From 0d6994fb5b1a40b3dbf12c6e4cf47bbf3b8e82c4 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 22 Jul 2019 21:30:07 -0400
Subject: small changes, added draft title and abstract, gnuplot plots

---
 main.tex | 222 ++++++++++++++++++++++++++++++++++++++++++++++++---------------
 1 file changed, 169 insertions(+), 53 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 2dac86f..de38d17 100644
--- a/main.tex
+++ b/main.tex
@@ -1,3 +1,4 @@
+
 \documentclass[aps,prl,reprint]{revtex4-1}
 \usepackage[utf8]{inputenc}
 \usepackage{amsmath,graphicx,upgreek,amssymb}
@@ -39,16 +40,27 @@
 
 \begin{document}
 
-\title{\urusi mft}
+\title{Elastic properties of \urusi are reproduced by modulated $\Bog$ order}
 \author{Jaron Kent-Dobias}
 \author{Michael Matty}
 \author{Brad Ramshaw}
-\affiliation{Laboratory of Atomic \& Solid State Physics, Cornell University, Ithaca, NY, USA}
+\affiliation{
+  Laboratory of Atomic \& Solid State Physics, Cornell University,
+  Ithaca, NY, USA
+}
 
 \date\today
 
 \begin{abstract}
-  blah blah blah its-a abstract
+  We develop a phenomenological theory for the elastic response of materials
+  with a \Dfh\ point group through phase transitions. The physics is
+  generically that of Lifshitz points, with disordered, uniform ordered, and
+  modulated ordered phases. Several experimental features of \urusi are
+  reproduced when the order parameter has $\Bog$ symmetry: the topology of the
+  temperature--pressure phase diagram, the response of the strain stiffness
+  tensor above the hidden-order transition, and the strain response in the
+  antiferromagnetic phase. In this scenario, the hidden order is a version of
+  the high-pressure antiferromagnetic order modulated along the symmetry axis.
 \end{abstract}
 
 \maketitle
@@ -104,38 +116,58 @@ of an OP coupled to strain and the effect that the OP symmetry has on the elasti
 in different symmetry channels. Our study finds that a single possible OP symmetry
 reproduces the experimental strain susceptibilities, and fits the experimental data well.
 
-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 order parameter, and their interaction. The most general quadratic free energy of the strain $\epsilon$ is $f_\e=\lambda_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the $\lambda$ 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
+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 order parameter, and their interaction.
+The most general quadratic free energy of the strain $\epsilon$ is
+$f_\e=\lambda_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the $\lambda$
+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
 \begin{equation}
   \begin{aligned}
     \epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth}
     \epsilon_\Aog^{(2)}=\epsilon_{33}               \\
     \epsilon_\Bog^{(1)}=\epsilon_{11}-\epsilon_{22} &&
-    \epsilon_\Btg^{(1)}=\epsilon_{12}               \\
-    \epsilon_\Eg^{(1)} =\{\epsilon_{11},\epsilon_{22}\}.
+    \epsilon_\Btg^{(1)}=2\epsilon_{12}               \\
+    \epsilon_\Eg^{(1)}=2\{\epsilon_{11},\epsilon_{22}\}.
   \end{aligned}
 \end{equation}
-Next, all quadratic combinations of these irreducible strains that transform like $\Aog$ are included in the free energy as
+Next, all quadratic combinations of these irreducible strains that transform
+like $\Aog$ are included in the free energy as
 \begin{equation}
   f_\e=\frac12\sum_\X\lambda_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)},
 \end{equation}
-where the sum is over irreducible representations of the point group and the $\lambda_\X^{(ij)}$ are
+where the sum is over irreducible representations of the point group and the
+stiffnesses $\lambda_\X^{(ij)}$ are
 \begin{equation}
   \begin{aligned}
     &\lambda_{\Aog}^{(11)}=\tfrac12(\lambda_{1111}+\lambda_{1122}) &&
     \lambda_{\Aog}^{(22)}=\lambda_{3333} \\
     &\lambda_{\Aog}^{(12)}=\lambda_{1133} &&
     \lambda_{\Bog}^{(11)}=\tfrac12(\lambda_{1111}-\lambda_{1122}) \\
-    &\lambda_{\Btg}^{(11)}=4\lambda_{1212} &&
-    \lambda_{\Eg}^{(11)}=4\lambda_{1313}.
+    &\lambda_{\Btg}^{(11)}=\lambda_{1212} &&
+    \lambda_{\Eg}^{(11)}=\lambda_{1313}.
   \end{aligned}
 \end{equation}
-The interaction between strain and the order parameter $\eta$ 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 the order parameter $\eta$ 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
 \begin{equation}
   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. We will therefore focus our attention on order parameter symmetries that produce linear couplings to strain.
+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. We will therefore focus
+our attention on order parameter symmetries that produce linear couplings to
+strain.
 
-If the order parameter transforms like $\Aog$, odd terms are allow in its free energy and any transition will be abrupt and not continuous without tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quartic free energy density is
+If the order parameter transforms like $\Aog$, odd terms are allowed in its
+free energy and any transition will be abrupt and not continuous without
+tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quartic
+free energy density is
 \begin{equation}
   \begin{aligned}
     f_\o=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2
@@ -145,29 +177,56 @@ If the order parameter transforms like $\Aog$, odd terms are allow in its free e
   \end{aligned}
   \label{eq:fo}
 \end{equation}
-where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$ and $\nabla_\perp=\partial_3$ transforms like $\Atu$. We'll take $D_\parallel=0$ since this does not affect the physics at hand. Neglecting interaction terms higher than quadratic order, the only strain relevant to the problem is $\epsilon_\X$, and this can be traced out of the problem exactly, since
+where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$ and
+$\nabla_\perp=\partial_3$ transforms like $\Atu$. We'll take $D_\parallel=0$
+since this does not affect the physics at hand. Neglecting interaction terms
+higher than quadratic order, the only strain relevant to the problem is
+$\epsilon_\X$, and this can be traced out of the problem exactly, since
 \begin{equation}
-  0=\frac{\delta F}{\delta\epsilon_{\X i}(x)}=\lambda_\X\epsilon_{\X i}(x)+\frac12b\eta_i(x)
+  0=\frac{\delta F}{\delta\epsilon_{\X i}(x)}=\lambda_\X\epsilon_{\X i}(x)
+    +\frac12b\eta_i(x)
 \end{equation}
-gives $\epsilon_\X(x)=-(b/2\lambda_\X)\eta(x)$. Upon substitution into the free energy, tracing out $\epsilon_\X$ has the effect of shifting $r$ in $f_\o$, with $r\to\tilde r=r-b^2/4\lambda_\X$.
+gives $\epsilon_\X(x)=-(b/2\lambda_\X)\eta(x)$. Upon substitution into the free
+energy, tracing out $\epsilon_\X$ has the effect of shifting $r$ in $f_\o$,
+with $r\to\tilde r=r-b^2/4\lambda_\X$.
 
-With the strain traced out \eqref{eq:fo} describes the theory of a Lifshitz point at $\tilde r=c_\perp=0$ \cite{lifshitz_theory_1942, lifshitz_theory_1942-1}. For a scalar order parameter ($\Bog$ or $\Btg$) it is traditional to make the field ansatz $\eta(x)=\eta_*\cos(q_*x_3)$. 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. 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
+With the strain traced out \eqref{eq:fo} describes the theory of a Lifshitz
+point at $\tilde r=c_\perp=0$ \cite{lifshitz_theory_1942,
+lifshitz_theory_1942-1}. For a scalar order parameter ($\Bog$ or $\Btg$) it is
+traditional to make the field ansatz $\eta(x)=\eta_*\cos(q_*x_3)$. 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. 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}=\frac{\tilde r_c-\tilde r}{3u}
+  \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{12D_\perp u}
+    =\frac{\tilde r_c-\tilde r}{3u}
 \end{equation}
-with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The transition between the uniform and modulated orderings is abrupt for a scalar field and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde r/5}$. For a vector order parameter ($\Eg$) we must also allow a relative phase between the two components of the field. 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 $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$ and
+with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The
+transition between the uniform and modulated orderings is abrupt for a scalar
+field and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde r/5}$. For a
+vector order parameter ($\Eg$) we must also allow a relative phase between the
+two components of the field. 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 $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$ and
 \begin{equation}
-  \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{16D_\perp u}=\frac{\tilde r_c-\tilde r}{4u}
+  \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{16D_\perp u}
+    =\frac{\tilde r_c-\tilde r}{4u}
 \end{equation}
-The uniform--modulated transition is now continuous. The schematic phase diagrams for this model are shown in Figure \ref{fig:phases}.
+The uniform--modulated transition is now continuous. The schematic phase
+diagrams for this model are shown in Figure \ref{fig:phases}.
 
 \begin{figure}[htpb]
   \includegraphics[width=0.51\columnwidth]{phases_scalar}\hspace{-1.5em}
   \includegraphics[width=0.51\columnwidth]{phases_vector}
-  \caption{Schematic phase diagrams for this model. Solid lines denote
-    continuous transitions, while dashed lines indicated abrupt transitions. (a)
-    The phases for a scalar ($\Bog$ or $\Btg$). (b) The phases for a vector
-    ($\Eg$).}
+  \caption{
+    Schematic phase diagrams for this model. Solid lines denote continuous
+    transitions, while dashed lines indicated abrupt transitions. (a) The
+    phases for a scalar ($\Bog$ or $\Btg$). (b) The phases for a vector
+    ($\Eg$).
+  }
   \label{fig:phases}
 \end{figure}
 
@@ -176,7 +235,8 @@ The susceptibility is given by
   \begin{aligned}
     &\chi_{ij}^{-1}(x,x')
     =\frac{\delta^2F}{\delta\eta_i(x)\delta\eta_j(x')} \\
-    &\quad=\Big[\big(\tilde r-c_\parallel\nabla_\parallel^2-c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+4u\eta^2(x)\big)\delta_{ij} \\
+    &\quad=\Big[\big(\tilde r-c_\parallel\nabla_\parallel^2
+      -c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+4u\eta^2(x)\big)\delta_{ij} \\
     &\qquad\qquad +8u\eta_i(x)\eta_j(x)\Big]\delta(x-x'),
   \end{aligned}
 \end{equation}
@@ -184,27 +244,37 @@ or in Fourier space,
 \begin{equation}
   \begin{aligned}
     \chi_{ij}^{-1}(q)
-      &=8u\sum_{q'}\tilde\eta_i(q')\eta_j(-q')+\bigg(\tilde r+c_\parallel q_\parallel^2-c_\perp q_\perp^2 \\
-      &\qquad+D_\perp q_\perp^4+4u\sum_{q'}\tilde\eta_k(q')\tilde\eta_k(-q')\bigg)\delta_{ij}.
+      &=8u\sum_{q'}\tilde\eta_i(q')\eta_j(-q')+\bigg(\tilde r
+        +c_\parallel q_\parallel^2-c_\perp q_\perp^2 \\
+      &\qquad+D_\perp q_\perp^4+4u\sum_{q'}\tilde\eta_k(q')\tilde\eta_k(-q')\bigg)
+        \delta_{ij}.
   \end{aligned}
 \end{equation}
 Near the unordered--modulated transition this yields
 \begin{equation}
   \begin{aligned}
-  \chi(q)
-  &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2+|\tilde r-\tilde r_c|} \\
-  &=\frac1{D_\perp}\frac{\xi_\perp^4}{1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2},
+    \chi_{ij}(q)
+    &=\frac{\delta_{ij}}{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
+      +|\tilde r-\tilde r_c|} \\
+    &=\frac{\delta_{ij}}{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}
-with $\xi_\perp=(|\tilde r-\tilde r_c|/D_\perp)^{-1/4}$ and $\xi_\parallel=(|\tilde r-\tilde r_c|/c_\parallel)^{-1/2}$.
+with $\xi_\perp=(|\tilde r-\tilde r_c|/D_\perp)^{-1/4}$ and
+$\xi_\parallel=(|\tilde r-\tilde r_c|/c_\parallel)^{-1/2}$.
 
-The elastic susceptibility (inverse stiffness) is given in the same way: we must trace over $\eta$ and take the second variation of the resulting free energy. Extremizing over $\eta$ yields
+The elastic susceptibility (inverse stiffness) is given in the same way: we
+must trace over $\eta$ and take the second variation of the resulting free
+energy. Extremizing over $\eta$ yields
 \begin{equation}
-  0=\frac{\delta F}{\delta\eta_i(x)}=\frac{\delta F_\o}{\delta\eta_i(x)}+\frac12b\epsilon_{\X i}(x),
+  0=\frac{\delta F}{\delta\eta_i(x)}=\frac{\delta F_\o}{\delta\eta_i(x)}
+    +\frac12b\epsilon_{\X i}(x),
   \label{eq:implicit.eta}
 \end{equation}
-which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this cannot be solved explicitly, we can make use of the inverse function theorem to write
+which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this
+cannot be solved explicitly, we can make use of the inverse function theorem to
+write
 \begin{equation}
   \begin{aligned}
     \bigg(\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}\bigg)^{-1}
@@ -214,7 +284,8 @@ which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this cann
   \end{aligned}
   \label{eq:inv.func}
 \end{equation}
-It follows from \eqref{eq:implicit.eta} and \eqref{eq:inv.func} that the susceptibility of the material to $\epsilon_\X$ strain is given by
+It follows from \eqref{eq:implicit.eta} and \eqref{eq:inv.func} that the
+susceptibility of the material to $\epsilon_\X$ strain is given by
 \begin{widetext}
 \begin{equation}
   \begin{aligned}
@@ -241,16 +312,13 @@ whose Fourier transform follows from \eqref{eq:inv.func} as
   \chi_{\X ij}(q)=\frac{\delta_{ij}}{\lambda_\X}+\frac{b^2}{4\lambda_\X^2}\chi_{ij}(q).
   \label{eq:elastic.susceptibility}
 \end{equation}
-At $q=0$, which is where the stiffness measurements used here were taken, this predicts a cusp in the elastic susceptibility of the form $|\tilde r-\tilde r_c|^\gamma$ for $\gamma=1$.
+At $q=0$, which is where the stiffness measurements used here were taken, this
+predicts a cusp in the elastic susceptibility of the form $|\tilde r-\tilde
+r_c|^\gamma$ for $\gamma=1$.
 
 \begin{figure}[htpb]
   \centering
-  \includegraphics[width=0.49\columnwidth]{stiff_a11.pdf}
-  \includegraphics[width=0.49\columnwidth]{stiff_a22.pdf}
-  \includegraphics[width=0.49\columnwidth]{stiff_a12.pdf}
-  \includegraphics[width=0.49\columnwidth]{stiff_b1.pdf}
-  \includegraphics[width=0.49\columnwidth]{stiff_b2.pdf}
-  \includegraphics[width=0.49\columnwidth]{stiff_e.pdf}
+  \includegraphics[width=\columnwidth]{fig-stiffnesses}
   \caption{
     Measurements of the effective strain stiffness as a function of temperature
     for the six independent components of strain from ultrasound. The vertical
@@ -259,40 +327,88 @@ At $q=0$, which is where the stiffness measurements used here were taken, this p
   \label{fig:data}
 \end{figure}
 
-We have seen that mean field theory predicts that whatever component of strain transforms like the order parameter will see a $t^{-1}$ softening in the stiffness that ends in a cusp. Ultrasound experiments \textbf{[Elaborate???]} yield the strain stiffness for various components of the strain; this data is shown in Figure \ref{fig:data}.  The $\Btg$ and $\Eg$ stiffnesses don't appear to have any response to the presence of the transition, exhibiting the expected linear stiffening with a low-temperature cutoff \textbf{[What's this called? Citation?]}. The $\Bog$ stiffness has a dramatic response, softening over the course of roughly $100\,\K$. There is a kink in the curve right at the transition. While the low-temperature response is not as dramatic as the theory predicts, mean field theory---which is based on a small-$\eta$ expansion---will not work quantitatively far below the transition where $\eta$ has a large nonzero value and higher powers in the free energy become important. The data in the high-temperature phase can be fit to the theory \eqref{eq:elastic.susceptibility}, with a linear background stiffness $\lambda_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is shown in Figure \ref{fig:fit}. The data and theory appear consistent.
+We have seen that mean field theory predicts that whatever component of strain
+transforms like the order parameter will see a $t^{-1}$ softening in the
+stiffness that ends in a cusp. Ultrasound experiments \textbf{[Elaborate???]}
+yield the strain stiffness for various components of the strain; this data is
+shown in Figure \ref{fig:data}.  The $\Btg$ and $\Eg$ stiffnesses don't appear
+to have any response to the presence of the transition, exhibiting the expected
+linear stiffening with a low-temperature cutoff \textbf{[What's this called?
+Citation?]}. The $\Bog$ stiffness has a dramatic response, softening over the
+course of roughly $100\,\K$. There is a kink in the curve right at the
+transition. While the low-temperature response is not as dramatic as the theory
+predicts, mean field theory---which is based on a small-$\eta$ expansion---will
+not work quantitatively far below the transition where $\eta$ has a large
+nonzero value and higher powers in the free energy become important. The data
+in the high-temperature phase can be fit to the theory
+\eqref{eq:elastic.susceptibility}, with a linear background stiffness
+$\lambda_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is
+shown in Figure \ref{fig:fit}. The data and theory appear consistent.
 
 \begin{figure}[htpb]
-  \includegraphics[width=\columnwidth]{cusp}
+  \includegraphics[width=\columnwidth]{fig-fit}
   \caption{
     Strain stiffness data for the $\Bog$ component of strain (solid) along with
-    a fit of \eqref{eq:elastic.susceptibility} to the data above $T_c$ (dashed). The fit gives $\lambda_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp q_*^4\simeq0.0038\,\K^{-1}$.
+    a fit of \eqref{eq:elastic.susceptibility} to the data above $T_c$
+    (dashed). The fit gives
+    $\lambda_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$,
+    $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp
+    q_*^4\simeq0.0038\,\K^{-1}$.
   }
   \label{fig:fit}
 \end{figure}
 
-Mean field theory neglects the effect of fluctuations on critical behavior, yet also predicts the magnitude of those fluctuations. This allows a mean field theory to undergo an internal consistency check to ensure the predicted fluctuations are indeed negligible. This is typically done by computing the Ginzburg criterion \cite{ginzburg_remarks_1961}, which gives the proximity to the critical point $t=(T-T_c)/T_c$ at which mean field theory is expected to break down by comparing the magnitude of fluctuations in a correlation-length sized box to the magnitude of the field, or since the correlation function is $k_BT\chi(x,x')$,
+Mean field theory neglects the effect of fluctuations on critical behavior, yet
+also predicts the magnitude of those fluctuations. This allows a mean field
+theory to undergo an internal consistency check to ensure the predicted
+fluctuations are indeed negligible. This is typically done by computing the
+Ginzburg criterion \cite{ginzburg_remarks_1961}, which gives the proximity to
+the critical point $t=(T-T_c)/T_c$ at which mean field theory is expected to
+break down by comparing the magnitude of fluctuations in a correlation-length
+sized box to the magnitude of the field, or since the correlation function is
+$k_BT\chi(x,x')$,
 \begin{equation}
   V_\xi^{-1}k_BT\int_{V_\xi}d^3x\,\chi(x,0)
   =\langle\delta\eta^2\rangle_{V_\xi}
   \lesssim\frac12\eta_*^2=\frac{|\Delta\tilde r|}{6u}
 \end{equation}
-with $V_\xi$ the correlation volume, which we will take to be a cylinder of radius $\xi_\parallel/2$ and height $\xi_\perp$. Upon substitution of \eqref{eq:susceptibility} and using the jump in the specific heat at the transition from
+with $V_\xi$ the correlation volume, which we will take to be a cylinder of
+radius $\xi_\parallel/2$ and height $\xi_\perp$. Upon substitution of
+\eqref{eq:susceptibility} and using the jump in the specific heat at the
+transition from
 \begin{equation}
-  c_V=-T\frac{\partial^2f}{\partial T^2}=\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c,\end{cases}
+  c_V=-T\frac{\partial^2f}{\partial T^2}
+    =\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c,\end{cases}
 \end{equation}
 this expression can be brought to the form
 \begin{equation}
-  \frac{2k_B}{\pi\Delta c_V\xi_{\perp0}\xi_{\parallel0}^2}\mathcal I(\xi_{\perp0} q_*|t|^{-1/4})\lesssim |t|^{13/4},
+  \frac{2k_B}{\pi\Delta c_V\xi_{\perp0}\xi_{\parallel0}^2}
+    \mathcal I(\xi_{\perp0} q_*|t|^{-1/4})
+  \lesssim |t|^{13/4},
 \end{equation}
-where $\xi=\xi_0t^{-\nu}$ defines the bare correlation lengths and $\mathcal I$ is defined by
+where $\xi=\xi_0t^{-\nu}$ defines the bare correlation lengths and $\mathcal I$
+is defined by
 \begin{equation}
-  \mathcal I(x)=\frac1\pi\int_{-\infty}^\infty dy\,\frac{\sin\tfrac y2}y\bigg(\frac1{1+(y^2-x^2)^2}-\frac{K_1(\sqrt{1+(y^2-x^2)^2})}{\sqrt{1+(y^2-x^2)^2}}\bigg)
+  \mathcal I(x)=\frac1\pi\int_{-\infty}^\infty dy\,\frac{\sin\tfrac y2}y
+    \bigg(\frac1{1+(y^2-x^2)^2}
+      -\frac{K_1(\sqrt{1+(y^2-x^2)^2})}{\sqrt{1+(y^2-x^2)^2}}\bigg)
 \end{equation}
 For large argument, $\mathcal I(x)\sim x^{-4}$, yielding
 \begin{equation}
   t_\G^{9/4}\sim\frac{2k_B}{\pi\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}^5q_*^4}
 \end{equation}
-Experiments give $\Delta c_V\simeq1\times10^5\,\J\,\m^{-3}\,\K^{-1}$ \cite{fisher_specific_1990}, and our fit above gives $\xi_{\perp0}q_*=(D_\perp q_*^4/aT_c)^{1/4}\sim2$. We have reason to believe that at zero pressure, very far from the Lifshitz point, $q_*$ is roughly the inverse lattice spacing \textbf{[Why???]}. Further supposing that $\xi_{\parallel0}\simeq\xi_{\perp0}$, we find $t_\G\sim0.04$, so that an experiment would need to be within $\sim1\,\K$ to detect a deviation from mean field behavior. An ultrasound experiment able to capture data over several decades within this vicinity of $T_c$ may be able to measure a cusp with $|t|^\gamma$ for $\gamma=\text{\textbf{???}}$, the empirical exponent \textbf{[Citation???]}. Our analysis has looked at behavior for $T-T_c>1\,\K$, and so it remains self-consistent.
+Experiments give $\Delta c_V\simeq1\times10^5\,\J\,\m^{-3}\,\K^{-1}$
+\cite{fisher_specific_1990}, and our fit above gives $\xi_{\perp0}q_*=(D_\perp
+q_*^4/aT_c)^{1/4}\sim2$. We have reason to believe that at zero pressure, very
+far from the Lifshitz point, $q_*$ is roughly the inverse lattice spacing
+\textbf{[Why???]}. Further supposing that $\xi_{\parallel0}\simeq\xi_{\perp0}$,
+we find $t_\G\sim0.04$, so that an experiment would need to be within
+$\sim1\,\K$ to detect a deviation from mean field behavior. An ultrasound
+experiment able to capture data over several decades within this vicinity of
+$T_c$ may be able to measure a cusp with $|t|^\gamma$ for
+$\gamma=\text{\textbf{???}}$, the empirical exponent \textbf{[Citation???]}.
+Our analysis has looked at behavior for $T-T_c>1\,\K$, and so it remains
+self-consistent.
 
 \begin{acknowledgements}
 
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