From ff30683bce54ac4c45f13c3d5e83ca127cd6f729 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Debias <jaron@kent-dobias.com>
Date: Tue, 22 Oct 2019 13:38:03 -0400
Subject: started clarifying some statements

---
 main.tex | 24 +++++++++++++++---------
 1 file changed, 15 insertions(+), 9 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 20d001b..3a43f91 100644
--- a/main.tex
+++ b/main.tex
@@ -115,7 +115,12 @@ strongly motivates future ultrasound experiments under pressure approaching the
 
 \section{Model}
 The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally
-respect this symmetry in the high-temperature phase. Our phenomenological free energy density contains three parts: the elastic free energy, the \op, and the interaction between strain and \op. The most general quadratic free energy of the strain $\epsilon$ is $f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, where the six irreducible components of strain are
+respect this symmetry in the high-temperature phase. Our phenomenological free
+energy density contains three parts: the elastic free energy, the \op, and the
+interaction between strain and \op. The most general quadratic free energy of
+the strain $\epsilon$ is $f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$. Linear
+combinations of the six independent components of strain form five irreducible
+components of strain as
 \begin{equation}
   \begin{aligned}
     \epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth}
@@ -147,13 +152,14 @@ The interaction between strain and an \op\ $\eta$ depends on the point group rep
 \begin{equation}
   f_\i=-b^{(i)}\epsilon_\X^{(i)}\eta.
 \end{equation}
-If the representation $\X$ is not present in the strain \brad{what does "present in the strain" mean?} there can be no linear
-coupling, and the effect of the \op\ condensing at a continuous phase
-transition is to produce a jump in the $\Aog$ elastic modului if $\eta$ is
-single-component \cite{luthi_sound_1970, ramshaw_avoided_2015,
-shekhter_bounding_2013}, and jumps in other elastic moduli if multicompenent \cite{ghosh_single-component_nodate}. Because we are interested
-in physics that anticipates the phase transition, we will focus our attention
-on \op s that can produce linear couplings to strain.  Looking at the
+If there doesn't exist a component of strain that transforms like the
+representation $\X$ there can be no linear coupling, and the effect of the \op\
+condensing at a continuous phase transition is to produce a jump in the $\Aog$
+elastic modului if $\eta$ is single-component \cite{luthi_sound_1970,
+ramshaw_avoided_2015, shekhter_bounding_2013}, and jumps in other elastic
+moduli if multicompenent \cite{ghosh_single-component_nodate}. Because we are
+interested in physics that anticipates the phase transition, we will focus our
+attention on \op s that can produce linear couplings to strain.  Looking at the
 components present in \eqref{eq:strain-components}, this rules out all of the
 \emph{u}-reps (which are odd under inversion) and the $\Atg$ irrep.
 
@@ -221,7 +227,7 @@ traditional to make the field ansatz
 $\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$ \brad{Why is it traditional to ignore any in-plane modulation (x1, x2)?}. For $\tilde r>0$ and $c_\perp>0$,
 or $\tilde r>c_\perp^2/4D_\perp$ and $c_\perp<0$, the only stable solution is
 $\eta_*=q_*=0$ and the system is unordered. For $\tilde r<0$ there are free
-energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and this system has uniform order \brad{probably helpful to specify what kind of order here - uniform $\Bog$ order, correct?}. For $c_\perp<0$ and $\tilde r<c_\perp^2/4D_\perp$ there are free
+energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and this system has uniform order of whatever . For $c_\perp<0$ and $\tilde r<c_\perp^2/4D_\perp$ there are free
 energy minima for $q_*^2=-c_\perp/2D_\perp$ and
 \begin{equation}
   \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{12D_\perp u}
-- 
cgit v1.2.3-70-g09d2

