Problem Set 3
1 \(N(\mathbf{k})\) in the Ground State
In second order perturbation theory, find the occupation number \(N(\mathbf{k})=\langle{0}\rvert a^\dagger_{\mathbf{k},s}a^{\vphantom{\dagger}}_{\mathbf{k},s} \lvert 0 \rangle\) 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
\[ C_{ss'}(\mathbf{p},\mathbf{p}')\equiv\langle n_{\mathbf{p},s} n_{\mathbf{p}',s'}\rangle-\langle n_{\mathbf{p},s}\rangle\langle n_{\mathbf{p}',s'}\rangle \]
Show that for the BCS state
\[ \begin{align} C_{\uparrow\downarrow}(\mathbf{p}_1,\mathbf{p}_2)&=\delta_{\mathbf{p}_1,-\mathbf{p}_2}u_{\mathbf{p}_1}^2v_{\mathbf{p}_1}^2=\delta_ {\mathbf{p}_1,-\mathbf{p}_2}\frac{|\Delta|^2}{4E_{\mathbf{p}_1}^2}\nonumber\\ C_{\uparrow\uparrow}(\mathbf{p}_1,\mathbf{p}_2)&=\delta_{\mathbf{p}_1,\mathbf{p}_2}\frac{|\Delta|^2}{4E_{\mathbf{p}_1}^2} \end{align} \]
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
\[ H = \frac{p^2}{2m}+\frac{1}{2}m\omega^2 x^2 + \alpha x\left(N_\uparrow+N_\downarrow\right), \]
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
\[ H = H_\text{Hubbard} + \sum_j \left[\frac{p_j^2}{2m}+\frac{1}{2}m\omega^2 x_j^2 + \alpha x_j\left(N_{j,\uparrow}+N_{j,\downarrow}\right)\right]. \]
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
\[ \lvert{\int f(x)g^*(x) dx}\rvert^2 \leq \int \lvert{f(x)}\rvert^2 dx \int \lvert{g(x)}\rvert^2 dx \]
to obtain the Onsager bound on the static structure factor
\[ \lim_{\mathbf{q}\to 0}\frac{S_\rho(\mathbf{q})}{\lvert{\mathbf{q}}\rvert}\leq \frac{N}{2mc} \]
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
\[ G_\mathbf{k}(\tau) = e^{-\xi(\mathbf{k})\tau}\begin{cases} 1-\langle N_\mathbf{k}\rangle & \tau>0\\ -\langle N_\mathbf{k}\rangle & \tau<0 \end{cases} \tag{1}\]
where \(\langle N_\mathbf{k}\rangle = n_\text{F}(\xi(\mathbf{p}))\), and \(n_\text{F}(\omega)=\frac{1}{e^{\beta\omega}+1}\) is the Fermi–Dirac distribution.
We also have
\[ G_\mathbf{k}(\tau) = T\sum_{\epsilon_n} \frac{e^{-i\epsilon_n \tau}}{-i\epsilon_n+\xi(\mathbf{k})}. \]
Evaluate the sum to find Equation 1. In the lecture, we used the auxillary function \(\tanh\left(\frac{\beta\epsilon}{2}\right)\) 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 \(1-n_\text{F}(\omega)\).