changes

1 files changed, 59 insertions, 11 deletions

diff --git a/monte-carlo.tex b/monte-carlo.tex
index 929fc65..c1c70ab 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -100,6 +100,36 @@
 
 \maketitle
 
+Spin systems are important in the study of statistical physics and phase
+transitions. Rarely exactly solvable, they are typically studied by
+approximation methods and numeric means. Monte Carlo methods are a common way
+of doing this, approximating thermodynamic quantities by sampling the
+distribution of systems states. For a particular system, a Monte Carlo
+algorithm is better the faster it arrives at a statistically independent
+sample. This is typically a problem at critical points, where critical slowing
+down \cite{wolff1990critical} results in power-law divergences of any dynamics. Celebrated cluster
+algorithms largely addressed this for many spin systems in the absence of
+external fields by using nonlocal updates \cite{janke1998nonlocal} whose clusters undergo a percolation
+transition at the critical point of the system \cite{coniglio1980clusters} and that in relatively small
+dynamic exponents \cite{wolff1989comparison,du2006dynamic,liu2014dynamic,wang1990cluster},
+including the Ising, $n$-component \cite{wolff1989collective}, and Potts
+\cite{swendsen1987nonuniversal,baillie1991comparison} models. These
+algorithms rely on the natural symmetry of the systems in question under
+global rotations, so the general application of external fields is not
+trivial. Some
+success has been made in extending these algorithms to systems in certain
+external fields based on applying the ghost site representation
+\cite{coniglio1989exact} of certain
+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
+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.
+
 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
@@ -132,20 +162,25 @@ well for these cases, but we will drop the additional index notation for clarity
 
 \begin{table*}[htpb]
   \begin{tabular}{l||ccccc}
-          & Spin states ($X$) & Symmetry ($R$)  &  Action ($g\cdot s$)  &
-    Coupling ($Z(s,t)$) & Field ($B(s)$) \\
+          & Spins ($X$) & Symmetry ($R$)  &  Action ($g\cdot s$)  &
+    Coupling ($Z(s,t)$) & Common Field ($B(s)$) \\
     \hline\hline
     Ising         & $\{-1,1\}$    & $\Z/2\Z$                 &  $0\cdot
     s\mapsto s$, $1\cdot s\mapsto -s$ & $st$ & $Hs$ \\
     $n$-component & $S^{n-1}$ & $\mathrm O(n)$ & $M\cdot s\mapsto Ms$ & $s^{\mathrm T}t$ & $H^{\mathrm T}s$\\
     Potts & $\mathbb Z/q\mathbb Z$ & $D_n$ & $r_m\cdot s=m+s$, $s_m\cdot
-    s=m-s$ & $\delta(s,t)$ & $\sum_mH_m\delta(m,s)$\\
+    s=-m-s$ & $\delta(s,t)$ & $\sum_mH_m\delta(m,s)$\\
     Clock & $\mathbb Z/q\mathbb Z$ & $D_n$ & $r_m\cdot s=m+s$, $s_m\cdot
     s=-m-s$ & $\cos(2\pi\frac{s-t}q)$ & $\sum_mH_m\cos(2\pi\frac{s-m}q)$\\
-    Roughening & $\mathbb Z$ & $D_\inf$ & $r_m\cdot s=m+s$, $s_m\cdot s=-m-s$
+    Discrete Gaussian & $\mathbb Z$ & $D_\inf$ & $r_m\cdot s=m+s$, $s_m\cdot s=-m-s$
     & $(s-t)^2$ & $Hs^2$\\
   \end{tabular}
-  \caption{Some models.}
+  \caption{Several examples of spin systems and the symmetry groups that act
+    on them. Common choices for the spin--spin coupling in these systems and
+    their external fields are also given. Other fields are possible, of course:
+    for instance, some are interested in modulated fields $H\cos(2\pi k\theta(s))$ for
+    integer $k$ and $\theta(s)$ giving the angle of $s$ to some axis applied
+    to $n$-component models \cite{jose1977renormalization}.}
   \label{table:models}
 \end{table*}
 
@@ -161,8 +196,13 @@ that state appearing, or
   =\frac{\int_{X^N}A(\vec s)e^{-\beta\H(\vec s)}\,\dd\mu(\vec s)}
   {\int_{X^N}e^{-\beta\H(\vec s)}\,\dd\mu(\vec s)}
 \]
+where for $Y_1\times\cdots\times Y_N=Y\subseteq X^N$ the measure
+$\mu(Y)=\mu(Y_1)\cdots\mu(Y_N)$ is the simple extension of the
+measure on $X$ to a measure on $X^N$.
 
-We will first describe a generalized version of the celebrated Wolff algorithm
+While any several related cluster algorithms can be described for this
+system, we will focus on the Wolff algorithm in particular
+\cite{wolff1989collective}. We will first describe a generalized version of the celebrated Wolff algorithm
 in the standard case where $B(s)=0$. After reflecting on the technical
 requirements of that algorithm, we will introduce a transformation to our
 system and Hamiltonian that allows the same algorithm to be applied with
@@ -191,8 +231,9 @@ When the stack is exhausted, a cluster of connected spins will have been
 rotated by the action of $r$. In order for this algorithm to be useful, it
 must satisfy ergodicity and detailed balance. The probability $P(\vec s\to\vec
 s')$ that the configuration $\vec s$ is brought to $\vec s'$ by the flipping
-of a cluster $C\subseteq E$ is related to the probability of the reverse process $P(\vec
-s'\to\vec s)$ by
+of a cluster  formed by accepting rotations of spins via bonds $C\subseteq E$
+and rejecting rotations via bonds $\partial C\subset E$ is related to the
+probability of the reverse process $P(\vec s'\to\vec s)$ by
 \begin{widetext}
 \[
   \begin{aligned}
@@ -380,17 +421,18 @@ interpretation as being two-dimensional vectors fixed with even spacing along
 the unit circle.
 
 
-\subsection{Roughening}
+\subsection{Discrete (or Continuous) Gaussian Model}
 
 Though not often thought of as a spin model, simple roughening of surfaces can
 be described in this framework. The set of states is the integers $\Z$ and its
 symmetry group is the infinite dihedral group $D_\infty=\{r_i,s_i\mid
 i\in\Z\}$, where the action of the symmetry on the spins $j\in\Z$ is given by $r_i\cdot
-j=i+j$ and $s_i\cdot j=i-j$. These are shifts by $i$ and reflection about the
+j=i+j$ and $s_i\cdot j=-i-j$. These are shifts by $i$ and reflection about the
 integer $i$, respectively. The elements of order two are the reflections
 $s_i$, which suffice to provide ergodicity as any integer can be taken to any
 other in one step of this kind. The coupling is usually taken to be
-$Z(i,j)=|i-j|$, though it may also be any function of the absolute difference.
+$Z(i,j)=(i-j)^2$, though it may also be any function of the absolute
+difference $|i-j|$.
 Because randomly choices of integer will almost always result in energy
 changes so big that the whole system is always flipped, it is better to select
 random reflections about integers close to the average state of the system.
@@ -410,6 +452,11 @@ one-dimensional space---are equally well described.
 %  \label{fig:correlation}
 %\end{figure}
 
+\section{Dynamic scaling}
+
+We measured the autocorrelation time using methods here
+\cite{geyer1992practical}.
+
 \begin{figure}
   \centering
   \input{fig_correlation_collapse-hL}
@@ -433,5 +480,6 @@ one-dimensional space---are equally well described.
 
 \bibliography{monte-carlo}
 
+
 \end{document}