Problem Set 3

H_{o} = ∑_{k} ε_{k}c_{kσ}^{†}c_{kσ}

N = ∑_{k} c_{kσ}^{†}c_{kσ}

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

ρ_{q} ≡ ∑_{kσ} c_{tgσ}^{†}c_{kσ}

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

a) zero temperature. Calculate the Fourrier 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}(o))⟩ in 3 dimensions and in 2 dimensions.

b) Finite temperature - if τ is the imaginary time it is convenient to use Fourrier series. G_{ρ}(iν_{n}, g) = ∫_{o}^{β} e^{iν_{n}τ}G_{ρ}(τ,_{t}^{o})dτ G_{p}(iν_{n}, q) = ∫_{o}^{β} e^{iν_{n}}τG_{p}(τ,_{t}^{o})dτ

\underline{Evaluate them} in 3 and 2 dimensions. Now G_{g} and G_{p} are defined by G_{ρ}(τ, q) = tre^{−β(H_{o}−μN)} ρ_{t}^{o}(ρ_{q}(o)) G_{p}(τ, q) = t_{r} e^{−β(H_{o}−μN)} Δ_{q}(ρ)Δ_{tq}(o)

c) What happens in 1 dimension?

Hints to solve problem set 3 Define ρ_{g}(t, t^{o}) = −i ⟨T (ρq(t)ρtq^{t}(o))⟩ ρ_{Δ}(t, q) = −i ⟨(Δ_{q}(t) Δ_{t}^{t}(o)⟩) ρ_{q}^{(t)} = ∑_{k} c_{−kq↑}c_{k↓}

a) Use Wick’s theorem to express ρ_{g}(t, q) and ρ_{Δ}(t, q) in terms of −i⟨T (c_{k}(t)c_{k}^{t}(o))⟩ = G(kt)

b) Compute G(k,w) = ∫_{∞}^{∞} e^{iwt}G(k, t)dt - show it as given by G(k, w) = \frac{1}{iw−ε_{k}tiρsign(|k|−k_{F})} ρ⟩0

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

ρ_{g}(w, q) = ∫_{−∞}^{∞} dte^{iwt}ρ_{g}(t, q) ρ_{Δ}(w, q) = ∫_{−∞}^{∞} dte^{iwt}ρΔ(t, q)

are given Bg expressions line ρ_{g}(w, q) − i ∫ dw′ ∑_{k} G(w′tw, ke)G(w′, k) ρ_{Δ}(w, q) − i ∫ dw′ ∑_{k} G(−w′tw, ktq)G(w′, k)

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

ρ_{g}(w, q) ∑_{k} \frac{[f(ε_{ktq})−f(ε_{k})]}{ε_{ktq}−ε_{k}−U−iσ} w > o ρ_{Δ}(w, q) ∑_{k} \frac{[f(ε_{ktq})−f(ε_{k})]}{ε_{ktq}+ε_{k}−w−iσ}w > o f(_{k}) = 1lfk < k_{F} 1 o k¡k_{F}

e) Now everything is reduced to a d dimensional integral over k. One way to do them is stay in the case of ρ_{g} ρ_{g} ∑_{k} f(ε_{k}[\frac{1}{ε_{k}−ε_{ktq}−w−iσ} − \frac{1}{ε_{ktq}−ε_{k}−w−iσ}]

Introduce polar ? and intergate over dσ and over dk oΔk, k_{F}; The 3d integral is done in Fetter and Walecva page 158-162.

If you want you can calculate the ρ_{g} and ρ_{g} separately. In 1d the integrals are even easier; 2d is more fun.

Repeat the same for ρ_{Δ}.

Good Luck