Problem Set 4, Due November 15th 2000

Problem 1:

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+q↑}^{†}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))⟩

write an expression it will involve a k - integration consider the case of d dimensions can you evaluate some of the integrals in d=1,2, or 3?

Problem 2 Repeat the first problem at finite temperature 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_{ρ} 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)

Problem 3: Evaluate the following Matsubara sums

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 why do the two sums differ