# Problem Set 3

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

## $$N(\bk)$$ 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 $$1-n_\text{F}(\omega)$$.

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