maany changes

1 files changed, 132 insertions, 26 deletions

diff --git a/monte-carlo.tex b/monte-carlo.tex
index 23ce92b..96e5f91 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -96,6 +96,14 @@
 \date\today
 
 \begin{abstract}
+  We introduce a generalization of the `ghost spin' representation of spin
+  systems that restores full symmetry group invariance in an
+  arbitrary external field via the introduction of a `ghost transformation.'
+  This offers a natural way to extend celebrated spin-cluster
+  Monte Carlo algorithms to systems in arbitrary fields by running the
+  ordinary cluster-flipping process on the new representation. For several
+  canonical systems, we show that this extension with field preserves the scaling of
+  dynamics so celebrated without field.
 \end{abstract}
 
 \maketitle
@@ -123,16 +131,17 @@ external fields based on applying the ghost site representation
 spin systems that returns global rotation invariance to spin Hamiltonians at
 the cost of an extra degree of freedom, but these results only allow the application of a narrow
 category of fields
-\cite{alexandrowicz1989swendsen,destri1992swendsen,lauwers1989critical,wang1989clusters}. We show, by a redefinition of the spin--spin coupling in a
-generic class of such systems, systems with arbitrary external fields applied
-can be treated using cluster methods. The scaling of correlation
+\cite{alexandrowicz1989swendsen,destri1992swendsen,lauwers1989critical,wang1989clusters}.
+We show that the scaling of correlation
 time near the critical point of several models suggests that this approach is
 a natural one, e.g., that it extends the celebrated scaling of dynamics in
-these algorithms at zero field to various non-symmetric perturbations.
+these algorithms at zero field to various non-symmetric perturbations. We also show, by a redefinition of the spin--spin coupling in a
+generic class of such systems, systems with arbitrary external fields applied
+can be treated using cluster methods. 
 
 Let $G=(V,E)$ be a graph, where the set of vertices $V=\{1,\ldots,N\}$
 enumerates the sites of a lattice and the set of edges $E$ contains pairs of
-neighboring sites. Let $R$ be a group and $X$ an $R$-set, with the action
+neighboring sites. Let $R$ be a group acting on a set $X$, with the action
 of group elements $r\in R$ on elements $s\in X$ denoted $r\cdot s$. $X$ is the
 set of states accessible by a spin, and $R$ is the \emph{symmetry group} of
 $X$. The set $X$ must admit a measure $\mu$ that is invariant under the action of $R$, e.g., for any
@@ -274,13 +283,13 @@ coupling $Z$ depends only on the relative orientation of the spins---global
 reorientations by acting by some rotation do not affect the Hamiltonian. The
 external field $B$ breaks this symmetry. However, this can be resolved. Define
 a new graph $\tilde G=(\tilde V,\tilde E)$, where $\tilde V=\{0,1,\ldots,N\}$
-and
+adds a new `ghost' site $0$ which is connected by
 \[
-  \tilde E=E\cup\big\{\{0,i\}\mid i\in V\big\}.
+  \tilde E=E\cup\big\{\{0,i\}\mid i\in V\big\}
 \]
-We have introduced a new site to the lattice that neighbors every other site.
-Instead of assigning this site a spin whose value comes from the set $X$, we
-will assign it values $s_0\in R$, symmetry group elements, so that the new
+to all other sites.
+Instead of assigning this ghost site a spin whose value comes from the set $X$, we
+will assign it values in the symmetry group $s_0\in R$, so that the new
 configuration space of the model is $R\times X^N$. We introduce a Hamiltonian
 $\tilde\H:R\times X^N\to\R$ defined by
 \[
@@ -375,7 +384,9 @@ To summarize, spin systems in a field may be treated in the following way.
     lattice, substituting $\tilde Z$ as defined in \eqref{eq:new.z} for $Z$.
 \end{enumerate}
 Ensemble averages of observables $A$ can then be estimated by sampling the
-value of $\tilde A$ on the new system.
+value of $\tilde A$ on the new system. In contrast with the simpler ghost spin
+representation, this form of the Hamiltonian mya be considered the ``ghost
+transformation'' representation.
 
 \section{Examples}
 
@@ -388,7 +399,9 @@ exactly the same as the spins themselves. The only nontrivial element is of
 order two. Because the symmetry group and the spins are described by the same
 elements, performing the algorithm on the Ising model in a field is very
 accurately described by simply adding an extra spin coupled to all others and
-running the ordinary algorithm.
+running the ordinary algorithm. The ghost spin version of the algorithm has
+been applied by several researchers previously
+\cite{wang1989clusters,ray1990metastability,destri1992swendsen,lauwers1989critical}
 
 \subsection{The $n$-component Model}
 
@@ -401,7 +414,9 @@ through the origin. Since the former generate the entire group, the set of
 reflections alone suffices to provide ergodicity. Computation of the coupling
 of ordinary spins with the external field and expectation values requires a
 matrix inversion, but since the matrices in question are orthogonal this is
-quickly accomplished by a transpose.
+quickly accomplished by a transpose. The ghost-spin version of the algorithm
+has been used to apply a simple vector field by previous researchers
+\cite{dimitrovic_finite-size_1991}.
 
 \subsection{The Potts \& Clock Models}
 
@@ -461,27 +476,118 @@ one-dimensional space---are equally well described.
 
 \section{Dynamic scaling}
 
-We measured the autocorrelation time using methods here
-\cite{geyer1992practical}.
+No algorithm worthwhile if it doesn't run efficiently. Our algorithm, being an
+extension of the Wolff algorithm, should be considered successful if it
+likewise extends its efficiency in the systems that algorithm succeeds. The
+Wolff algorithm succeeds at 
 
-\begin{figure}
-  \centering
-  \input{fig_correlation_collapse-hL}
-  \caption{The correlation time $\tau$ as a function of the renormalization
-    invarient $hL^{-\beta\delta/\nu}$ for the $N=L\times L$ square lattice
-    Ising model for $L=8$, $16$, $32$, $64$, $128$, and $256$. $z=0.3$
-  }
-\end{figure}
+
+Cluster algorithms were celebrated for their small dynamic exponents $z$,
+which with the correlation time $\tau$ scales like $L^z$, where $L=N^{-D}$. In
+the vicinity of the critical point, the renormalization group predicts scaling
+behavior for the correlation time of the form
+\[
+  \tau=t^{-z\nu}\mathcal T(ht^{-\beta\delta},Lt^\nu)
+  =h^{-z\nu/\beta\delta}\mathcal T'(ht^{-\beta\delta},Lt^\nu).
+\]
+If a given dynamics for a system at zero field results in scaling like
+$t^{-z\nu}$, one should expect its natural extension in the presence of a
+field to scale like $h^{-z\nu/\beta\delta}$.  We measured the autocorrelation
+time for the 2D square-lattice model at a variety of system sizes,
+temperatures, and fields using methods here
+\cite{geyer1992practical}. The resulting scaling behavior, plotted in
+Fig.~\ref{fig:correlation_time-collapse}, is indeed consistent with the
+zero-field scaling behavior.
 
 \begin{figure}
   \centering
+  \input{fig_correlation_collapse-hL}
   \input{fig_correlation-temp}
-  \caption{The correlation time $\tau$ as a function of the renormalization
-    invarient $ht^{-\beta\delta}$ for the $N=128\times 128$ square lattice
-    Ising model. $z=0.3$
+  \caption{Collapses of the correlation time $\tau$ of the 2D square lattice
+    Ising model (top) 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 $hL^{\beta\delta/\nu}$ and (bottom) in the low-temperature phase
+    at $L=128$ for various temperatures as a function of the invariant
+    $ht^{-\beta\delta}$.
   }
+  \label{fig:correlation_time-collapse}
 \end{figure}
 
+Since the formation and flipping of clusters is the hallmark of the Wolff
+dynamics, another way to ensure that the dynamics with field scale like those without is
+to analyze the distribution of cluster sizes. The success of the algorithm at
+zero field is related to the way that clusters formed undergo a percolation
+transition at models' critical point.
+According to the scaling theory of percolation \cite{stauffer_scaling_1979}, the distribution of cluster sizes in a full decomposition of the system scales
+consistently near the critical point if it has the form
+\[
+  P_{\text{SW}}(s)=s^{-\tau}f(ts^\sigma,th^{-1/\beta\delta},tL^{1/\nu}).
+\]
+The distribution of cluster sizes in the Wolff algorithm can be computed from
+this using the fact that the algorithm selects clusters with probability
+proportional to their size, or
+\[
+  \begin{aligned}
+    \avg{s_{\text{\sc 1c}}}&=\sum_ssP_{\text{\sc
+    1c}}(s)=\sum_ss\frac sNP_{\text{SW}}(s)\\
+    &=t^{-\gamma}g(th^{-1/\beta\delta},tL^{1/\nu})\\
+    &=L^{\gamma/\nu}\mathcal G(ht^{-\beta\delta},hL^{\beta\delta/\nu})
+  \end{aligned}
+\]
+
+For the Ising model, an additional scaling relation can be written. Since in
+that case the average cluster size is the average squared magnetization, it
+can be related to the scaling functions of the magnetization and
+susceptibility per site by (with $ht^{-\beta\delta}$ dependence dropped)
+\[
+  \begin{aligned}
+    \avg{s_{\text{\sc 1c}}}
+    &=L^{D}\avg{M^2}=\beta\avg\chi+L^{D}\avg{M}^2\\
+    &=L^{\gamma/\nu}\big[(hL^{\beta\delta/\nu})^{-\gamma/\beta\delta}\beta \mathcal
+      Y(hL^{\beta\delta/\nu})\\
+      &\hspace{7em}+(hL^{\beta\delta/\nu})^{2/\delta}\mathcal
+    M(hL^{\beta\delta/\nu})\big].
+  \end{aligned}
+\]
+We therefore expect that, for the Ising model, $\avg{s_{\text{\sc 1c}}}$
+should go as $(hL^{\beta\delta})^{2/\delta}$ for large argument. We further
+conjecture that this scaling behavior should hold for other models whose
+critical points correspond with the percolation transition of Wolff clusters.
+This behavior is supported by our numeric work along the critical isotherm for various Ising, Potts, and
+$\mathrm O(n)$ models, shown in Fig.~\ref{fig:cluster_scaling}. As can be
+seen, the average cluster size collapses for each model according to the
+scaling hypothesis, and the large-field behavior likewise scales as we expect
+from the na\"ive Ising conjecture.
+
+\begin{figure*}
+  \input{fig_clusters_ising2d}
+  \caption{Collapses of rescaled average Wolff cluster size $\avg s_{\text{\sc
+    1c}}L^{-\gamma/\nu}$ as
+    a function of field scaling variable $hL^{\beta\delta/\nu}$ for a variety
+    of models. Critical exponents $\gamma$, $\nu$, $\beta$, and $\delta$ are
+    model-dependant. Colored lines and points depict values as measured by the
+    extended algorithm. Solid black lines show a plot of $f(x)=x^{2/\delta}$
+    for each model.
+  }
+  \label{fig:cluster_scaling}
+\end{figure*}
+
+We have taken several disparate extensions of cluster methods to models in an
+external field and generalized them to any model of a broad class. This new
+algorithm has an elegant statement that involves the introduction of not a
+ghost spin, but a ghost transformation. We provided evidence that extensions
+deriving from this method are the natural way to extend cluster methods tithe
+presence of a field, in the sense that it appears to reproduce the scaling
+of the dynamics in a field that would be expected from renormalization group
+predictions.
+
+In addition to uniting several extensions of cluster methods under a single
+description, our approach allows the application of fields not possible under
+prior methods. Instead of simply applying a spin-like field, this method
+allows for the application of \emph{arbitrary functions} of the spins. For
+instance, theoretical predictions for the effect of symmetry-breaking
+perturbations on spin models can be tested numerically
+\cite{jose1977renormalization}.
 
 \begin{acknowledgments}
 \end{acknowledgments}