From 70d55c8e13ab4a153c644c016a470b0386e5787e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 23 Aug 2019 15:11:37 -0400
Subject: started removing eta indices, going to specialize to scalar field

---
 main.tex | 20 ++++++++------------
 1 file changed, 8 insertions(+), 12 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 706b74f..a53e0ff 100644
--- a/main.tex
+++ b/main.tex
@@ -267,22 +267,18 @@ diagrams for this model are shown in Figure \ref{fig:phases}.
 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'),
+    &\chi^{-1}(x,x')
+      =\frac{\delta^2F}{\delta\eta(x)\delta\eta(x')}
+      =\big(\tilde r-c_\parallel\nabla_\parallel^2-c_\perp\nabla_\perp^2 \\
+    &\qquad\qquad+D_\perp\nabla_\perp^4+12u\eta^2(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}
+    \chi^{-1}(q)
+    =\tilde r+c_\parallel q_\parallel^2-c_\perp q_\perp^2+D_\perp q_\perp^4
+      +12u\sum_{q'}\tilde\eta_{q'}\tilde\eta_{-q'}.
 \end{equation}
 Near the unordered--modulated transition this yields
 \begin{equation}
-- 
cgit v1.2.3-70-g09d2

