5. Response functions
5.1 Random Phase Approximation
The truncation of the equation of motion for the Greens function with the
Hartree Fock mean field approximation transforms the Hamiltonian into
an effective single particle operator; this too simple treatment of the influence of interactions leads to a lack of realism. A better approach than
the truncation of the equation of motion hierarchy is the approximate evaluation also of the higher order equations of motion. This allows to better
account for the interaction as the full Hamiltonian enters the equation of
motion at each step.
The so called random phase approximation (RPA) involves replacement of
operator products by averages as in the mean field approximation (neglect
of the fluctuation of averages). For the single particle Greens function
G*kσ (ω) = hhc*kσ ; c*† ii
(5.1)
kσ
the result is again the Hartree-Fock result.
The magnetic susceptibility is an important quantity
describing
the re*
*
−(ω+iδ)t
sponse of a material to an external magnetic field H(t) = H0 e
:
X Z∞
0
hMα i(t) = hMα i −
dt 0 hhMα (t); Mβ (t 0 )iiHβ0 e−i(ω+iδ)t (5.2)
β
−∞
with cartesian directions α, β. As the Greens function is homogeneous in
time we have
X Z∞
0
dt 0 hhMα ; Mβ (t 0 − t)iiHβ0 e−i(ω+iδ)(t −t) e−i(ω+iδ)t
hMα i(t) = hMα i −
β
= hMα i −
−∞
X Z∞
β
−∞
0
dt 0 hhMα ; Mβ (t 0 )iie−i(ω+iδ)t Hβ (t)
(5.3)
45
with a field independent part of the magnetization hMα i which is important for ferromagnetic systems and the second part describing the magnetization induced by an external magnetic field. This part is proportional to
H(t); the prefactor represents the magnetic susceptibility χ:
X
hMα i(t) = hMα i +
χαβ (ω)Hβ (t)
(5.4)
β
Comparison yields
Z∞
χαβ (ω) =
dt e−i(ω+iδ)t hhMα ; Mβ (t)ii ≡ −hhMα ; Mβ ii(ω−iδ) (5.5)
−∞
Thus, χαβ (ω) is given by the Fourier transform of the Greens function.
*
We can now
replace
the
magnetic
moment
operator
M
i at site i by a spin
*
operator Si to obtain
µ
µ ν
ν
2 2
χµν
ij (ω) = − hhMi ; Mj ii(ω) = − g µB hhSi ; Sj ii(ω)
(5.6)
with gyromagnetic factor g and Bohr magneton µB . Of particular interest
are the longitudinal susceptibility
2 2
z z
χzz
ij (ω) = − g µB hhSi ; Sj ii(ω)
(5.7)
and the transversal susceptibility
y
2 2
+ −
±
x
χ+−
ij (ω) = − g µB hhSi ; Sj ii(ω) where Si = Si ± iSi
(5.8)
−
The operators Szi , S+
i and Si can again be replaced by creation and annihilation operators:
1
Szi = (ni↑ − ni↓ ) ,
2
†
S+
i = ci↑ ci↓ ,
†
S−
i = ci↓ ci↑
(5.9)
which leads to
1 2 2
χzz
ij (ω) = − g µB (2δσσ 0 − 1)hhniσ ; njσ 0 ii(ω)
4
(5.10)
†
†
2 2
χ+−
ij (ω) = − g µB hhci↑ ci↓ ; ci↓ ci↑ ii(ω)
(5.11)
and
linking the susceptibilities to special two particle Greens functions. We
will now apply the random phase approximation to the calculation of the
46
transversal magnetic susceptibility within the Hubbard model;
this appro*
ximation will be valid in the limit of weak interactions. In k space we have
for the susceptibility
X †
*
χ+− (q, ω) = − g2 µ2B
hhc* c*k+*q↓ ; c*† 0 c*k 0 −*q↑ ii(ω)
(5.12)
**
kk 0
k↓
k↑
The equation of motion for this two-particle Greens function is
ωhhc*† c*k+*q↓ ; c*† 0 c*k 0 −*q↑ ii(ω) = hn*k↑ i − hn*k+*q↓ i δ*k,*k+*q +
k↓
k↑
+ ε*k+*q − ε*k hhc*† c*k+*q↓ ; c*† 0 c*k 0 −*q↑ ii(ω)+
k↓
k↑
X
U
+
hh(c*† c*† c*k −*q ↑ c*k+*q−*q ↓ − c*† * c*† c*k +*q ↓ c*k+*q↓ ); c*† 0 c*k 0 −*q↑ ii(ω)
1
k↑ k1 ↑ 1 1
k+q1 ↑ k1 ↓ 1 1
k↓
N* *
k1 ,q1
(5.13)
According to the principles of the random phase approximation the excess
operators in the higher order Greens functions are replaced by averages:
c*† c*† c*k −*q ↑ c*k+*q+*q ↓ ≈ hn*k ↑ iδ*q1 ,0 c*† c*k+*q↓ −hn*k↑ iδ*k *k −*q c*†
k↑ k1 ↑
1
1
1
1
k↑
1 1
1
c*
* *
*
k−q1 ↑ k−q−q1 ↓
(5.14)
Here, conservation of momentum and spin was used:
hc*† ck*0 σ 0 i = hn*kσ iδ*kk*0 δσσ 0
(5.15)
kσ
Also
c*k+*q ↑ c*k ↓ c*k +*q ↓ c*k+*q↓ ≈ hn*k ↓ iδ*q1 ,0 c*† c*k+*q↓ −hn*k+*q↓ iδ*k ,*k+*q c*†
1
1
1
1
1
k↑
1
c*
* *
*
k+q1 k+q+q1 ↓
(5.16)
This yields
h
†
†
ω − ω* (q) hhc* c*k+*q↓ ; c *0 ck*0 −*q↑ ii(ω) = hn*k↑ i − hn*k+*q↓ i
k
k↑
k↓
i
X
U
†
†
hhc *00 ck*00 +*q↓ ; c *0 ck*0 −*q↑ ii(ω)
(5.17)
−
k ↑
k↓
N *
↑↓
*
k 00
with the Stoner single particle excitation spectrum
0 *
ω*σσ (q) = ε*k+*q − ε*k + U(n−σ 0 − n−σ )
k
47
(5.18)
*
Now we divide Eq. (5.17) by ω and sum over k:
X †
X hn*k↑ i − hn*k+*q↓ i
†
*
*
hhc* ck+*q↓ ; c *0 ck 0 −*q↑ ii(ω) =
δ*k 0 ,*k+*q
↑↓ *
k↑
k
↓
ω − ω* (q)
*
*
k
k
k
−
X hn*k↑ i − hn*k+*q↓ i U X
*
k
↑↓
*
ω − ω* (q) N
k
*
k 00
hhc†*00 ck*00 +*q↓ ; c†*0 ck*0 −*q↑ ii(ω) (5.19)
k ↑
k↓
After renaming summations indices this means
X
*
†
k
↑↓
k↓
δ*k 0 ,*k+*q
k
k
1+
*
ω − ω* (q)
*
†
hhc* c*k+*q↓ ; c *0 ck*0 −*q↑ ii(ω) =
k↑
X hn*k↑ i − hn*k+*q↓ i
U X hn*k 00 ↑ i − hn*k 00 +*q↓ i
N
(5.20)
ω − ω*↑↓00 (q)
*
*
k 00
k
with the susceptivility of the Stoner model
0
*
χ (q, ω) =
g2 µ2B
X hn*k↑ i − hn*k+*q↓ i
ω − ω*↑↓ (q)
*
*
(5.21)
k
k
This yields for the transverse susceptibility
*
χ
+−
χ0 (q, ω)
(q, ω) =
*
1 − NgU2 µ2 χ0 (q, ω)
*
(5.22)
B
The denominator in this RPA expression for the susceptibility can become
*
small for certain q and ω values so that the susceptibility becomes big.
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