Problem Set 4, Due November 15th 2000

H_{0} = ∑_{kσ} ε_{k}c_{kσ}^{†}c_{kσ}

N = ∑_{kσ} c_{kσ}^{†}c_{kσ}

| ψ_{0}⟩ = ∏_{k<k_{F}} c_{k↑}^{†}c_{k↓}^{†} | 0⟩

ρ_{q} ≡ ∑_{kσ} c_{k+qσ}^{†}c_{kσ}

Δ_{q} ≡ ∑_{k} c_{−k+qσ↑}^{†}c_{k↓}^{†}

a) zero temperature. Calculate the Fourier transform of G_{ρ}(t, q), G_{p}(t, q) in time.

G_{ρ}(t, q) = ⟨T (ρ_{q}(t)ρ_{−q}(0))⟩
G_{p}(t, q) = ⟨T (Δ_{q}(t)Δ_{−q}(0))⟩

while a general expression involves integration for the case of d dimensions can you evaluate some of the integrals?

b) Finite temperature - if τ is the imaginary time it is convenient to use Fourier series

G_{ρ}(iν_{n}, g) = ∫_{o}^{β} e^{iν_{n}τ}G_{ρ}(τ, q)dτ
G_{p}(iν_{n}, q) = ∫_{o}^{β} e^{iν_{n}}τG_{p}(τ, q)dτ.

Evaluate them in 3 and 2 dimensions.

Now G_{g} and G_{p} are defined by

G_{ρ}(τ, q) = \frac{1}{Z} tre^{−β(H_{o}−μN)}ρ_{q}(τ)ρ_{−q}(o))

G_{ρ}(τ, q) = \frac{1}{Z} tre^{−β(H_{o}−μN)}Δ_{q}(τ)Δ_{−q}(o))

c) Calculate

T ∑_{n} \frac{e^{iω_{n}0^{+}}}{iω_{n} − ε_{k}} and T ∑_{n} \frac{e^{iω_{n}0^{−}}}{iω_{n} − ε_{k}} .

Interpret the result.

Hints to solve problem set 3

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

set.

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