small changes, added draft title and abstract, gnuplot plots

12 files changed, 23471 insertions, 59 deletions

diff --git a/data/c11mc12.dat b/data/c11mc12.dat
new file mode 100644
index 0000000..3db97a0
--- /dev/null
+++ b/data/c11mc12.dat
@@ -0,0 +1,3868 @@
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\ No newline at end of file
diff --git a/data/c11pc12.dat b/data/c11pc12.dat
new file mode 100644
index 0000000..f96bbab
--- /dev/null
+++ b/data/c11pc12.dat
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\ No newline at end of file
diff --git a/data/c13.dat b/data/c13.dat
new file mode 100644
index 0000000..2827f96
--- /dev/null
+++ b/data/c13.dat
@@ -0,0 +1,3868 @@
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\ No newline at end of file
diff --git a/data/c33.dat b/data/c33.dat
new file mode 100644
index 0000000..02bf917
--- /dev/null
+++ b/data/c33.dat
@@ -0,0 +1,3868 @@
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\ No newline at end of file
diff --git a/data/c44.dat b/data/c44.dat
new file mode 100644
index 0000000..10d8c48
--- /dev/null
+++ b/data/c44.dat
@@ -0,0 +1,3868 @@
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\ No newline at end of file
diff --git a/data/c66.dat b/data/c66.dat
new file mode 100644
index 0000000..4ce92b5
--- /dev/null
+++ b/data/c66.dat
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\ No newline at end of file
diff --git a/fig-fit.gplot b/fig-fit.gplot
new file mode 100644
index 0000000..f35fde7
--- /dev/null
+++ b/fig-fit.gplot
@@ -0,0 +1,27 @@
+
+set terminal epslatex size 8.68cm, 5.5cm standalone
+set output "fig-fit.tex"
+
+set tmargin 0.3
+set lmargin 7
+set rmargin 1.5
+set bmargin 3
+
+set nokey
+set xrange [-25:300]
+set format x '\scriptsize $%g$'
+set format y '\scriptsize $%g$'
+
+set arrow 1 from 17.26,graph 0 to 17.26,graph 1 nohead lw 3 lc 'red' dt 3
+
+set xlabel '\footnotesize $T / \mathrm K$'
+set ylabel '\footnotesize $\tilde\lambda_{\mathrm B_{1\mathrm g}} / \mathrm{GPa}$' offset 2
+set yrange [65.05:65.7]
+
+plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc 'medium-blue'
+
+lam(T) = 71.1083 - 0.0103972 * T
+
+plot lam(x) / (1 + 6.24582 / (1 + 0.0037839 * abs(x - 17.26)) / lam(x)) dt 4 lc 'goldenrod' lw 4
+
+
diff --git a/fig-fit.pdf b/fig-fit.pdf
new file mode 100644
index 0000000..592942e
--- /dev/null
+++ b/fig-fit.pdf
diff --git a/fig-stiffnesses.gplot b/fig-stiffnesses.gplot
new file mode 100644
index 0000000..15ae95c
--- /dev/null
+++ b/fig-stiffnesses.gplot
@@ -0,0 +1,58 @@
+
+set terminal epslatex size 8.68cm, 7.5cm standalone
+set output "fig-stiffnesses.tex"
+
+set multiplot  layout 3, 2  margins 0.185, 0.97, 0.15, 0.998  spacing 0.13, 0.015
+
+set nokey
+set xrange [-25:300]
+set arrow 1 from 17.26,graph 0 to 17.26,graph 1 nohead lw 3 lc 'red' dt 3
+
+
+set format x ""
+set xtics (0, 100, 200, 300)
+set ylabel '\footnotesize $\tilde\lambda / \mathrm{GPa}$' offset 3
+set format y '\scriptsize $%.1f$'
+
+set yrange [217.5:220.75]
+set ytics 218,1,220
+set title '\footnotesize (a) $\tilde\lambda^{(11)}_{\mathrm{A_{1\mathrm g}}}$' offset -1,-5.2
+plot "data/c11pc12.dat" using 1:(100 * $2) with lines lw 3
+
+set ylabel ''
+
+set yrange [306:317]
+set ytics 308,2,316
+set title '\footnotesize (b) $\tilde\lambda^{(22)}_{\mathrm{A_{1\mathrm g}}}$' offset -1,-5.2
+plot "data/c33.dat"     using 1:(100 * $2) with lines lw 3
+
+set ylabel '\footnotesize $\tilde\lambda / \mathrm{GPa}$'
+
+set yrange [112.6:114]
+set ytics 112.8,0.3,113.9
+set title '\footnotesize (d) $\tilde\lambda^{(12)}_{\mathrm{A_{1\mathrm g}}}$' offset -1,-5.2
+plot "data/c13.dat"     using 1:(100 * $2) with lines lw 3
+
+set ylabel ''
+
+set yrange [65.05:65.7]
+set ytics 62.1,0.1,65.6
+set title '\footnotesize (c) $\tilde\lambda^{(11)}_{\mathrm{B_{1\mathrm g}}}$' offset -1,-5.2
+plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3
+
+set format x '\scriptsize $%.0f$'
+set ylabel '\footnotesize $\tilde \lambda / \mathrm{GPa}$'
+set xlabel '\footnotesize $T / \mathrm K$'
+
+set yrange [140:145]
+set ytics 141,1,144
+set title '\footnotesize (e) $\tilde\lambda^{(11)}_{\mathrm{B_{2\mathrm g}}}$' offset -1,-5.2
+plot "data/c66.dat"     using 1:(100 * $2) with lines lw 3
+
+set ylabel ''
+
+set yrange [101:106]
+set ytics 102,1,105
+set title '\footnotesize (f) $\tilde\lambda^{(11)}_{\mathrm{E_{\mathrm g}}}$' offset -1,-5.2
+plot "data/c44.dat"     using 1:(100 * $2) with lines lw 3
+
diff --git a/fig-stiffnesses.pdf b/fig-stiffnesses.pdf
new file mode 100644
index 0000000..9baa3d1
--- /dev/null
+++ b/fig-stiffnesses.pdf
diff --git a/hidden_order.bib b/hidden_order.bib
index 9d289d5..3057a7c 100644
--- a/hidden_order.bib
+++ b/hidden_order.bib
@@ -26,19 +26,20 @@
 }
 
 @article{fisher_specific_1990,
-  title = {Specific Heat of {{URu}}{$_{2}$}{{Si}}{$_{2}$}: {{Effect}} of Pressure and Magnetic Field on the Magnetic and Superconducting Transitions},
+  title = {Specific Heat of {{URu}}{$_{2}$}{{Si}}{$_2$}: {{Effect}} of Pressure and Magnetic Field on the Magnetic and Superconducting Transitions},
   volume = {163},
   issn = {0921-4526},
   shorttitle = {Specific Heat of {{URu2Si2}}},
   abstract = {Specific heats were measured in the range 0.3 {$\leqslant$}T{$\leqslant$}30 K for 0{$\leqslant$}H{$\leqslant$}7T and P=0, and for H=0 and 0{$\leqslant$}P{$\leqslant$}6.3 kbar. For H=0 and P=0, the measurements were extended to 0.15K. Above the superconducting transition the H=0 and 7T data can be superimposed. For the magnetic transition near T0 = 18K, T0 increased with increasing P accompanied by a broadening and attenuation of the specific heat anomally. The superconducting transition near Tc = 1.5 K was broadened, attenuated and shifted to lower temperatures for both increasing P and H. The superconducting transition is similar to that of UPt3, and both the temperature dependence of the superconducting state specific heat and the derived parameters are consistent with an unconventional polar-type pairing.},
   number = {1},
   journal = {Physica B: Condensed Matter},
-  doi = {10/ck6dj9},
+  doi = {10.1016/0921-4526(90)90229-n},
   author = {Fisher, R. A. and Kim, S. and Wu, Y. and Phillips, N. E. and McElfresh, M. W. and Torikachvili, M. S. and Maple, M. B.},
   month = apr,
   year = {1990},
+  keywords = {_tablet},
   pages = {419-423},
-  file = {/home/pants/.zotero/data/storage/HHVDKMSP/Fisher et al. - 1990 - Specific heat of URu2Si2 Effect of pressure and m.pdf}
+  file = {/home/pants/.zotero/data/storage/HHVDKMSP/Fisher et al. - 1990 - Specific heat of URu₂Si₂ Effect of pressure and m.pdf}
 }
 
 @article{hornreich_lifshitz_1980,
@@ -52,6 +53,7 @@
   author = {Hornreich, R. M.},
   month = jan,
   year = {1980},
+  keywords = {_tablet},
   pages = {387-392},
   file = {/home/pants/.zotero/data/storage/FQWHY9TF/Hornreich - 1980 - The Lifshitz point Phase diagrams and critical be.pdf}
 }
@@ -62,7 +64,7 @@
   journal = {Proceedings of the USSR Academy of Sciences Journal of Physics},
   author = {Lifshitz, EM},
   year = {1942},
-  keywords = {⛔ No DOI found},
+  keywords = {_tablet,⛔ No DOI found},
   pages = {251},
   file = {/home/pants/.zotero/data/storage/TAA9G46H/Lifshitz - 1942 - On the theory of phase transitions of the second o.pdf}
 }
@@ -73,7 +75,7 @@
   journal = {Proceedings of the USSR Academy of Sciences Journal of Physics},
   author = {Lifshitz, EM},
   year = {1942},
-  keywords = {⛔ No DOI found},
+  keywords = {_tablet,⛔ No DOI found},
   pages = {61},
   file = {/home/pants/.zotero/data/storage/D9BYG3FK/Lifshitz - 1942 - On the theory of phase transitions of the second o.pdf}
 }
@@ -86,7 +88,8 @@
   author = {Ginzburg, V. L.},
   year = {1961},
   keywords = {⛔ No DOI found},
-  pages = {1824-1834}
+  pages = {1824-1834},
+  file = {/home/pants/.zotero/data/storage/JVMTIZGB/Ginzburg - 1961 - Some Remarks on Phase Transitions of the Second Ki.pdf}
 }
 
 
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}