1 files changed, 53 insertions, 1 deletions
diff --git a/main.tex b/main.tex
index af39f7f..e5c2d81 100644
--- a/main.tex
+++ b/main.tex
@@ -45,7 +45,7 @@
 
 \begin{document}
 
-\title{Elastic properties of hidden order in \urusi\ reproduced by modulated $\Bog$ order}
+\title{An exact relation between response functions of linearly coupled fields:}
 \author{Jaron Kent-Dobias}
 \author{Michael Matty}
 \author{Brad Ramshaw}
@@ -125,6 +125,58 @@ applied pressure in comparison with recent x-ray scattering experiments
 \cite{choi_pressure-induced_2018}. Finally, we discuss our conclusions and the
 future experimental and theoretical work motivated by our results.
 
+\begin{equation}
+  F[\varphi,\vartheta]=F_1[\varphi]+F_2[\vartheta]-\int dx\,b\varphi(x)\vartheta(x)
+\end{equation}
+\begin{equation}
+  \begin{aligned}
+    0&=\frac{\delta F[\varphi,\vartheta]}{\delta\vartheta(x)}\bigg|_{\vartheta=\vartheta_\star}
+    =\frac{\delta F_2[\vartheta]}{\delta\vartheta(x)}\bigg|_{\vartheta=\vartheta_\star}-b\varphi(x)
+  \end{aligned}
+\end{equation}
+\begin{equation}
+  \vartheta_\star^{-1}[\vartheta](x)=\frac1b\frac{\delta F_2[\vartheta]}{\delta\vartheta(x)}
+\end{equation}
+\begin{equation}
+  \bigg(\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}\bigg)^\recip
+  =\frac{\delta\vartheta_\star^{-1}[\vartheta](x)}{\delta\vartheta(x')}\bigg|_{\vartheta=\vartheta_\star[\varphi]}
+  =\frac1b\frac{\delta^2F_2[\vartheta]}{\delta\vartheta(x)\delta\vartheta(x')}\bigg|_{\vartheta=\vartheta_\star[\varphi]}
+\end{equation}
+\begin{widetext}
+\begin{equation}
+  \begin{aligned}
+    \frac{\delta^2F[\varphi,\vartheta_\star(\varphi)]}{\delta\varphi(x)\delta\varphi(x')} 
+    &=\frac{\delta^2F_1[\varphi]}{\delta\varphi(x)\delta\varphi(x')}-
+    2b\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}
+    -b\int dx''\,\frac{\delta^2\vartheta_\star[\varphi](x)}{\delta\varphi(x')\delta\varphi(x'')}\varphi(x'') +\int dx''\,\frac{\delta\vartheta_\star[\varphi](x'')}{\delta\varphi(x)\delta\varphi(x')}\frac{\delta F_2[\vartheta]}{\delta\vartheta(x'')}\bigg|_{\vartheta=\vartheta_\star[\varphi]} \\
+    &\qquad\qquad+\int dx''\,dx'''\,\frac{\delta\vartheta_\star[\varphi](x'')}{\delta\varphi(x)}\frac{\delta\vartheta_\star[\varphi](x''')}{\delta\varphi(x')}\frac{\delta^2F_2[\vartheta]}{\delta\vartheta(x'')\delta\vartheta(x''')}\bigg|_{\vartheta=\vartheta_\star[\varphi]}
+    \\ 
+    &=\frac{\delta^2F_1[\varphi]}{\delta\varphi(x)\delta\varphi(x')}
+    -2b\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}-b\int dx''\,\frac{\delta^2\vartheta_\star[\varphi](x)}{\delta\varphi(x')\delta\varphi(x'')}\varphi(x'') +\int dx''\,\frac{\delta\vartheta_\star[\varphi](x'')}{\delta\varphi(x)\delta\varphi(x')}(b\varphi(x'')) \\
+    &\qquad\qquad+b\int dx''\,dx'''\,\frac{\delta\vartheta_\star[\varphi](x'')}{\delta\varphi(x)}\frac{\delta\vartheta_\star[\varphi](x''')}{\delta\varphi(x')}\bigg(\frac{\partial\vartheta_\star[\varphi](x'')}{\partial\varphi(x''')}\bigg)^\recip \\ 
+    &=\frac{\delta^2F_1[\varphi]}{\delta\varphi(x)\delta\varphi(x')}-
+    2b\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}
+    +b\int dx''\,\delta(x-x'')\frac{\delta\vartheta_\star[\varphi](x'')}{\delta\varphi(x')} 
+    =\frac{\delta^2F_1[\varphi]}{\delta\varphi(x)\delta\varphi(x')}-
+    b\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}.
+  \end{aligned}
+\end{equation}
+since $\vartheta_\star[\langle\varphi\rangle]=\langle\vartheta\rangle$,
+\begin{equation}
+  \bigg(\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}\bigg)^\recip\bigg|_{\varphi=\langle\varphi\rangle}
+  =\frac1b\frac{\delta^2F_2[\vartheta]}{\delta\vartheta(x)\delta\vartheta(x')}\bigg|_{\vartheta=\langle\vartheta\rangle}
+  =\frac1b\chi_\vartheta^\recip(x,x')+\frac{\delta\varphi_\star[\vartheta](x)}{\delta\vartheta(x')}\bigg|_{\vartheta=\langle\vartheta\rangle}
+  =\frac1b\chi_\vartheta^\recip(x,x')+\frac1b\frac{\delta^2F_1[\varphi]}{\delta\varphi(x)\delta\varphi(x')}\bigg|_{\varphi=\langle\varphi\rangle}
+\end{equation}
+\begin{equation}
+  \begin{aligned}
+    \chi_\varphi^\recip(x,x')=
+    \frac{\delta^2F[\varphi,\vartheta_\star(\varphi)]}{\delta\varphi(x)\delta\varphi(x')} \bigg|_{\varphi=\langle\varphi\rangle}
+    =
+      \end{aligned}
+\end{equation}
+\end{widetext}
+
 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.