new data, references

1 files changed, 27 insertions, 22 deletions

diff --git a/monte-carlo.tex b/monte-carlo.tex
index 96e5f91..222c5e8 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -115,23 +115,23 @@ 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
+down \cite{wolff_critical_1990} 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
+external fields by using nonlocal updates \cite{janke_nonlocal_1998} whose clusters undergo a percolation
+transition at the critical point of the system \cite{coniglio_clusters_1980} and that in relatively small
+dynamic exponents \cite{wolff_comparison_1989,du_dynamic_2006,liu_dynamic_2014,wang_cluster_1990},
+including the Ising, $\mathrm O(n)$ \cite{wolff_collective_1989}, and Potts
+\cite{swendsen_nonuniversal_1987,baillie_comparison_1991} 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
+\cite{coniglio_exact_1989} 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}.
+\cite{alexandrowicz_swendsen-wang_1989,destri_swendsen-wang_1992,lauwers_critical_1989,wang_clusters_1989}.
 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
@@ -147,10 +147,10 @@ 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
 $A\subseteq X$ and $r\in R$, $\mu(r\cdot A)=\mu(A)$. This trait is shared by
 the counting measure on any discrete set, or by any group acting by isometries
-on a Riemannian manifold, such as $O(n)$ on $S^{n-1}$ in the $n$-component
-model. Finally, the subset of elements in $R$ of order two must act
+on a Riemannian manifold, such as $\mathrm O(n)$ on $S^{n-1}$ in the $\mathrm O(n)$
+model \cite{caracciolo_wolff-type_1993}. Finally, the subset of elements in $R$ of order two must act
 transitively on $X$. This property, while apparently obscure, is shared by any
-symmetric space \cite{loos1969symmetric} or by any transitive, finitely generated isometry group. In fact, all the examples listed here have spins spaces with natural
+symmetric space \cite{loos_symmetric_1969} or by any transitive, finitely generated isometry group. In fact, all the examples listed here have spins spaces with natural
 metrics whose symmetry group is the set of isometries of the spin spaces.
 We put one spin at each site of the lattice described by $G$, so that the
 state of the entire spin system is described by elements $\vec s\in X\times\cdots\times
@@ -176,7 +176,7 @@ well for these cases, but we will drop the additional index notation for clarity
     \hline\hline
     Ising         & $\{-1,1\}$    & $\Z/2\Z$                 &  $0\cdot
     s\mapsto s$, $1\cdot s\mapsto -s$ & $st$ & $Hs$ \\
-    $n$-component & $S^{n-1}$ & $\mathrm O(n)$ & $M\cdot s\mapsto Ms$ & $s^{\mathrm T}t$ & $H^{\mathrm T}s$\\
+    $\mathrm O(n)$ & $S^{n-1}$ & $\mathrm O(n)$ & $M\cdot s\mapsto Ms$ & $s^{\mathrm T}t$ & $H^{\mathrm T}s$\\
     Potts & $\mathbb Z/q\mathbb Z$ & $D_n$ & $r_m\cdot s=m+s$, $s_m\cdot
     s=-m-s$ & $\delta(s,t)$ & $\sum_mH_m\delta(m,s)$\\
     Clock & $\mathbb Z/q\mathbb Z$ & $D_n$ & $r_m\cdot s=m+s$, $s_m\cdot
@@ -189,7 +189,7 @@ well for these cases, but we will drop the additional index notation for clarity
     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}.}
+  to $\mathrm O(n)$ models \cite{jose_renormalization_1977}.}
   \label{table:models}
 \end{table*}
 
@@ -222,7 +222,7 @@ $\vec s'$ in the ensemble).
 
 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
+\cite{wolff_collective_1989}. 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
@@ -401,11 +401,11 @@ 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. The ghost spin version of the algorithm has
 been applied by several researchers previously
-\cite{wang1989clusters,ray1990metastability,destri1992swendsen,lauwers1989critical}
+\cite{wang_clusters_1989,ray_metastability_1990,destri_swendsen-wang_1992,lauwers_critical_1989}
 
-\subsection{The $n$-component Model}
+\subsection{The $\mathrm O(n)$ Model}
 
-In the $n$-component model, spins are described by vectors on the $(n-1)$-sphere,
+In the $\mathrm O(n)$ model, spins are described by vectors on the $(n-1)$-sphere,
 so that $X=S^{n-1}$. The symmetry group of this model is $O(n)$, $n\times n$
 orthogonal matrices. The symmetry group acts on the spins by matrix
 multiplication. The elements of $O(n)$ that are order two are reflections
@@ -460,7 +460,8 @@ 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.
 Continuous roughening models---where the spin states are described by real
 numbers and the symmetry group is $\mathrm E(1)$, the Euclidean group for
-one-dimensional space---are equally well described.
+one-dimensional space---are equally well described. A variant of the algorithm has been
+applied without a field before \cite{evertz_stochastic_1991}.
 
 
 %\begin{figure}
@@ -494,8 +495,8 @@ 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
+temperatures, and fields $B(s)=hs/\beta$ using methods here
+\cite{geyer_practical_1992}. The resulting scaling behavior, plotted in
 Fig.~\ref{fig:correlation_time-collapse}, is indeed consistent with the
 zero-field scaling behavior.
 
@@ -554,7 +555,10 @@ 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
+$\mathrm O(n)$ models, shown in Fig.~\ref{fig:cluster_scaling}. Fields for the
+Potts and $\mathrm O(n)$ models take the form
+$B(s)=(h/\beta)\sum_m\cos(2\pi(s-m)/q)$ and $B(s)=(h/\beta)[1,0,\ldots,0]s$
+respectively. 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.
@@ -587,7 +591,8 @@ 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}.
+\cite{jose_renormalization_1977}
+\cite{blankschtein_fluctuation-induced_1982,bruce_coupled_1975,manuel_carmona_$n$-component_2000}.
 
 \begin{acknowledgments}
 \end{acknowledgments}