Drude conductivity and Diffuson in a non-interacting system

In this subsection the disorder averaged single-particle Green’s function of a non-interacting system is calculated. We have already introduced the mean free path l which characterizes transport of particles through a disordered medium. l is the average distance travelled by an electron between two scattering events. The mean free path gives rise to a characteristic time τ =l/v, the collision time. v is the group velocity of the electron wave. Although the energy is conserved during a scattering event the momentum of the scattered particle changes and hence a plane wave |ki has only a finite lifetime τ. The collision time τ can be estimated from the lifetimeτ_k of a plane wave |kiby using Fermi’s golden rule:

1

τ = 2πX

k⁰

|hk|V|k⁰i|²δ(k−k⁰) (2.13) HereV denotes the Gaussian disorder field. In the beginning of this section we have introduced the Gaussian white-noise model with variance given by V(r)V(r⁰) = Ddisδ(r−r⁰) and zero mean value. Performing the disorder average in Fermi’s golden rule, equation (2.13) we have to insert the relation Ω|hk|V|k⁰i|² =D_dis. Where Ω is the volume of the system andD_dis is the disorder strength. Hence the average lifetime of a state with energy k is given by:

1

τ = 2πν(_k)

Ω D_dis (2.14)

3see [8]

Here we introduced the density of states at energy _k: ν(_k) :=P

k⁰δ(_k−_k⁰). In order to describe the evolution of a plane wave through disordered media we need to disorder average the single-particle Green’s function G(ω,k). The diagrammatic expansion of the Green’s function is given in figure (2.2): The corresponding formula is:

Figure 2.2: Diagrammatic expansion of the single particle Green’s functionG(ω,k) in a disordered medium with a specific random fieldV(r). The dotted line represents a scattering event, the solid line on the right hand side is the electron Green’s functionG₀ of a clean system.

G(ri,r_f, t) =G0(ri,r_f, t) + Z

G0(ri,r, t)V(r)G0(r,r_f, t)dr+. . .

=G0(ri,rf, t) + Z

G0(ri,r, t)V(r)G(r,rf, t)dr (2.15) This is known as the Dyson equation.

After performing the disorder average: V(r) = 0, V(r)V(r⁰) = D_disδ(r−r⁰) we get the following diagrammatic expansion, see Fig (2.3).

Figure 2.3: Diagrammatic expansion of the full disorder averaged single particle Green’s function G(ω,k) in a disordered medium.

Disorder averaging is represented by a connection of two dotted lines.

We reformulate the diagrammatic expansion (2.15) in energy-momentum space:

G(ω,k) =G₀(ω,k) +G₀(ω,k)Σ(ω,k)G(ω,k) (2.16) Where Σ(ω,k) denotes the self energy, that means the sum of all irreducible diagrams without external links:

Σ(ω,k) =

(2.17) Now we solve the Dyson equation (2.16) and write the full disordered Green’s function in terms of the self energy.

G(iω_n,k) = G₀(iω_n,k)

1−G0(iωn,k)Σ(iωn,k) = 1

iωn−ξ(k)−Σ(iωn,k) (2.18) Where we usedG⁻¹₀ (iωn,k) =iωn−ξ(k). iωn is a Matsubara frequency. Note that we arrive at the corresponding retarded expression by puttingiω_n→ω+iδ, whereδ is an infinitesimal.

2.1 Review of known results in disordered systems 37

It can be shown⁴ that in dimensions higher than one all crossing diagrams are smaller by a factor of 1/k_Flthan the non-crossing diagrams. Thus in two and three dimensions the third diagram on the right hand side of equation (2.17) is much smaller compared to the second one. For the remaining part of this section we exclude the discussion about disorder effects in 1D and postpone it to the section about disorder in Luttinger liquid.

We are mainly interested in the imaginary part of the self energy since the real part ReΣ yields only a renormalization of the energy(k). Furthermore we only account for the non-crossing diagrams and neglect contributions from the crossed ones. This is known as the Born approximation [10](p.220). As a further approximation we calculate the self energy by taking into account only the first diagram of (2.17).

Σ(iωn,k)≈ni

X

k⁰

|V_k−k⁰|² 1

iω_n−ξ_k⁰ ≈v₀²niν(_k) Z

dz 1

iω_n−z (2.19) Here we used that the scattering potential V_k−k⁰ is smooth on the Fermi momentum shell and thus: v0≈Vk−k⁰.

Performing the integral in Eq. (2.19) and using Eq. (2.14) we obtain the self energy:

Σ(iω_n,k) =−i·sign(ω_n) 1

2τ (2.20)

Finally, we get the following expression for the disorder averaged single particle Green’s function:

G(iω_n,k) = 1

iωn−ξ(k) +sign(ωn)_2τⁱ (2.21) We get the retarded Green’s function from the Matsubara Green’s function by performing the analytic continuation: iω_n→ω+iδ.

To conclude, the discussion of the single particle Green’s function, we give the retarded Green’s function in energy-momentum and space-energy representation [6].

G^R(,k) = 1

−ξ(k) + _2τⁱ (2.22)

G^R(r,r⁰, ) = G^R₀(r,r⁰, )e^{−|r−r0|2l} (2.23)

Equation (2.23) shows that correlations are decaying on a length scale of the order of the mean free path. In the last chapter we already introduced the Kubo formula for calculating the current density, Eq. (1.138). In contrast to equation (1.144), the current operator, written in terms of electron fields ˆψ, ˆψ^† is given by:

ˆj(r, t) = e

2mi(∇_r− ∇_r⁰) ˆψ^†(r⁰, t) ˆψ(r, t)

r⁰=r (2.24)

Thus, the current-current correlation function in Eq. (1.138) yields a four Fermion correla-tion funccorrela-tion. Using Wick’s theorem the four Fermion correlacorrela-tion funccorrela-tion can be expressed as a product of two Green’s functions. Diagrammatically, the product of two Green’s func-tions corresponds to a bubble, see Fig. (2.4). The expansion of the conductivity in terms of diagrams is given in figure (2.4). Furthermore this product of Green’s functions has to

4 [10] p. 223

be disorder averaged. In general the disorder average over a product of Green’s functions G₀(p)G₀(p⁰) is different from the product of two averaged Green’s functionsG₀(p)·G₀(p⁰).

We remind that the productG^R(r,r⁰, )G^A(r⁰,r, −ω) represents the probability of quantum diffusion P(r,r⁰, ω) which is the probability for a wave packet to travel from rtor⁰ [6]. The so called Drude-Boltzmann approximation is to approximate the average of the product of two Green’s functions by the product of two averaged Green’s functions:

G^R(r,r⁰, )G^A(r⁰,r, −ω)≈G^R(r,r⁰, )·G^A(r⁰,r, −ω) (2.25) Since the right hand side of (2.25) corresponds to an empty bubble of two disorder averaged Green’s functions, this approximation neglects diagrams where upper and lower Green’s func-tions of the bubble are connected by an impurity line, such as the second bubble in Fig (2.4).

There are also higher order diagrams in impurity lines possible. These diagrams are called particle-hole ladder diagrams, see Fig. (2.5,c).

Although Eq. (2.25) seems to be a crude approximation, it can be shown, [6] p.277, that for isotropic impurity scattering the contribution of the particle-hole ladder to the current van-ishes. The physical picture behind this cancellation is that a scattered electron has completely lost its memory about the direction of the current as long as the impurity has an isotropic scattering potential. Hence higher correlations of scattering events do not contribute.

Figure 2.4: Classical contributions to the conductivity. The solid line is the disorder averaged Green’s functionG(p), the dashed line is a scattering event. The empty bubble on the right hand side represents the Drude-Boltzmann approximation of incoherent collisions. The second bubble is the first order contribution in the disorder strengthDdis. It contributes to the so called particle-hole ladder.

The particle-hole ladder diagrams, Fig (2.5), are called Diffuson contribution and they do matter in a calculation of the density-density correlation function. In figure (2.5 b) the meaning of the Diffuson contribution is shown in terms of propagating particles and holes. The ladder approximation corresponds to the weak disorder limitkl1 [6](p.102). Moreover it is an important contribution to the probabilityP(r,r⁰, ω) which we mentioned above. Together with the empty bubble the Diffuson yields the classical contributions. The first two Green’s functions |G^R(r,r₁)|² = G^R(r,r₁)G^A(r₁,r) are the probability for a particle to propagate from rtor₁ without any scattering event, [6] p.96. The evolution of the wave packet fromr₁ tor2 is described by Γ(r1,r2) in the diagrammatic language. The vertex function Γ takes into account all possible ways to get from r₁ tor₂ with an arbitrary number of scattering events.

Furthermore, the probability to go from point r₂ to r⁰ is computed by Green’s functions in the same way as from r to r1. Finally we integrate over all possible points r1 and r2. The computation of the vertex function is depicted in (2.5 c). Γ can be evaluated self consistently by the so called Bethe-Salpeter equation which is a Dyson equation for vertex functions. It can be shown that after a large number of collisions, i.e. tτ, the vertex function fulfills a classical diffusion equation [6](4.5).

2.1 Review of known results in disordered systems 39

Figure 2.5: Diffuson contribution. a) the diagrammatic representation of the Diffuson contribution to the pair bubble is shown in terms of disorder averaged Green’s functionsGand the vertex function Γ. The corresponding intuitive picture of propagating particles and holes is depicted in b). Solid lines represent G^R and dashed lines describe G^A. The structure of the vertex function Γ is shown in c).

The diagrams that contribute to Γ are called particle-hole ladder diagrams.