From b08d7d2e6f0083e426ab738e0fb0f729335271ab Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 5 Nov 2019 16:23:14 -0500
Subject: line lengths and equation indentation in the first half of the paper

---
 main.tex | 71 +++++++++++++++++++++++++++++++++++++++++++++-------------------
 1 file changed, 50 insertions(+), 21 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index d20c7f4..bca1d8c 100644
--- a/main.tex
+++ b/main.tex
@@ -125,12 +125,38 @@ T$_{\mathrm{HO}}=17.5~$ K \brad{find old citations for this data}; and two, a
 $\Bog$ nematic distortion is observed by x-ray scattering under sufficient
 pressure to destroy the \ho\ state \cite{choi_pressure-induced_2018}.
 
-Recent \emph{resonant ultrasound spectroscopy} (\rus) measurements examined the thermodynamic discontinuities in the elastic moduli at T$_{\mathrm{HO}}$ \cite{ghosh_single-component_nodate}. The observation of discontinues only in compressional, or $\Aog$, elastic moduli requires that the point-group representation of \ho\ is one-dimensional. This rules out a large number of order parameter candidates \brad{cite those ruled out} in a model-free way, but still leaves the microscopic nature of \ho~ undecided. 
-
-Recent X-ray experiments discovered rotational symmetry breaking in \urusi\ under pressure \cite{choi_pressure-induced_2018}. Above 0.13--0.5 $\GPa$ (depending on temperature), \urusi\ undergoes a $\Bog$ nematic distortion. While it is still unclear as to whether this is a true thermodynamic phase transition, it may be related to the anomalous softening of the $\Bog$ elastic modulus---$(c_{11}-c_{12})/2$ in Voigt notation---that occurs over a broad temperature range at zero-pressure \brad{cite old ultrasound}. Motivated by these results, hinting at a $\Bog$ strain susceptibility associated with the \ho\ state, we construct a phenomenological mean field theory for an arbitrary \op\ coupled to strain, and the determine the effect of its phase transitions on the elastic response in different symmetry channels. 
-
-We find that only one \op\ symmetry reproduces the anomalous $\Bog$ elastic modulus, which softens in a Curie-Weiss like manner from room temperature, but which cusps at T$_{\mathrm{HO}}$. That theory associates \ho\ with a $\Bog$ \op\ \emph{modulated along the $c$- axis}, the \afm\ state with uniform $\Bog$ order, and the triple point between them with a Lifshitz point. Besides the agreement with ultrasound data across a broad temperature range, the theory predicts uniform $\Bog$ strain at high pressure---the same distortion which was recently seen in x-ray scattering experiments \cite{choi_pressure-induced_2018}. This theory
-strongly motivates future ultrasound experiments under pressure approaching the Lifshitz point, which should find that the $(c_{11}-c_{12})/2$ diverges once the uniform $\Bog$ strain sets in.
+Recent \emph{resonant ultrasound spectroscopy} (\rus) measurements examined the
+thermodynamic discontinuities in the elastic moduli at T$_{\mathrm{HO}}$
+\cite{ghosh_single-component_nodate}. The observation of discontinues only in
+compressional, or $\Aog$, elastic moduli requires that the point-group
+representation of \ho\ is one-dimensional. This rules out a large number of
+order parameter candidates \brad{cite those ruled out} in a model-free way, but
+still leaves the microscopic nature of \ho~ undecided. 
+
+Recent X-ray experiments discovered rotational symmetry breaking in \urusi\
+under pressure \cite{choi_pressure-induced_2018}. Above 0.13--0.5 $\GPa$
+(depending on temperature), \urusi\ undergoes a $\Bog$ nematic distortion.
+While it is still unclear as to whether this is a true thermodynamic phase
+transition, it may be related to the anomalous softening of the $\Bog$ elastic
+modulus---$(c_{11}-c_{12})/2$ in Voigt notation---that occurs over a broad
+temperature range at zero-pressure \brad{cite old ultrasound}. Motivated by
+these results, hinting at a $\Bog$ strain susceptibility associated with the
+\ho\ state, we construct a phenomenological mean field theory for an arbitrary
+\op\ coupled to strain, and the determine the effect of its phase transitions
+on the elastic response in different symmetry channels. 
+
+We find that only one \op\ symmetry reproduces the anomalous $\Bog$ elastic
+modulus, which softens in a Curie-Weiss like manner from room temperature, but
+which cusps at T$_{\mathrm{HO}}$. That theory associates \ho\ with a $\Bog$
+\op\ \emph{modulated along the $c$- axis}, the \afm\ state with uniform $\Bog$
+order, and the triple point between them with a Lifshitz point. Besides the
+agreement with ultrasound data across a broad temperature range, the theory
+predicts uniform $\Bog$ strain at high pressure---the same distortion which was
+recently seen in x-ray scattering experiments
+\cite{choi_pressure-induced_2018}. This theory strongly motivates future
+ultrasound experiments under pressure approaching the Lifshitz point, which
+should find that the $(c_{11}-c_{12})/2$ diverges once the uniform $\Bog$
+strain sets in.
 
 
 \emph{Model.}
@@ -185,9 +211,12 @@ 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.
 
-If the \op\ transforms like $\Aog$ (e.g. a fluctuation in valence number), odd terms are allowed in its free energy and any transition will be first order and not continuous without fine-tuning. Since the \ho\ phase transition is second-order \brad{cite something}, we will henceforth rule out $\Aog$ \op s as well. 
-For the \op\ representation $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general
-quadratic free energy density is
+If the \op\ transforms like $\Aog$ (e.g. a fluctuation in valence number), odd
+terms are allowed in its free energy and any transition will be first order and
+not continuous without fine-tuning. Since the \ho\ phase transition is
+second-order \brad{cite something}, we will henceforth rule out $\Aog$ \op s as
+well.  For the \op\ representation $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the
+most general quadratic free energy density is
 \begin{equation}
   \begin{aligned}
     f_\op=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2
@@ -275,9 +304,10 @@ between the two components of the \op. In this case the uniform ordered phase
 is only stable for $c_\perp>0$, and the modulated phase is now characterized by
 helical order with $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$.
 The uniform to modulated transition is now continuous. This does not reproduce
-the physics of \ho, which has a first order transition between \ho\ and \afm, and so we will henceforth neglect
-the possibility of a multicomponent order parameter. The schematic phase
-diagrams for this model are shown in Figure~\ref{fig:phases}.
+the physics of \ho, which has a first order transition between \ho\ and \afm,
+and so we will henceforth neglect the possibility of a multicomponent order
+parameter. The schematic phase diagrams for this model are shown in
+Figure~\ref{fig:phases}.
 
 \emph{Results.}
 We will now derive the \emph{effective elastic tensor} $C$ that results from
@@ -295,10 +325,9 @@ The generalized susceptibility of a single component ($\Bog$ or $\Btg$) \op\ is
 \begin{equation}
   \begin{aligned}
     &\chi^\recip(x,x')
-    =\frac{\delta^2F[\eta,\epsilon_\star[\eta]]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\langle\eta\rangle}
+      =\frac{\delta^2F[\eta,\epsilon_\star[\eta]]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\langle\eta\rangle}
       =\big[\tilde r-c_\parallel\nabla_\parallel^2 \\
-    &\qquad\qquad-c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+12u\langle\eta(x)\rangle^2\big]
-    \delta(x-x'),
+        &\qquad\qquad-c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+12u\langle\eta(x)\rangle^2\big]\delta(x-x'),
   \end{aligned}
   \label{eq:sus_def}
 \end{equation}
@@ -308,7 +337,7 @@ where $\recip$ indicates a \emph{functional reciprocal} in the sense that
 \end{equation}
 Taking the Fourier transform and integrating over $q'$ we have
 \begin{equation}
-    \chi(q)
+  \chi(q)
     =\big(\tilde r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+D_\perp q_\perp^4
     +12u\sum_{q'}\langle\tilde\eta_{q'}\rangle\langle\tilde\eta_{-q'}\rangle\big)^{-1}.
 \end{equation}
@@ -316,10 +345,10 @@ Near the unordered to modulated transition this yields
 \begin{equation}
   \begin{aligned}
     \chi(q)
-    &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
-      +|\Delta\tilde r|} \\
-    &=\frac1{D_\perp}\frac{\xi_\perp^4}
-      {1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2},
+      &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
+        +|\Delta\tilde r|} \\
+      &=\frac1{D_\perp}\frac{\xi_\perp^4}
+        {1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2},
   \end{aligned}
   \label{eq:susceptibility}
 \end{equation}
@@ -327,7 +356,7 @@ with $\xi_\perp=(|\Delta\tilde r|/D_\perp)^{-1/4}=\xi_{\perp0}|t|^{-1/4}$ and
 $\xi_\parallel=(|\Delta\tilde
 r|/c_\parallel)^{-1/2}=\xi_{\parallel0}|t|^{-1/2}$, where $t=(T-T_c)/T_c$ is
 the reduced temperature and $\xi_{\perp0}=(D_\perp/aT_c)^{1/4}$ and
-$\xi_{\parallel0}=(c_\parallel/aT_c)^{1/2}$ are the bare correlation lengths \brad{needs a descriptor like "in and perpendicular to the x-y plane" or something like that}.
+$\xi_{\parallel0}=(c_\parallel/aT_c)^{1/2}$ are the bare correlation lengths.
 Notice that the static susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde
 r|)^{-1}$ does not diverge at the unordered to modulated transition. Though it
 anticipates a transition with Curie--Weiss-like divergence at $\Delta\tilde
-- 
cgit v1.2.3-70-g09d2

