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%%from  Rudro Rana Biswas
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bookmarks={false}, pdfauthor={John McGreevy}, pdftitle={Yay, physics!}]{hyperref}
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\begin{document}
\begin{center}

University of California at San Diego -- 
Department of Physics --
Prof.~John McGreevy

{\Large\bf  Physics 215C QFT Spring 2025}\\
{\Large\bf Assignment 8  \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Wednesday, May 22, 2025}




\bigskip
\hrule
\begin{enumerate}





\def\dslash{\rlap{{\hskip0.08em/}}\partial}
\def\gs{{\text{gs}}}


\item{\bf Diagrammatic understanding of BCS instability of Fermi liquid theory.}



\begin{enumerate}

\item Recall that 
only the four-fermion interactions with special kinematics 
are marginal.  Keeping only these interactions, show 
that cactus diagrams (like this: \parfig{.05}{fig-one-cactus.png}) dominate.


\item To sum the cacti, we can make bubbles with a corrected propagator.
Argue that this correction to the propagator is innocuous and can be ignored.


\item Armed with these results, compute diagrammatically the Cooper-channel susceptibility (two-particle Green's function), 
$$\chi(\omega_0) \equiv 
\vev{ \CT  \psi^\dagger_{\vec k, \omega_3, \down} \psi^\dagger_{-\vec k, \omega_4, \up}\psi^\nd_{\vec p, \omega_1, \down}\psi^\nd_{-\vec p, \omega_2, \up}} $$
as a function of 
$ \omega_0 \equiv \omega_1 + \omega_2$, the frequencies of the incoming particles.
Think of $\chi$ as a two point function of the Cooper pair field 
$ \Phi = \eps_{\alpha\beta} \psi_\alpha \psi_{\beta} $
at zero momentum.
%$\vev{ \Phi(\omega_0, \vec 0)^\dagger \Phi(\omega_0, \vec 0)}$ 

Sum the geometric series in terms of a (one-loop) integral kernel.



\item Do the integrals.  
In the loops, restrict the range of 
momenta to $|\eps(k)| < E_D$, 
%energies to $ |\omega| < E_D$ (or $ |\eps(k)| < E_D$), 
the Debye energy, 
since it is electrons with these energies that experience attractive interactions.

Consider for simplicity a round Fermi surface.
For doing integrals of functions singular near a round Fermi surface, approximate the dispersion relation as 
$ \eps(k) \simeq  v_F (|k| - k_F)$, so that 
$ d^d k \simeq k_F^{d-1} { d \xi \over v_F} d\Omega_{d-1} $.
I recommend doing to the frequency integral first (by residues).




\item Show that when $V<0$ is attractive, $\chi(\omega_0)$ has a pole.
Does it represent a bound-state?
Interpret this pole in the two-particle Green's function 
%as indicating the presence of a two-particle bound-state
%which lead to 
%Interpret %the diverging susceptibility associated with the pole 
%this as 
as indicating an instability of the Fermi liquid to superconductivity.
Compare the location of the pole to the energy $E_\text{BCS}$ where the Cooper-channel interaction becomes strong.



\item {\bf Cooper problem.} [optional] We can compare this result to Cooper's influential analysis 
of the problem of two electrons interacting with each other in the presence of an inert Fermi sea.  
Consider a state with two electrons with antipodal momenta and opposite spin
$$ \ket{\psi} = \sum_k a_k  \psi_{k, \up}^\dagger \psi_{-k, \down}^\dagger \ket{F} $$
where $\ket{F} = \prod_{k<k_F} \psi_{k, \up}^\dagger \psi_{k, \down}^\dagger \ket{0}$ is a filled Fermi sea.
Consider the Hamiltonian 
$$ H = \sum_k \eps_k \psi_{k,\sigma}^\dagger \psi_{k, \sigma}   + \sum_{k,k'} V_{k,k'} \psi_{k,\sigma}^\dagger \psi_{k, \sigma} \psi_{k',\sigma'}^\dagger \psi_{k', \sigma'}. $$
Write the Schr\"odinger equation as 
$$ (\omega - 2 \eps_k ) a_k = \sum_{k'} V_{k,k'} a_{k'} . $$
Now assume (following Cooper) that the potential has the following form: 
$$ V_{k,k'} = V w_{k'}^\star w_{k} , ~~~~ w_k = \begin{cases} 1,&  0 < \eps_k < E_D \\   0,& \text{else} \end{cases} . $$
Defining $C \equiv \sum_k \omega_k^\star a_k $, show that the Schr\"odinger equation requires 
\be\label{eq:cooper} 1 = V \sum_k  { |w_k|^2 \over \omega - 2 \eps_k } .\ee
Assuming $V$ is attractive, find a bound state.
Compare \eqref{eq:cooper} to the condition for a pole found from the bubble chains above.


\end{enumerate}



\item{\bf Fermion propagator in a metal.} [bonus problem]

Starting from
\be G(p,t) = -{1\over 2 \pi \ii } \bra{\gs} \CT c_p^\nd(t) c_p^\dagger(0)  \ket{\gs} \ee
and using the free fermion time evolution operator, and the fact that 
the groundstate has all levels filled up to the Fermi level:
\be\bra{\gs}  c_p^\dagger c_p^\nd \ket{gs} = \begin{cases} 1, & \eps_p< 0 \\ 0, & \eps_p >0 \end{cases} \ee
show that the free fermion propagator can be written as 
\be G(p, \omega) =  { a  \over \omega - \eps_p - \ii \eta b \text{sgn}(\eps_p) } \ee
or 
\be G(p, \omega) =  { a'  \over \omega(1 + \ii b' \eta)  - \eps_p }  \ee
where $\eta = 0^+$ is an infinitesimal for some constants $a,b,a',b'$ to be determined.

\end{enumerate}

\end{document}