Problem Set 3

\[ \nonumber \newcommand{\cN}{\mathcal{N}} \newcommand{\br}{\mathbf{r}} \newcommand{\bp}{\mathbf{p}} \newcommand{\bk}{\mathbf{k}} \newcommand{\bq}{\mathbf{q}} \newcommand{\bv}{\mathbf{v}} \newcommand{\pop}{\psi^{\vphantom{\dagger}}} \newcommand{\pdop}{\psi^\dagger} \newcommand{\Pop}{\Psi^{\vphantom{\dagger}}} \newcommand{\Pdop}{\Psi^\dagger} \newcommand{\Phop}{\Phi^{\vphantom{\dagger}}} \newcommand{\Phdop}{\Phi^\dagger} \newcommand{\phop}{\phi^{\vphantom{\dagger}}} \newcommand{\phdop}{\phi^\dagger} \newcommand{\aop}{a^{\vphantom{\dagger}}} \newcommand{\adop}{a^\dagger} \newcommand{\bop}{b^{\vphantom{\dagger}}} \newcommand{\bdop}{b^\dagger} \newcommand{\cop}{c^{\vphantom{\dagger}}} \newcommand{\cdop}{c^\dagger} \newcommand{\Nop}{\mathsf{N}^{\vphantom{\dagger}}} \newcommand{\bra}[1]{\langle{#1}\rvert} \newcommand{\ket}[1]{\lvert{#1}\rangle} \newcommand{\inner}[2]{\langle{#1}\rvert #2 \rangle} \newcommand{\braket}[3]{\langle{#1}\rvert #2 \lvert #3 \rangle} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\tr}{tr} \newcommand{\abs}[1]{\lvert{#1}\rvert} \newcommand{\brN}{\br_1, \ldots, \br_N} \newcommand{\xN}{x_1, \ldots, x_N} \newcommand{\zN}{z_1, \ldots, z_N} \]

\(N(\bk)\) in the Ground State

In second order perturbation theory, find the occupation number \(N(\bk)=\braket{0}{\adop_{\bk,s}\aop_{\bk,s}}{0}\) in the ground state of the interacting Fermi gas. Compare with the quantity \(z_\bk\) introduced in the lecture.

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 \(\bp\) states

\[ C_{ss'}(\bp,\bp')\equiv\langle n_{\bp,s} n_{\bp',s'}\rangle-\langle n_{\bp,s}\rangle\langle n_{\bp',s'}\rangle \]

Show that for the BCS state

\[ \begin{align} C_{\uparrow\downarrow}(\bp_1,\bp_2)&=\delta_{\bp_1,-\bp_2}u_{\bp_1}^2v_{\bp_1}^2=\delta_ {\bp_1,-\bp_2}\frac{|\Delta|^2}{4E_{\bp_1}^2}\nonumber\\ C_{\uparrow\uparrow}(\bp_1,\bp_2)&=\delta_{\bp_1,\bp_2}\frac{|\Delta|^2}{4E_{\bp_1}^2} \end{align} \]

Interpret these two expressions.

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!).

An Inequality for the Static Structure factor

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

\[ \abs{\int f(x)g^*(x) dx}^2 \leq \int \abs{f(x)}^2 dx \int \abs{g(x)}^2 dx \]

to obtain the Onsager bound on the static structure factor

\[ \lim_{\bq\to 0}\frac{S_\rho(\bq)}{\abs{\bq}}\leq \frac{N}{2mc} \]

\(S_\rho(q,\omega)\) for 1D Fermi Gas

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

\(S_\rho(q)\) for the Elastic Chain

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

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.

Limits of the Polarization

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

Explicit Evaluation of Green’s Functions

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

\[ G_\bk(\tau) = e^{-\xi(\bk)\tau}\begin{cases} 1-\langle N_\bk\rangle & \tau>0\\ -\langle N_\bk \rangle & \tau<0 \end{cases} \label{Gexp} \]

where \(\langle N_\bk\rangle = n_\text{F}(\xi(\bp))\), and \(n_\text{F}(\omega)=\frac{1}{e^{\beta\omega}+1}\) is the Fermi–Dirac distribution.

We also have

\[ G_\bk(\tau) = T\sum_{\epsilon_n} \frac{e^{-i\epsilon_n \tau}}{-i\epsilon_n+\xi(\bk)}. \]

Evaluate the sum to find \(\eqref{Gexp}\). 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 \(1-n_\text{F}(\omega)\).