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\begin{document}
\begin{center}

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

{\Large\bf  Physics 230 Quantum Phases of Matter, Spr 2024}\\
{\Large\bf Assignment 5 \answer{--~~~ Solutions} }
\end{center}


\noindent
\hfill {\bf Due 11pm {Thursday, May 9, 2024}} 


%Thanks for following the submission guidelines on \htmladdnormallink{hw01}{https://mcgreevy.physics.ucsd.edu/s21/2021S-211C-hw01.pdf}.  Please ask me by email if you have any trouble.


%I will add another problem later today.
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\begin{enumerate}

\item {\bf Charges of quasiparticles in abelian CS EFT.}

In an abelian CS theory with $K$-matrix $K$, show that a quasiparticle with charge $\ell^I$ under CS gauge field $a^I$ has electric charge
$$ q_l  = t K^{-1} l . $$


\item {\bf Quasiparticle wavefunctions.} 

\begin{enumerate}
\item Use the flux-threading argument starting from the Laughlin $\nu = {1\over m}$ state 
to construct wavefunctions for the quasihole and quasiparticle.  
That is, write down a wavefunction of $N$ electrons with the property 
that it acquires a phase $e^{ \pm \ii \theta }$ when the coordinate $z_i$ of {\it any} electron is taken around the point $z_i=w$ by an angle $\theta$: $ z_i - w \to e^{ \ii \theta} (z_i -w )$, $\forall i=1..N$. 

In the latter case, don't forget to project onto the lowest Landau level.

Alternatively, you can try to use the parton construction, 
\ie~add or remove a single parton.  



\item Using the plasma analogy, show that your quasihole wavefunction produces a localized charge deficit of charge $1/m$.  


\item Construct a wavefunction with {\it two} quasiholes and use it to verify their statistics (by adiabatically moving them around each other and computing the resulting Berry phase).


\end{enumerate}


\end{enumerate}
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