Exchange vertex correction

3.3.2.1 Spin part of exchange diagram The spin part of the exchange diagram yields

P_ex = ^X

η_l⁰ηl,α⁰α

σ_η⁰

lηl·σ_α⁰_µσ_ηlη⁰

l·σ_µ⁰_α σ_η⁰_η ·σ_αα⁰

=^X

α⁰α



σ_η⁰_η·σ_αα^{0 X}

η⁰_lηl

σ_α⁰_µ·σ_η⁰

lηl σ_ηlη⁰

l·σ_µ⁰_α



 ; (3.71)

moreover,

X

η⁰_lηl

σα⁰µ·σ_η⁰

lηlσ_ηlη⁰

l·σµ⁰α=^X

ab

X

η_l⁰ηl

σ^a_α⁰_µσ_η^a0

lη_lσ_η^b

lη_l⁰σ^b_µ⁰_α

=^X

ab

σ_α^a0_µσ_µ^b0_αTr[σ^aσ^b]

| {z }

=2δab

= 2σ_α⁰_µ·σ_µ⁰_α . (3.72) Thus,P_ex, Eq. (3.71), simplifies to

P_ex = 2^X

α⁰α

σ_η⁰_η·σ_αα⁰ σ_α⁰_µ·σ_µ⁰_α = 2^X

ab

X

αα⁰

σ^a_η⁰_η(σ^aσ^b)αµσ_µ^b0_α

= 2^X

ab

σ_η^a0_η^X

α

σ_µ^b0_α δabδαµ+ i^X

c

abcσ^c_αµ

!!

= 2^X

ab

σ_η^a0_ησ_µ^b0_µδab+ 2i^X

abc

σ_η^a0_ηabc

X

α

σ_µ^b0_ασ_αµ^c

| {z }

=(σ^bσ^c)_µµ0=δbcδ_µµ0+iP

dbcdσ^d

µ0µ

= 2σ_η⁰_η·σ_µ⁰_µ+ 2i^X

abc

σ^a_η⁰_{η abc}δ_bcδ_µµ⁰

| {z }

=0

+2i·i^X

abcd

σ_η^a0_{η abcbcd}σ^d_µ⁰_µ , (3.73)

which by using the identity (AppendixA), X

bc

_abcbcd =^X

bc

_abcdbc = 2δ_ad , yields

P_ex= 2ση⁰η·σµ⁰µ−2^X

ad

2δad σ_η^a0_ησ^d_µ⁰_µ

= 2σ_η⁰_η·σ_µ⁰_µ−4σ_η⁰_η·σ_µ⁰_µ

=−2ση⁰η·σµ⁰µ . (3.74)

3.3.2.2 Real and imaginary parts of the exchange diagram

The real and imaginary parts of the exchange diagram can be obtained by decomposing the Green’s functions and the f-susceptibility into their imaginary and real parts and collecting factors of the real and imaginary parts of the impurity susceptibility. A detailed but

straight-forward calculation using the explicit form the Green’s functions andf-susceptibility, yields the result

Γex =J J₀²(gµ_B)²W π

1 π(mk_F

2π )²ε_F T_K

sin(2%)ν+ 2i sin²(%)|ν|

%² Θ(T_K− |Ω|) , (3.75) whereν = Ω/εF. One observes that in the relevant energy interval, Ω∼ O(TK)D0, where the RGβ-function is non-vanishing (see section 3.4), the direct and exchange vertex corrections satisfy|Γex/Γdir| ∼ ^T_D^K

0 1, and thus, the exchange vertex correction can be safely neglected.⁹

Explicit Calculations The explicit form of the exchange vertex correction is obtained by performing the Matsubara summation,¹⁰

Γex = +2J J₀²

Z dε^hnF(ε+ω)Ac(x−x_l, ε+ω)G^R_c(x_l−x, ε+ω+ Ω)χ^A_f(ε) +n_F(ε+ω+ Ω)G^R_c(x−x_l, ε+ω)A_c(x_l−x, ε+ω+ Ω)χ^A_f(ε)

+n_B(ε+ω+ Ω)G^R_c(x−x_l, ε+ω)G^R_c(x_l−x, ε+ω+ Ω)1

πχ^A00_f (ε)ⁱ . (3.77) To reduce the notational clutter, we drop the position coordinates in the following so that

Γex= 2J J₀²

Z dεⁿn_F(ε+ω)A_c(ε+ω)G^R_c(ε+ω+ Ω)χ^A_f(ε)

+nF(ε+ω+ Ω)G^A_c(ε+ω)Ac(ε+ω+ Ω)χ^A_f(ε) (3.78) +n_B(ε)G^R_c(ε+ω)G^R_c(ε+ω+ Ω)1

πχ^R00_f (ε)^o . (3.79) By decomposing χ_f into its real and imaginary parts, we can further expand the relation for Γex:

χ^A0_f (ε)≡χ^A0_f (ε) + iχ^A00_f (ε)

=χ^R0_f (ε)−iχ^R00_f (ε) , (3.80)

9 This can be compared to the Migdal’s theorem in the context of electron-phonon interaction, which states that phonon correction (renormalization) to the electron-phonon vertex is suppressed at least by a factor

pme

M_i ∼10⁻², where me andMi represent the mass of an electron and an ion (for a detailed discussion, consult, e.g., Ref. [122,123]). Here the impurity vertex correction to the electronic vertex is suppressed by a factor ^T_D^K

0 ∼10⁻³.

10 Note that the following relations are used

Reχ^A_f(ε) = Reχ^R_f(ε),

Imχ^Af(ε) =−Imχ^Rf(ε). (3.76)

so that Γex= 2J J₀²

Z dε^hnF(ε+ω)Ac(ε+ω)G^R_c(ε+ω+ Ω) +nF(ε+ω+ Ω)G^A_c(ε+ω)Ac(ε+ω+ Ω)χ^R0_f (ε)

−in_F(ε+ω)A_c(ε+ω)G^R_c(ε+ω+ Ω) +n_F(ε+ω+ Ω)G^A_c(ε+ω)A_c(ε+ω+ Ω)χ^R00_f (ε) +n_B(ε)G^R_c(ε+ω)G^R_c(ε+ω+ Ω)1

πχ^R00_f (ε)ⁱ. (3.81)

Therefore, the expression for Γex can be decomposed into two major terms i) A factor of Reχ^R_f which reads

F⁰ :=^Z dε^hn_F(ε+ω)A_c(ε+ω)G^R_c(ε+ω+ Ω)

+nF(ε+ω+ Ω)G^A_c(ε+ω)Ac(ε+ω+ Ω)ⁱχ^R0_f (ε) . (3.82) By decomposing the Green’s functions into their real and imaginary parts,

G^R(ε) =G^R0(ε) + iG^R00(ε) =G^R0(ε)−iπA(ε)

G^A(ε) =G^A0(ε) + iG^A00(ε) =G^A0(ε) + iπA(ε) , (3.83) and applying the transformationε7→ε−ω, the factor becomes

F⁰ =^Z dε^hn_F(ε)Ac(ε)G^R0_c (ε+ Ω) +n_F(ε+ Ω)G^R0_c (ε)A_c(ε+ Ω)

−iπ(n_F(ε)−n_F(ε+ Ω))A_c(ε)A_c(ε+ Ω)ⁱχ^R0_f (ε−ω) . (3.84) This is so far exact.

ii) A factor of Imχ^R_f which reads

F⁰⁰=−i^Z dε^hnF(ε+ω)Ac(ε+ω)G^R_c(ε+ω+ Ω) +n_F(ε+ω+ Ω)G^A_c(ε+ω)Ac(ε+ω+ Ω) + in_B(ε)G^R_c(ε+ω)G^R_c(ε+ω+ Ω)1

π

iχ^R00_f (ε) . (3.85) By decomposing the Green’s functions to their real and imaginary parts as before, and applying the transformationε7→ε−ω, the factor becomes

F⁰⁰=−i^Z dε^h(n_F(ε+ω) +n_B(ε−ω))Ac(ε)G^R0_c (ε+ Ω) + (n_F(ε+ Ω) +n_B(ε−ω))G^R0_c (ε)A_c(ε+ Ω)

+ (−iπ) (n_F(ε)−n_F(ε+ Ω) +n_B(ε−ω))A_c(ε)A_c(ε+ Ω) + i

πnB(ε−ω)G^R0_c (ε)G^R0_c (ε+ Ω)ⁱχ^R00_f (ε−ω) . (3.86) This is exact up to here.

The integrations above cannot be performed analytically. In order to make progress, we need to use the asymptotic form of the impurity susceptibility,χ_f. Furthermore, we will let the energy of the incoming electron to be on the Fermi level,ω= 0.^!

Dimensionless variables Before the actual calculation, we introduce the dimensionless variables used in this section:

x:= ε ε_F , ν := Ω

ε_F ,

%:=k_Fr ,

τK :=TK/D0 1 , d₀ := ε_F

T_K ≈ D₀

T_K 1 . (3.87)

3.3.2.3 Asymptotic form of the impurity susceptibility

The susceptibility for the f-pseudo-fermions was given before, in section 3.2. The real part of the impurity susceptibility reads

Reχ^R_f(ε)≡χ^R00_f = (gµB)²W π

1 T_K

1

p1 + (ε/T_K)² ,

where W is the Wilson ratio, g is the Landé g-factor andµ_B is the Bohr magneton. Let us define

a⁰ := (gµ_B)²W

π , (3.88)

to absorb the constants and reduce the notational clutter. Then, the asymptotic approximation to the real part reads

Reχ^R_f(ε)≈







a⁰

TK ;|ε| T_K

a⁰ T_K

1

|ε/T_K| ;|ε| TK

. (3.89)

The corresponding imaginary part (which can be obtained from the real part by the Kramers-Kronig relations) reads

Imχ^R_f(ε)≡χ^R00_f (ε) = 2 π

(gµ_B)²W π

! 1 T_K

arcsinh(ε/T_K) p1 + (ε/TK)² . We can absorb the constant factors in a⁰⁰,

a⁰⁰ := 2 π

(gµB)²W

π = 2

πa⁰ . (3.90)

The asymptotic approximation for Imχ_f is¹¹ Imχ^R_f(ε)≈







a⁰⁰ T_K

ε

T_K ;|ε| TK a⁰⁰

TK

ln|ε/T_K|

ε/TK ;|ε| TK

. (3.95)

We need these asymptotic approximations in the explicit calculations in the next part.

11 The asymptotic form of arcsinh function can be obtained as below:

x:= sinh⁻¹(y)⇒y= sinh(x) = e^x−e^−x

2 . (3.91)

Ifz≥0 is defined asz:=e^x⇔x= lnz, then z=y±p

1 +y^{2 z≥0}→ z=y+p 1 +y²

→x= lnz= ln(y+p

1 +y²). (3.92)

Ify→0,

y→0limy+p

1 +y² =→1 +y

→ln(y+p

1 +y²)→ln(1 +y

|{z}

>0

) =y+O(y²)

→x≡sinh⁻¹(y)≈y . (3.93)

Ify→+∞,

lim

y→+∞y+p

1 +y² =→1 +|y|^y>0= 2y≈y

→x≡sinh⁻¹(y)→ln(2y)≈ln(y). (3.94)

Ify→ −∞,

y→−∞lim y+p

1 +y² =→1 +|y|(1 + 1

y²)¹² ≈y+|y|(1 + 1 2y²)

= y+|y|

| {z }

y<0

=y−y=0

+ 1 2|y|= 1

2|y|

→x= sinh⁻¹(y)→ln( 1

2|y|) =−ln(2|y|)≈ −ln|y|. Therefore,

lim

y→±∞sinh⁻¹(y) =±ln|y|=|y|

y ln|y|. Finally,

arcsinh(y) p1 +y² =

(y ;y→0

|y|

y ln|y|

|y| ;y→ ±∞ .

3.3.2.4 Factors of the real part off-susceptibility

The factor of the real part of the impurity susceptibility reads F⁰ =^Z dε^hn_F(ε)Ac(ε)G^R0_c (ε+ Ω)

+n_F(ε+ Ω)G^R0_c (ε)A_c(ε+ Ω)

−iπ(n_F(ε)−n_F(ε+ Ω)A_c(ε)A_c(ε+ Ω)ⁱχ⁰_f(ε) . (3.96)

First integral The first integral on the RHS of Eq. (3.96) is

Z dε n_F(ε)A_c(ε)G^R0_c (ε+ Ω)χ⁰_f(ε) . (3.97) At low energies, the factor containing statistical distribution in the integrand merely determines the range of integration:

nF(ε)^T→0+≈ Θ(−ε) ; (3.98)

therefore,ε∈[−D₀,0]→x∈[−1,0]. The first integral will be

−1 π(m

2π)² a⁰ T_K

ε_F

r² I₁⁰ , (3.99)

with

I₁⁰ :=^{Z 0}

−1dx sin(%√

1 +x) cos(%√

1 +x)−1

2%νsin(%√ 1 +x)

√1 +x

! 1

p1 + (d₀x)² , (3.100) where we have used the dimensionless variables defined in Eq. (3.87). The integral cannot be performed analytically. We have to use the asymptotic form of the impurity susceptibility, introduced in section3.3.2.3, and consider all the possible cases

i) When Ω>0∧Ω> TK →ε∈[−D₀,0]:

I₁⁰ =−1

2sin(2%)τ_Klnτ_K+O(τ_K) . (3.101) ii) When Ω>0∧Ω< TK →ε[−D₀,0]:

I₁⁰ =−1

2sin(2%)τ_Klnτ_K+O(τ_K) . (3.102) iii) When Ω<0∧ |Ω|> T_K ≡Ω<−T_K →ε∈[−D₀,0]:

I₁⁰ =−1

2sin(2%)τKlnτK+O(τK) . (3.103)

iv) When Ω<0∧ |Ω|< T_K ≡Ω>−T_K →ε∈[0,−Ω]:

I₁⁰ =−1

2sin(2%)τKlnτK+O(τK) . (3.104) Second integral The second integral on the RHS of Eq. (3.96) is

Z dε n_F(ε+ Ω)G^R0_c (ε)A_c(ε+ Ω)χ⁰_f(ε). (3.105) At low energies, the factor containing statistical distribution in the integrand merely determines the range of integration:

nF(ε+ Ω)^T→0+≈ Θ(−(ε+ Ω)) ; (3.106) therefore, ε∈[−D₀,−Ω]→x∈[−1,−ν]. Then the integral will be

−1 π(m

2π)² a⁰ TK

ε_F

r² I₂⁰ , (3.107)

with

I₂⁰ :=^{Z −ν}

−1 dx cos(%√

1 +x) sin(%√

1 +x+ν) 1 p1 + (d0x)²

=^{Z −ν}

−1 dx cos(%√

1 +x) sin(%√

1 +x) +1

2%νcos(%√ 1 +x)

√1 +x

! 1

p1 + (d0x)² , (3.108) where in the second line a Taylor-expansion is performed with respect toν 1. The integ-ral cannot be performed analytically. We have to use the asymptotic form of the impurity susceptibility, introduced in section3.3.2.3, and consider all the possible cases

i) When Ω>0∧Ω> T_K→ε∈[−D₀,−Ω]:

I₂⁰ =O(τK) . (3.109)

ii) When Ω>0∧Ω< T_K→ε[−D₀,−Ω]:

I₂⁰ =−1

2sin(2%)ν+O(ν²) +O(τ_K) . (3.110) iii) When Ω<0∧ |Ω|> T_K ≡Ω<−T_K →ε∈[−D₀,−Ω]:

I₂⁰ =−sin(2%)τKlnτK+ 1

2sin(2%)τKlnν+O(τK) . (3.111) iv) When Ω<0∧ |Ω|< T_K ≡Ω>−T_K →ε∈[0,−Ω]:

I₂⁰ =−1

2sin(2%)τKlnτK−1

2νsin(2%) +O(τK). (3.112)

Third integral The third integral on the RHS of Eq. (3.96) is

Z dε (nF(ε)−nF(ε+ Ω))Ac(ε)Ac(ε+ Ω)χ⁰_f(ε) . (3.113) At low energies, the factor containing statistical distribution in the integrand merely determines the range of integration:

n_F(ε)−n_F(ε+ Ω)^T→0≈⁺ Θ(ε+ Ω)−Θ(ε) ; (3.114) therefore, for Ω>0,

ε∈[−Ω,0]→x∈[−ν,0], and for Ω<0,

ε∈[0,−Ω]→x∈[0,−ν]. Then the proper integral will be

(m 2π)² a⁰

T_K εF

r² I₃⁰ (3.115)

with

I₃⁰ :=^Z dx sin(%√

1 +x) sin(%√

1 +x+ν) 1 p1 + (d₀x)²

=^Z dxsin(%√

1 +x) sin(%√

1 +x) +1

2%νcos(%√ 1 +x)

√1 +x

! 1

p1 + (d₀x)² , (3.116) where in the second line a Taylor-expansion is performed with respect to ν 1. The integ-ral cannot be performed analytically. We have to use the asymptotic form of the impurity susceptibility, introduced in section3.3.2.3, and consider all the possible cases:

i) When Ω>0∧Ω> T_K →ε∈[−Ω,0]:

I₃⁰ = sin²(%)τKln|ν/τ_K|+O(τK) . (3.117) ii) When Ω>0∧Ω< TK →ε[−Ω,0]:

I₃⁰ = sin²(%)ν+O(ν³) . (3.118) iii) When Ω<0∧ |Ω|> T_K ≡Ω<−T_K →ε∈[0,−Ω]:

I₃⁰ = sin²(%)τ_Kln|ν/τ_K|+O(τ_K) . (3.119) iv) When Ω<0∧ |Ω|< T_K ≡Ω>−T_K →ε∈[0,−Ω]:

I₃⁰ =−sin²(%)ν+O(ν³) . (3.120)

Final result for factors of real part of f-susceptibility Putting the integral pieces together, one obtains

F⁰=−1 π(m

2π)² a⁰ T_K

ε_F r²I₁⁰

− 1 π(m

2π)² a⁰ TK

ε_F r²I₂⁰

−iπ(1 π)²(m

2π)² a⁰ T_K

εF

r²I₃⁰

=−1 π(m

2π)² a⁰ T_K

ε_F

r² (I₁⁰ +I₂⁰ + iI₃⁰)

| {z }

:=I⁰

(3.121)

i) When Ω>0∧Ω> TK→ε∈[−Ω,0]:

I⁰ =−1

2sin(2%)τ_Klnτ_K+ i sin²(%)τ_Kln|ν/τ_K|. (3.122) ii) When Ω>0∧Ω< TK→ε[−Ω,0]:

I⁰ =−1

2sin(2%)τ_Klnτ_K−1

2sin(2%)ν+ i sin²(%)ν . (3.123) iii) When Ω<0∧ |Ω|> T_K ≡Ω<−T_K →ε∈[0,−Ω]:

I⁰= 1

2sin(2%)τ_Kln|ν/τ_K|+ i sin(2%) ln|ν/τ_K|. (3.124) iv) When Ω<0∧ |Ω|< T_K ≡Ω>−T_K →ε∈[0,−Ω]:

I⁰ =−sin(2%)τ_Klnτ_K−ν

2sin(2%)−iνsin²(%) . (3.125) 3.3.2.5 Factors of the imaginary part of χ_f

The factor ofχ^R00_f , at low temperatures (T →0⁺) and low frequencies (ω →0), is F⁰⁰=−i^Z dε^h(n_F(ε+ Ω)−n_F(ε))G^R0_c (ε)A_c(ε+ Ω)

−iπ(−n_F(ε+ Ω))A_c(ε)A_c(ε+ Ω)

− i

πn_F(ε)G^R0_c (ε)G^R0_c (ε+ Ω)ⁱχ^R00_f (ε) . (3.126) The three integrations will be considered separately.

First Integral The first integral on the RHS is

Z dε (nF(ε+ Ω)−nF(ε))G^R0_c (ε)Ac(ε+ Ω)χ^R00_f (ε). (3.127) At low temperatures, the factor containing the Fermi-Dirac distributions merely limits the range

of integration:

n_F(ε+ Ω)−n_F(ε) = Θ(ε)−Θ(ε+ Ω)≡ −(Θ(ε)−Θ(ε+ Ω)) ; (3.128) thus, if Ω≥0,

ε∈[−Ω,0]→x∈[−Ω

ε_F,0]≡[−ν,0] , (3.129) otherwise, Ω<0, and

ε∈[0,−Ω]→x∈[0,−Ω

εF]≡[0,−ν]. (3.130)

In the proper integration range, we have to consider the integral Z dε G^R0_c (ε)A_c(ε+ Ω)χ^R00_f (ε)

= −1 π (m

2π)²ε_F r²

Z dx cos(%√

1 +x) sin(%√

1 +x+ν)χ⁰⁰_f(d₀x), (3.131) where the previously-defined dimensionless variables, Eq. (3.87), are used. Regarding Eq. (??), we should consider the dimensionless integral,

I₁⁰⁰:=^Z dx cos(%√

1 +x) sin(%√

1 +x+ν)χ⁰⁰_f(d₀x) . (3.132) At low frequencies, ν1, one can Taylor-expand the integrand in ν to obtain

I₁^00ν1= ^Z dx cos(%√

1 +x)(sin(%√

1 +x) +1

2%νcos(%√ 1 +x)

√1 +x χ⁰⁰_f(d₀x). (3.133) The integral cannot be performed analytically. We have to use the asymptotic form of the impurity susceptibility, introduced in section3.3.2.3, and consider all the possible cases:

i) When Ω>0∧Ω> TK →ε∈[−Ω,0]:

I₁⁰⁰=−1

2sin(2%)τ_Kln|ν/τ_K|+O(τ_K) . (3.134)

ii) When Ω>0∧Ω< T_K →ε[−Ω,0]:

I₁⁰⁰= −sin(2%)

4τ_K ν²+O(ν³). (3.135)

iii) When Ω<0∧ |Ω|> TK ≡Ω<−T_K →ε∈[0,−Ω]:

I₁⁰⁰= 1

2sin(2%)τ_Kln|ν/τ_K|+O(ν³) . (3.136)

iv) When Ω<0∧ |Ω|< T_K ≡Ω>−T_K →ε∈[0,−Ω]:

I₁⁰⁰=O(τK)−1

2sin(2%)τKln|ν/τ_K|. (3.137) Second Integral The second integral on the RHS of Eq. (3.126) is

Z dε(−n_F(ε+ Ω))A^R0_c (ε)A_c(ε+ Ω)χ^R00_f (ε) . (3.138) At low temperatures, the factor containing the Fermi-Dirac distributions merely limits the range of integration:

−n_F(ε+ Ω) =−Θ(−(ε+ Ω)) ; (3.139) thus, for any Ω,

ε∈[−D₀,−Ω]→x∈[−1,−ν]. (3.140) In the proper integration range, we have to consider the integral

Z dε A^R0_c (ε)A_c(ε+ Ω)χ^R00_f (ε)

= ( m

2π²)²a⁰⁰ε_F r²

Z dx sin(%√

1 +x) sin(%√

1 +x+ν)χ⁰⁰_f(d₀x) , (3.141) where previously defined dimensionless variables are used. Consider the dimensionless integral,

I₂⁰⁰:=^Z dx sin(%√

1 +x) sin(%√

1 +x+ν)χ⁰⁰_f(d0x) . (3.142) At low frequencies,ν 1, one can Taylor-expand the integrand in ν to obtain

I₂^00ν1= ^Z dxsin(%√

1 +x) sin(%√

1 +x) +1

2%νcos(%√ 1 +x)

√1 +x

!

χ⁰⁰_f(d₀x) . (3.143) The integral cannot be performed analytically. We have to use the asymptotic form of the impurity susceptibility, introduced in section 3.3.2.3, and consider all the possible cases:

i) When Ω>0∧Ω> TK→ε∈[−D₀,−Ω]:

I₂⁰⁰=−1

2sin(2%)τ_Kln|ν|+O(τ_K) . (3.144) ii) When Ω>0∧Ω< T_K→ε[−D₀,−Ω]:

I₂⁰⁰ =−1

2(1−cos(2%))τKlnτK+O(τK). (3.145) iii) When Ω<0∧ |Ω|> TK ≡Ω<−T_K →ε∈[−D₀,−Ω]:

I₂⁰⁰=−1

2(1−cos(2%))τ_Kln|ν|+O(τ_K) . (3.146)

iv) When Ω<0∧ |Ω|< T_K ≡Ω>−T_K →ε∈[0,−Ω]:

I₂⁰⁰=O(τK)−1

2(1−cos(2%))τKlnτK . (3.147)

Third integral The third integral on the RHS of Eq. (3.126) is

Z dε(−n_F(ε))G^R0_c (ε)G^R0_c (ε+ Ω)χ^R00_f (ε) . (3.148) At low temperatures, the factor containing the Fermi-Dirac distributions merely limits the range of integration:

n_F(ε) = Θ(−ε) ; (3.149)

thus, for any Ω,

ε∈[−D₀,0]→x∈[−D₀,0]. (3.150) In the proper range, we have to consider the integral

Z dε G^R0_c (ε)A_c(ε+ Ω)χ^R00_f (ε)

= (m 2π)²εF

r²

Z dx cos(%√

1 +x) cos(%√

1 +x+ν)χ⁰⁰_f(d0x) , (3.151) where we have used the previously defined dimensionless variables, Eq. (3.87). Consider the dimensionless integral,

I₃⁰⁰:=^Z dx cos(%√

1 +x) cos(%√

1 +x+ν)χ⁰⁰_f(d₀x) . (3.152) At low frequencies, ν1, one can Taylor-expand the integrand in ν to obtain

I₃^{00 ν1}= ^Z dx cos(%√

1 +x) cos(%√

1 +x)−1

2%νsin(%√ 1 +x)

√1 +x

!

χ⁰⁰_f(d0x) . (3.153) The integral cannot be performed analytically. We have to use the asymptotic form of the impurity susceptibility, introduced in section3.3.2.3, and consider all the possible cases:

i) When Ω>0∧Ω> T_K →ε∈[−D₀,0]:

I₃⁰⁰= 1

2(1 + cos(2%))τ_Klnτ_K+O(τ_K). (3.154)

ii) When Ω>0∧Ω< TK →ε[−D₀,0]:

I₃⁰⁰= 1

2(1 + cos(2%))τ_Klnτ_K+O(τ_K). (3.155)

iii) When Ω<0∧ |Ω|> T_K ≡Ω<−T_K →ε∈[−D₀,0]:

I₃⁰⁰= 1

2(1 + cos(2%))τKlnτK+O(τK) . (3.156) iv) When Ω<0∧ |Ω|< T_K ≡Ω>−T_K →ε∈[0,−Ω]:

I₃⁰⁰= 1

2(1 + cos(2%))τ_Klnτ_K+O(τ_K) . (3.157) Final result for the factors of the imaginary part of f-susceptibility Putting all the integral pieces together, one obtains

F⁰⁰ =−1 π(m

2π)²ε_F r²

a⁰⁰

T_K(−i)(−I₁⁰⁰) + (−i)(−iπ)( m

2π²)²ε_F r²

a⁰⁰ TK

I₂⁰⁰ + (−i)(−i

π )(m 2π)²εF

r² a⁰⁰ T_KI₃⁰⁰

= a⁰⁰ π (m

2π)²ε_F T_K

1

r² −I₂⁰⁰−I₃⁰⁰−iI₁⁰⁰

| {z }

:=I⁰⁰

. (3.158)

Now we can obtain I⁰⁰ for different ranges of Ω:

i) When Ω>0∧Ω> T_K→ε∈[−D₀,0]:

I⁰⁰= sin(2%)τKlnν+O(ν²) . (3.159) ii) When Ω>0∧Ω< TK→ε[−D₀,0]:

I⁰⁰=O(ν²). (3.160)

iii) When Ω<0∧ |Ω|> T_K ≡Ω<−T_K →ε∈[−D₀,0]:

I⁰⁰=O(ν²). (3.161)

iv) When Ω<0∧ |Ω|< TK ≡Ω>−T_K →ε∈[0,−Ω]:

I⁰⁰=O(ν²). (3.162)

3.3.2.6 Explicit expression for the exchange vertex correction

Ultimately, we have obtained an approximate expression for the exchange diagram, Γex, Γex(Ω) = 2J J₀²factors ofχ^R0_f + factors ofχ^R00_f . (3.163)

By defining a constant to simplify the notation, c0 := 1

π(m 2π)² 1

r² εF

TK

, (3.164)

we obtain

i) When Ω>0∧Ω> T_K:

Γex =O(ν²) . (3.165)

ii) When Ω>0∧Ω< TK:

Γex = 2J J₀² 1

2c₀a⁰sin(2%)−2i sin²(%)ν+O(ν²) . (3.166) iii) When Ω<0∧ |Ω|> T_K:

Γex =O(ν²) . (3.167)

iv) When Ω<0∧ |Ω|< T_K:

Γex = 2J J₀² 1

2c0a⁰sin(2%) + 2i sin²(%)ν+O(ν²) . (3.168) Therefore, the ultimate result of the previous calculation is the simple expression

Γex=J J₀² 3π

2 n_eN(ε_F)1

%² 1 TK

a⁰sin(2%)ν+ 2i sin²(%)|ν|·Θ(T_k− |Ω|) , (3.169) wheren_e= ^N_V^e is the average density of the electrons and

n_eN(ε_F) = mk_F⁴

6π⁴ . (3.170)

Therefore, ultimately, we reach the expected expression, Γex =J J₀²(gµB)²W

π 1 π(mkF

2π )²εF

T_K

sin(2%)ν+ 2i sin²(%)|ν|

%² Θ(TK− |Ω|) . (3.171) One observes that in the relevant energy interval, Ω ∼ O(T_K) D₀, where the RG β -function is non-vanishing (see section 3.4), the direct and exchange vertex corrections satisfy

|Γex/Γdir| ∼ ^T_D^K

0 1, and thus, the exchange vertex correction can be safely neglected.¹²

12 This can be compared to the Migdal’s theorem in the context of electron-phonon interaction and supercon-ductivity, which states that phonon correction (renormalization) to the electron-phonon vertex is suppressed at least by a factor pme

M_i ∼10⁻², whereme andMi represent the mass of an electron and an ion (for a detailed discussion, consult, e.g., Ref. [122,123]). Here the impurity vertex correction to the electronic vertex is suppressed by a factor ^T_D^K

0 ∼10⁻³.

3.4 Renormalization group analysis for the RKKY-modified Kondo

Im Dokument Quantum Phase Transitions in Multi-Impurity and Lattice Kondo Systems (Seite 60-75)