Problem Set 3

1 $N(\mathbf{k})$ in the Ground State

In second order perturbation theory, find the occupation number $N(\mathbf{k})=⟨0|a_{\mathbf{k},s}^{†}{a}_{\mathbf{k},s}^{\phantom{†}}|0⟩$ in the ground state of the interacting Fermi gas. Compare with the quantity $z_{\mathbf{k}}$ introduced in the lecture.

2 Pair Correlations in the BCS State

As well as the average occupancy of a given momentum state we can consider the correlations between the occupancy of different $\mathbf{p}$ states

$$C_{s{s}^{\prime }}(\mathbf{p},{\mathbf{p}}^{\prime })\equiv ⟨n_{\mathbf{p},s}{n}_{{\mathbf{p}}^{\prime },s^{\prime }}⟩-⟨n_{\mathbf{p},s}⟩⟨n_{{\mathbf{p}}^{\prime },s^{\prime }}⟩$$

Show that for the BCS state

$$\begin{array}{rl}{C}_{\uparrow \downarrow }({\mathbf{p}}_1,{\mathbf{p}}_2)& ={\delta }_{{\mathbf{p}}_1,-{\mathbf{p}}_2}{u}_{{\mathbf{p}}_1}^2{v}_{{\mathbf{p}}_1}^2={\delta }_{{\mathbf{p}}_1,-{\mathbf{p}}_2}\frac{|\mathrm{\Delta }{|}^2}{4E_{{\mathbf{p}}_1}^2}\\ C_{\uparrow \uparrow }({\mathbf{p}}_1,{\mathbf{p}}_2)& ={\delta }_{{\mathbf{p}}_1,{\mathbf{p}}_2}\frac{|\mathrm{\Delta }{|}^2}{4E_{{\mathbf{p}}_1}^2}\end{array}$$

Interpret these two expressions.

3 A Very Simple Model for Phonon Mediated Attraction

An optical phonon mode gives rise to an oscillating charge that couples to the electrons at a site. We model this by the Hamiltonian

$$H=\frac{p^2}{2m}+\frac{1}{2}m{\omega }^2{x}^2+\alpha x(N_{\uparrow }+N_{\downarrow }),$$

where $N_s=0,1$ are the number of electrons of spin $s$ at the site. Since $N_s$ is conserved you can solve the model exactly.

Next, introduce an oscillator at each site of a Fermi Hubbard model

$$H=H_{\text{Hubbard}}+\sum _j[\frac{p_j^2}{2m}+\frac{1}{2}m{\omega }^2{x}_j^2+\alpha x_j(N_{j,\uparrow }+N_{j,\downarrow })].$$

If the energy $\omega$ of the oscillators is larger than other scales, you can use the technique from Lecture 7 to derive an effective Hamiltonian. What form does this take?

The physics behind this mechanism is a very simple consequence of living in an elastic medium, and is not really a quantum effect at all. The fact that two heavy spheres on a stretched horizontal rubber sheet will roll towards each other is a nearly perfect analogy for this effect (as well as a very poor one for gravitational attraction in GR!).

4 An Inequality for the Static Structure factor

Use the f-sum and compressibility sum rules, together with the Cauchy-Schwarz inequality

$$|\int f(x)g^{\ast }(x)dx{|}^2\le \int |f(x){|}^{2}dx\int |g(x){|}^{2}dx$$

to obtain the Onsager bound on the static structure factor

$$\underset{\mathbf{q}\to 0}{lim}\frac{S_{\rho }(\mathbf{q})}{|\mathbf{q}|}\le \frac{N}{2mc}$$

5 $S_{\rho }(q,\omega )$ for 1D Fermi Gas

Find the dynamical structure factor for a 1D Fermi gas, and verify the Onsager bound.

6 $S_{\rho }(q)$ for the Elastic Chain

In Lecture 3 we found the static structure factor of the elastic chain. Verify the Onsager bound.

7 Ground State Energy of Jellium in Perturbation Theory

In Lecture 9 we found the corrections to the eigenenergies of a Fermi gas to second order in the interaction. Show that for Jellium the correction to the ground state energy has an infrared divergence.

8 Limits of the Polarization

Check the two limits for the polarization described at the end of Lecture 12.

9 Explicit Evaluation of Green’s Functions

Starting from the definition of the fermion Green’s function, show that

$$\begin{array}{}\text{(1)}& G_{\mathbf{k}}(\tau )=e^{-\xi (\mathbf{k})\tau }\{\begin{array}{ll}1-⟨N_{\mathbf{k}}⟩& \tau >0\\ -⟨N_{\mathbf{k}}⟩& \tau <0\end{array}\end{array}$$

where $⟨N_{\mathbf{k}}⟩=n_{\text{F}}(\xi (\mathbf{p}))$, and $n_{\text{F}}(\omega )=\frac{1}{e^{\beta \omega }+1}$ is the Fermi–Dirac distribution.

We also have

$$G_{\mathbf{k}}(\tau )=T\sum _{{ϵ}_n}\frac{e^{-i{ϵ}_n\tau }}{-i{ϵ}_n+\xi (\mathbf{k})}.$$

Evaluate the sum to find Equation 1. In the lecture, we used the auxillary function $\tanh \left(\frac{\beta ϵ}{2}\right)$ to turn the Matsubara sum into an integral. Here, to be able to deform the contour in a useful way, we must use $n_{\text{F}}(\omega )$ or $1-n_{\text{F}}(\omega )$.