some changes to the text

1 files changed, 95 insertions, 1 deletions

diff --git a/monte-carlo.tex b/monte-carlo.tex
index e30acd1..d202ef5 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -369,7 +369,31 @@ interest include $(n+1)$-dimensional spherical harmonics
 \cite{jose_renormalization_1977} and cubic fields
 \cite{bruce_coupled_1975,blankschtein_fluctuation-induced_1982}, which can be
 applied with the new method. The method is
-quickly generalized to spins whose symmetry groups other compact Lie groups
+quickly generalized to spins whose symmetry groups other compact Lie groups.
+
+At low temperature or high field, selecting reflections uniformly becomes
+inefficient because the excitations of the model are spin waves, in which the
+magnetization only differs by a small amount between neighboring spins. Under
+these conditions, most choices of reflection plane will cause a change in
+energy so great that the whole system is always flipped, resulting in many
+highly correlated and inefficiently generated samples. To ameliorate this, one
+can draw reflections from a distribution that depends on how the first spin
+flip is transformed. We implement this in the following way. Say that the seed
+of the cluster is $s$. Generate a vector $t$ taken uniformly from the space of
+unit vectors orthogonal to $s$. Let the plane of reflection that whose normal
+is $n=s+\zeta t$, where $\zeta$ is drawn from a normal distribution of mean
+zero and variance $\sigma$. It follows that the tangent of the angle between
+$s$ and the plane of reflection is also distributed normally with zero mean
+and variance $\sigma$. Since the distribution of reflection planes only
+depends on the angle between $s$ and the plane and that angle is invariant
+under the reflection, this choice preserves detailed balance. The choice of
+$\sigma$ can be inspired by mean field theory. At high field or low
+temperature, spins are likely to both align with the field and each other and
+the model is asymptotically equal to a simple Gaussian one, with in the limit
+of large $L$ the expected square angle between neighbors being
+\[
+  \avg{\theta^2}\simeq\frac{(n-1)T}{D+H/2}
+\]
 
 \subsection{The Potts model} In the $q$-state Potts model spins are described
 by elements of $\{1,\ldots,q\}$. Its symmetry group is the symmetric group
@@ -528,11 +552,81 @@ perturbations on spin models can be tested numerically
 \cite{jose_renormalization_1977, blankschtein_fluctuation-induced_1982,
 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}
 
 \bibliography{monte-carlo}
 
+
 \end{document}