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%%from  Rudro Rana Biswas
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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 5  \answer{--~~~ Solutions} }
\end{center}


\noindent
%Posted September 17, 2015 
\hfill {\bf Due  11:59pm Wednesday, May 7, 2025}




\bigskip
\hrule
\begin{enumerate}
\item {\bf Order parameter exponent at the Wilson-Fisher fixed point.} [bonus problem]
In lecture we outlined the computation of $\eta$ using position space diagrams.  
Find the coefficient $c$ in $\eta = c \eps^2 + \CO(\eps^3)$ 
as a function of $n$ at the $\gO(n)$ Wilson-Fisher fixed point.
Check that the factors of $r_2$ drop out.

[Hint: The answer for the Ising model ($n=1$) is $\eta = {\eps^2\over 54}$.]


\item {\bf OPE.} [bonus problem]
Consider the Gaussian fixed point with $\gO(n)$ symmetry.  
Compute the OPE coefficients for the operators $ \CO_2 \equiv :\phi_a \phi_a:$, 
$\CO_4 \equiv :( \phi_a \phi_a)^2:$, and the identity operator (here $a=1..n$ and the repeated index is summed).
Use this information to compute the beta function, 
find the Wilson-Fisher fixed point and the correlation length critical exponent $\nu$
there.




\item {\bf RG analysis of less-symmetric spin systems.} 


Suppose that we break the rotation symmetry of the $\gO(n)$ model
to the subgroup of $\pi/2$ rotations, \ie~the cubic symmetry,
(for example, for $n=2$, $ (s_1, s_2) \to (s_2, -s_1) $.)
If the spins live on the cubic lattice, a spin-orbit coupling
could do this.

\begin{enumerate}

\item {\cor Don't look at the next part of the problem yet!}
What functions of an $n$-vector $\phi_a$ are invariant under $ \pi/2$ rotations, but not general
rotations?


$$\vdots$$

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$$\vdots$$

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$$\vdots$$



\vfill\eject
\item Show that 
in addition to the usual  $\gO(n)$-symmetric interaction
$$\int d^d x ~ \sum_{a,b=1}^n u \phi_a^2\phi_b^2, $$
the LG free energy should include a term of the form
$$\int d^d x ~ \sum_{a=1}^n v \phi_a^4 .$$
Argue that this is the only new term 
(preserving cubic symmetry but not the full $\gO(n)$ symmetry)
which can be a relevant perturbation of 
the Gaussian fixed point near $ d=4$.

\item 
Treating $ \CO(u) = \CO(v) = \CO(\eps)$, 
redo the analysis of the running couplings in $d=4-\eps$ dimensions to derive 
beta functions for $ u $ and $v$ up to corrections of order $ \CO(u^3) = \CO(v^2u) = ... = \CO(\eps^3) $.

\item Your answer to the previous part will be of the form
\bea
- s \partial_s u &=&  -\eps u + A_1 u^2 + A_2  uv + A_3 v^2 + \CO(u^3 ) \cr 
- s \partial_s v &=& - \eps v + B_1 u^2 + B_2  uv + B_3 v^2 + \CO(u^3 ) .
\eea
You should find that $ A_3= B_1 = 0$.
Find four fixed points: 
\begin{itemize}
\item The gaussian fixed point.
\item A fixed point where only $u\neq 0$.  
\item A fixed point where only $v\neq 0$.  Describe the physics of this fixed point.  (Hint: the action is a sum of $n$ terms.)
\item A fixed point where both $(u,v)$ are nonzero.
\end{itemize}
(In every case, the assumption of $ u \sim v \sim \eps$ is self-consistent.)



\item Analyze the stability of these fixed points 
(by computing the matrix of derivatives of the beta functions at each fixed point).
Draw the phase diagram. 
Which fixed point dominates the critical behavior?
You will want to consider different cases depending on whether $n>4$ or $n<4$.

\item When $ n>4$ you may find that $v$ wants to become negative.  This means that 
the effective potential for $m$ becomes unbounded,  within our approximation.  
What have we left out that will restore sanity?  
What does this mean for the order of the phase transition?  
(Notice that mean field theory predicts a continuous transition, 
so any change in this conclusion is a dramatic effect of the fluctuations, 
more dramatic than just changing the values of critical exponents by a little.)



\end{enumerate}



\end{enumerate}
\end{document}
