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


\noindent
\hfill {\bf Due 11pm {Thursday, May 2, 2024}} 


\bigskip
\hrule


\begin{enumerate}

\item {\bf A simple avatar of the Lieb-Schulz-Mattis theorem.}
Consider the effective theory describing a system living in the continuum that spontaneously forms a solid, say a cubic lattice in $d$ dimensions.  
Since translation symmetry is spontaneously broken, the degrees of freedom must include a collection of Goldstone bosons $\theta^I$, where $I=1..d$ runs over the spatial dimensions.  
$\theta^I(x)$ is the shift of the atom at location $x$ in the $I$ direction relative to its equilibrium position.
These fields live on a circle, because if I shift all the atoms by the lattice spacing, I get back the original lattice.  
\begin{enumerate}
\item Convince yourself that the effective action takes the form
\be\label{eq:elastic} S_\text{elastic}[\theta^I] = \int d^{d+1} x  ~\kappa^{ijKL} \partial_i \theta^K \partial_j \theta^L + \text{terms with more derivatives} ,\ee
where the coupling constant $\kappa^{ijKL}$ is the elasticity tensor.  With various symmetries imposed, it can be decomposed further into various tensors with names from the 19th century.
These tensors describe things like bending moduli -- the rigidity of the solid to various kinds of strain.

\item Now suppose that the number of atoms is a conserved quantity. 
That is, consider a situation where there is also a $\gU(1)$ symmetry.  
So we can couple the system to a background gauge field $A_\mu$ for this $\gU(1)$ symmetry.  
We'll assume this $\gU(1)$ symmetry is not spontaneously broken.  
What are the leading terms in the (local!) effective action $S_\text{eff}[\theta^I, A_\mu]$
that preserve gauge invariance and translation symmetry?

\item Consider the case of $d=1$.  In addition to the terms involving $dA$, one interesting term is 
\be\label{eq:top-term1}S_\nu[\theta, A] \equiv { \nu \over 2\pi} \int A \wedge d \theta = {\nu\over 2\pi} \int dx dt A_\mu \partial_\nu \theta \eps^{\mu\nu} .\ee
One point to notice about it is that it is not obviously gauge invariant, because it depends explicitly on $A$ and not just the gauge-invariant object $F$.
Show that $e^{ \ii S_\nu}$ is gauge invariant if $\nu $ is an integer.  


\item What does the new term \eqref{eq:top-term1} do?  Well, the first question we should ask about an effective action for a background gauge field is:
what is the resulting charge density:
$$ \rho(x) = { \delta S \over \delta A_0(x) } ~?$$
Interpret your result.  


\item What is the analog of \eqref{eq:top-term1} in $d$ dimensions? 
(That is, find a term in $d$ spatial dimensions involving a single power of $A$ and derivatives of the $\theta^I$ that can be written without using the metric.)
Show that its coefficient $\nu$ is quantized to be an integer.
What contribution does it make to the density?  

\item We can identify the goldstone field $\theta$ with the phase field describing 
the displacements of the atoms from their equilibrium positions:
$$ u^i(x,t) = {1\over 2\pi} a^i_I \theta^I(x,t) - x^i $$
where $\vec a_I$ are generators of the lattice $\Gamma$.  
Then the equilibrium configuration is actually $\theta^I(x,t) = K^I_i x^i $ 
where $ K^I_i\(  { a \over 2 \pi } \)_I^j = \delta_i^j $, so $K^I_i$ is the matrix 
whose columns are the reciprocal lattice generators.




\item The conclusion you should find 
by the gauge invariance argument above, under the present assumptions, 
is that $\nu$, and hence the equilibrium density must be an integer.  
This is an avatar of the Lieb-Schulz-Mattis-Oshikawa-Hastings (LSMOH) theorem.
Now, you may say to yourself, why can't I make a system at some filling which is not an integer?
Indeed, I can take 20007 particles and place them in a volume with 20004 unit cells, and the system must have some groundstate.  What gives?

\end{enumerate}



\item {\bf Edge modes of CS theory.}
Now we return to abelian Chern-Simons theory (for an extra challenge, redo this part in the non-Abelian case).  If there is a boundary of spacetime, something must be done
to fix up the fact that the action is not invariant under would-be gauge transformations that are nontrivial at the boundary.  Consider the case where 
$\Sigma = \IR \times \text{UHP} $ where $\IR$ is the time direction,
and UHP is the upper half-plane $y>0$.
One way to fix the problem is simply to declare that 
the would-be gauge transformations
which do not vanish at $y=0$ are 
not redundancies.   This means that 
they represent physical degrees of freedom.
\begin{enumerate}
\item First consider the simplest case of $\gU(1)$ CS theory at level $k$.  
Choose $a_0 = 0$ gauge, and plug the solution of the bulk equations of motion 
$a= \tilde d\phi$ 
(where $\phi(x, y \to 0) \equiv \phi(x)$ is a 
scalar field, and $\tilde d$ is the exterior derivative on the spatial manifold) 
into the Chern-Simons action
to find the resulting action for $\phi$.



\item We can also add local terms at the boundary to the action.
Consider adding $ \Delta S =   g \int_{\partial \Sigma} a_x^2 $
(for some coupling constant $g$).
Find the equations of motion for $\phi$.
%Alternatively, we could {\it add} some other degrees of 
%freedom to the theory to cancel the variation of the action.
%Show that this is accomplished by adding a 
%{\it chiral} scalar 


\answer{This term evaluates to 
$ \Delta S = \int_{\partial \Sigma} v \( \partial_x \phi\)^2 .$
Altogether we now have
$$ S_{\text{edge}}[\phi] = \int_{y=0} dx dt  \partial_x \phi \(  {k \over 4 \pi} \partial_t \phi + g \partial_x \phi \)  . $$
The EoM is then 
$$ { \delta \over \delta \phi(x) } S_{\text{edge}}[\phi] 
=  \partial_t \( {k \over 4 \pi} \partial_t \phi + g \partial_x \phi\) $$
which is solved if 
$ {k \over 4 \pi} \partial_t \phi + g \partial_x \phi = 0 $.
This describes a dispersionless wave which moves only in the $ \text{sign}{k}$ direction -- a chiral bosonic edge mode.
}


Interpretation: the Chern-Simons theory 
on a space with boundary necessarily produces a chiral edge mode.

\item If you feel like it, redo the previous parts for the general $K$-matrix theory.

\end{enumerate}


\end{enumerate}
\end{document}

