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@@ -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.