76
P. Krüger
3.7.1 Green’s Functions
Quasiparticles can be described using many-body Green’s function techniques. We
introduce the retarded one-electron Green’s function
G(r, r
, t − t
) = −iθ(t − t
)0|{(rt), ,
+
(r
t
)}|0,
(3.33)
where
+
(rt) is a Heisenberg field operator which creates an electron at point r and
time t, and destroys one. |0 is the many-particle ground state and {A, B} = AB +
B A denotes the anti-commutator. Note that we have suppressed spin for convenience.
This Green’s function, or ‘propagator’, gives the probability amplitude for an electron
to be found at rt if one was added at r
t
. The one-electron removal and addition
spectrum is given by the spectral function
A(k, ω) = −
1
π
ImG(k, ω) ,
(3.34)
where G(k, ω) is the space and time Fourier transform of (3.33). In the following,
we focus on a perfect crystal and suppress the band index. The Hamiltonian of the
non-interacting system is then given by H 0 =
k k ˆ
n k , where k labels the Bloch
eigenstates with energy k and ˆ
n k is the corresponding occupation number operator.
It is easy to see that in this non-interacting case, Green’s and spectral functions are
given by
G 0 (k, ω) = (ω − k + iη)
−1
, A(k, ω) = δ(ω − k ) .
(3.35)
Thus, the photoemission peaks are delta functions, meaning that Bloch states are exact
excitations of energy k (band energy) and infinite lifetime. As mentioned above, due
to electron interaction, the true photoemission peaks are shifted, broadened and may
have satellite structures. In quasiparticle theory, these effects are described by the
so-called self-energy , which is essentially the difference between the inverses of
the exact and the free Green’s function. The self-energy is defined through the Dyson
equation
G = G 0 + G 0 G ⇔ G
−1
= G
−1
0 − .
(3.36)
For a single band in a crystal, we have
G
−1
(k, ω) = ω − k − k (ω) .
(3.37)
It is clear from (3.35) and (3.37) that Re describes a shift of the eigenvalues k
(band energy) and Im results in peak broadening, i.e. it reflects the finite lifetime
τ = /Im of the quasiparticle. There are various methods to find (approximate)
self-energies. For the electron correlation effect, two of the most popular methods
are the so-called GW approximation and dynamical mean-field theory.
P. Krüger
3.7.1 Green’s Functions
Quasiparticles can be described using many-body Green’s function techniques. We
introduce the retarded one-electron Green’s function
G(r, r
, t − t
) = −iθ(t − t
)0|{(rt), ,
+
(r
t
)}|0,
(3.33)
where
+
(rt) is a Heisenberg field operator which creates an electron at point r and
time t, and destroys one. |0 is the many-particle ground state and {A, B} = AB +
B A denotes the anti-commutator. Note that we have suppressed spin for convenience.
This Green’s function, or ‘propagator’, gives the probability amplitude for an electron
to be found at rt if one was added at r
t
. The one-electron removal and addition
spectrum is given by the spectral function
A(k, ω) = −
1
π
ImG(k, ω) ,
(3.34)
where G(k, ω) is the space and time Fourier transform of (3.33). In the following,
we focus on a perfect crystal and suppress the band index. The Hamiltonian of the
non-interacting system is then given by H 0 =
k k ˆ
n k , where k labels the Bloch
eigenstates with energy k and ˆ
n k is the corresponding occupation number operator.
It is easy to see that in this non-interacting case, Green’s and spectral functions are
given by
G 0 (k, ω) = (ω − k + iη)
−1
, A(k, ω) = δ(ω − k ) .
(3.35)
Thus, the photoemission peaks are delta functions, meaning that Bloch states are exact
excitations of energy k (band energy) and infinite lifetime. As mentioned above, due
to electron interaction, the true photoemission peaks are shifted, broadened and may
have satellite structures. In quasiparticle theory, these effects are described by the
so-called self-energy , which is essentially the difference between the inverses of
the exact and the free Green’s function. The self-energy is defined through the Dyson
equation
G = G 0 + G 0 G ⇔ G
−1
= G
−1
0 − .
(3.36)
For a single band in a crystal, we have
G
−1
(k, ω) = ω − k − k (ω) .
(3.37)
It is clear from (3.35) and (3.37) that Re describes a shift of the eigenvalues k
(band energy) and Im results in peak broadening, i.e. it reflects the finite lifetime
τ = /Im of the quasiparticle. There are various methods to find (approximate)
self-energies. For the electron correlation effect, two of the most popular methods
are the so-called GW approximation and dynamical mean-field theory.