a) the noninteracting electron Green’s function and

(b) the DMFT electron Green’s function. The vertices carry a factor

equal to the

-ph coupling,

(

where

are phonon energies, while

(

) are equilibrium

occupation numbers for phonons and

(

) for electrons

at temperature

. When DFT is used to obtain the band

structure

, we denote this self-energy approximation as

-ph@DFT.

In systems where DFT fails to describe the electronic

structure, several methods are used to correct the QP band

dispersion and

-ph coupling, including

[

], DFT

[

], and DFT with hybrid functionals [

]. Therefore, as

previously shown in the literature, the

-ph self-energy in

Eq. (

) can be computed using the “best available” QP

band structure and

-ph couplings from one of these meth-

ods [

–

]. Here, for example, by fitting the QP peaks

of the DMFT spectral functions we obtain an improved

DMFT band structure and use it to compute the self-energy

in Eq. (

). Below we refer to this level of theory as

-ph@DMFT bands.

However, even with these improved schemes, the standard

-ph self-energy in Eq. (

) misses key features of the elec-

tronic structure in correlated materials. These include the QP

peak broadening, finite QP weight, and any satellite peaks,

background, or spectral weight redistribution outside the QP

peak, all of which are encoded in the electron spectral func-

tion. Improving the QP band structure or

-ph couplings in

Eq. (

) addresses only part of these renormalization effects

because it places the full spectral weight on the electronic

band states, neglecting spectral weight redistribution from

interactions. The inadequacy of this approximation is appar-

ent in correlated systems like SRO, where the QP weight

of the

bands is only

≈

2–0

3. Going beyond these

limitations requires computing the

-ph self-energy directly

from the electronic spectral function dressed by electron

correlations.

To address this challenge, we develop a method to calcu-

late the

-ph self-energy from the DMFT electron Green’s

function,

DMFT

. Starting from the Lehmann representa-

tion [

], and using a band-diagonal DMFT self-energy,

we write

DMFT

(

)

∫

DMFT

(

)

−

(5)

where

DMFT

(

)

=−

Im[

DMFT

(

)]

/π

is the DMFT spectral

function. We substitute this expression in Eq. (

) and fol-

low the usual derivation of the lowest-order

-ph self-energy

[

]. After analytic continuation to the real frequency axis,

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FIG. 3. Imaginary part of the

-ph and

self-energies in SRO, plotted on cuts of the Fermi surface at 77 K (left) and 310 K (right). We

plot the

-ph self-energy obtained using (a) the DFT electronic structure (

-ph@DFT), (b) the DMFT band dispersion (

-ph@DMFT bands),

and (c) the DMFT spectral function using Eq. (

)(

-ph@

DMFT

). (d) DMFT

self-energy, shown here for comparison. The corresponding

self-energies at 310 K are plotted in (e)–(h), respectively.

→

, we obtain

-ph

(

ω,

)

∑

,ν,

(

)

∫

DMFT

(

)

[

(

)

(

)

−

(

)

−

(

)

−

]

(6)

This

-ph self-energy, written as an integral of the DMFT

spectral function, captures on equal footing the key factors

mentioned above—band renormalization, finite QP weight

and broadening, and background or satellite contributions

caused by electron correlations—all of which are included

in the DMFT spectral function. This approximation for the

-ph self-energy, here referred to as

-ph@

DMFT

,isshown

diagrammatically in Fig.

2(b)

. Its numerical evaluation is dis-

cussed in the Appendix. [Note that in this work we do not

renormalize the

-ph coupling,

(

), using DMFT; the

role of that renormalization is discussed in Sec.

C. Electron-phonon and DMFT self-energies

We analyze the

-ph self-energy obtained with these dif-

ferent approximations and compare it with the DMFT

self-energy. Figure

shows these quantities on a cut of the

Fermi surface in the

plane. (The real and imaginary fre-

quency dependent self-energy at select

points is plotted

in the Supplemental Material [

].) In each plot, the middle

band with stronger

-ph and

coupling has

character,

while the other two bands have

and

characters. The

-ph self-energy, which depends on both electronic orbital and

momentum

, varies by a factor of

∼

5 on the Fermi surface

at 77 K and 310 K. This

dependence in the self-energy

also changes as a function of temperature in some parts of the

Brillouin zone. For example, in the

band the self-energy is

greater in magnitude near the

(

) axis at low (high)

temperature, due to a change of available

-ph inter- and

intraband scattering processes. Our DMFT self-energy also

depends on orbital character, but it is local and thus

in-

dependent by construction; it can accurately capture electron

correlations in SRO [

Comparing our different

-ph self-energy approximations

sheds light on the physics they capture. In the picture of

renormalized QP bands [

], the Fermi velocity decreases

by a factor of 1

(in SRO, 1

∼

3for

and

∼

5for

bands [

]) and the QP weight also decreases

by the same factor, leaving the density of states (DOS) at

the Fermi energy unchanged. This renormalized QP picture

holds reasonably well in SRO. The

-ph self-energy from DFT

bands (

-ph@DFT), shown in Fig.

3(a)

at 77 K and in Fig.

3(e)

at 310 K, misses all these renormalization effects and treats

the material as weakly correlated. Computing the

-ph self-

energy in Eq. (

) with the renormalized DMFT band structure

(

-ph@DMFT bands), which is obtained by fitting the QP

peaks of DMFT spectral functions, artificially enhances the

-ph self-energy by a factor of 3–5, as shown in Fig.

3(b)

77 K and in Fig.

3(f)

at 310 K. This artifact is a consequence

of using a band structure with Fermi velocity decreased by

a factor of 1

≈

3–5, which increases the DOS at the Fermi

energy and thus also increases Im

−

in Eq. (

)bythesame

factor.

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However, the DOS at the Fermi energy is nearly unchanged

in the renormalized electronic structure because the decrease

in Fermi velocity is compensated by a corresponding decrease

in QP weight. This physics is correctly described by Eq. (

where the imaginary part of the

-ph self-energy is roughly

proportional to the DOS at Fermi energy written as

(

)

∝

∑

∫

(

)

(

−

), which captures changes in the

QP weights. Therefore, as we show in Fig.

3(c)

at 77 K

and in Fig.

3(g)

at 310 K, the more accurate

-ph self-

energy computed from the DMFT spectral functions using

Eq. (

)(

-ph@

DMFT

method) removes the artificial enhance-

ment introduced by using the “best-available band structure”

approach.

Overall, the

-ph self-energy obtained from DFT bands in

Fig.

3(a)

and DMFT Green’s functions in Fig.

3(c)

shows

similar magnitude and trends at 77 K mainly because the

DOS at Fermi energy is nearly unchanged in DFT and DMFT.

However, at higher temperatures the effects of broadening

in the DMFT spectral functions become more pronounced;

for example, at 310 K we find a decrease in

dependence

in the

-ph self-energy from DFMT Green’s function, as

shown by comparing Figs.

3(e)

and

3(g)

. We attribute this

difference to smearing of sharp features in the electronic

structure by the spectral width, which is not captured by the

DFT-based calculation; this effect is more pronounced in the

strongly correlated

band. Our results show the importance

of computing the

-ph self-energy from the electron spectral

functions in correlated materials to capture the subtle interplay

-ph and

interactions.

D. Spectral functions

We obtain the spectral function including both

and

-ph

interactions using the Green’s function in Eq. (

(

ω,

)

=−

/π

)Im

(

ω,

)

(7)

To evaluate this expression, our most accurate approximation

consists of first computing the DMFT

self-energy starting

from DFT and then obtaining the

-ph self-energy from the

DMFT spectral function. This

-ph@

DMFT

approach is jus-

tified by the adiabatic approximation: the fast

interactions

renormalize the electronic states and the slow nuclear motions

governing

-ph coupling occur in this renormalized ground

state.

Figures

4(a)

and

4(b)

show the electron spectral functions

computed in SRO with this approach and plotted in the

→

direction at 77 K and 310 K. The main features are consistent

with photoemission and transport measurements [

]—

in particular, the spectral functions remain fairly sharp near

the Fermi energy below

∼

100 K, indicating well-defined QP

excitations at low temperature [

]. The spectral broadening

increases at higher temperatures, further demonstrating the

importance of computing

-ph interactions beyond the band

picture. Figures

4(c)

and

4(d)

compare the spectral func-

tions and their QP peak broadening at a fixed

point (

75 M) with and without the inclusion of

-ph interactions.

We find that the

-ph interactions broaden the QP peaks only

slightly, consistent with the overall weak

-ph coupling in

SRO. Increasing the temperature, rather than including

-ph

interactions, is the main factor responsible for broadening. We

FIG. 4. Electronic spectral functions in the

→

direction

computed using the DMFT

plus

-ph@

DMFT

self-energies at

(a) 77 K and (b) 310 K. (c) Spectral functions at

75M showing

DMFT quasiparticle peaks (dashed lines) broadened by the inclusion

-ph interactions (solid lines) at two temperatures. (d) Calculated

full width at half maximum for the spectral functions in (c), showing

quantitatively the small

-ph broadening effect.

find similar trends when analyzing spectral functions in the

→

direction [

]. These results point to a dominant role

interactions in SRO.

E. Transport

We study electrical transport in SRO in the characteristic

bad metallic regime [

], focusing on understanding the roles

and

-ph interactions. We compute the optical conduc-

tivity from our combined DMFT plus

-ph spectral function

in Eq. (

), using Green-Kubo theory without current-vertex

corrections [

αβ

(

ω,

)

∫

(

)

−

(

ω,

)

∑

(

)

(

ω,

)

(8)

where

are band velocities,

and

are Cartesian direc-

tions, and

is the unit cell volume; the dc resistivity tensor

is obtained as

(

)

−

(

→

). This approximation

has been used (separately) to study

-ph limited transport in

oxides and organic crystals [

] and

limited transport

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FIG. 5. Transport properties as a function of temperature in SRO. (a) In-plane resistivity, comparing DMFT

-ph, DMFT-only, and

-ph-

only calculations to experimental data from Ref. [

]. (b) Same as (a), but plotted as a fraction of the experimental resistivity. (c) Effective

scattering rate and (d) effective plasma frequency. In the legend, DMFT only and

-ph only refer to using, respectively, only the DMFT or

-ph

self-energies in the Green’s function, spectral function, and Green-Kubo formula.

in correlated metals [

–

]; different from these previous

studies, here we focus on combining the

and

-ph interac-

tions and studying their interplay.

Figure

5(a)

shows the resistivity in the

plane as a

function of temperature. We compare the resistivity limited

interactions alone, computed from the Green’s func-

tion including only the DMFT self-energy, with calculations

including both

and

-ph interactions in the Green’s func-

tion. When both interactions are included, we compare results

from the more accurate

-ph@

DMFT

approach with calcu-

lations using Eq. (

) with the best-available band structure

(

-ph@DMFT bands method). Experimental values are also

shown for comparison [

]. For convenience, Fig.

5(b)

shows

the same results expressed as a fraction of the experimental

resistivity.

The calculation including only DMFT

interactions cor-

rectly predicts the order of magnitude of the experimental

resistivity, providing a resistivity smaller than experiment by

a factor of 2 in the entire temperature range studied here

(50–310 K). This calculation is comparable with results from

Deng

et al.

[

], who used a different scheme to construct

the correlated subspace in the DMFT calculation and obtained

somewhat larger values for the

limited resistivity due to

stronger interaction parameters

and

. In contrast, the re-

sistivity limited by the

-ph interactions alone (computed with

the

-ph@

DMFT

approach) is only

∼

10% of the experimental

value.

When combining

and

-ph interactions through the

ph@

DMFT

approach, the resulting resistivity is the sum of the

individual

and

-ph limited resistivities, a behavior known

as Matthiessen’s rule [

]. Therefore, adding

-ph interactions

improves the agreement with experiment and predicts a re-

sistivity equal to

∼

60% of the experimental value between

50 and 310 K. In contrast, using only an improved QP band

dispersion (

-ph@DMFT bands) artificially enhances the

ph interactions, leading to an incorrect prediction that the

-ph contribution is 30% of the experimental resistivity [see

Fig.

5(b)

]. Note that our most accurate calculation still under-

estimates the experimental resistivity by

∼

40%. The possible

origin of this “missing resistivity” is discussed below.

To better understand the contributions to the resistivity, we

analyze our transport results using an effective Drude model

[

], writing the resistivity as

π/

[

∗

(

∗

)

], where 1

/τ

∗

is an effective scattering rate and

∗

is an effective plasma

frequency. We extract these quantities from our computed

optical conductivity

(

)using[

]

∗

=−

πσ

∫

∞

∂

Re[

(

)]

∂ω

(9)

and

(

∗

)

−

∫

∞

∂

Re[

(

)]

∂ω

(10)

Figure

5(c)

shows the effective scattering rate 1

/τ

∗

and

Fig.

5(d)

the plasma frequency (

∗

)

obtained from this

analysis. These results confirm that

interactions dominate

electron scattering and lead to a decreased Fermi velocity (and

thus plasma frequency) by renormalizing the band structure.

The effective

-ph scattering rate is significantly smaller (and

the plasma frequency greater) than for

interactions, con-

sistent with the small

-ph contribution to the resistivity and

negligible band renormalization (relative to DFT) from

-ph

interactions. In addition, most of the temperature dependence

of the resistivity comes from the effective scattering rates,

while the effective plasma frequency depends weakly on tem-

perature.

Figure

5(c)

additionally compares the effective scattering

rates with the Planckian limit

[

]. Interestingly,

the

-ph scattering rate is lower than

and approaches it near

310 K, whereas the scattering from

interactions exceeds

above

∼

100 K, reaching

∼

at 310 K. The weaker

-ph

scattering in SRO contrasts the cases of insulating oxides such

as SrTiO

and CoO [

], where pronounced polaron effects

are present—both materials exhibit polaron satellites, and in

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COMBINING ELECTRON-PHONON AND DYNAMICAL ...

PHYSICAL REVIEW MATERIALS

, 093801 (2023)

SrTiO

the effective

-ph scattering rate exceeds the Planckian

limit [

IV. DISCUSSION

Our most accurate transport calculation underestimates the

in-plane resistivity in SRO by

∼

40% relative to experiments.

The origin of this discrepancy deserves a detailed discussion.

First, we verify that our lowest-order (Fan-Migdal) treatment

-ph interactions is sufficient in SRO by recomputing the

resistivity with an

-ph cumulant method that can describe

higher-order

-ph interactions and transport in the presence

of polarons [

]. Consistent with the effective screening

of polar phonons in metals, we find that the

-ph limited

transport is nearly identical in the lowest-order and cumulant

calculations [

Second, our Green-Kubo calculations neglect current-

vertex corrections, which typically play a small role in

-ph

limited transport in metals. While vertex corrections are dif-

ficult to compute in Green-Kubo theory [

], they can

be quantified in the semiclassical limit using the Boltzmann

transport equation (BTE). In particular, including vertex cor-

rections corresponds to a full solution of the BTE, obtained

here with an iterative approach (ITA) [

], while neglecting

vertex corrections is equivalent to the relaxation time approx-

imation (RTA), to which the Green-Kubo formula reduces in

the weak-coupling limit [

]. Therefore, we can estimate the

effect of vertex corrections by comparing our

-ph limited

resistivity computed with Green-Kubo to BTE calculations

using the ITA and RTA, which respectively include and ne-

glect vertex corrections [

]. From low temperature up to

310 K, the ITA resistivity is

∼

20% greater than the RTA

value, while the RTA and Green-Kubo calculations agree to

within 2% at 310 K [

]. These results indicate that vertex

corrections have a modest effect on

-ph limited transport.

Third, most DMFT studies of transport neglect vertex cor-

rections from

interactions, which is strictly correct only

in the limit of infinite dimensions [

]. However, their role

is difficult to quantify and often non-negligible [

]. Fourth,

although our DMFT calculations produce band dispersions

in excellent agreement with photoemission data [

], DMFT

results can in some cases be sensitive to the method used to

construct the local site [

], and neglecting nonlocal interac-

tions may remove relevant

scattering mechanisms which

affect the calculated spectral width. Studying these effects is

an active area of DMFT research and is beyond the scope of

this work.

Finally, recent work has shown that DFPT based on semilo-

cal functionals can lead to underestimated

-ph coupling in

correlated materials [

]. Yet much of that work has fo-

cused on materials with nonlocal correlations, so it is difficult

to estimate the size of the effect in SRO, where correlations

are dominantly local [

]. It is possible that the

-ph coupling

and its role in transport are somewhat stronger than captured

by our DFPT calculations. However, an enhancement of the

-ph coupling consistent with those reported to date would

not change our main conclusion that the

-ph contribution to

the resistivity is relatively small—for example, assuming the

limit case of a doubling of

-ph coupling strength

would

still give only an

∼

20%

-ph resistivity contribution.

A qualitative comparison between

and

-ph interactions

is also interesting. The

dependence of the

-ph interactions

clearly contrasts the local (

-independent)

interactions.

Extending this comparison to other correlated materials may

highlight key differences between materials with dominant

-ph interactions. We also observe that our

-ph@

DMFT

self-energy generally has a weaker

dependence than pre-

dicted by the

-ph@DFT method, suggesting that electron

correlations suppress nonlocal

-ph interactions. This effect

may have interesting implications for materials with strong

or long-range

-ph interactions, including cuprates and corre-

lated insulators.

Lastly, our development of the

-ph@

DMFT

method fits

within efforts in computational many-body physics to obtain

accurate components of Feynman diagrams. For

-ph inter-

actions, this includes improving the accuracy of the electron

propagator

, phonon propagator

, and

-ph coupling

When viewed this way, there is a clear analogy with the

use of different levels of theory in the

method, where

the accuracy can be improved by using the “best

” and

“best

.” In

-ph calculations, methods for Hubbard- [

]

or DMFT-corrected phonons [

], as well as anharmonic

lattice dynamics [

], address the phonon propagator

while Hubbard-corrected DFPT [

perturbation the-

ory (GWPT) [

], and DFT

DMFT deformation potentials

[

] aim to improve the

-ph coupling

. These effects

can be treated at a suitable level of theory for each ma-

terial; for example, if the lattice is strongly anharmonic,

the phonon propagator can be computed using renormalized

phonon frequencies or one can use a full frequency dependent

phonon propagator analogous to our use of the DMFT Green’s

function.

Our work improves the accuracy of the electron propagator

entering the

-ph self-energy diagram by dressing it with the

DMFT self-energy to capture dynamical electron correlations.

This provides a scheme where the

-ph self-energy is com-

puted from the DMFT spectral functions. Our results highlight

the importance of using the spectral functions to compute

-ph

interactions in correlated materials and caution against using

just the best available QP band structure—if the dispersion

is renormalized but the QP weight is not adjusted, the

-ph

self-energy becomes inaccurate as we have shown for SRO.

V. CONCLUSION

We demonstrated a method to combine strong dynamical

correlations described by DMFT with first-principles

-ph

calculations. This advance enables a quantitative treatment of

both

and

-ph interactions and their effect on transport in

correlated materials. Our approach computes

-ph interactions

from the DMFT Green’s function, which captures band struc-

ture and spectral weight renormalization from strong electron

correlations. In SRO, where

-ph interactions are relatively

weak, we have shown that transport is governed by

inter-

actions, with

-ph interactions contributing only

∼

10% of the

experimental resistivity in a wide temperature range.

Our work expands the reach of first principles

-ph calcula-

tions to strongly correlated materials by treating the electronic

structure at the level of the spectral function rather than nonin-

teracting bands. Note that our calculations can also use more

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