many text and figure changes

1 files changed, 69 insertions, 24 deletions

diff --git a/monte-carlo.tex b/monte-carlo.tex
index 370ac0f..b24fa9d 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -89,7 +89,7 @@ extra degree of freedom, allowing the method to be used in a subcategory of
 interesting fields \cite{alexandrowicz_swendsen-wang_1989, wang_clusters_1989,
 ray_metastability_1990}. Static fields have also been applied by including a
 separate metropolis or heat bath update step after cluster formation
-\cite{destri_swendsen-wang_1992, lauwers_critical_1989}, and other categories
+\cite{destri_swendsen-wang_1992, lauwers_critical_1989, ala-nissila_numerical_1994}, and other categories
 of fields have been applied using replica methods
 \cite{redner_graphical_1998,chayes_graphical_1998,machta_replica-exchange_2000}.
 We show that the scaling of correlation time near the critical point of
@@ -192,9 +192,12 @@ in the following way.
 \begin{enumerate}
   \item Pick a random site $m_0$ and add it to the stack.
 
-  \item Pick a random rotation $r\in R$ of order two from the set of such
-    rotations with a probability distribution of the form
-    $P_{m_0}(r)=f(\J(m_0,r\cdot m_0))$.
+  \item Select a rotation $r\in R$ of order two. For detailed balance it is
+    sufficient to ensure that probability distribution $f_{m_0}(r\mid \set s)$ the
+    rotation is sampled from depend only on $Z(s,r\cdot s)$ for every site,
+    since these numbers are invariant under a cluster flip, e.g., $Z(s',r\cdot
+    s')=Z(r\cdot s,r\cdot(r\cdot s))=Z(r\cdot s,(rr)\cdot s)=Z(r\cdot
+    s,s)=Z(s,r\cdot s)$.
   \item While the stack isn't empty,
     \begin{enumerate}
       \item pop site $m$ from the stack.
@@ -204,7 +207,7 @@ in the following way.
           \item For every $j$ such that $\{m,j\}\in E$, add site $j$ to the
             stack with probability
             \[
-              p_r(s_m,s_j)=\min\{0,1-e^{\beta(\J(r\cdot s_m,s_j)-\J(s_m,s_j))}\}.
+              p_r(s_m,s_j\mid q)=\min\{0,1-qe^{\beta(\J(r\cdot s_m,s_j)-\J(s_m,s_j))}\}.
             \]
           \item Take $s_m\mapsto r\cdot s_m$.
       \end{enumerate}
@@ -225,12 +228,12 @@ C\subset E$ is related to the probability of the reverse process $P(\set{s'}\to\
 \[
   \begin{aligned}
     \frac{P(\set s\to\set{s'})}{P(\set{s'}\to\set s)}
-    &=\frac{P_{m_0}(r)}{P_{m_0}(r^{-1})}\prod_{\{i,j\}\in
+    &=\frac{f_{\set s}(r\mid m_0)}{f_{\set s'}(r^{-1}\mid m_0)}\prod_{\{i,j\}\in
   C}\frac{p_r(s_i,s_j)}{p_{r^{-1}}(s_i',s_j')}\prod_{\{i,j\}\in\partial
   C}\frac{1-p_r(s_i,s_j)}{1-p_{r^{-1}}(s'_i,s'_j)}\\
   &=\prod_{\{i,j\}\in\partial
   C}e^{\beta(\J(r\cdot s_i,s_j)-\J(s_i,s_j))}
-  =\frac{p_r(s_i,s_j)}{p_{r}(s_i,s_j)}\frac{e^{-\beta\H(\set
+  =\frac{e^{-\beta\H(\set
   s)}}{e^{-\beta\H(\set{s'})}},
 \end{aligned}
 \]
@@ -340,6 +343,7 @@ representation, this form of the Hamiltonian might be considered the `ghost
 transformation' representation.
 
 \section{Examples}
+\label{sec:examples}
 
 Several specific examples from Table~\ref{table:models} are described in the
 following.
@@ -357,7 +361,10 @@ wang_clusters_1989, ray_metastability_1990}. The algorithm has been
 implemented by one of the authors in an existing interactive Ising
 simulator at \texttt{https://mattbierbaum.github.io/ising.js} \cite{bierbaum_ising.js_nodate}.
 
-\subsection{The $\mathrm O(n)$ model} In the $\mathrm O(n)$ model spins are described by vectors on the
+\subsection{The $\mathrm O(n)$ model}
+\label{sec:examples:on}
+
+In the $\mathrm O(n)$ model spins are described by vectors on the
 $(n-1)$-sphere $S^{n-1}$. Its symmetry group is $O(n)$, $n\times n$ orthogonal
 matrices, which act on the spins by matrix multiplication. The elements of
 $O(n)$ of order two are reflections about hyperplanes through the origin and
@@ -394,6 +401,11 @@ of large $L$ the expected square angle between neighbors being
 \[
   \avg{\theta^2}\simeq\frac{(n-1)T}{D+H/2}
 \]
+We shall in in the numeric experiments of Section \ref{sec:performance} that
+this choice ameliorates the problem but probably is not the best. A more
+subtle technique---for instance by matching statistics of the effective
+Gaussian model that results in these circumstances to the cluster
+statistics---may result in better performance.
 
 \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
@@ -428,6 +440,7 @@ of the system.  A variant of the algorithm has been applied without a field
 \cite{evertz_stochastic_1991}.
 
 \section{Performance}
+\label{sec:performance}
 
 No algorithm is worthwhile if it doesn't run efficiently. This algorithm,
 being an extension of the Wolff algorithm into a new domain, should be
@@ -454,24 +467,19 @@ one should expect its natural extension in the presence of a field to scale
 roughly like $h^{-z\nu/\beta\delta}$ and collapse appropriately as a function
 of $hL^{\beta\delta/\nu}$.  We measured the autocorrelation time for the $D=2$
 square-lattice model at a variety of system sizes, temperatures, and fields
-$B(s)=hs/\beta$ using standard methods \cite{geyer_practical_1992}. The
+$B(s)=hs/\beta$ using standard methods \cite{ossola_dynamic_2004}. The
 resulting scaling behavior, plotted in
 Fig.~\ref{fig:correlation_time-collapse}, is indeed consistent with an
 extension to finite field of the behavior at zero field.
 
-\begin{figure}
-  \centering
-  \input{fig_correlation_collapse-hL}
-  \caption{Collapse of the correlation time $\tau$ of the 2D square lattice
-    Ising model along the critical isotherm at various systems sizes
-    $N=L\times L$ for $L=8$, $16$, $32$, $64$, $128$, and $256$ as a function
-    of the renormalization invariant $hL^{\beta\delta/\nu}$. The exponent
-    $z=0.30$ is taken from recent measurements at zero field
-  \cite{liu_dynamic_2014}. The solid black line shows a plot of
-  $(hL^{\beta\delta/\nu})^{-z\nu/\beta\delta}$.
+\begin{figure*}
+  \include{fig_correlation-times}
+  \caption{
+    Correlation times $\tau$ scaled by the average cluster size as a function
+    of external field for various models at various system sizes. 
   }
   \label{fig:correlation_time-collapse}
-\end{figure}
+\end{figure*}
 
 Since the formation and flipping of clusters is the hallmark of Wolff
 dynamics, another way to ensure that the dynamics with field scale like those
@@ -538,10 +546,50 @@ conjecture.
   \include{fig_generator-times}
 \end{figure}
 
+\section{Applying Nonlinear Fields to the xy Model}
+
+This far our numeric work has quantified the performance of existing
+techniques. Here, we apply our general framework in a new way:
+harmonic perturbations to the low-temperature {\sc xy}, or \twodee O(2),
+model. We consider fields of the form $B_n(s)=h_n\cos(n\theta(s))$, where
+$\theta$ is the angle made between $s$ and the $x$-axis, say. Corrections of
+these types are expected to appear in realistic models of systems na\"ively
+expected to exhibit Kosterlitz--Thouless critical behavior due to the presence
+of the lattice or substrate. Whether these fields are relevant or irrelevant
+in the renormalization group sense determines whether those systems spoil or
+admit that critical behaviour. Among many fascinating
+\cite{jose_renormalization_1977, kankaala_theory_1993,
+ala-nissila_numerical_1994, dierker_consequences_1986, selinger_theory_1988}
+results that emerge from systems with one or more of these fields applied,
+it is predicted that $h_4$ is relevant while $h_6$ is not at some
+sufficiently high temperatures below the Kosterlitz--Thouless point
+\cite{jose_renormalization_1977}.
+
 \begin{figure}
   \include{fig_harmonic-susceptibilities}
+  \caption{Susceptibilities as a function of system size for a \twodee O(2)
+    model at $T=0.7$ and with (top) fourfold symmetric and (bottom) eightfold
+    symmetric perturbing fields. Different field strengths are shown in
+    different colors.
+  }
+  \label{fig:harmonic-susceptibilities}
 \end{figure}
 
+We made a basic investigation of this result using our algorithm. Since we ran
+the algorithm at fairly high fields we did not choose reflections though the
+origin uniformly. Instead, we choose the planes of reflection first by
+rotating our starting spin by $2\pi m/n$ degrees for $m$ uniformly taken from
+$1,\ldots,n$ and generating a normal to the plane from that direction as
+described in Section \ref{sec:examples:on}. The resulting susceptibilities as
+a function of system size are shown for various field strengths in
+Fig.~\ref{fig:harmonic-susceptibilities}. In the fourfold case, for each field
+strength there is a system size at which the divergence in the susceptibility
+is cut off, while for the sixfold case we measured no such cutoff, even up to
+strong fields. This confirms the expected result, that even in a strong field
+the sixfold perturbations preserve the critical behavior.
+
+\section{Conclusions}
+
 We have taken several disparate extensions of cluster methods to spin models
 in an external field and generalized them to work for any model of a broad
 class.  The resulting representation involves the introduction of not a ghost
@@ -560,7 +608,7 @@ perturbations on spin models can be tested numerically
 \cite{jose_renormalization_1977, blankschtein_fluctuation-induced_1982,
 bruce_coupled_1975, manuel_carmona_$n$-component_2000}.
 
-<<<<<<< HEAD
+
 \appendix
 
 \section{$\mathrm O(n)$ model at high field}
@@ -629,9 +677,6 @@ whence
 \end{align}
 
 \section{Calculating autocorrelation time}
-\begin{figure*}
-  \include{fig_correlation-times}
-\end{figure*}
 
 \begin{acknowledgments}
   This work was supported by NSF grant NSF DMR-1719490.