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

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10 R. Shankar 3 Large-N Approach to Fermi Liquids Not only did Landau say we could describe Fermi liquids with an F function, he also managed to compute the response functions at small ω and q in terms of the F function even when it was large, say 10, in dimensionless units. Again the RG gives us one way to understand this. To this end we need to recall the the key ideas of “large-N” theories. These theories involve interactions between N species of objects. The largeness of N renders fluctuations (thermal or quantum) small, and enables one to make approximations which are not perturbative in the coupling constant, but are controlled by the additional small parameter 1/N . As a specific example let us consider the Gross-Neveu model [5] whichis one of the simplest fermionic large-N theories. This theory has N identical massless relativistic fermions interacting through a short-range interaction. The Lagrangian density is L = N� i=1 ¯ψi �∂ψi − λ � �2 N� ¯ψiψi N i=1 (21) Note that the kinetic term conserves the internal index, as does the interaction term: any index that goes in comes out. You do not have to know much about the GN model to to follow this discussion, which is all about the internal indices. Figure 4 shows the first few diagrams in the expression for the scattering amplitude of particle of isospin index i and j in the Gross-Neveu theory. The “bare” vertex comes with a factor λ/N. The one-loop diagrams all share a factor λ 2 /N 2 from the two vertices. The first one-loop diagram has a free internal summation over the index k that runs over N values, with the contribution being identical for each value of k. Thus, this one-loop diagram acquires a compensating factor of N which makes its contribution of order λ 2 /N ,thesame order in 1/N as the bare vertex. However, the other oneloop diagrams have no such free internal summation and their contribution i i j i j i j j i i j k i = + + + i j + j i j i k j i j j Fig. 4. Some diagrams from a large-N theory i j ...
RG for Interacting Fermions 11 is indeed of order 1/N 2 . Therefore, to leading order in 1/N , one should keep only diagrams which have a free internal summation for every vertex, that is, iterates of the leading one-loop diagram, which are called bubble graphs. For later use remember that in the diagrams that survive (do not survive), the indices i and j of the incoming particles do not (do) enter the loops. Let us assume that the momentum integral up to the cutoff Λ for one bubble gives a factor −Π(Λ, qext), where qext is the external momentum or frequency transfer at which the scattering amplitude is evaluated. To leading order in large-N the full expression for the scattering amplitude is Γ (qext) = 1 λ N 1+λΠ(Λ, qext) (22) Once one has the full expression for the scattering amplitude (to leading order in 1/N ) one can ask for the RG flow of the “bare” vertex as the cutoff is reduced by demanding that the physical scattering amplitude Γ remain insensitive to the cutoff. One then finds (with t =ln(Λ0/Λ)) dΓ(qext) dt =0⇒ dλ dt = −λ2 dΠ(Λ, qext) dt (23) which is exactly the flow one would extract at one loop. Thus the one-loop RG flow is the exact answer to leading order in a large-N theory. All higher-order corrections must therefore be subleading in 1/N . 3.1 Large-N Applied to Fermi Liquids Consider now the ¯ ψψ − ¯ ψψ correlation function (with vanishing values of external frequency and momentum transfer). Landau showed that it takes the form χ = χ0 1+F0 , (24) where F0 is the angular average of F (θ) andχ0 is the answer when F =0. Note that the answer is not perturbative in F . Landau got this result by working with the ground-state energy as a functional of Fermi surface deformations. The RG provides us with not just the ground-state energy, but an effective hamiltonian (operator) for all of lowenergy physics. This operator problem can be solved using large N-techniques. The value of N here is of order KF /Λ, and here is how it enters the formalism. Imagine dividing the annulus in (Fig. 1) intoN patches of size (Λ) in the radial and angular directions. The momentum of each fermion ki is a sum of a “large” part (O(kF )) centered on a patch labelled by a patch index i =1,...N and a “small” momentum (O(Λ) within the patch [2]. Consider a four-fermion Green’s function, as in (Fig. 4). The incoming momenta are labelled by the patch index (such as i) and the small momentum is not shown but implicit. We have seen that as Λ → 0, the two outgoing
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: RG for Interacting Fermions 9 These
Page 23 and 24: RG for Interacting Fermions 13 the
Page 25 and 26: RG for Interacting Fermions 15 this
Page 27 and 28: RG for Interacting Fermions 17 equa
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
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B // Semiconductor Few-Electron Qua
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Semiconductor Few-Electron Quantum
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a RESERVOIR 200 nm Semiconductor Fe
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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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6.7 Unresolved Issues Semiconductor
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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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158 C.W.J. Beenakker its momentum).
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