The Kubo formula of the electric conductivity We consider an electron under the influence of an electric field E(t). This system may be described by a Hamiltonian of the form ∑ Hˆ (t) = Hˆ0 − e Ei(t)xi, (1) i

with Hˆ0 as reference Hamiltonian. For simplicity, we assume a spatially homogeneous field. We further stipulate that Ei = Ekδik and set exk = Dˆ. The time evolution of the system is given by the Liouville-von Neumann equation:

∂ρˆ(t) 1 = [Hˆ (t), ρˆ(t)]. (2) ∂t i~ We split the density operatorρ ˆ into a equilibrium part,ρ ˆ0, and a time- dependent part ∆ˆρ(t), according to (ˆρ =ρ ˆ0 + ∆ˆρ(t)). We then approximate ∂∆ˆρ 1 ≈ ([Hˆ , ∆ˆρ(t)] − E (t)[D,ˆ ρˆ ]. (3) ∂t i~ 0 k 0

where use has been made of the relation [ˆρ0, Hˆ0] = 0. This equation is solved by ∫ t 1 ′ ′ ′ ∆ˆρ(t) = − [Dˆ(t − t), ρˆ0]Ek(t )dt . (4) i~ −∞ ˆ ≡ ˆ † ˆ ˆ We operate here in the interaction picture, implying that D(t) U0 (t)DU0(t), ˆ i ˆ where the propagator U0(t) is defined as exp(− ~ H0t). The change in an ob- servable Oˆ as a function of time can then be found from ∫ t 1 ′ ′ ′ ∆O(t) ≡ T r{∆ˆρOˆ} = − [Dˆ(t − t), ρˆ0]OEˆ k(t )dt (5) i~ −∞ We extend the upper integration limit to positive infinity by introducing the temporally non-local function γ: 1 γ (t) = − Θ(t)T r{[Dˆ(−t), ρˆ ]Oˆ}, (6) OˆDˆ i~ 0 such that ∫ +∞ ′ ′ ′ ∆O(t) = γ(t − t )Ek(t )dt (7) −∞ We factorize γ to separate the theta distribution:

γOˆDˆ (t) = θ(t)αOˆDˆ (t) (8) and observe [1]

1 1 1 α (t) = T r{[ˆρ , Dˆ(−t)]Oˆ} = T r{ρˆ [Dˆ(−t), Oˆ]} = T r{ρˆ [Dˆ(0), Oˆ(t)]} OˆDˆ i~ 0 i~ 0 i~ 0 (9)

1 In the following step, we identify the operator Oˆ with the current density operator ˆjl. Setting jl(t = 0) = 0, we obtain from Eqs. (7) - (9): ∫ +∞ − ′ − ′ ′ ′ jl(t) = θ(t t )αOˆDˆ (t t )Ek(t )dt (10) −∞ and 1 α (t − t′) = T r{ρˆ [Dˆ(0), ˆj (t − t′)]}. (11) OˆDˆ i~ 0 l We find an expression for the conductivity from the Fourier transform of 10, ∫ ∞ where the Fourier transform of f(t) is defined as f˜(ω) = + f(t) exp(−iωt)dt, ∫ −∞ 1 +∞ ˜ and the back transform as f(t) = 2π −∞ f(ω) exp(iωt)dω. Thus

∫ +∞ ˜jl(ω) = jl(t) exp(−i(ω + iη)t)dt −∞ ∫ ∫ +∞ +∞ − ′ − ′ − − ′ ′ − ′ ′ = Θ(t t )αOˆDˆ (t t ) exp( i(ω + iη)(t t ))dt Ek(t ) exp( i(ω + iη)t )dt −∞ −∞ ∫ +∞ ′ ′ ′ = σ(ω)Ek(t ) exp(−i(ω + iη)t )dt . −∞ (12)

We include here an infinitesimal increment iη to ensure that the interaction vanishes exponentially as t, t′ → −∞. Summarizing:

˜jl(ω) =σ ˜lk(ω)E˜k(ω), (13) with ∫ +∞ ′ ′ ′ E˜k(ω) = Ek0(t ) exp(−i(ω + iη)t )dt , (14) −∞ and ∫ +∞ 1 { ˆ ˆ } − σ˜lk(ω) = ~ T r ρˆ0[D(0), jl(t)] exp( i(ω + iη)t)dt. (15) i 0 We bring the right-hand side of this equation into a more symmetric form, using the relation i α˜ (ω) = α˜ (ω), (16) AˆBˆ ω CˆBˆ where

dAˆ Cˆ(t) = (t). (17) dt

2 To justify this result, we operate with the complete bases |m⟩⟨m| and |n⟩⟨n|, and set

ρˆ0 = |m⟩pm⟨m|. (18) Thus, we have ∑ 1 − α (t) = p (A B e iωmnt − B A eiωmnt), (19) AˆBˆ i~ m mn nm nm mn m,n − ≡ (Em En) where ωmn ~ . For the Fourier transform of αAˆBˆ (t) we find

2π ∑ α˜ (ω) = p (A B δ(ω + ω ) − B A δ(ω − ω )). (20) AˆBˆ i~ m mn nm mn mn nm mn m,n

ˆ dAˆ 1 ˆ ˆ If C = dt = i~ [A, H0], then

2π ∑ α˜ (ω) = p (−ω A B δ(ω + ω ) − ω B A δ(ω − ω )), CˆBˆ i m nm mn nm mn mn mn nm mn m,n (21) from which we obtain Eq.(16). We identify Aˆ with the dipole moment operator and derive the following relation between Dˆ and ˆjk:

1 dDˆ ˆj (t) = (t). (22) k ad dt where ad is the d-dimensional real space volume per particle. As we use Eq. (17) to substitute for Dˆ(0) in Eq. (15), we obtain: ∫ d +∞ a { ˆ ˆ } − σ˜lk(ω) = ~ T r ρˆ0[jk(0), jl(t)] exp( i(ω + iη) t)dt. (23) ω 0

This expression for σlk(ω) is referred to as Kubo’s formula of conductivity. Using the identity e ˆj = vˆ , (24) l ad l with

i ˆ i ˆ vˆl = ~[H, xl] = ~[H0, xl] (25) we express the current density operators ˆj through velocity operators vˆ. Further, we consider the transverse conductance in the two-dimensional case,d = 2, as relevant for the quantum Hall effect in a graphene sheet. With k = x and l = y, this yields:

3 ∫ e2 +∞ σ˜ (ω) = T r{ρˆ [ˆv (0), vˆ (t)]} exp(−i(ω + iη)t)dt. (26) xy ~ 2 0 x y ωa 0 In order to carry out the trace, we adopt the basis {|m⟩}, where

Hˆ0|m⟩ = Em|m⟩. (27) We note that

ρˆ0|m⟩ = F (Em)|m⟩, (28) where F denotes the Fermi-Dirac distribution, and

† i ⟨n|vˆ (t)|m⟩ = ⟨n|Uˆ (t)ˆv Uˆ (t)|m⟩ = exp(− (E − E )t)⟨n|vˆ |m⟩. (29) y 0 y 0 ~ m n y With the help of the latter statements, and after inserting a complete basis |n⟩⟨n| = 1ˆ between the operatorsv ˆx andv ˆy, the integration in Eq. (26) can be performed directly. This results in [2]

e2 ∑ ⟨m|vˆ |n⟩⟨n|vˆ |m⟩ σ˜ (ω) = (F (E ) − F (E )) x y . (30) xy iωad m n ~ω + i~η + E − E m,n m n Expanding 1 up to first order with respect to ~ω, we obtain ~ω+i~η+Em−En i~e2 ∑ ⟨m|vˆ |n⟩⟨n|vˆ |m⟩ σ˜ (ω) = (F (E ) − F (E )) x y . (31) xy ad m n ~ω − i~η + E − E m,n m n Note that the zeroth-order term of this expansion does not contribute, as the sum over m, n vanishes for this contribution. Considering the DC conductivity (ω = 0) at zero temperature, and setting η = 0, we find

i~e2 ∑ ⟨m|vˆ |n⟩⟨n|vˆ |m⟩ σ˜ (ω) = (Θ(E − E ) − Θ(E − E )) x y . (32) xy ad F m F n (E − E )2 m,n m n

The factor following the summation sign is non-zero only if Em < EF < En or En < EF < Em. Summing over these two configurations results in

2 ∑ i~e ⟨m|vˆx|n⟩⟨n|vˆy|m⟩ − ⟨m|vˆy|n⟩⟨n|vˆx|m⟩ σ˜xy = d 2 . (33) a (Em − En) EmEF

In the following, we focus on electronic states in lattices, as described by Bloch functions (see (3.116)), assuming |m⟩, |n⟩ to be located in the first Bril- louin zone. From Eq. (25), we conclude

4 ⟨n|xiHˆ0|m⟩ − ⟨n|Hˆ0xi|m⟩ Em − En Em − En ∂ ⟨n|vˆi|m⟩ = = ⟨n|xi|m⟩ = ⟨n| |m⟩, i~ i~ ~ ∂ki (34) where i = x, y. Likewise:

En − Em ∂ ⟨m|vˆi|n⟩ = − ⟨ m|n⟩. (35) ~ ∂ki The latter relation involves a shift of the derivative ∂ from the ket to the ∂ki bra position of the scalar product. This is legitimate, as the position operator in k-space (=i ∂ ) is hermitian in the region considered here, namely a Brillouin ∂ki zone, and thus a space with periodic boundary conditions. By Eqs. (34) and (35), it holds that

2 ∑ ie ⟨ ∂ | ⟩⟨ | ∂ ⟩ − ⟨ ∂ | ⟩⟨ | ∂ ⟩ σ˜xy = d m n n m m n n m . (36) ~a ∂kx ∂ky ∂ky ∂kx EmEF

The sum over the unoccupied states can be expressed through the sum over the occupied states by employing the completeness relation: ∑ ∑ |n⟩⟨n| = 1 − |n⟩⟨n|. (37)

En>EF En

2 ∑ ∑ ie ⟨ ∂ | ∂ ⟩ − ⟨ ∂ | ∂ ⟩ σ˜xy = d m m m m . (38) ~a ∂kx ∂ky ∂ky ∂kx k Em

The contribution toσ ˜xy due to the second term on the right-hand side of Eq. (37) yields an expression that is odd with respect to the exchange of the indices m and n and thus vanishes upon summing over these indices. In what follows, we will consider the two-dimensional case which is of relevance for the planar graphene sheet. In order to take formula (38) into its ultimate form we make two further steps. First, we average the conductivity as given by Eq. (38) over the first Brillouin zone (BZ), involving integration with respect to kx, ky 2 (2π) 1 over the Brillouin zone while simultaneously dividing through its area, ad : ∫ 2 ∑ BZ ie dkxdky ⟨ ∂ | ∂ ⟩ − ⟨ ∂ | ∂ ⟩ σ˜xy = m m m m . (39) h BZ 2π ∂kx ∂ky ∂ky ∂kx Em

Secondly, we introduce the vector field Ak by defining

1Each k-space area cell contributes dkxdky to the electron density, see p.91 of the main (2π)2 text.

5 ∑ ∂ ∑ ∂ Ak,x = i ⟨m| m⟩,Ak,y = i ⟨m| m⟩. (40) ∂kx ∂ky Em

6 Bibliography

[1] T.O.Stadelmann, Antidot Superlattices in InAsGaSb Double Heterostruc- tures: Transport Studies, Ph.D. Thesis, Oxford 2006.

[2] G.Tkachov, Topological Insulators, Pan Stanford Publishing, 2016 [3] D.J.Thouless, M.Kohmoto, M.P.Nightingale, M.de Nijs, Phys.Rev.Lett.49, 405 (1982)

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