Problem Set 5, If you have handed in 4 homeworks, you need not bother with this

Problem set 5: a new look way of looking at problem set 4

Now that you know about Wick’s theorem, you have another way way of redoing the first part of the previous homework by a direct application of that idea. Rederive your expressions using the following hints:

Define

D_{ρ}(t, q) = −i⟨T (ρ_{q}(t)ρ_{q}^{†}(0))⟩
D_{Δ}(t, q) = −i⟨T (Δ_{q}(t)Δ_{q}^{†}(0))⟩

ρ_{q}(t) = ∑_{kσ} c_{k+q}^{†σ}c_{kσ}; Δ^{†}_{q}(t) = ∑_{k} c_{k+q↓}c_{k↑}; ⟨...⟩ is with respect to a Fermi

sea.

a) Use Wick’s theorem to express D_{ρ}(t, q) and D_{Δ}(t, q) in terms of −i⟨T (c_{k}(t)c_{k}^{†}(0))⟩ = G(k, t)

b) Compute

G(k, ω) = ∫_{−∞}^{∞} e^{iωt}G(k, t)dt

- show it is given by

G(k, ω) = \frac{1}{iω − ε_{k} + iδsign(| k | −k_{F})}, δ > 0, δ → 0.

c) Combine (a) + (b) and use the fact that the Fourier transform of the product is the solution of the Fourier transform to show that

D_{ρ}(ω, q) = ∫_{−∞}^{∞} dte^{iωt}D_{ρ}(t, q)
D_{Δ}(ω, q) = ∫_{−∞}^{∞} dte^{iωt}D_{Δ}(t, q)

are given by expressions like

D_{ρ}(ω, q) ∼ −i ∫ dω′ ∑_{k} G(ω′ + ω, k + q)G(ω′, k)

D_{Δ}(ω, q) ∼ −i ∫ dω′ ∑_{k} G(−ω′ + ω, k + q)G(ω′, k)

d) Now carry out the frequency integral using the residue theorem. Get expressions of the functions

D_{ρ}(ω, q) ∼ ∑_{k} \frac{[f(ε_{k+q}) − f(ε_{k})]}{ε_{k+q} − ε_{k} − ω − iδ} , ω > 0

D_{Δ}(ω, q) ∼ ∑_{k} \frac{[1 − (f(ε_{k+q}) + f(ε_{k}))]}{ε_{k+q} + ε_{k} − ω − iδ}, ω > 0

f(ε_{k}) = \{ \begin{matrix}1 & if k < k_{F} \\ 0 & if k > k_{F} \end{matrix}

e) Now everything is reduced to a d dimensional integral over k. One way to do them is say in the case of D_{ρ}

D_{ρ} ∼ ∑_{k} f(ε_{k}) [\frac{1}{ε_{k} − ε_{k+q} − ω − iδ} − \frac{1}{ε_{k+q} − ε_{k} − ω − iσ} ]

Introduce polar coordinates and integrate over dθ and over dk, 0 ≤ k ≤ k_{F}. The 3d integral is done in Fetter and Walecka page 158-162.

If you want, you can calculate the D_{ρ} and ImD_{ρ} separately. In 1d the integrals are even easier; 2d is more fun.

Repeat the same for D_{Δ}.

Good Luck