Clarifying text changes to construction of elastic and interaction free

energies.

1 files changed, 25 insertions, 23 deletions

diff --git a/main.tex b/main.tex
index a809312..07b9366 100644
--- a/main.tex
+++ b/main.tex
@@ -126,15 +126,15 @@ strongly motivates future \rus\ experiments that could either further support
 or falsify it.
 
 The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally
-respect this symmetry. We will introduce a phenomenological free energy density
-in three parts: that of the strain, the \op, and their interaction.
-The most general quadratic free energy of the strain $\epsilon$ is
+respect this symmetry. Our phenomenological free energy density contains three
+parts: the free energy for the strain, the \op, and their interaction.  The
+most general quadratic free energy of the strain $\epsilon$ is
 $f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the bare strain
-stiffness tensor $C$ tensor is constrained by both that $\epsilon$ is a
-symmetric tensor and by the point group symmetry \cite{landau_theory_1995}. The
-latter can be seen in a systematic way. First, the six independent components
-of strain are written as linear combinations that behave like irreducible
-representations under the action of the point group, or
+stiffness tensor $C$ tensor is constrained by both the index symmetry of the
+strain tensor and by the point group symmetry \cite{landau_theory_1995}. The
+six independent components of strain can written as linear combinations that
+each behave like irreducible representations under the action of the point
+group, or
 \begin{equation}
   \begin{aligned}
     \epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth}
@@ -145,13 +145,13 @@ representations under the action of the point group, or
   \end{aligned}
   \label{eq:strain-components}
 \end{equation}
-Next, all quadratic combinations of these irreducible strains that transform
-like $\Aog$ are included in the free energy as
+All quadratic combinations of these irreducible strains that transform like
+$\Aog$ are included in the free energy,
 \begin{equation}
   f_\e=\frac12\sum_\X C_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)},
 \end{equation}
 where the sum is over irreducible representations of the point group and the
-stiffnesses $C_\X^{(ij)}$ are
+bare stiffnesses $C_\X^{(ij)}$ are
 \begin{equation}
   \begin{aligned}
     &C_{\Aog}^{(11)}=\tfrac12(C_{1111}+C_{1122}) &&
@@ -162,20 +162,22 @@ stiffnesses $C_\X^{(ij)}$ are
     C_{\Eg}^{(11)}=C_{1313}.
   \end{aligned}
 \end{equation}
-The interaction between strain and the \op\ $\eta$ depends on the
-representation of the point group that $\eta$ transforms as. If this
-representation is $\X$, then the most general coupling to linear order is
+The interaction between strain and an \op\ $\eta$ depends on the representation
+of the point group that $\eta$ transforms as. If this representation is $\X$,
+the most general coupling to linear order is
 \begin{equation}
-  f_\i=b^{(i)}\epsilon_\X^{(i)}\eta
+  f_\i=b^{(i)}\epsilon_\X^{(i)}\eta.
 \end{equation}
-If $\X$ is a representation not present in the strain there can be no linear
-coupling, and the effect of $\eta$ going through a continuous phase transition
-is to produce a jump in the $\Aog$ strain stiffness \cite{luthi_sound_1970,
-ramshaw_avoided_2015, shekhter_bounding_2013}. We will therefore focus our
-attention on \op\ symmetries that produce linear couplings to strain. Looking
-at the components present in \eqref{eq:strain-components}, this rules out all
-of the u-reps (odd under inversion) and the $\Atg$ irrep as having any
-anticipatory response in the strain stiffness.
+If the representation $\X$ is not present in the strain 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 $\Aog$ strain stiffness if $\eta$ is
+single-component \cite{luthi_sound_1970, ramshaw_avoided_2015,
+shekhter_bounding_2013}, and jumps in other strain stiffnesses 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\ symmetries that can produce linear couplings to strain.  Looking at the
+components present in \eqref{eq:strain-components}, this rules out all of the
+u-reps (which are odd under inversion) and the $\Atg$ irrep.
 
 If the \op\ transforms like $\Aog$, odd terms are allowed in its
 free energy and any transition will be abrupt and not continuous without