The Kubo Formula for the Conductivity II . . . [10]

University of Regensburg Summer Term 2018

Quantum Theory of Condensed Matter I

Prof. John Schliemann

Dr. Paul Wenk, M.Sc. Martin Wackerl Mo. 08:00-10:00 c.t., PHY 5.0.21

Sheet 8

The Kubo Formula for the Conductivity II . . . [10]

We continue Ex.1, sheet 7. In the following we are going to rewrite the Kubo formula in a form which is often the starting point for treating the effect of disorder when calculating conductivity in wires.

(a)(5P) In the first step we assume to have the spectrum n of the Hamiltonian H = P

nna^†_nan with according eigenstates|ni. Using the previous result from sheet 7, show that we can further rewrite σ^βα(ω) as follows:

σ^βα(ω) = i~VX

nm

f(n)−f(m)

(m−n)(m−n+~ω+ i0⁺)hn|˜j^α|mi hm|˜j^β|ni. (1) In this expression ˜j^α=−ep^α/(mV) with momentump^αis an operator in the Schr¨odinger represen- tation (in first quantization) and f() is the Fermi-Dirac distribution function.

Hint: Remember, in second quantization the current operator is written as j^α(0) =P

nmhn|˜j^α|mia^†_nam. You’ll need the relation

Tr[ρa^†_mana^†_paq] =δmqδnpf(m)(1−f(n)). (2) Furthermore, expressθ(t−t⁰) = (i/2π)R∞

−∞dxexp(−ix(t−t⁰))/(x+ i0⁺).

(b)(5P) The real part of the conductivity gives the dissipative contribution. Compute the real part of the longitudinal conductivity Eq. (1) in the DC limit, i.e.,

Re[σ^xx(0)] = πe² V m²

Z ∞

−∞

dE

−∂f(E)

∂E

Tr[p^xδ(E−H)p^xδ(E−H)]. (3)

2. Two Coupled Spins . . . [10P]

Once again we use the Heisenberg model. However, this time we only consider two spin 1/2 particles, S1 = S2= 1/2, in an external magnetic fieldB:

H= −J(S₁⁺S₂⁻+S₁⁻S⁺₂ + 2S₁^zS₂^z)−g_Jµ_BB

~

(S₁^z+S^z₂). (4)

A complete system of equations of motion is given by using the following (retarded or advanced) Green’s functions:

G11(t, t⁰) =hhS₁⁻(t);S₁⁺(t⁰)ii=+ (5) G₂₁(t, t⁰) =hhS₂⁻(t);S₁⁺(t⁰)ii₌₊ (6) Γ₁₂(t, t⁰) =hhS₁^z(t)S₂⁻(t);S₁⁺(t⁰)ii=+ (7) Γ21(t, t⁰) =hhS₂^z(t)S₁⁻(t);S₁⁺(t⁰)ii=+ (8)

Show that the Green’s functions can be written as G11(E) =

3

X

i=1

ai

E−E_i, G21(E) =

3

X

i=1

bi

E−E_i (9)

and

Γ₁₂(E)−Γ₂₁(E) =

3

X

i=2

c_i E−Ei

. (10)

Determine the coefficientsai,biandci, which are functions ofhS^z₁i=hS₂^zi ≡ hS^ziandρ12≡ hS⁺₁S₂⁻i+ 2hS₁^zS₂^zi, and the polesEi.