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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 6 \answer{--~~~ Solutions} }
\end{center}


\noindent
\hfill {\bf Due 11pm {Thursday, May 16, 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.
\bigskip
\hrule


\begin{enumerate}

\item {\bf Hall plateaux as a crazy manifestation of quantum oscillations.} 
Check the claim that the hierarchy states at fillings $ \nu = { \nu^\star \over 2\nu^\star \pm 1}$ 
for $\nu^\star \in \IZ$ can be regarded as an extreme version of quantum oscillations in the HLR state at $\nu =\half$.

\item {\bf Quantum Hall states of quasiparticles.} 
In lecture we explained how to find incompressible states with filling fraction 
$ \nu = { 1 \over k - { 1 \over \tilde k}} $ 
by placing the quasiparticle excitations of a $\nu = 1/k$ FQH state in a $\tilde \nu = 1/\tilde k$ FQH state. 
Check this relation.  
When $\tilde k = 2$, this reproduced one branch of the composite fermion states 
we found previously.  Explain how to get the other branch.  



\item {\bf Excitations of hierarchy states.}
Find the torus groundstate degeneracy, and the \htmladdnormallink{charges}{https://www.nature.com/articles/20384} and statistics of the quasiparticle excitations of the 
abelian incompressible FQH state
at $\nu = {2\over 5}$ (for example, using the description in terms of the $K$-matrix CS theory).


\item {\bf Boson Integer Quantum Hall State from Partons.}  
%\begin{enumerate}
Consider a system made from two species of bosons, $b_\up, b_\down$.  They could be distinguished by living in two layers.
We'll assume that only the total boson number, acting by $ (b_\up, b_\down) \to e^{ \ii \alpha} (b_\up, b_\down) $ is conserved (so that if the label is a layer label, the particles are able to tunnel between layers), and couple to a background field $\CA$ for that symmetry.

\begin{enumerate}
\item

Consider the parton ansatz:
$$ b_\up = f_0 f_\up, ~~b_\down = f_0 f_\down f_1 f_2 $$
where all the $f$s are fermionic partons.
There are three $\gU(1)$ gauge fields that glue these partons back together, and the charge assignments are 
as follows:
\begin{table}[h!]
\begin{center}
\begin{tabular}{ |c|c | c | c| c|c|c| c|}
   \hline                      
  & $a_1$ & $a_2$ & $a_3$ & $\CA$ & Chern $\# $ in & Chern $\#$ in &  Chern $\#$ in \\ 
& & & & & Phase 1 & Phase 2 & Phase 3 \\ \hline
$f_\up$ & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\ \hline
$f_\down$ & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\  \hline  
$f_0$ & -1 & 0 & 0 & 0 & -1 & -1 & -1 \\  \hline  
$f_1$ & 0 & -1 & 1 & 0 & -1 & -1  & -1 \\  \hline  
$f_2$ & 0 & 0 & -1 &0 & -1 & 0 & 1\\  \hline  
\end{tabular}
\end{center}
%\caption{The gauge charges and the Chern number assignments for various partons in the three distinct phases studied in this paper.}
\label{table1}
\end{table}

Also in the table are the Chern numbers of the bands filled by each of the partons in three distinct phases.
(Only the Chern number of $f_2$ changes.)
%\begin{enumerate}
Identify the three phases, and describe the critical theories separating them.

Hint: I recommend describing the parton currents in terms of dynamical gauge fields $j^{(\alpha)}_\mu = {1\over 2\pi} \eps_{\mu\nu\rho} \partial_\nu b_\rho^{(\alpha)}$, where $\alpha = \up,\down,0,1,2$.





\item For this part of the problem, let's retreat to the continuum.  
Consider the simpler parton ansatz:
$$ b_\up = f_0 f_\up, ~~b_\down = f_0 f_\down $$
where all the $f$s are fermionic partons.
Choose the $\gU(1)_\CA$ to be charges $ q_0 = 2, q_\up = -1, q_\down = -1$.

Consider an equal number $N$ of $b_\up$ and $b_\down$ particles, so that the total filling fraction is $\nu=2$.
How many $f_0$ particles are there, and how many $f_\down, f_\up$ particles are there?





Write a candidate groundstate wavefunction $\Psi(r_i^\up, r_i^\down)$ for the bosons.



\item Bonus question: why does the simpler ansatz of the previous part produce a wavefunction in the same phase 
as one of the phases of the first part?



\item Actually, here is a simpler description of the same phase diagram, closer to what I said in lecture.
Consider a single species of boson, with the simple parton ansatz with $ b = d_1 d_2$
in terms of two fermions.  
Let $d_1$ and $d_2$ fill Chern bands with total Chern number $c_1$ and $c_2$.
Fix $c_1 = -1$.  Consider what happens when $c_2 = 2$.

Describe the effective field theory of $d_2$ filling two bands with chern number $1$
by introducing two gauge fields each with CS term $ {1\over 4 \pi} b_a d b_a $.



\end{enumerate}



\end{enumerate}
\end{document}

