From 9a2dc6ae681139b5a27b1b1b467a5c744fa4778a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 16 Sep 2024 12:41:00 +0200
Subject: Started appendix for squared calculation.

---
 topology.tex | 78 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 78 insertions(+)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 3801eae..65cede5 100644
--- a/topology.tex
+++ b/topology.tex
@@ -1225,6 +1225,84 @@ orders in $N$, since for odd-dimensional manifolds the Euler characteristic is
 always zero. When $N+M+1$ is even, we have $\overline{\chi(\Omega)}=2$ to
 leading order in $N$, as specified in the main text.
 
+\section{The root mean square Euler characteristic}
+\label{sec:rms}
+
+Here we calculate $\overline{\chi(\Omega)^2}$, the average of the squared Euler
+characteristic. This is accomplished by taking two copies of the integral, with
+\begin{equation}
+  \chi(\Omega)^2
+  =\int d\pmb\phi_1\,d\pmb\sigma_1\,d\pmb\phi_2\,d\pmb\sigma_2\,
+  e^{\int d1\,[L(\pmb\phi_1(1),\pmb\sigma_1(1))+L(\pmb\phi_2(1),\pmb\sigma_2(1))]}
+\end{equation}
+The same steps as in the derivation of the Euler characteristic follow. The result is the same as \eqref{eq:post.hubbard-strat}, but with the substitutions
+\begin{align}
+  \mathbb Q\mapsto\begin{bmatrix}
+    \mathbb Q_{11} & \mathbb Q_{12} \\
+    \mathbb Q_{21} & \mathbb Q_{22}
+  \end{bmatrix}
+  &&
+  \mathbb M\mapsto\begin{bmatrix}
+    \mathbb M_1 \\
+    \mathbb M_2
+  \end{bmatrix}
+\end{align}
+where we have defined
+\begin{align}
+  \mathbb Q_{ij}(1,2)=\frac1N\pmb\phi_i(1)\cdot\pmb\phi_j(2)
+  &&
+  \mathbb M_i(1)=\frac1N\pmb\phi_i(1)\cdot\mathbf x_0
+\end{align}
+Expanding the superindices and applying the Dirac $\delta$-functions implied by
+the Lagrange multipliers associated with the spherical constraint, we arrive at an expression
+\begin{equation}
+  \frac1N\log\overline{\chi(\Omega)^2}
+  =-\hat m_1-\hat m_2
+  -\frac\alpha2\log\frac{\det A_1}{\det A_2}
+  -\frac{\alpha V_0^2}2\begin{bmatrix}
+    0 & 1 & 0 & 1
+  \end{bmatrix}
+  A_1^{-1}
+  \begin{bmatrix}
+    0 \\ 1 \\ 0 \\ 1
+  \end{bmatrix}
+  +\frac12\log\frac{\det A_3}{\det A_4}
+\end{equation}
+with the matrices $A_1$, $A_2$, $A_3$, and $A_4$ defined by
+\begin{align}
+  A_1&=\begin{bmatrix}
+    D_{11}f'(1) & iR_{11}f'(1) & D_{12}f'(C_{12})+\Delta_{12}f''(C_{12}) & i R_{21}f'(C_{12}) \\
+    i R_{11}f'(1) & f(1) & i R_{12}f'(C_{12}) & f(C_{12}) \\
+    D_{12}f'(C_{12}) + \Delta_{12}f''(C_{12}) & iR_{12}f'(C_{12}) & D_{22} & iR_{22}f'(1) \\
+    iR_{21}f'(C_{12}) & f(C_{12}) & iR_{22}f'(1) & f(1)
+  \end{bmatrix}
+  \\
+  A_2&=\begin{bmatrix}
+    0 & R_{11}f'(1) & 0 & -G_{21}f'(C_{12}) \\
+    -R_{11}f'(1) & 0 & G_{12}f'(C_{12}) & 0 \\
+    0 & -G_{12}f'(C_{12}) & 0 & R_{22}f'(1) \\
+    G_{21}f'(C_{12}) & 0 & -R_{22}f'(1) & 0
+  \end{bmatrix}
+  \\
+  A_3&=\begin{bmatrix}
+    1-m_1^2 & i(R_{11}-m_1\hat m_1) & C_{12}-m_1m_2 & i(R_{21}-m_1\hat m_2) \\
+    i(R_{11}-m_1\hat m_1) & D_{11}+\hat m_1^2 & i(R_{12}-m_2\hat m_1) & D_{12}+\hat m_1\hat m_2 \\
+    C_{12}-m_1m_2 & i(R_{12}-m_2\hat m_1) & 1-m_2^2 & i(R_{22}-m_2\hat m_2) \\
+    i(R_{21}-m_1\hat m_2) & D_{12}+\hat m_1\hat m_2 & i(R_{22}-m_2\hat m_2) & D_{22}+\hat m_2^2
+  \end{bmatrix}
+  \\
+  A_4&=\begin{bmatrix}
+    0 & R_{11} & 0 & -G_{21} \\
+    -R_{11} & 0 & G_{12} & 0 \\
+    0 & -G_{12} & 0 & R_{22} \\
+    G_{21} & 0 & -R_{22} & 0
+  \end{bmatrix}
+\end{align}
+and where $\Delta_{12}=G_{12}G_{21}-R_{12}R_{21}$. The expression must be
+extremized over all the order parameters. We look for solutions in two regimes
+that are commensurate with the solutions found for the Euler characteristic.
+These correspond to $m_1=m_2=0$ and $C_{12}=0$, and $m_1=m_2=m^*$ and
+$C_{12}=1$.
 
 \section{The quenched shattering energy}
 \label{sec:1frsb}
-- 
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