Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

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16 R. Shankar dependence on the energy of αβγδ. In the spinless case, the first two terms on the right hand side make equal contributions and produce the constant charging energy in the Universal Hamiltonian of (26), while in the spinful case they produce the charging and exchange terms. The final term of (30) produces the Cooper interaction of (26). Thus the UH contains the rotationally invariant part of the Landau interaction. The others, i.e., those that do not survive ensemble averaging, are dropped because they are of order 1/g. But we have seen before in the BCS instability of the Fermi liquid that a term that is nominally small to begin with can grow under the RG. That this is what happens in this case was shown by the RG calculation of Murthy and Mathur. There was however one catch. The neglected couplings could overturn the UH description for couplings that exceeded a critical value. However the critical value is of order unity and so one could not trust either the location or even the very existence of this critical point based on their perturbative one-loop calculation. Their work also gave no clue as to what lay on the other side of the critical point. Subsequently Murthy and I [24] showed that the methods of the large N theories (with g playing the role of N) were applicable here and could be used to show nonperturbatively in the interaction strength that the phase transition indeed exists. This approach also allowed us to study in detail the phase on the other side of the transition, as well as what is called the quantum critical region, to be described later. Let us now return to Murthy and Mathur and ask how the RG flow is derived. After integrating some of the g states within ET , we end up with g ′ = ge −t states. Suppose we compute a scattering amplitude Γαβγδ for the process in which two fermions originally in states αβ are scattered into states γδ. This scattering can proceed directly through the vertex Vαβγδ(t), or via intermediate virtual states higher order in the interactions, which can be classified by a set of Feynman diagrams, as shown in Fig. 5. All the states in the diagrams belong to the g ′ states kept. We demand that the entire amplitude be independent of g ′ , meaning that the physical amplitudes should be the same in the effective theory as in the original theory. This will lead to a set of flow γ α δ β γ δ γ δ δ ν = + + + + µ µ α β α β α β Fig. 5. Feynman diagrams for the full four-point amplitude Γαβγδ ν γ γ α µ ν β δ ...
RG for Interacting Fermions 17 equations for the Vαβγδ. In principle this flow equation will involve all powers of V but we will keep only quadratic terms (the one-loop approximation). Then the diagrams are limited to the ones shown in Fig. 5, leading to the following contributions to the scattering amplitude Γαβγδ Γαβγδ = Vαβγδ + � µ,ν ′ NF (ν) − NF (µ) εµ − εν − � ′ 1 − NF (µ) − NF (ν) µν εµ + εν � VανµδVβµνγ − VανµγVβµνδ VαβµνVνµγδ � (31) where the prime on the sum reminds us that only the g ′ remaining states are to be kept and where NF (α) is the Fermi occupation of the state α. We will confine ourselves to zero temperature where this number can only be zero or one. The matrix element Vαβγδ now explicitly depends on the RG flow parameter t. Now we demand that upon integrating the two states at ±g ′ ∆/2 we recover the same Γαβγδ. Clearly, since g ′ = ge −t , d δ = −g′ dt δg ′ (32) The effect of this differentiation on the loop diagrams is to fix one of the internal lines of the loop to be at the cutoff ±g ′ ∆/2, while the other one ranges over all smaller values of energy. In the particle-hole diagram, since µ or ν can be at +g ′ ∆/2 or−g ′ ∆/2, and the resulting summations are the same in all four cases, we take a single contribution and multiply by a factor of 4. The same reasoning applies to the Cooper diagram. Let us define the energy cutoff Λ = g ′ ∆/2 to make the notation simpler. Since we are integrating out two states we have δg ′ =2 0= dVαβγδ dt − g′ � ′ 4 2 µ=Λ,ν NF (ν) − NF (µ) εµ − εν + g′ � 4 2 εµ + εν µ=Λ,ν � ′ 1 − NF (µ) − NF (ν) VανµδVβµνγ − VανµγVβµνδ VαβµνVνµγδ � (33) where µ = Λ means εµ = Λ and so on. The changed sign in front of the 1-loop diagrams reflects the sign of (32) So far we have not made any assumptions about the form of Vαβγδ, and the formulation applies to any finite system. In a generic system such as an atom, the matrix elements depend very strongly on the state being integrated over, and the flow must be followed numerically for each different set αβγδ kept in the low-energy subspace.
Page 1 and 2: Lecture Notes in Physics Editorial
Page 3 and 4: W. Dieter Heiss (Ed.) Quantum Dots:
Page 5: Preface Nanoscale physics, nowadays
Page 9 and 10: Contents A Guide for the Reader ...
Page 11 and 12: A Guide for the Reader Quantum dots
Page 13 and 14: The Renormalization Group Approach
Page 15 and 16: RG for Interacting Fermions 5 where
Page 17 and 18: where � is a shorthand: � ≡ 3
Page 19 and 20: RG for Interacting Fermions 9 These
Page 21 and 22: RG for Interacting Fermions 11 is i
Page 23 and 24: RG for Interacting Fermions 13 the
Page 25: RG for Interacting Fermions 15 this
Page 29 and 30: � p � dθp = g 2π RG for Inter
Page 31 and 32: RG for Interacting Fermions 21 the
Page 33 and 34: References RG for Interacting Fermi
Page 35 and 36: Semiconductor Few-Electron Quantum
Page 39 and 40: 1.2 Implementations Semiconductor F
Page 43 and 44: GaAs n-AlGaAs AlGaAs GaAs Semicondu
Page 49 and 50: ε1 ε0 e Semiconductor Few-Electro
Page 53 and 54: 250 Ω twisted pair 20 MΩ powder
Page 61 and 62: 10 5 0 -5 Semiconductor Few-Electro
Page 63 and 64: G ( e / h) 2 2 1 0 Semiconductor Fe
Page 67 and 68: V R (V) V R (V) Semiconductor Few-E
Page 69 and 70: V L (V) -1.084 -1.082 -1.080 -1.078
Page 71 and 72: B // Semiconductor Few-Electron Qua
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a RESERVOIR 200 nm Semiconductor Fe
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Semiconductor Few-Electron Quantum
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4.5 QPC Versus SET Semiconductor Fe
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a reservoir dot Semiconductor Few-E
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a b c V P (mV) 10 5 0 ∆I QPC E F
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Spin-down counts 200 100 a Semicond
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6.7 Unresolved Issues Semiconductor
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98 M. Pustilnik and L.I. Glazman Co
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100 M. Pustilnik and L.I. Glazman F
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132 C.W.J. Beenakker e e N I N S Fi
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134 C.W.J. Beenakker 2 Andreev Refl
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136 C.W.J. Beenakker F E x 8d/hv ga
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138 C.W.J. Beenakker and we have de
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140 C.W.J. Beenakker A stroboscopic
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142 C.W.J. Beenakker largely indepe
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144 C.W.J. Beenakker The effective
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146 C.W.J. Beenakker Fig. 6. Ensemb
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148 C.W.J. Beenakker 1.4 1.2 1.0 0.
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150 C.W.J. Beenakker 〈ρ(E)〉 =
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152 C.W.J. Beenakker P 0.5 0.4 0.3
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154 C.W.J. Beenakker P(x) 0.4 0.3 0
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156 C.W.J. Beenakker tunnel/ Egap 4
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158 C.W.J. Beenakker its momentum).
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160 C.W.J. Beenakker E00 = π� =
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162 C.W.J. Beenakker 〈ρ(E)〉 δ
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164 C.W.J. Beenakker As described i
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166 C.W.J. Beenakker 0.30 � Egap
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168 C.W.J. Beenakker The τE-depend
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170 C.W.J. Beenakker that a fully m
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172 C.W.J. Beenakker (τ−,τ+), w
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174 C.W.J. Beenakker 61. P. G. Silv
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