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


\noindent
\hfill {\bf Due 11pm {Thursday, April 18, 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.


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\begin{enumerate}



\item {\bf Anyons in the toric code.}  

\begin{enumerate}

\item Show that when acting on a toric code groundstate the operator 
$$ W_C = \prod_{\ell \in C} X_\ell $$
creates a state which violates only the star operators at the sites in the boundary of $C$, $\partial C$, a pair of $e$-particles.


\item Show that when acting on a toric code groundstate the operator 
$$ V_{\check C} = \prod_{\ell \perp \check C} Z_\ell $$
creates a state which violates only the plaquette operators in the boundary of $\check C$, $\partial \check C$.



\item Show that a boundstate of an $e$ particle and an $m$ particle in the 2d toric code
must be a fermion.






\end{enumerate}

\item {\bf Toric code as $\IZ_2$ gauge theory with matter.}
Consider a model with qubits on the links of a lattice (with Pauli operators $X_\ell, Z_\ell$, $X_\ell Z_\ell = - Z_\ell X_\ell$)
and qubits on the sites of the lattice (with Pauli operators $\sigma^x_j, \sigma^z_j$, $\sigma^x_j \sigma^z_j = - \sigma^z_j \sigma^x_j$).  


\begin{enumerate}
\item Show that the operator
$$ G_j \equiv A_j \sigma^z_j $$ 
(where $A_j$ is the star operator) generates the gauge transformation 
\be\label{eq:gauge} \sigma^x_j \to (-1)^{s_j} \sigma^x_j, ~~ X_{ij} \to (-1)^{s_i} X_{ij} (-1)^{s_j} , ~ \sigma^z_j \to \sigma^z_j, ~ Z_{ij} \to Z_{ij}\ee
(where $i, j$ are the sites at the ends of the link labelled $ij$).  
By {\it generates} here I mean that an operator $\CO$ transforms as 
$$ \CO \to \CG_s^\dagger \CO \CG_s, ~~ \CG_s \equiv \prod_j G_j^{s_j} $$
with $s_j =0,1$.  



\item Show that the Hamiltonian
$$ \HH = - \sum_j G_j  -\sum_p B_p - h \sum_{ij} \sigma^x_i X_{ij} \sigma^x_j  - g \sum_{\vev{ij}} Z_{ij} $$
is gauge invariant.  


\end{enumerate}
Here we can identify $\sigma^x_j$ as the operator which creates an $e$ particle at site $j$.
And we can identify $\sigma^z_j = (-1)^{n_j}$ as the parity of the number operator.

\begin{enumerate}[resume]

\item Show that if we set $\sigma^x_j = 1$ and $\sigma^z_j=1$ for all $j$ we get back the (perturbed) toric code.

Bonus problem: interpret this operation as a choice of gauge in the model where $G_j=1$ is imposed as a constraint on physical states.


\end{enumerate}



\item{\bf A topological qubit.}  Consider the toric code on this cell complex:
$$ \parfig{.3}{fig-bc-smooth-rough.png} $$
Recall that rough boundary conditions means that plaquette terms get truncated,
such as the term $ - X_1 X_2 X_3$,
while smooth boundary conditions mean that star terms get truncated,
such as the term $ - Z_4 Z_5 Z_6 $.  

Show that there is a two-dimensional space of groundstates.
A good way to do this is using the algebra of string operators which terminate at the various components of the boundary
without creating excitations.



\item{\bf Duality wall.} [Bonus problem] Show that the following hamiltonian realizes a {\it duality wall} in the toric code.
$$
\parfig{.6}{fig-duality-wall.png}
$$
What this means is that when crossing the wall, an $e$ particle turns into an $m$ particle and vice versa.
(More precisely, there is a string operator which transports an $e$ particle to the wall, and can be completed by a string operator
transporting an $m$ particle away from the wall, without creating any excitations.)

To be more precise about the figure: the dotted line carries no degrees of freedom.
The terms in the hamiltonian along the wall are of the form
$$ H = ... - X_2 X_5 X_3 Z_7 - Z_3 X_7 X_8 X_9 $$
and there is a term at the end of the wall (the little blue circle) of the form $ - Z_6 Y_{17} X_{18}X_{19}X_{20}$.
Show that these terms commute with each other and all the usual star and plaquette terms, such as 
$A =  Z_1 Z_2 Z_3Z_4$ and $B = X_8X_{10} X_{11}X_{12}$.

What can you say about the end of the duality wall (the little blue circle)?

















\end{enumerate}

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