Influence of the Electron–Phonon Interaction on the Topological Phase. . .
307
that would rely on the robust topologically protected metallic surface states, such as
spintronics, quantum computing, or topological transistors.
2 Electron–Phonon Interaction from First-Principles
In this section, we briefly present how EPI can be computed from first-principles.
For more details, we refer our readers to the review by Giustino [2]. We use the
Hartree atomic unit system ( ¯
h = m e = e = 1). Let us start from the Hamiltonian:
H =
kn
ε
0
kn c
†
kn c kn +
qν
ω qν
a
†
qν a qν +
1
2
+
knn ,qν
g knn ,qν c
†
k+qn c kn
a qν + a
†
−qν
.
(1)
The first term describes Bloch electrons, with wavevector k, band index n, and bare
electronic eigenenergy ε 0
kn . The second accounts for the phonons, with wavevector
q, branch index ν, and frequency ω qν . The last term captures the interaction between
the kn-electron and the qν-phonon, with EPI vertex g knn ,qν . c
†
kn , c kn and a
†
qν , a qν
are, respectively, the creation and annihilation operators for electrons and phonons.
Within many-body perturbation theory, the temperature-dependent correction to
the electronic eigenenergies corresponds to the energy of the quasiparticle peak in
the real part of the frequency-dependent electron–phonon self-energy, Σ EPI
kn (T , ω).
Applying the on-the-mass-shell approximation, thus evaluating the self-energy at
the poles of the Green’s function, namely at the bare electronic eigenvalues, the
renormalized eigenenergies take the form:
ε kn (T ) ≈ ε
0
kn + Re
Σ EPI
kn
T , ω = ε
0
kn
.
(2)
We work within the Migdal approximation, where the EPI vertex corrections
can be neglected because of the large mass difference between electrons and ions.
Expanding the third term of Eq. (1) to the lowest order, one obtains a contribution
known in the literature as the Fan term (Fig. 1, middle):
Σ Fan
kn
T , ω = ε
0
kn
=
qν
n
1
2ω qν
ψ kn | δV
(1)
qν
ψ k+qn
2
×
n q,ν (T ) + f k+q,n (T )
ω − ε 0
k+qn + ω qν + iη k
+
n q,ν (T ) + 1 − f k+q,n (T )
ω − ε 0
k+qn − ω qν + iη k
.
(3)
In this expression, |ψ kn is a static lattice eigenstate and η k is an infinitesimal
parameter introduced to maintain causality. δV
(1)
qν is the first-order variation of
307
that would rely on the robust topologically protected metallic surface states, such as
spintronics, quantum computing, or topological transistors.
2 Electron–Phonon Interaction from First-Principles
In this section, we briefly present how EPI can be computed from first-principles.
For more details, we refer our readers to the review by Giustino [2]. We use the
Hartree atomic unit system ( ¯
h = m e = e = 1). Let us start from the Hamiltonian:
H =
kn
ε
0
kn c
†
kn c kn +
qν
ω qν
a
†
qν a qν +
1
2
+
knn ,qν
g knn ,qν c
†
k+qn c kn
a qν + a
†
−qν
.
(1)
The first term describes Bloch electrons, with wavevector k, band index n, and bare
electronic eigenenergy ε 0
kn . The second accounts for the phonons, with wavevector
q, branch index ν, and frequency ω qν . The last term captures the interaction between
the kn-electron and the qν-phonon, with EPI vertex g knn ,qν . c
†
kn , c kn and a
†
qν , a qν
are, respectively, the creation and annihilation operators for electrons and phonons.
Within many-body perturbation theory, the temperature-dependent correction to
the electronic eigenenergies corresponds to the energy of the quasiparticle peak in
the real part of the frequency-dependent electron–phonon self-energy, Σ EPI
kn (T , ω).
Applying the on-the-mass-shell approximation, thus evaluating the self-energy at
the poles of the Green’s function, namely at the bare electronic eigenvalues, the
renormalized eigenenergies take the form:
ε kn (T ) ≈ ε
0
kn + Re
Σ EPI
kn
T , ω = ε
0
kn
.
(2)
We work within the Migdal approximation, where the EPI vertex corrections
can be neglected because of the large mass difference between electrons and ions.
Expanding the third term of Eq. (1) to the lowest order, one obtains a contribution
known in the literature as the Fan term (Fig. 1, middle):
Σ Fan
kn
T , ω = ε
0
kn
=
qν
n
1
2ω qν
ψ kn | δV
(1)
qν
ψ k+qn
2
×
n q,ν (T ) + f k+q,n (T )
ω − ε 0
k+qn + ω qν + iη k
+
n q,ν (T ) + 1 − f k+q,n (T )
ω − ε 0
k+qn − ω qν + iη k
.
(3)
In this expression, |ψ kn is a static lattice eigenstate and η k is an infinitesimal
parameter introduced to maintain causality. δV
(1)
qν is the first-order variation of