lots of progress on writing the theory sections

13 files changed, 276 insertions, 30 deletions

diff --git a/.gitignore b/.gitignore
index a43f0be..3108da8 100644
--- a/.gitignore
+++ b/.gitignore
@@ -5,5 +5,6 @@
 *.fls
 *.log
 *.synctex.gz
-*.pdf
+main.pdf
+hidden-order.pdf
 mainNotes.bib
diff --git a/cusp.pdf b/cusp.pdf
index e566319..b0500ef 100644
--- a/cusp.pdf
+++ b/cusp.pdf
diff --git a/hidden-order.tex b/hidden-order.tex
index 364602e..b326d0d 100644
--- a/hidden-order.tex
+++ b/hidden-order.tex
@@ -69,7 +69,8 @@ Consider a generic order parameter $\eta$. To write down its free energy, we in
 \[
   f_{\X}=\tfrac12b^{(i)}\epsilon_{\X}^{(i)}\cdot\eta
   +\tfrac12e^{(i)}\epsilon_{\Aog}^{(i)}\eta^2
-V\]
+  +\tfrac12h^{(ij)}\epsilon_{\Aog}^{(i)}\epsilon_{\X}^{(j)}\eta
+\]
 The total  free energy is 
 \[
   F=\int d^3x\,(f_{\mathrm e}+f_{\mathrm o}+f_{\X})
@@ -90,12 +91,12 @@ The most general quartic free energy density (discounting total derivatives) is
 independent of the symmetry of $\eta$. In principle we could have $D_\parallel\neq D_\perp$, but this does not affect the physics at hand. This is the free energy for a Lifshitz point, and so we expect to see that phenomenology in $\eta$. 
 Before doing anything, we can minimize the free energy with respect to strain alone to find the strain in terms of $\eta$ exactly. We have
 \[
-  0=\frac{\delta F}{\delta\epsilon_{\mathrm X}^{(1)}(x)}=\lambda_{\mathrm X}^{(11)}\epsilon_{\mathrm X}^{(1)}(x)+\frac12b^{(1)}\eta(x)
+  0=\frac{\delta F}{\delta\epsilon_{\mathrm X}^{(1)}(x)}=\lambda_{\mathrm X}^{(11)}\epsilon_{\mathrm X}^{(1)}(x)+\frac12b^{(1)}\eta(x)+\frac12h^{(i)}\epsilon_{\Aog}^{(i)}\eta
 \]
 whence we immediately have $\epsilon_{\mathrm X}^{(1)}=-\frac{b^{(1)}}{2\lambda_{\mathrm X}^{(11)}}\eta(x)$. We also have
 \[
   0=\frac{\delta F}{\delta\epsilon_{\Aog}^{(i)}(x)}
-  =\lambda_{\Aog}^{(ij)}\epsilon_{\Aog}^{(j)}(x)+\frac12 e^{(i)}\eta^2(x)
+  =\lambda_{\Aog}^{(ij)}\epsilon_{\Aog}^{(j)}(x)+\frac12 e^{(i)}\eta^2(x)+\frac12h^{(i)}\epsilon_\X\eta
 \]
 which is a linear system whose solutions are
 \begin{align*}
diff --git a/hidden_order.bib b/hidden_order.bib
index 9822fa0..8cbe971 100644
--- a/hidden_order.bib
+++ b/hidden_order.bib
@@ -1,18 +1,64 @@
 
+@book{landau_theory_1995,
+  series = {Landau and {{Lifshitz Course}} of {{Theoretical Physics}}},
+  title = {Theory of {{Elasticity}}},
+  author = {Landau, Lev Davidovich and Lifshitz, Eugin M and Berestetskii, VB and Pitaevskii, LP},
+  year = {1995},
+  keywords = {_tablet},
+  file = {/home/pants/.zotero/data/storage/AQ7G8AHB/Landau et al. - 1995 - Theory of Elasticity.pdf}
+}
+
 @article{fisher_specific_1990,
-	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 {URu}2Si2},
-	url = {http://www.sciencedirect.com/science/article/pii/092145269090229N},
-	doi = {10/ck6dj9},
-	abstract = {Specific heats were measured in the range 0.3 ?T?30 K for 0?H?7T and P=0, and for H=0 and 0?P?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},
-	urldate = {2019-06-17},
-	journal = {Physica B: Condensed Matter},
-	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},
-	pages = {419--423},
-	file = {Fisher et al. - 1990 - Specific heat of URu2Si2 Effect of pressure and m.pdf:/home/pants/.zotero/data/storage/HHVDKMSP/Fisher et al. - 1990 - Specific heat of URu2Si2 Effect of pressure and m.pdf:application/pdf}
-}
\ No newline at end of file
+  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},
+  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},
+  pages = {419-423},
+  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,
+  title = {The {{Lifshitz}} Point: {{Phase}} Diagrams and Critical Behavior},
+  volume = {15-18},
+  issn = {0304-8853},
+  shorttitle = {The {{Lifshitz}} Point},
+  abstract = {The Lifshitz multicritical point (LP) divides the phase diagram of a magnetic system into paramagnetic, uniform (ferro- or antiferromagnetic) and modulated (spiral or helicoidal) phases, which coexist at the LP. It can occur in a variety of different systems, including magnetic compounds and alloys, liquid crystals, charge-transfer salts, and structurally incommensurate materials. Theoretical studies, including renormalization group, exact spherical model and high temperature series expansion calculations, are reviewed with emphasis on possible experimental (including Monte Carlo) verifications of the theoretical predictions in three and two dimensional systems. Some promising materials for further research are indicated.},
+  journal = {Journal of Magnetism and Magnetic Materials},
+  doi = {10/ccgt88},
+  author = {Hornreich, R. M.},
+  month = jan,
+  year = {1980},
+  pages = {387-392},
+  file = {/home/pants/.zotero/data/storage/FQWHY9TF/Hornreich - 1980 - The Lifshitz point Phase diagrams and critical be.pdf}
+}
+
+@article{lifshitz_theory_1942-1,
+  title = {On the Theory of Phase Transitions of the Second Order {{II}}. {{Phase}} Transitions of the Second Order in Alloys},
+  volume = {6},
+  journal = {Proceedings of the USSR Academy of Sciences Journal of Physics},
+  author = {Lifshitz, EM},
+  year = {1942},
+  keywords = {⛔ 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}
+}
+
+@article{lifshitz_theory_1942,
+  title = {On the Theory of Phase Transitions of the Second Order {{I}}. {{Changes}} of the Elementary Cell of a Crystal in Phase Transitions of the Second Order},
+  volume = {6},
+  journal = {Proceedings of the USSR Academy of Sciences Journal of Physics},
+  author = {Lifshitz, EM},
+  year = {1942},
+  keywords = {⛔ 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}
+}
+
+
diff --git a/main.tex b/main.tex
index 3334127..794f30d 100644
--- a/main.tex
+++ b/main.tex
@@ -1,7 +1,31 @@
 \documentclass[aps,prl,reprint]{revtex4-2}
 \usepackage[utf8]{inputenc}
+\usepackage{amsmath,graphicx}
 
-\newcommand{\urusi}{URu$_2$Si$_2\ $}
+% Our mysterious boy
+\def\urusi{URu$_2$Si$_2\ $}
+
+\def\e{{\mathrm e}} % "elastic"
+\def\o{{\mathrm o}} % "order parameter"
+\def\i{{\mathrm i}} % "interaction"
+
+\def\Dfh{D$_{4\mathrm h}$}
+
+% Irreducible representations (use in math mode)
+\def\Aog{{\mathrm A_{1\mathrm g}}}
+\def\Atg{{\mathrm A_{2\mathrm g}}}
+\def\Bog{{\mathrm B_{1\mathrm g}}}
+\def\Btg{{\mathrm B_{2\mathrm g}}}
+\def\Eg {{\mathrm E_{ \mathrm g}}}
+\def\Aou{{\mathrm A_{1\mathrm u}}}
+\def\Atu{{\mathrm A_{2\mathrm u}}}
+\def\Bou{{\mathrm B_{1\mathrm u}}}
+\def\Btu{{\mathrm B_{2\mathrm u}}}
+\def\Eu {{\mathrm E_{ \mathrm u}}}
+
+% Variables to represent some representation
+\def\X{\mathrm X}
+\def\Y{\mathrm Y}
 
 \begin{document}
 
@@ -20,19 +44,193 @@
 \maketitle
 
 \begin{enumerate}
-    \item \urusi HO intro paragraph/discuss the phase diagram
-    \item Strain/OP coupling discussion/RUS
-    \item Discussion of experimental data
-    \item Analogy of lack of divergence/AFM w/ FM $\chi$
-    \item We look at MFT's for OP's of various symmetries
-    
-    \item Introduce various pieces of free energy
-    
-    \item MFT piece
+  \item Introduction
+    \begin{enumerate}
+      \item \urusi hidden order intro paragraph, discuss the phase diagram
+      \item Strain/OP coupling discussion/RUS
+      \item Discussion of experimental data
+      \item Analogy of lack of divergence/AFM w/ FM $\chi$
+      \item We look at MFT's for OP's of various symmetries
+    \end{enumerate}
+      
+  \item Theory
+    \begin{enumerate}
+      \item Introduce various pieces of free energy
+      
+      \item Summary of MFT results
+    \end{enumerate}
     
     \item Data piece
     
     \item Talk about more cool stuff like AFM C4 breaking etc
 \end{enumerate}
 
+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}\}.
+  \end{aligned}
+\end{equation}
+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
+\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}.
+  \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
+\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 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
+\begin{equation}
+  \begin{aligned}
+    f_\o=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2
+      +c_\perp(\nabla_\perp\eta)^2 \\
+      &\quad+D_\parallel(\nabla_\parallel^2\eta)^2
+      +D_\perp(\nabla_\perp^2\eta)^2\big]+u\eta^4
+  \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
+\begin{equation}
+  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 $r\to\tilde r=r-b^2/4\lambda_\X$.
+
+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, hornreich_lifshitz_1980}. 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}
+\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
+\begin{equation}
+  \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}.
+
+\begin{figure}
+  \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
+    ($\Eg$).}
+  \label{fig:phases}
+\end{figure}
+
+The susceptibility is given by
+\begin{equation}
+  \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} \\
+    &\qquad\qquad +8u\eta_i(x)\eta_j(x)\Big]\delta(x-x'),
+  \end{aligned}
+\end{equation}
+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}.
+  \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},
+  \end{aligned}
+\end{equation}
+with $\xi_\perp=(|\tilde r-\tilde r_c|/D_\perp)^{-1/4}$ and $\xi_\parallel=(|\tilde r-\tilde r_c|/c_\parallel)^{-1/2}$.
+
+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),
+  \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
+\begin{equation}
+  \begin{aligned}
+    \bigg(\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}\bigg)^{-1}
+    &=\frac{\delta\eta_j^{-1}[\eta](x)}{\delta\eta_i(x')}
+    =-\frac2b\frac{\delta^2F_\o}{\delta\eta_i(x)\delta\eta_j(x')} \\
+    &=-\frac2b\chi^{-1}(x,x')-\frac{b}{2\lambda_\X}\delta(x-x')
+  \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
+\begin{widetext}
+\begin{equation}
+  \begin{aligned}
+    \chi_{\X ij}^{-1}(x,x')
+    &=\frac{\delta^2F}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\
+    &=\lambda_\X\delta(x-x')+
+    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
+    +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_{\X i}(x')\delta\epsilon_{\X j}(x'')} \\
+    &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o}{\delta\eta_k(x'')\delta\eta_\ell(x''')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')}
+    +\int dx''\,\frac{\delta F_\o}{\delta\eta_k(x'')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\ 
+    &=\lambda_\X\delta(x-x')+
+    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
+    -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta_k(x'')}{\partial\epsilon_{\X\ell}(x''')}\bigg)^{-1}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')} \\ 
+    &=\lambda_\X\delta(x-x')+
+    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
+    -\frac12b\int dx''\,\delta_{i\ell}\delta(x-x'')\frac{\delta\eta_\ell(x'')}{\delta\epsilon_{\X j}(x')} 
+    =\lambda_\X\delta(x-x')+
+    \frac12b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')},
+  \end{aligned}
+\end{equation}
+\end{widetext}
+whose Fourier transform follows from \eqref{eq:inv.func} as
+\begin{equation}
+  \chi_{\X ij}(q)=\frac1{\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 $\gamma=1$.
+
+\begin{figure}
+  \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}
+  \caption{
+    Measurements of the effective strain stiffness as a function of temperature
+    for the six independent components of strain from ultrasound. The vertical
+    dashed lines show the location of the hidden order transition.
+  }
+\end{figure}
+
+\begin{figure}
+  \includegraphics[width=\columnwidth]{cusp}
+  \caption{
+    Strain stiffness data for the $\Bog$ component of strain (solid) along with
+    a fit of \eqref{eq:elastic.susceptibility} (dashed).
+  }
+  }
+\end{figure}
+
+\begin{acknowledgements}
+
+\end{acknowledgements}
+
+\bibliography{hidden_order}
+
 \end{document}
diff --git a/phases_scalar.pdf b/phases_scalar.pdf
new file mode 100644
index 0000000..9203f7f
--- /dev/null
+++ b/phases_scalar.pdf
diff --git a/phases_vector.pdf b/phases_vector.pdf
new file mode 100644
index 0000000..fc4cb37
--- /dev/null
+++ b/phases_vector.pdf
diff --git a/stiff_a11.pdf b/stiff_a11.pdf
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index 0000000..89891b7
--- /dev/null
+++ b/stiff_a11.pdf
diff --git a/stiff_a12.pdf b/stiff_a12.pdf
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index 0000000..404e775
--- /dev/null
+++ b/stiff_a12.pdf
diff --git a/stiff_a22.pdf b/stiff_a22.pdf
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index 0000000..92a4a99
--- /dev/null
+++ b/stiff_a22.pdf
diff --git a/stiff_b1.pdf b/stiff_b1.pdf
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index 0000000..e2bbeb9
--- /dev/null
+++ b/stiff_b1.pdf
diff --git a/stiff_b2.pdf b/stiff_b2.pdf
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index 0000000..93bb660
--- /dev/null
+++ b/stiff_b2.pdf
diff --git a/stiff_e.pdf b/stiff_e.pdf
new file mode 100644
index 0000000..3aa2ef1
--- /dev/null
+++ b/stiff_e.pdf