University of Groningen Conserving Approximations in Nonequilibrium

University of Groningen

Conserving approximations in nonequilibrium green function theory Stan, Adrian

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Download date: 23-09-2021 Chapter 2 Theory of many-particle systems. The Green function method.

Abstract

In this chapter we introduce the fundamental concepts behind the Green Function Theory. We start from a description of the second quantization method and then, after introducing the ensemble average of an operator, we deﬁne the main object of the theory, the Green function. Further on, after obtaining the equation of motion for the Green function, we introduce the self-energy operator and we discuss the Hartree-Fock approximation, the second-Born approximation and the GW approximation to the self-energy.

11 12 THE GREEN FUNCTION METHOD

2.1 Second quantization one can easily write the general Hamiltonian in the second quantization form Let us consider a system of N identical, non-relativistic particles. The physical state of each particle j is de- Hˆ = dxψˆ†(x)h(r,t)ψˆ(x) (2.5) scribed by the label x . The wave function for this j Z system is Ψ(x1 ...xN ). If the particles are identical, 1 † † 2 + dx1dx2ψˆ (x1)ψˆ (x2)w(r1, r2)ψˆ(x2)ψˆ(x1), then the probability density Ψ(x1 ...xN ) must be un- 2 | | changed under arbitrary exchanges of the labels we use ZZ 1 where w(r1, r2)= and where the one-body part to identify each particle. If the system is described by a |r1−r2| scalar1 Ψ must transform according to a 1-D represen- of the Hamiltonian is tation of the permutation group. Hence, for the system 1 2 under consideration, we have two cases: a) Ψ is even h(r,t)= + v(r,t) µ. (2.6) − 2 ∇ − under permutation (P Ψ = Ψ); and b) Ψ is odd under permutation (P Ψ= Ψ). The latter case represents a In Eq. 2.6 we introduced the chemical potential µ and − system of bosons and the former a system of fermions. the external potential v(r,t) which will be switched on Since in this work we are only concerned with fermionic at t = t0. Hence, for tquantum mechanics, it proves to be quite to study the time-evolution of a many-particle fermion cumbersome. Moreover, the usual representation is system, after it has been perturbed from the ground suited for problems with a fixed particle number N and state, or statistical equilibrium. We have seen that does not allow for fluctuations. This can constitute a the formalism of the second quantization is appropri- too tight constrain and so it is convenient to work in ate since it allows for extending quantum mechanics a grand canonical formulation [2], where the particle to macroscopic number of particles and to describe the number is allowed to fluctuate. In line with this last dynamical response and internal correlations of large constrain, for practical purposes, one would should be systems. In the next section we will briefly go through able to determine the amplitude for adding a particle the steps to determine the time-dependent expectation to the system at a certain space-time coordinate, say, value for an operator in the grand canonical ensemble. (x1,t1), followed by its removal at (x2,t2). The method which removes all these shortcomings is the second quantization [3, 4]. Within this method the 2.2 Evolution of ensembles operator ψˆ†(x) is called creation operator and the operator ψˆ(x) is called annihilation operator. By applying In the grand canonical ensemble [5, 2], the expectation ˆ a creation operator, a particle is added to the system value of an operator O, for a system at temperature T and by applying an annihilation operator, the particle is with the chemical potential µ, is removed [3]. These operators satisfy the commutation −βEi Ψi Oˆ Ψi e relations Oˆ = ih | | i , (2.7) −βEi h i i e ψˆ†(x)ψˆ(x′)+ ψˆ(x′)ψˆ†(x)= δ(x x′), (2.1) P − ˆ ˆ ′ ˆ ′ ˆ where Ψi is a completeP set of states in the Fock ψ(x)ψ(x )+ ψ(x )ψ(x) = 0. (2.2) {| i} space, β = 1/kB T , with kB the Boltzmann constant Using the definitions of the field operators and their and Ei are the eigenvalues of the Hamiltonian Hˆ . Note commutation relations 2.1 and 2.2 we can express the that the chemical potential is included in the one-body usual operators in terms of field operators. If we con- part. If the system evolves in time, the expectation sider a general one-body operator value becomes [2] ˆ ˆ† ˆ O = dxψ (x)o(x)ψ(x), (2.3) Oˆ(t) = Tr ρÔˆ(t) , (2.8) Z h i { } and the two-particle interaction where the statistical operatorρ ˆ acts as a weight function for the Heisenberg representation of the operator ˆ 1 ˆ† ˆ† ˆ ˆ W = dx1dx2ψ (x1)ψ (x2)w(r1, r2)ψ(x2)ψ(x1), ˆ 2 O, i.e. ZZ (2.4) OˆH (t)= Uˆ (t0,t)OÛˆ(t,t0), (2.9)

1For higher order objects, such as tensors, higher order and where Uˆ is an evolution operator. representations of the permutation group are possible. Because the time-dependent quantities are switched on 2.4. SELF-ENERGY APPROXIMATIONS 13

hilation operator

† Tr Uˆ(t0 iβ,t0)TC [ψˆH (1)ψˆ (2)] G(1, 2) = i { − H } − Tr Uˆ(t0 iβ,t0) { − } † = i TC [ψˆH (1)ψˆ (2)] , (2.13) − h H i

where TC denotes the time-ordering operator on the contour and where we used the compact notation 1 = (x1,t1) and 2=(x2,t2). The average is taken over the grand canonical ensemble [2]. The Green function pro- vides us with the all ground state expectation values of one-particle operators, as well as the total energy. If we consider the Green function at time t1 = t0 iβ − and use the cyclic property of the trace, we ﬁnd that Green function deﬁned in Eq. (2.13) obeys the bound- Figure 2.1: The Keldysh contour. ary conditions

G(x1t0, 2) = G(x1t0 iβ, 2), (2.14) at t = t0, our Hamiltonian is time-independent, for − − G(1, x2t0) = G(1, x2t0 iβ). (2.15) times t

−βHˆ 0 These boundary conditions are sometimes referred as e the Kubo-Martin-Schwinger boundary conditions [9, ρˆ = ˆ , (2.10) Tr e−βH0 { } 10, 11]. If we explicitly write the time-ordering operator for the Green function, we obtain where by Hˆ 0 we denote the Hamiltonian for times t < ′ > ′ < t0 and where we have included the chemical potential G(1, 2) = θ(t,t )G (1, 2) + θ(t ,t)G (1, 2).(2.16) in the one-body part of the Hamiltonian. Now, the ˆ operator e−βH0 can be seen as an evolution operator in with θ(t1,t2) being contour step functions generalized imaginary time [6] and the expectation value becomes to time arguments on the contour i.e. θ(t1,t2) = 1 if t1 >t2 and θ(t1,t2) = 0 otherwise [12]. > < Tr Uˆ(t0 iβ,t0)Uˆ(t0,t)OÛˆ (t,t0) The greater and lesser components, G and G respec- Oˆ(t) = { − } . (2.11) h i Tr Uˆ(t0 iβ,t0) tively, in Eq. 2.16, have the explicit form { − } > ˆ ˆ† This expression can be interpreted as follows: on the G (1, 2) = i ψH (1)ψ (2) , (2.17) − h H i real axis, the system evolves from an initial time t0 to < ˆ† ˆ G (1, 2) = i ψH (2)ψH (1) . (2.18) a time t when the operator Oˆ is applied and the system h i evolves back from t to t0 and along the imaginary track, The greater component can be interpreted as a from t0 to t0 iβ. This time-contour can be imagined propagation of a particle in the system and the lesser − as lying in the complex time plane. Equation 2.11 can component, as the propagation of a hole [13]. By also be written in terms of contour-ordered products as taking the Fourier transform of the G< component, we obtain a function that is peaked at the removal −i R dt¯Hˆ (t¯) Tr TC e C Oˆ(t) Oˆ(t) = { } , (2.12) energies and by taking the Fourier transform of the −i R dt¯Hˆ (t¯) > h i Tr TC e C G component we get a function peaked at addition { } energies (affinities) [14]. The corresponding contour for Eq. 2.11 and 2.12 is called the Keldysh contour (see Fig. 2.1). It was orig- inally introduced in the works of Schwinger [8] and Keldysh [7]. 2.4 Self-energy approximations 2.3 The Green function By taking the commutators [ψˆ(x), Oˆ] and [ψˆ(x), Wˆ ], 2 The one-particle Green function is defined as a we can immediately obtain the commutators of the countour-ordered product of a creation and an anni- Green functions differ in the nature of the averaging process 2Different types of Green functions have been applied in performed in order to obtain them, in the time arguments quantum field theory and in many body theory. The types of on which they depend and in their analytic properties. 14 THE GREEN FUNCTION METHOD

field operators with the Hamiltonian, [ψˆ(x), Hˆ ] and such that iG2v = ΣG [14]. In other words, we define † − [ψˆ (x), Hˆ ]. Further we have the equation of motion the self-energy Σ and its adjoint Σˆ by for the field operators ′ + + ′ d2Σ(1, 2)G(2, 1 )= i d2w(1 , 2)G2(1, 2, 2 , 1 ), i∂ ψˆ(1) = [ψˆ(1), Hˆ (1)] = h(1)ψˆ(1) − t1 Z Z † ′ ′+ + ′ ˆ ˆ ˆ d2G(1, 2)Σ(2ˆ , 1 )= i d2w(1 , 2)G2(1, 2, 2 , 1 ). + d2w(1, 2)ψ (2)ψ(2)ψ(1), (2.19) − Z Z Z † † † i∂t ψˆ (1) = [ψˆ (1), Hˆ (1)] = h(1)ψˆ (1) 1 − One can show that for initial equilibrium conditions the † self-energy operator Σ and its adjoint Σˆ are the same. d2w(1, 2)ψˆ (1)ψˆ(2)ψˆ(2), (2.20) − From here on we will assume that this is the case. Z With the considerations above, the equations of motion with w(1, 2) = δ(t1,t2)/ r1 r2 being the Coulomb | − | of the Green function will be written as [12] interaction. If we take the derivative of the Green function 2.16, in respect to t1 and t2, and we use 2.19 i∂t1 G(1, 2) = δ(1, 2) + h(1)G(1, 2) and 2.20 respectively, we obtain the equation of motion for the Green function + d2Σ(1, 3)G(3, 2), (2.26) Z i∂t2 G(1, 2) = δ(1, 2) + h(2)G(1, 2) i∂t1 G(1, 2) = δ(1, 2) + h(1)G(1, 2) (2.21) −

+ † † + d2G(1, 3)Σ(3, 2). (2.27) i d3w(1 , 3) TC [ψˆ(1)ψˆ (2)ψˆ (3)ψˆ(3)] , − h i Z Z If we take the functional derivative of the Green function in respect to the external field, we find that the i∂t G(1, 2) = δ(1, 2) + h(2)G(1, 2) (2.22) − 2 time ordered product of four field operators equals the + † † functional derivative of the one-particle Green function i d3w(2 , 3) TC [ψˆ (2)ψˆ(1)ψˆ(3)ψˆ (3)] . − h i Z plus the product of the expectation value of the density operator and the one-particle Green function The product of field operators in the last term of 2.21 and 2.22 is the two particle Green function which de- † † δG(1, 2) i TC [ψˆ(1)ψˆ (2)ψˆ (3)ψˆ(3)] = i (2.28) scribes the propagation of two particles, two holes or a − h i δv(3) particle and a hole through the system. We rewrite the + nH (3) G(1, 2). Eqs. 2.21-2.22 as3 h i This follows directly from the equation of motion for the i∂ G(1, 2) = δ(1, 2) (2.23) t1 evolution operator. Making use of the definition of the + + inverse Green function G(1, 2)G−1(2, 1) = δ(1, 2), we +h(1)G(1, 2) i d3w(1 , 3)G2(1, 3, 3 , 2), − obtain an expression for the functional derivative of the Z R one-particle Green function in respect to the external potential i∂t2 G(1, 2) = δ(1, 2) (2.24) − −1 + + δG(1, 2) δG (4, 5) +h(2)G(1, 2) i d3w(2 , 3)G2(1, 3, 3 , 2). = d4d5G(1, 4) G(5, 2)(2.29). − δv(3) − δv(3) Z Z where G2 is the two particle Green function was defined If we define from the n-particle Green function δG−1(4, 5) ′ ′ = Γ(45; 3) (2.30) Gn (1,...,n, 1 ,...,n ) = (2.25) − δv(3) n † ′ † ′ ( i) TC [ψˆ(1),..., ψˆ(n), ψˆ (1 ),..., ψˆ (n )] , − h i to be the vertex function. Further, making use Eq. 2.29 and of It can be proved by induction that the n-body Green function depends on the n+1-body Green function [10, −1 G (1, 2)=(i∂t h(1))δ(1, 4) Σ(1, 4), (2.31) 3]. At the first level, the truncation of this hierarchy 1 − − is performed by introducing the self-energy operator Σ we write the vertex function Γ in the form

3The notation + means that the limit is taken from above δΣ(1, 4) + x Γ(14; 3) = δ(1, 2)δ(1, 4) + . (2.32) on the contour, i.e., 1 = 1, t1 + δ. δv(3) 2.4. SELF-ENERGY APPROXIMATIONS 15

By inserting 2.29, with 2.32, into 2.28, we can rewrite These terms are usually referred to as the second-order the equation of motion for the one-particle Green func- direct and exchange terms. They can be represented tion as diagramatically as the two diagrams to second order in the two-particle interaction [15, 16]. ′ ′ [i∂t h(1)]G(1, 1 )= δ(1, 1 ) 1 − Further iterations of Eq. 2.35 lead to higher order approximations. + d4 i d2d3G(1, 3)w(1+, 2)Γ(34; 2) Z h Z +δ(1, 4)δ(3, 2)w(4, 3) nˆ(3) G(4, 1′). (2.33) 2.4.2 The GW approximation h i i In the GW approximation the self-energy is expanded Using the expression for the density nˆ(3) = + h i in terms of the Green function and the dynamically iG(3, 3 ) together with Eqs. 2.26 and 2.23, we can − screened interaction W [17]. The screened interaction identify the self-energy as W describes the dynamical shielding in the Coulomb potential. Including the effective interaction of the elec- Σ(1, 4) = i d2d3 G(1, 3)w(1, 2)Γ(34; 2) trons, the screened interaction is defined as Z h δ(1, 2)δ(3, 2)w(1, 3)G(3, 3+) . (2.34) δV (2) W (1, 2) = d3w(1, 3) , (2.40) − δv(3) i Z We can now use the expression for the vertex Γ in Eq. 2.32 together with Eq. 2.34 to generate higher order where approximations. If we insert 2.32 in 2.34 we have V (2) = v(2) + d4w(2, 4) nˆ(4) , (2.41) h i Σ(1, 4) = iG(1, 2)w(1+, 2) Z and describes the effects of a test charge at point 2 on iδ(1, 2) d3w(1, 3)G(3, 3+) − the potential at point 1. By substituting 2.41 in 2.40 Z δΣ(3, 2) and taking the functional derivative, we can use the + i d4d3G(1, 3)w(1+, 4) .(2.35) identity δv(4) δ δV (2) δ Z = d2 , (2.42) v(1) δv(1) δV (2) If we consider in the equation 2.35 only the first order Z terms in w, we obtain the Hartree-Fock self-energy to obtain

Σ(1, 2) = iG(1, 2)w(1+, 2) iδ(1, 2) d3w(1, 3)G(3, 3+). W (1, 2) = d3w(1, 3) (2.43) − Z Z (2.36) δ nˆ(4) δV (5) This approximation is the lowest order approximation δ(2, 3) + d4d5w(2, 4) h i , × δV (5) δv(3) for the many-body effects. " Z # In Eq.2.43 we identify the irreducible polarizability 2.4.1 The second Born approxima- δ nˆ(4) P (4, 5) = h i , (2.44) tion δV (5) Inserting 2.36 back into 2.35 leads to the second Born as the density response to the effective field. We take approximation to the self-energy. By keeping the terms the functional derivative of the Green function with re- to second-order in w we have spect to V (2) and we obtain Σ(1, 2) = ΣHF (1, 2) + Σ(2)(1, 2), (2.37) δG(1, 2) δG−1(4, 5) = d4d5G(1, 4) G(1, 5), (2.45) HF δV (3) − δV (3) where Σ is the HF part of the self energy 2.36 and Z Σ(2) = Σ(2a) + Σ(2b) is the sum of the two terms where, using the definition of the inverse Green function, we define the vertex Γ as Σ(2a) (1, 2) = i2G(1, 2) d3 d4 w(1, 3) − Z δG−1(1, 2) δΣ(1, 2) G(3, 4)G(4, 3)w(4, 2), (2.38) Γ(12; 3) = = δ(1, 2)δ(1, 3) + . × − δV (3) δV (3) Σ(2b) (1, 2) = i2 d3 d4 G(1, 3)w(1, 4)G(3, 4) (2.46) Note that this vertex is different form the vertex in Z G(4, 2)w(3, 2). (2.39) Eq. 2.32. From the definition of the density nˆ(4) = × h i 16 THE GREEN FUNCTION METHOD

iG(4, 4+) we can rewrite the irreducible polarizability 2.5 Conserving − as approximations δG−1(5, 6) P (4, 7) = i d5d6G(4, 5) G(6, 4), (2.47) V (7) Z One of the most important issues when making an ap- and we can use the vertex function 2.46 to write it in proximation is the satisfaction of the conservation laws. the form In Green Function Theory it is possible to guarantee that the macroscopic conservation laws, such as those P (4, 7) = i d5d6G(4, 5)Γ(56, 7)G(6, 4). (2.48) of particle, momentum and energy conservation, are Z obeyed. From Eq. 2.21 and 2.28 we identify the self-energy as Baym showed [19] how the macroscopic conservation

+ laws are related to the invariant properties of a func- Σ(1, 2) = iδ(1, 2) d3w(1, 3)G(3, 3 ) (2.49) 4 − tional Φ . If we know the Green function, we can Z calculate the density and the current density from δG−1(4, 2) i d3d4w(1, 3)G(1, 4) − δv(3) nˆ(1) = iG(1, 1+) (2.55) Z h i where the first term on the right hand side is the ′ ˆ 1 1 + Hartree term and the second term represents the j(1) = i ∇ ∇ G(1, 1 ) .(2.56) h i − 2i − 2i ′ + exchange-correlation part. We can express the self- » – ff1 =1 energy in terms of screened interaction and vertex func- These two quantities must satisfy the continuity equation by using the identity 2.42. We obtain tion ˆ ∂t1 nˆ(1) + j(1) = 0, (2.57) Σ(1, 2) = iδ(1, 2) d3w(1, 3)G(3, 3+) h i ∇·h i Z which relates the accumulation of charge in a spacial region to the current flow in that region. This can be +i d3d4W (1, 3)G(1, 4)Γ(42; 3).(2.50) readily proved by subtracting 2.26 and 2.27 (with 2 Z + → Finally, we apply the chain rule in the last term of 1 )

Eq. 2.46 and we obtain a self-consistent expression for + + [i∂1 i∂ + h(1) + h(1 )]G(1, 1 ) = (2.58) the vertex function − 1 − δG−1(1, 2) d1[Σ(1¯ , 1)¯ G(1¯, 1+) G(1, 1)Σ(¯ 1¯, 1+)]. Γ(12; 3) = = δ(1, 2)δ(1, 3) (2.51) − − δV (3) Z δΣ(1, 2) If the system is perturbed by a potential corresponding + d4d5d6d7 G(4, 6)Γ(67; 3)G(7, 5), δG(4, 5) to a gauge transformation, the one body part of the Z Hamiltonian becomes which together with the self-energy 2.50, the irreducible polarizability 2.48 and the screened interaction 1 2 ∂Λ(1) hg (1) = [ i Λ] + i + v(1) µ. (2.59) 2 ∇− ∇ ∂t − W (1, 2) = w(1, 2) + i d3d4d5d7w(2, 4)W (1, 7)P (4, 7) Z With this form of the single-particle Hamiltonian, and (2.52) the conservation of particles at vertices [16], we can constitute the Hedin equations [17]. We have achieved 4Luttinger and Ward [20] have constructed a functional a systematic expansion of the self-energy in G and W . Φ of G by summing over all irreducible self-energy diagrams This is preferable for extended systems with long range closed with an additional Green function and multiplied by interactions, where W is usually much smaller than speciﬁc numerical factors. This functional can be written as the bare interaction w [18]. These equations are to be 1 (n) solved iteratively to obtain the self-energy Σ. However, Φ[G]= X Tr[Σ G] (2.53) 2n k such a calculation is, in principle, not possible even for n,k very simple systems, mainly due to the presence of the where n indicates the number of interaction lines and the vertex. We can take the simplest approximation to index k labels topologically diﬀerent self-energy diagrams. the vertex function 2.51, i.e., Γ(12; 3) = δ(1, 2)δ(1, 3) The trace implies a summation over all indices and a fre- and obtain the GW approximation [17]. With this ap- quency integration. In other words they proved that for the proximation to the vertex Γ, the self-energy depends on exact self-energy there is a functional Φ of G such that the self-consistent Green function and the screened po- δΦ Σ(1, 2) = . (2.54) tential W , calculated from the full Green function and δG(2, 1) therefore the equations must be solved self-consistently. 2.5. REFERENCES 17

show that for the interacting Green function and the where Hˆ0 represents the system without the added self-energy we have the transformations ﬁeld Hˆ0 = Hˆ Vˆ . In Ref. [19], Baym has shown h i h − i that the condition for a self-energy approximation −iΛ(1) iΛ(2) G(1, 2; Λ) = e G(1, 2)e , (2.60) to be conserving is that is has to be Φ-derivable. Σ(1, 2; Λ) = e−iΛ(1)Σ(1, 2)eiΛ(2). (2.61) This means that for the approximate self-energy there is a functional Φ such that the equation (2.54) This solutions obey the boundary condition if we as- is obeyed. Such approximations to the self-energy sume Λ(t0)=Λ(t0 iβ). A ﬁrst order change in G due are called conserving or Φ-derivable approximations. − to the gauge transformation is Well-known conserving approximations are the Hartree approximation – where Φ = 0 –, the Hartree-Fock ap- δG(1, 2) = i(δΛ(1) δΛ(2))G(1, 2). (2.62) proximation [16], the second Born [15] approximation, − − the GW approximation [17] and the T -matrix [15] The exponential factors cancel at the internal vertices of approximation. Σ because factors from the particles entering a vertex cancel the factors from outgoing particles. The only factors remaining are those at external vertices. The corresponding change in the functional Φ is given by References

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