From 8239c93cd0888531cb0af098bde6aedabb65b0a3 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 27 Jun 2019 00:11:50 -0400
Subject: lots of progress on writing the theory sections

---
 hidden-order.tex | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

(limited to 'hidden-order.tex')

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*}
-- 
cgit v1.2.3-70-g09d2

