TKM II: Equation of motion technique - Anderson model and screening (RPA)

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1 TKM II: Equation of motion technique - Anderson model and screening RPA Dr. Jens Michelsen Contents 1 Preliminaries: Bands etc 2 2 Anderson s model for magnetic impurities Single level coupled to continuum Short on analytical continuation Full Anderson Model Self-consistency: Formation of local magnetic moment Screening effects in metals Dielectric response Non-interacting case Lindhard function Limits Interacting system - RPA approximation Plasma oscillations Screened potential Diagrammatic representation of the mean-field approximations Anderson model without on-site interaction Anderson model with on-site interaction RPA

2 1 Preliminaries: Bands etc Let us consider the non-interacting Hamiltonian in second quantized form H 0 = d 3 r [ ] p ˆψ σr 2 V 2m µ + V r ˆψ σ r 1 σ where V r is a periodic crystal potential. The eigenfunctions φ nkσ r = u n rχ σ e ik r of the single-particle Hamiltonian are characterized by band index n, crystal momentum k 1st B.Z. and spin index σ. The energy spectrum is separated into bands ɛ nk. If ɛ F = µt = 0 crosses one of the bands the material is a metal and the band is referred to as conduction band. The bandwidth is defined as D n = max[ɛ nk ] min[ɛ nk ]. The field operator can be expanded in the complete set of eigenstates ˆψ σ r = φ nkσ rĉ nkσ 2 nk The Hamiltonian in this basis takes the form H 0 = nkσ ɛ nk ĉ nkσĉnkσ 3 For metals it is common to restrict oneself to the conduction band and adopt an effective mass approximation φ nkσ r 1 k 1.B.Z. V e ik r χ σ k unrestricted m m nkσ ɛ 4 nkĉ nkσĉnkσ kσ ɛ kĉ kσĉkσ We will, however, for the following discussion of the Anderson Model, assume that the spectrum has a certain bandwidth D. The single particle Matsubara thermal Green s function is given by G σσ rτ, r τ = T τ ˆψσ r, τ ˆψ σ r, τ 5 where ˆψ σ r, τ = e H0τ ˆψσ re H0τ, ˆψ σ r, τ = e H0τ ˆψ σ re H0τ 6 It is usually easier to work with the GF in momentum space which is related to the GF in position space through G kσ,k σ τ, τ = T τ ĉ kσ τĉ k σ τ 7 G σσ rτ, r τ = 1 T τ ĉ V kσ τĉ k σ τ e ik r ik r χ σ χ σ k,k = 1 8 G kσ,k V σ τ, τ e ik r ik r χ σ χ σ k,k If the system is invariant under spin-rotational symmetry, then the GF in Eq. 5 will be proportional to δ σσ, and if we have translational periodic invariance in space the GF s will only 2

3 depend on the relative coordinate r r. Finally, if the Hamiltonian does not explicitly depend on imaginary time τ, the GF will depend only on the relative coordinates τ τ. Thus under these conditions we may restrict ourselves to Single particle GF Fermions G k,σ τ = T τ ĉ kσ τĉ kσ 0 9 The single particle Matsubara thermal Green s function is defined as G k,σ τ = T τ ĉ kσ τĉ kσ 0 = [ Θτ ĉ kσ τĉ kσ 0 Θ τ ĉ kσ 0ĉ kσ τ ] 10 with We note, in particular that ĉ k τ = e τh ĉ k e τh, ĉ k τ = eτh ĉ k e τh, 11 G kσ 0 = n kσ 12 The retarded and advanced GFs are obtained by analytical continuation of the Fouriertransform G R kσω = G kσ ω + iδ, G A kσω = G kσ ω iδ 13 The momentum- and spin resolved spectral function is given by A kσ ω = i [ G R kσω G A kσω ] = 2ImG R kσω 14 For a non-interacting system, H = H 0 = k ξ kĉ kσĉkσ, with ξ k = ɛ k µ, we have ĉ kσ τ = e ξ kτ ĉ kσ, ĉ kσ τ = eξ kτ ĉ kσ. 15 The Matsubara GF s can then be explicitly written as [ ] G 0,kσ τ = Θτ1 n F ɛ k Θ τn F ɛ k e ξ kτ 16 The unperturbed GF satisfies the equation of motion or in frequency domain τ ξ k G 0,kσ τ = δτ 17 iω n ξ k G 0,kσ iω n = 1 18 which yields 1 G 0,kσ iω n = 19 iω n ξ k Exercise: Check that G 0,kσ τ = 1 β ω n G 0,kσ iω n e iωnτ. The spectral density for the unperturbed system is given by A 0,kσ ω = 2πδω ξ k 20 3

4 2 Anderson s model for magnetic impurities The Anderson model has been used to describe the appearance of a magnetic moment of impurities of certain magnetic ions embedded in a non-magnetic host metal. It has also been used in the description of coulomb blockade in quantum dots. It is a natural starting point to describe the Kondo effect in both metals and quantum dots. The host metal is described as indicated above, i.e. by an effective mass Hamiltonian describing the conduction band. The impurity atoms are assumed to have only one spin-degenerate state in the active shell usually the d-shell, hence the subsequent notation d for the impurity states. The system is described in the eigenbasis of the individual Hamiltonians of the metal and the impurity atoms, φ kσ r and φ dσ r, with corresponding energies and creation-/annihilation operators ξ k = ɛ k µ, ĉ kσ, ĉ kσ, ξ d = ɛ d µ, ĉ dσ, ĉ dσ 21 Once the impurity atoms are embedded into the metal, the crystal- and impurity potentials are perturbed and the basis φ kσ, φ dσ is no longer an eigen-basis of the full single particle Hamiltonian. Expanding the field-operators in this eigen-basis then does no longer lead to a diagonal form but will include overlaps between the states of the conduction band and the states of the impurities. These terms are usually called hopping terms and describe a probability amplitude of transitions from the impurity atoms to the conduction band and vice versa. Furthermore, because of the highly localized wave functions at the impurity atom, the Coulomb interaction is much stronger between electrons occupying these states. Consequently, there is an additional energy cost associated with having both spin-degenerate impurity states occupied. This is modeled by a so-called on-site interaction with characteristic energy U. The full Hamiltonian of this model is then given by H = H c + H t + H d + H U 22 where H c = k,σ ξ k ĉ kσĉkσ,, H d = σ ξ d ĉ dσĉdσ 23 describe the individual unperturbed Hamiltonians of the conduction band and the impurity atom and H t = t k ĉ kσĉdσ + t kĉ dσĉkσ 24 kσ kσ describes the hopping between the two subsystems. Finally the last term H U = U ˆn ˆn 25 describes the on-site Coulomb interaction. The d-electron energy ɛ d is below the chemical potential i.e. ξ d < 0 and from the kinetic energy point of view it is favorable to fill the orbital by two electrons. However, this costs potential energy U, which is not possible if 2ɛ d + U > 2µ i.e. if 2ξ d + U > 0. Single occupancy of the atom would appear to lead, quite naturally, to a magnetic moment since only one of the spin-degenerate states must be occupied. On the other hand, the hopping between the impurity states and the conduction band implies that the direction of the spin may fluctuate strongly, and average out. The efficiency of such processes depends on the strength of the hopping. But strong hopping also increases the energy the contribution H t to the Hamiltonian so there is a trade-off between on-site interaction and hopping. 4

5 It will turn out, that for certain values of the parameters it is energetically favorable for the system to have a magnetic moment and thus minimizing the on-site interaction energy while for other values there is no magnetic moment. 2.1 Single level coupled to continuum To get a better feel for the system let us first consider only the effect of the hopping between the impurity states and the conduction band states and, for the time being, neglect the on-site interaction. H = H c + H t + H d 26 The operators obey the equations of motion τ ĉ dσ τ = [H, ĉ dσ τ] = ξ d ĉ dσ τ t kĉ kσ k 27 τ ĉ kσ τ = [H, ĉ kσ τ] = ξ k ĉ kσ τ t k ĉ dσ Analogous to the conduction band Green s functions, we define the GF of the impurity states G dσ τ = T τ ĉ dσ τĉ dσ 0 28 This GF obeys the equation of motion τ ξ d G dσ τ = δτ τ k = δτ τ + k t k T τ ĉ kσ τĉ dσ 0 t kf kσ τ 29 where we also defined This GF satisfies the equation of motion F kσ τ = T τ ĉ kσ τĉ dσ τ ξ k F kσ τ = t k G dσ τ 31 This gives us a system of coupled equations which in Fourier-space can be written iω n ξ d G dσ iω n = 1 + t kf kσ iω n k iω n ξ k F kσ iω n = t k G dσ iω n 32 Using G 1 0,kσ iω n = iω n ξ k we can write the second equation in the form Inserting into the first gets us F kσ iω n = G 0,kσ iω n t k G dσ iω n 33 G dσ iω n = G 0,dσ iω n + G 0,dσ iω n k t kg 0,kσ iω n t k G dσ iω n 34 5

6 If we define Σ σ iω n = k t k 2 G 0,kσ iω n 35 we have with the solution for G dσ iω n : G dσ iω n = G 0,dσ iω n + G 0,dσ iω n Σ σ iω n G dσ iω n 36 1 G dσ iω n = G 1 0,dσ iω n + Σ σ iω n = 1 iω n ξ d + Σ σ iω n We have Σ σ iω n = t k 2 G 0,kσ iω n = t k 2 38 iω n ξ k k k Let us assume that t k and ξ k both only depend on k = k such that Σ σ iω n = t k 2 d 3 k t k 2 = V iω n ξ k 2π 3 = V iω n ξ k 2π 2 dkk 2 t k 2 iω n ξ k k = dξ Nξ t kξ 2 39 iω n ξ dξ ξ = π iω n ξ where ξ = πnξ t kξ 2 is the hybridization energy and the density of states Nξ is defined such that 1 V Nξ = 1 2π 2 k 2 dk = νξ 40 dξ k=kξ Note: Here we distinguish between the density of states Nξ and the density of states per unit volume νξ compare dimensions. To simplify matters, let us assume that the chemical potential lies in the middle of the gap and thus ξ [ D, D] where D is the bandwidth. We also assume that the hybridization energy is constant across the band ξ =. We then have Σiω n = π Short on analytical continuation D D [ ] 1 dξ iω n ξ = ln iωn + D iω n D Note lnz = lnre iϕ = lnr + ln e iϕ = lnr + iϕ, not periodic w.r.t. ϕ such that even though z = re iϕ and z = re iϕ+2π represent the same complex number, the logarithm does not give the same result. lnz is a multivalued function in the complex plane. To avoid this problem a principal branch is chosen as ϕ π, π and a branch cut is introduced at Rez, 0. The logarithm is discontinuous across this branch cut and the function is analytic everywhere else. For a complex number with an infinitesimally small imaginary part, z = x ± iδ, δ > 0, we have that the absolute value is r x and that the phase is always either ϕ ±πθ x { ln x, x > 0 lnx ± iδ ln x + iπθ x = 42 ln x ± iπ, x <

7 D 0 D D 0 D D 0 D D 0 D Figure 1: Left: Real part of Σz. Right: Imaginary part of Σz. Thus we have Σω ± iδ = [ ] lnω ± iδ + D lnω ± iδ D π = [ ] ln ω + D ln ω D ± iπθ ω D iπθ ω + D π = π ln ω + D ω D iθd ω 43 We note that for ω D we have ln ω + D/ω D ω/d 0 Having G R dσ ω = G dσω + iδ we can write the spectral function A dσ ω = 2ImG R ω = ω ξ d Figure 2: Effect of hybridization between d electron orbitals and conduction band. 2.2 Full Anderson Model Let us now attack the full model H = H c + H t + H d + H U 45 7

8 The equation of motion of the operators is given by τ ĉ dσ τ = ξ d ĉ dσ τ t kĉ kσ + [H U, ĉ dσ tτ] k 46 τ ĉ kσ τ = ξ k ĉ kσ τ t k ĉ dσ which is the same as before except for the extra term [ ] [H U, ĉ dσ τ] = U ˆn d τˆn d τ, ĉ dσ τ = U ˆn d, σ τĉ dσ τ 47 which follows from [ˆn d σ τ, ĉ dσ τ] = 0 and [ˆn dσ τ, ĉ dσ τ] = ĉ dσ τ. We then have for the GFs τ ξ d G dσ τ = δτ + k τ ξ k F kσ τ = t k G dσ τ t kf kσ τ U T τ ˆn d, σ τĉ dσ τĉ dσ 0 48 The last term of the top equation complicates the equation of motion as we need also the equation of motion of the two particle GF, which in turn depends on the three-particle GF etc. The straight forward route would be to perform perturbation theory. This, however, is only meaningful for weak interaction, and can not directly cover non-perturbative effects such as bound states and the appearance of local moments which we are interested in here. Instead we make a so-called mean-field approximation: T τ ˆn d, σ τĉ dσ τĉ dσ 0 = ĉ d, σ τĉ d, σ τĉ dσ τĉ dσ 0 T τ ĉ d, σ τĉ d, σ τ T τ ĉ dσ τĉ dσ 0 = ˆn d, σ G dσ τ 49 The equation of motion of the GFs are then in Fourier space given by using from now on the notation n σ = ˆn dσ iω n ξ d Un σ G dσ iω n = 1 + t kf kσ iω n k iω n ξ k F kσ iω n = t k G dσ iω n 50 Solving again for the second one and inserting into the first one we get G dσ iω n = 1 iω n ξ d Un σ + Σiω n 51 With the same approximations as in the previous section we find A dσ ω = ω ξ d Un σ

9 2.3 Self-consistency: Formation of local magnetic moment Using the relationship derived from the Lehmann-representation dω dω n σ = 2π n F ωaω = 2π n F ω ω ξ d Un σ we obtain a self-consistency equation. At T = 0, i.e. n F ω = Θ ω, we get 1 n σ = π arctan ξd + Un σ 54 or using cot π 2 arctanx = x cotπn σ = ξ d + Un σ Introducing the total occupation N = n + n and the magnetization M = n n we may write this as N = 1 ξd + U 2 N σm arccot π σ 56 M = 1 ξd + U 2 N σm σarccot π σ Note that there is always a solution with M = 0 since the second equation of 56 is trivially satisfied. The total occupation is then given by N = 2n where n is the occupation of one of the levels and given by 55 with index σ removed. It is useful to consider a special case where the states are half-filled on average. To find the condition for we set n = 1/2 and get the condition Half-filling, N = 1: ξ d = U 2, for M = 0, i.e. n = n σ = The interesting thing about the Anderson model is, however, the existence of a solution with finite magnetization. To see that such a solution exists, let us try to find the condition for complete magnetization 2 M = 1 which implies also that N = M = 1. Effectively this means that only one state is occupied and the other is empty. Inserting this condition into 56 and adding/subtracting the two equations we get ξd ξd + U π = arccot, 0 = arccot 58 Using the half-filling condition from before ξ d = U/2 the left and right equations become the same and lead to the condition ξ d = U = cotπ = Using the indefinite integral g dx x a 2 +g 2 2 Note that M [ 1, 1] since n σ 0, 1 = arctan a x g 9

10 This result should not be surprising since if the on-site interaction is infinitely strong compared to hybridization only one of the states should be occupied and processes where the electron occupying this state is exchanged with an electron from the conduction band are suppressed. To find the critical values of the parameters for a finite magnetization we may linearize the left hand side of Eq. 56 by using arccota + bx = arccota bx 1+a 2 + Ox2 to obtain N 2 π arccot ξd + U 2 N M 1 π ξd + U 2 N 2 U M 60 Thus, using again the half-filling condition ξ d = U/2 we get N = 1 from the first equation while the second equation requires, for non-zero M, the following condition to be satisfied U = π 61 which is the critical value for the ratio of interaction strength versus hybridization in order to form a local moment. M π U Figure 3: Top figures: Spectral density on x-axis, Energy on y-axis. Left: U = = 0. Middle: U < π. Right: U > π. Bottom: Magnetic moment as a function of U/. 10

11 3 Screening effects in metals The electric potential φr, t in a system can be separated into two parts, the externally applied potential φ ext r, t, as well as an induced potential φ ind r, t. φr, t = φ ext r, t + φ ind r, t 62 In the absence of an externally applied electric potential a system at equilibrium has a vanishing electric potential 3 φr, t = φ ind r, t = 0 63 The induced electric potential is related to the induced charge through the Poisson equation 2 φ ind r, t = 1 ρ ind r, t φr, t = d 3 r V c r r ρ ind r, t 64 ɛ 0 where V c r r = 1 4πɛ 0 1 r r is the Coulomb interaction. At equilibrium the charge density from the electrons is cancelled over macroscopic distances by the charge density of the ions, i.e. at equilibrium we have charge neutrality and ρ ind r, t = 0. The induced charge ρ ind is due to an induced displacement of the electron density δnr, t. In the linear response regime, the induced charge density can be written as ρ ind r, t = d 3 r dt χ R rt, r t φ ext r, t 66 where χ R is known as the polarizability function. 3.1 Dielectric response The relationship between the external potential and the total potential defines the dielectric function ε φ ext r, t = d 3 r dt εrt, r t φr, t 67 Usually it is more interesting/useful to talk about the inverse relationship φr, t = d 3 r dt ε 1 rt, r t φ ext r, t 68 Using the definitions from the previous section we can write this as ε 1 rt, r t = δr r δt t + d 3 r V c r r χ R r t, r t In principle it only needs to be constant, but such a potential is not observable since the electro-magnetic field depends on the gradient and time derivative of the potential - Gauge invariance 11

12 Translational invariance For a translationally invariant system the polarizability depends only on the relative coordinates χ R rt, r t = χ R r r ; t t and therefore so does the dielectric function and its inverse. It then makes sense to talk about the Fourier transformed quantities which are related through φk, ω = ɛ 1 k, ωφ ext k, ω 70 with the dielectric function being related to the polarizability ε 1 k, ω = 1 + V c kχ R k, ω 71 Using E ext k, ω = ikφ ext k, ω together with the continuity equation iωρk, ω + ik jk, ω = 0 we can obtain a relationship to the conductivity defined through or jk, ω = σk, ωe ext k, ω ɛ 1 k, ω = 1 i k2 σk, ω 72 ω σk, ω = iω k 2 V ckχ R k, ω 73 In a quantum mechanical system the coupling to the electric potential is described by adding a perturbation H ext = d 3 rˆρr, tφ ext r, t = e d 3 r ˆψ r, tψr, tφ ext r, t 74 to the Hamiltonian. Note: in this section we shall neglect spin, which only leads to a factor of two in the end-result. The linear response is then written as ρr, t = i d 3 r dt Θt t [ˆρr, t, ˆρr, t ] φ ext r t 75 which identifies the polarizability function χ R rt, r t = iθt t [ˆρr, t, ˆρr, t ] 76 In the basis of φ k r = 1 V e ik r the density can be written as and we have ˆρr, t = e ĉ k V tĉ k te ik k r = e ĉ k V tĉ te iq r k+q 77 kk kq χ R rt, r t = iθt t e2 V 2 [ĉ k tĉ k+q t, ĉ k t ĉ k +q t ] e iq r e iq r 78 kk qq 12

13 For a translationally invariant system the response function should only depend on the relative coordinate r r, which means that q = q. Introducing the notation ˆρ q t = e k ĉ k tĉ k+q t 79 we have or in momentum space χ R r r ; t = iθt 1 V 2 [ˆρ q t, ˆρ q 0] e iq r r q 80 χ R q; t = iθt 1 V [ˆρ qt, ˆρ q 0] 81 In the following sections we shall try to evaluate this response function by first deriving the Fourier transform of the corresponding imaginary time GF χq, iω n = β 0 dτχk, τe iωnτ, χk, τ = 1 V T τ ˆρ q τˆρ q 0 82 and then performing an analytical continuation to obtain the retarded GF χ R k, ω, i.e. the polarizability function. 3.2 Non-interacting case We start by investigating the non-interacting case where the unperturbed Hamiltonian is given by H 0 = k ξ k ĉ kĉk 83 and the equation of motion of the operators is τ ĉ k τ = [H, ĉ k ] = ξ k ĉ k τ τ ĉ k τ = [H, ĉ k ] = ξ kĉ k τ 84 It is useful to rewrite the two-particle GF as the index 0 denotes non-interacting where from now on setting e = 1 for notational brevity χ 0 q, τ = 1 χ 0 k, q, τ, 85 V k χ 0 k, q; τ = T τ ĉ k τĉ k+q τˆρ q0 86 It is then useful to investigate the time derivative τ ĉ k τĉ k+q τ = [H 0, ĉ k τĉ k+q τ] = ξ k+q ξ k ĉ k τĉ k+q τ 87 The equation of motion of the two-particle GF is then given by remember to take into account the discontinuity due to the time ordering τ ξ k+q ξ k χk, q; τ = δτ [ĉ kĉk+q, ˆρ q] 88 13

14 The commutator on the right hand side is [ĉ kĉk+q, ˆρ q] = ĉ kĉk ĉ k+qĉk+q = n F ξ k n F ξ k+q 89 In Fourier space the equation of motion then becomes iω n ξ k+q ξ k χ 0 k, q; iω n = n F ξ k n F ξ k+q 90 or The polarizability function is then given by χ 0 k, q; iω n = n F ξ k n F ξ k+q iω n ξ k+q ξ k χ R 0 q, ω = χ 0 q, ω + iδ = 1 V k n F ξ k n F ξ k+q ω + iδ ξ k+q ξ k The right hand side of this equation is known as the Lindhard function Lindhard function The Lindhard function can be evaluated exactly but for most purposes it is sufficient to consider the low wave-length approximation ξ k+q ξ k + q v k with v k = k ξ k, which leads to χ R 0 q, ω = d 3 k n F ξ k n F ξ k+q 2π 3 ω + iδ ξ k+q ξ k d 3 k 2π 3 q v k ω + iδ q v k dn F dɛ 93 The imaginary part of the response is given by Imχ R d 3 k 0 q, ω = 2π 3 q v kπδω q v k dn F dɛ 1 ω k2 F dcos θδω qv F cos θ 4πv F 1 = ω kf 2 Θ ω v F q v F q 4πv F ω = πν0 2v F q Θ ω v F q 94 We may use the Kramers-Kronig relation to obtain the real part [ Reχ R 0 q, ω = ν0 1 ω ] 2v F q ln ω + v F q ω v F q 95 with ν0 = k F m/2π 2. 14

15 Figure 4: Absorbtion imaginary part of χ R 0 q, ω at T = 0 derived from full Lindhard-function. Dashed line indicating ω = qv F Limits For ω = 0 static polarizability - compressibility we get Imχ R 0 q, 0 = 0 and χ R 0 q, 0 = Reχ R 0 q, 0 = ν0. For small v F q/ω the imaginary part of the polarizability goes to zero and for the real part we get to second order using ln 1+x 1 x = 2x + 2x3 /3 + Ox 4 χ R 0 q, ω = Reχ R 0 q, ω ν vf q 2 = ν0 ω 3 vf q 2 n = ω m q 2 ω 2 96 where we used the mean electron density n = d 3 k 2π n 3 F ξ k = 1 4πk 3 2π 3 F 3 = 1 k 3 2π 2 F 3 = ν0 3 mv2 F. This result could have been obtained more directly by expanding the integrand to first nonvanishing order in q to get χ R d 3 k q v k 2 0 q, ω 2π 3 ω 2 dn F = n q 2 97 dɛ m ω 2 Note: First order expansion in q cancels for isotropic medium. 3.4 Interacting system - RPA approximation In this section we extend our analysis to the case of interacting electrons by adding the interaction Hamiltonian H int = 1 d 3 rd 3 r 2 ˆψ r ˆψr V c r r ˆψr ˆψr 98 15

16 where V c r r is again the Coulomb interaction vertex. In momentum space the Hamiltonian takes the form H int = 1 V c qĉ 2V k+qĉ k qĉk ĉ k 99 kk,q 0 The equation of motion of the operator ĉ k τĉ k+q τ is then given by τ ĉ k τĉ k+q τ = [H 0, ĉ k τĉ k+q τ] + [H int, ĉ k τĉ k+q τ] 100 The first term is the same as before, while the last term is [H int, ĉ k τĉ k+q τ] = 1 2V k,q 0 { V q ĉ k+q ĉ k qĉk ĉ k+q + ĉ k +q ĉ k q ĉ k+q ĉ k ĉ k +q ĉ kĉk+q+q ĉ k ĉ kĉ k q ĉ k ĉ k+q q } 101 We now apply what is known as the Random Phase Approximation, consisting in replacing pairs of operators by their expectation values. In essence this is also a type of mean-field approximation. The procedure requires us to take all the pairings between annihilation- and creation operators. For instance we get for the first term ĉ k+q ĉ k qĉk ĉ k+q ĉ k+q ĉ k+q ĉ k qĉk + ĉ k+q ĉ k+q ĉ k qĉk = ĉ k+q ĉ k+q n F ξ k δ q,0 + n F ξ k δ q,q ĉ k qĉk 102 Doing this for all terms and summing over q leads to [H int, ĉ 1 k τĉ k+q τ] = V cq n F ξ k+q n F ξ k ĉ k qĉk 103 V k Collecting everything we then get τ ξ k+q ξ k χk, q; τ = n F ξ k n F ξ k+q 1 + V q 1 χk, q; τ V k which in Fourier space becomes iω n ξ k+q ξ k χk, q; iω n = n F ξ k n F ξ k+q 1 + V c q 1 χk, q; iω n V k Using the previous result for the non-interacting response χ 0 k, q; iω n we can write χk, q; iω n = χ 0 k, q; iω n 1 + V c q 1 χk, q; iω n V k Summing over k we get χq, iω n = χ 0 q, iω n 1 + V c qχq; iω n

17 Solving for χq, iω n yields χq, iω n = χ 0 q, iω n 1 V c qχ 0 q, iω n Since we have the analytical continuation of χ R 0 q, ω we can insert this into the expression for 3.5 Plasma oscillations χ R q, ω = 108 χ R 0 q, ω 1 V c qχ R 0 q, ω 109 Inserting the expression for the polarizability in the RPA approximation into the definition of the dielectric function we have ε 1 V c k, ω 1 k, ω = V c kχ R 0 k, ωχr 0 k, ω = 1 V c kχ R 0 k, ω 110 or inversely Thus we can write the polarizability as εk, ω = 1 V c kχ R 0 k, ω. 111 χ R k, ω = χr 0 k, ω εk, ω 112 which can be interpreted as a screened response to the external potential with the inverse of the dielectric function representing the screening. For very small k we have χ R 0 k, ω = n k 2 m ω. Then there exists a zero of the dielectric function at 2 ω = ω p =. This zero indicates a resonance associated with plasma oscillations. e 2 n ɛ 0m 3.6 Screened potential Alternatively, we can write where ε 1 k, ω = 1 + V c k, ω 1 V c kχ R 0 k, ωχr 0 k, ω = 1 + Ṽckχ R 0 k, ω 113 Ṽ c k, ω = V ck 1 V ckχ R 0 k,ω = V ck εk, ω represents a screened Coulomb interaction. For ω = 0 we have χ R 0 k, 0 = ν0, and thus Ṽ c k, 0 = V c k 1 + ν0v c k = 1 = ɛ 0 k ν0 ɛ 0 k ɛ 0 k 2 + κ 2 where κ = ɛ 1 0 ν0 is the Thomas-Fermi screening length. The Fourier transform of this interaction gives a short range interaction Ṽ c r r = e κ r r 4πɛ 0 r r

18 Figure 5: Absorbtion as calculated by imaginary part of χ R 0 q, ω at T = 0 together with the plasma mode ωq = ω p + 3 v F 10 ωp q 2. 4 Diagrammatic representation of the mean-field approximations 4.1 Anderson model without on-site interaction Recall that the GF equation for the Anderson Model in the absence of on-site interaction can be written as G dσ iω n = G 0,dσ iω n + G 0,dσ iω n t kg 0,kσ iω n t k G dσ iω n 116 This may be expressed diagrammatically by introducing the following representations of the propagators G dσ =, G 0,dσ =, G 0,kσ = 117 and the vertices k t k =, t k = 118 It is then easy to see that the diagrammatic representation of the Anderson model without on-site interaction is given by = By inserting this equation into itself we can generate an infinite series of diagrams. Needless to say, it is a big advantage to sum up these diagrams up by defining Σ = = k t kg 0,kσ t k 120 which justifies the use of the term self-energy for this object. 18

19 4.2 Anderson model with on-site interaction When we include on-site interaction in the Anderson model we also need to include the corresponding diagrams. To identify which diagrams are kept within the approximations used in this text, let us first look at the case when the hopping is absent, t k = 0. The equation of motion can then be written on the form Here n σ = G dσ τ = 0 = 1 β the vertex G dσ iω n = G 0,dσ iω n + G 0,dσ iω n [Un σ ] G dσ iω n 121 ω n G dσ iω n e iωn0. Using the diagrammatical representation of U = 122 The term Un dσ can then be represented by a tadpole diagram Un dσ = 123 and the equation of motion for the Greens function is given by = If we include also hopping we may write [ G dσ iω n = G 0,dσ iω n + G 0,dσ iω n Un σ + k t kg 0,kσ iω n t k ] G dσ iω n or where = = + = Σ tot 126 Note that since the tadpole diagram contains the full propagator the self-energy has to be evaluated self-consistently together with the Dyson equation. 4.3 RPA We write the unperturbed polarizability χ 0 χ 0 =

20 while we write the full response function as χ = 128 Using the equation of motion for the polarizability which we can represent diagrammatically as χq, iω n = χ 0 q, iω n + χ 0 q, iω n V c qχq; iω n 129 = If we use diagrammatic representation Ṽ c =, V c = 131 for the screened- and unscreened Coulomb potential and rewrite the equation for the screened potential as Ṽ c q, ω = V c q + V c qχ 0 q, ωṽcq, ω, which we can represent as =