Renormalization Group Analysis

In the following the total tunneling probabilities|T_k^r|² from a given reservoir into the double dot are assumed to bek-independent for energies close to the Fermi level of the reservoirs T_k^r = T_r. Then the Kondo coupling constants J_µν^kk0 are independent of the wave vectorsk andk⁰. Their indices are therefore omitted: J_µν^kk0 → J_µν. We also consider the case of symmetric coupling to both reservoirs η_k^s =η_k^dand omit the reservoir’s index of the tunneling angle.

To calculate the renormalization corrections to the coupling constants in first non-vanishing order one has to evaluate one loop diagrams, like the typical diagram shown in Fig. 2.5.

Using the propagators of the Fermi liquid in the reservoir G(iω_n) = 1

iω_n−ξ_k,

and combining them with the definition of the bi-spinor fields (2.6) one ob-tains the propagators of the fields ˆΨ_k

G(iω, k) =hΨ^†_kΨki= 1

iω−ξ_kP (2.11)

with the matrix

P= cos²η_k cosη_ksinη_k cosη_ksinη_k sin²η_k

!

⊗1₂ (2.12)

describing the non-trivial part of the coupling between the reservoirs and the dots. The propagator of the fields ˆΦ in the double dot reads

D(iω_n, k) = 1

iω_n1₂⊗1₂. (2.13)

Evaluating the diagram shown in Fig. 2.5 gives the following correction to the Kondo Hamiltonian (2.10)

δH_K¹ =

Z d^dk⁰ (2π)^d

Z dω⁰

2π(σ^µ⊗σ^ν)G(ω⁰−ω, k⁰)(σ^µ0⊗σ^ν0)(τ^µ0⊗τ^ν0)D(ω⁰)(τ^µ⊗τ^ν)

=

Z d^dk⁰ (2π)^d

Z dω⁰

2π(σ^µ⊗σ^ν)P(σ^µ0 ⊗σ^ν0)(τ^µ0 ⊗τ^ν0)(τ^µ⊗τ^ν)×

× 1

ω⁰−ω−ξ_k+i0 sgn(ω⁰−ω)

1

ω⁰+i0 sgnω⁰ (2.14) Sinceξ_k =ξ_k(|~k|) only depends on the absolute value of~kandPonly depends on the angle of ~k it is useful to split the integration over the wave vector k in two parts by

Z

d^dk =

Z

dΩ

Z

k^d−1dk

where dΩ denotes the d− 1 dimensional solid angle element. Then 2.14 becomes

δH_K¹ =

Z

dΩ(σ^µ⊗σ^ν)P(σ^µ0 ⊗σ^ν0)(τ^µ0 ⊗τ^ν0)(τ^µ⊗τ^ν)×

×

Z d^dk⁰k^0d−1 (2π)^d

Z dω⁰ 2π

1

ω⁰−ω−ξ_k+i0 sgn(ω⁰−ω)

1

ω⁰+i0 sgnω⁰ (2.15)

2.3.1 Integrating out the energy shell

The first integral gives the matrix structure of the correction and will be discussed later. To integrate out an upper energy shell from Λ−δΛ to Λ one has to perform the ω integration in the second integral of Eq. (2.14).

Substituting δΛ by Λl to keep the relative size δΛ/Λ of the energy shell constant all over the scaling and assuming Λ ≫ |ω| the integral can be

approximated by

Z Λ Λ−δΛ

dω⁰ 2π

1

ω⁰−ω−ξk+i0 sgn (ω⁰−ω)

1 ω⁰+i0 sgnω⁰

≈ Λl 2π

1 Λ−ξ_k+i0

1 Λ +i0.

For the remaining k integration the density of states is assumed to be constant at the Fermi level _(2π)^ddk0d →ν₀dξ_k. Using the principle value theorem one gets

Z d^dk⁰ (2π)^d

Λl 2π

1 Λ−ξk+i0

1 Λ +i0

= l 2π

Z ν0dξk

Λ−ξ_k+i0 = l

2π(−2πi)ν₀

=−ilν₀.

An analogous calculation for the lower energy shell from −Λ + δΛ to −Λ contributes also−ilν₀.

Evaluating all other possible one loop diagrams one gets the following results for the integrals over ω and ξk

^−2ilν0

^2ilν0

^−2ilν0

^2ilν0

2.3.2 Integrating over the solid angle of the wave vec-tor

The matrix structure P (2.12) of the propagator depends on the tunneling angle η_k which in turn depends only on the angular part of the wave vector k. To integrate overk in (2.14) therefore involves averaging over all possible angles.

hPi=

* cosη²_k cosη_ksinη_k cosη_ksinη_k sin²η_k

!+

⊗1₂ (2.16)

= hcosη²_ki hcosη_ksinη_ki hcosηksinηki hsin²ηki

!

⊗1₂ (2.17)

Decomposing (2.17) in the set of the Pauli matrices σⁱ and the unity matrix gives

hPi=

1

2hsin²η_k+ cos²η_ki1₂+hsinη_kcosη_kiσ¹+ 1

2hcos²η_k−sin²η_kiσ³

⊗1₂ (2.18)

= b₀1₂+

3

X

i=1

b_iσⁱ

!

⊗1₂

with

b₀ = 1

2 (2.19)

b₁ =hsinη_kcosη_ki (2.20)

b₂ = 0 (2.21)

b₃ = 1

2hcos²η_k−sin²η_ki (2.22) In the case of symmetric dots, which is assumed in this work, b₃ also van-ishes. The fluctuations of the tunneling angle are now described by a single parameter b₁. The limiting values for b₁ can be interpreted in the following way:

• In the case of strictly separated reservoirs the tunneling angle is either η_k= 0 orη_k =π/2 for the different modes. In both cases either the sine or the cosine in the definition of b₁ is zero and therefore b₁ = 0. The strong fluctuations of the tunneling angle will enable an SU(4) Kondo effect for this value of b₁.

• If all modes couple equally to both dots, which means one common reservoir for the double dot, the tunneling angle is fixed to η_k = π/4.

This results in b₁ = 1/2. The orbital Kondo effect is suppressed and the system will be in a spin SU(2) Kondo regime.

Because of this interpretation of the parameter b₁ it will be called mixing parameter in what follows. Summing up all diagrams results in a commutator like structure of the correction to the Kondo-Hamiltonian HK

δH_K =πν₀(δΛ/Λ)J_µνJ_µ⁰_ν⁰

×Φ^†h(τ^µ⊗τ^ν)τ^µ0⊗τ^ν0−τ^µ0 ⊗τ^ν0(τ^µ⊗τ^ν)ⁱΦ

×Ψ^†h(σ^µ⊗σ^ν)Pσ^µ0 ⊗σ^ν0−σ^µ0 ⊗σ^ν0P(σ^µ⊗σ^ν)ⁱΨ

(2.23)

This correction obviously has another structure than the original Hamiltonian (2.10)

The reason for this difference is that the original Hamiltonian is not gen-eral enough to hold during the scaling process. Differently from the scaling calculations in section 1.3 for the SU(2) spin Kondo effect new interactions are generated by the renormalization of the coupling constants.

The choice of our basis (2.8), (2.9) determines the matrix structure of the propagator in the reservoirs P, Eq. (2.19). The basis is chosen in a way that the non-trivial part ofP just operates on σ^µ and σ^µ0 in the tensor products of the Pauli matrices in (2.23). Therefore the indices µ, µ⁰ of the tensor product will be calledorbital indices and ν, ν⁰ spin indices. It is just the non trivial mixing of the orbital indices that generates new interactions in the renormalization process. With this consideration in mind, one can suppose that not all fourµ, µ⁰ and ν, ν⁰ indices are needed to describe the correction to the Kondo Hamiltonian, but just three are enough to reflect the physical structure of the system.

This motivates the trial to re-write the original Kondo Hamiltonian (2.10) for the double dot system in a more general form using two independent orbital indices and one spin index:

H_K =

3

X

µ,λ,ν=0

J_µλν^hΦ^†(τ^µ⊗τ^ν)Φ^{i h}Ψ^†(σ^λ⊗τ^ν)Ψⁱ (2.24) and hope that this form is general enough to respect the newly generated interactions during the scaling.

To get the flow equations for the correction to the Kondo Hamiltonian one has to redo the calculations leading to correction in the first place, but this time taking the more general “three index” variant of the Kondo Hamil-tonian (2.24) as starting point. If no new interactions are generated for this

generalized Kondo Hamiltonian, the assumption of three independent indices is justified and the flow equations for the coupling constantsJ_µλν can be read off the correction to the Hamiltonian.

2.3.3 Decomposition of the commutator into the gen-erators of the SU(4)

Redoing the calculations for the generalized Kondo Hamiltonian (2.24) one ends up with a commutator like structure, similar to (2.23). To find the flow equations for the coupling constants Jµλν the correction to the Hamiltonian has to be expanded into a direct product of Pauli matrices. To do this we use the relation for the product of the Pauli matricesσⁱ extended by σ⁰ =1₂ σⁿσ^m =δ_mnσ⁰+δ_n0β_m0σ^m+δ_m0β_n0σⁿ+i_nmkσ^k, (2.25) where

βij = 1−δij

and the Levi-Civita tensor _ijk are utilized. First, we rewrite (2.23) into a tensor product of matrix products by

(σ^µ⊗σ^ν)(σ^µ0⊗σ^ν0) = (σ^µσ^µ0)⊗(σ^νσ^ν0)

and apply identity (2.25) to the result. This gives for the commutator (2.23) (τ^µ⊗τ^ν)(τ^µ0 ⊗τ^ν0)−(τ^µ0 ⊗τ^ν0)(τ^µ⊗τ^ν) = (2.26)

−2i^h(τ^λ⊗τ^ν0)β_0ν⁰δ_0νµλµ⁰ + (τ^λ⊗τ^ν)β_0νδ_0ν⁰_µλµ⁰ + (τ^λ⊗τ⁰)δ_νν⁰_µλµ⁰ +(τ^µ0⊗τ^λ)β_0µ⁰δ_0µνλν⁰ + (τ^µ⊗τ^λ)β_0µδ_0µ⁰_νλν⁰ + (τ⁰⊗τ^λ)δ_µµ⁰_νλν⁰ⁱ, where λ is a summation index.

The non trivial part of the correction to the Kondo Hamiltonian (2.23) is given by the commutator resulting from the propagator in the reservoirs. It

expands to

(σ^µ⊗σ^ν)P(σ^µ0 ⊗σ^ν0)−(σ^µ0 ⊗σ^ν0)P(σ^µ⊗σ^ν) = (2.27)

−2ib₀^h(σ^λ⊗σ^ν0)β_0ν⁰δ_0νµλµ⁰ + (σ^λ⊗σ^ν)β_0νδ_0ν⁰_µλµ⁰ + (σ^λ⊗σ⁰)δ_νν⁰_µλµ⁰ +(σ^µ0⊗σ^λ)β_0µ⁰δ_0µνλν⁰ + (σ^µ⊗σ^λ)β_0µδ_0µ⁰_νλν⁰ + (σ⁰⊗σ^λ)δ_µµ⁰_νλν⁰ⁱ +2ib₁^h(σ⁰⊗σ⁰)δ_νν⁰_1µ⁰_µ+ (σ⁰⊗σ^ν)β_0νδ_0ν⁰_1µ⁰_µ+ (σ⁰⊗σ^ν0)β_0ν⁰δ_0ν1µ⁰_µ

+(σ⁰⊗σ^λ)(δ0µδ1µ⁰νν⁰λ+δ0µ⁰δ1µνν⁰λ) +(σ¹⊗σ^λ)(2δ_0µ⁰δ_0µνν⁰_λ−δ_µ⁰_µνν⁰_λ)

+(σ^µ0⊗σ^λ)β_0µ⁰δ_1µνν⁰_λ + (σ^µ⊗σ^λ)β_0µδ_1µ⁰_νν⁰_λ +(σ^λ⊗σ⁰)(δ_0µδ_νν⁰_1µ⁰_λ−δ_0µ⁰δ_νν⁰_1µλ)

+(σ^λ⊗σ^ν)(β_0νδ_0µδ_0ν⁰_1µ⁰_λ−β_0νδ_0µ⁰δ_0ν⁰_1µλ) +(σ^λ⊗σ^ν0)(β_0ν⁰δ_0µδ_0ν1µ⁰_λ−β_0ν⁰δ_0µ⁰δ_0ν1µλ)ⁱ.

Exploiting these identities for all indices and sorting them by the direct prod-uct of Pauli matrices one can read off the flow equations for the coupling con-stants. The full set of these is shown in appendix B. Since no disambiguations in the corrections to the coupling constants appear the generalized Kondo Hamiltonian (2.24) has proven to hold during the scaling.

2.3.4 The flow equations

The flow of the coupling constants differs for spin index ν = 0 and ν = 1,3.

The scaling equations reduce consistently to two sets of equations for these cases as shown in appendix C. In the following notation the value of the index ν is therefore restricted to ν = 1,3 and the case ν = 0 is explicitly denoted by a zero as spin index.

Assuming that coupling constants with non diagonal orbital indices start to flow with zero as starting value, it turns out thatJ_10ν, J_01ν andJ₁₀₀are the only non-diagonal coupling constants generated during the scaling. Further, the coupling constants with diagonal orbital indices J_µµν, µ = 2,3 have the same flow and will be denoted byJ_⊥0 respectiveJ_⊥ν. This eventually results

in the following set of flow equations:

dJ_00ν

dΛ = 8^hb₁(2J_00νJ_01ν + 2J_11νJ_10ν) +b₀(J_00ν² +J_11ν² + 2J_⊥ν² +J_01ν² +J_10ν² )ⁱ J₁₁₀

dΛ = 8b₀^hJ_⊥0² + 3J_⊥ν^{2 i} J_11ν

dΛ = 8 [b₁(2J_11νJ_01ν + 2J_00νJ_10ν) +b₀(2J_00νJ_11ν+ 2J_⊥0J_⊥ν + 2J_01νJ_10ν)]

J⊥0

dΛ = 8 [b₀(J₁₀J⊥0+ 3J_11νJ⊥ν) +b₁(J⊥0J₁₀₀+ 3J⊥νJ_10ν)]

dJ⊥ν

dΛ = 8 [b₀(J_11νJ⊥0+ 2J_00νJ⊥ν+J₁₀J⊥ν) +b₁(2J⊥νJ_01ν +J⊥νJ₁₀₀+J⊥0J_10ν)]

J_01ν

dΛ = 8^hb₀(2J00νJ01ν + 2J11νJ10ν) +b₁(J_00ν² +J_11ν² −2J_⊥ν² +J_01ν² +J_10ν² )ⁱ dJ₁₀₀

dΛ =−8b₁^hJ_⊥0² + 3J_⊥ν^{2 i} dJ_10ν

dΛ = 8 [b₀(2J_11νJ_01ν + 2J_00νJ_10ν) +b₁(2J_00νJ_11ν−2J_⊥0J_⊥ν + 2J_01νJ_10,ν)]

This coupled set of differential equations is integrated numerically from the starting values

J00ν =J0, J110=J0, J11ν =J0, J⊥0=J0, J⊥ν =J₀, J_01ν = 0, J₁₀₀ = 0, J_10ν = 0.

This assumes all non-diagonal coupling constants start to flow as zero, as they are just generated by the renormalization process. J₀ is assumed to be a small constant, which depends on the properties of the considered metal of the reservoirs.

The result of the numerical integration is that all coupling constants are monotonically increasing and diverge all at the same critical value Λ_c. An example of this behavior is shown in Fig. 2.6

Integrating the flow equations for different values of the mixing param-eters b1 gives the dependence of Λ_c of b1. The Kondo temperature TK is proportional to exp(−J₀Λ_c). The resulting dependency T_K(b₁), normalized to the SU(4) point at b₁ = 0 is shown in Fig. 2.7

Im Dokument Kondo Effect in Double Quantum Dots (Seite 36-43)