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@@ -675,75 +675,6 @@ perturbations on spin models can be tested numerically
 bruce_coupled_1975, manuel_carmona_$n$-component_2000}.
 
 
-\appendix
-
-\section{$\mathrm O(n)$ model at high field}
-
-
-\[
-  \H=-\sum_r\sum_{i=1}^D\sum_{j=1}^ns_r^js_{r+e_i}^j
-     -\sum_r\sum_{j=1}^nH^js_r^j
-\]
-under the constraint
-\[
-  1=\sum_{j=1}^ns_r^js_r^j
-\]
-Let $s=m+t$ for $|m|=1$ (usually $m=H/|H|$). Suppose without loss of
-generality that $m=e_1$.
-\[
-  1=|s|^2=1+2m\cdot t+|t|^2
-\]
-whence $m\cdot t=-\frac12|t|^2$. Then
-\begin{align}
-  s_1\cdot s_2
-  &=1+m\cdot(t_1+t_2)+t_1\cdot t_2\\
-  &=1-\frac12(|t_1|^2+|t_2|^2)+t_1\cdot t_2
-\end{align}
-and
-\[
-  H\cdot s=|H|(1+m\cdot t)=|H|(1-\frac12|t|^2)
-\]
-For small perturbations, there are only $n-1$ degrees of freedom. We must have
-(for $t$ in the same hemisphere as $m$)
-\[
-  t_\parallel=\sqrt{1-|t_\perp|^2}-1
-\]
-\[
-  t_1\cdot t_2=t_{1\perp}\cdot t_{2\perp}+O(t^4)
-\]
-Since there are $2D$ nearest neighbor bonds involving each spin,
-\[
-  \H
-  \simeq\H_0
-  -\sum_{\langle ij\rangle}t_{i\perp}\cdot t_{j\perp}
-  +(D+|H|/2)\sum_i|t_{i\perp}|^2
-\]
-Taking a discrete Fourier transform on the lattice, we find
-\[
-  \H
-  \simeq\H_0
-  -\sum_k|\tilde t_{k\perp}|^2(D+|H|/2-\sum_{i=1}^D\cos(2\pi k_i/L))
-\]
-It follows from equipartition (and the fact that $t_{k\perp}$ is an $n$-1
-component complex number) that
-\[
-  \avg{|\tilde t_{k\perp}|^2}=\frac
-  {n-1}2T\bigg(D+\frac{|H|}2-\sum_{i=1}^D\cos(2\pi k_i/L)\bigg)^{-1}
-\]
-whence
-\begin{align}
-  \avg{\theta^2}
-  &=\avg{\cos^{-1}s_i\cdot s_j}
-  \simeq2(1-\avg{s_i\cdot s_j})\\
-  &=2(\avg{|t|^2}-\avg{t_i\cdot t_j})
-  \simeq2(\avg{|t_\perp|^2}-\avg{t_{i\perp}\cdot t_{j\perp}})\\
-  &=2\sum_k(1-\cos(2\pi k_1/L))\avg{|\tilde t_{k\perp}|^2}\\
-  &=\frac{(n-1)T}{L^D}\sum_k\frac{1-\cos(2\pi
-  k_1/L)}{D+|H|/2-\sum_{i=1}^D\cos(2\pi k_i/L)}\\
-\end{align}
-
-\section{Calculating autocorrelation time}
-
 \begin{acknowledgments}
   This work was supported by NSF grant NSF DMR-1719490.
 \end{acknowledgments}