many changes

1 files changed, 229 insertions, 275 deletions

diff --git a/monte-carlo.tex b/monte-carlo.tex
index 222c5e8..bff5ffb 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -88,7 +88,7 @@
 
 \begin{document}
 
-\title{An efficient cluster algorithm for spin systems in a symmetry-breaking field}
+\title{A natural extension of cluster algorithms in arbitrary symmetry-breaking fields}
 \author{Jaron Kent-Dobias}
 \author{James P.~Sethna}
 \affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA}
@@ -96,78 +96,80 @@
 \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.
+  We generalize the `ghost spin' representation of spin systems to restore
+  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-building process on the new
+  representation. For several canonical systems, we show that this extension
+  preserves the scaling of dynamics celebrated in the absence of a field.
 \end{abstract}
 
 \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{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{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
+approximation and numeric methods. Monte Carlo techniques are a common way of
+doing this, approximating thermodynamic quantities by sampling the
+distribution of systems states. These Monte Carlo algorithms are better the
+faster they arrive at a statistically independent sample. This typically
+becomes a problem near critical points, where critical slowing down
+\cite{wolff_critical_1990} results in power-law divergences of dynamic
+timescales. Celebrated cluster algorithms largely addressed this for many spin
+systems in the absence of symmetry-breaking fields by using nonlocal updates
+\cite{janke_nonlocal_1998} whose eponymous clusters undergo a percolation
+transition at the critical point of the system \cite{coniglio_clusters_1980}
+and result 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{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{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
-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. 
+global rotations of spins. Some success has been made in extending these
+algorithms to systems in certain external fields by applying the `ghost site'
+representation \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{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 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 spin systems, \emph{arbitrary} external fields can be treated
+using cluster methods. Rather than the introduction of a `ghost spin,' our
+representation relies on introducing a `ghost transformation.'
 
 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 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
+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
-$A\subseteq X$ 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 $\mathrm O(n)$ 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{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 $\vec s\in X\times\cdots\times
-X=X^N$. 
+$X$. 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 $\mathrm O(n)$ 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{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 their set of isometries.  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 $\vec s\in X\times\cdots\times X=X^N$. 
 
 The Hamiltonian of this system is a function $\H:X^N\to\R$ defined by
 \[
   \H(\vec s)=-\!\!\!\!\sum_{\{i,j\}\in E}\!\!\!\!Z(s_i,s_j)-\sum_{i\in V}B(s_i),
 \]
-where $Z:X\times X\to\R$ couples adjacent spins and
-$B:X\to\R$ is an external field. $Z$ must be symmetric in its arguments and
-invariant under the action of any element of $R$ applied to the entire lattice, that is, for any $r\in R$ and
-$s,t\in X$, $Z(r\cdot s,r\cdot t)=Z(s,t)$. 
-One may also allow $Z$ to also be a function of the edge---for modelling
-random-bond, long-range, or anisotropic interactions---or allow $B$ to be a
-function of site---for applying arbitrary boundary conditions or modelling random fields. All the formal results of this paper hold equally
-well for these cases, but we will drop the additional index notation for clarity.
+where $Z:X\times X\to\R$ couples adjacent spins and $B:X\to\R$ is an external
+field. $Z$ must be symmetric in its arguments and invariant under the action
+of any element of $R$ applied to the entire lattice, that is, for any $r\in R$
+and $s,t\in X$, $Z(r\cdot s,r\cdot t)=Z(s,t)$.  One may also allow $Z$ to also
+be a function of edge---for modelling random-bond, long-range, or anisotropic
+interactions---or allow $B$ to be a function of site---for applying arbitrary
+boundary conditions or modelling random fields. The formal results of this
+paper hold equally well for these cases, but we will drop the additional index
+notation for clarity.
 
 \begin{table*}[htpb]
   \begin{tabular}{l||ccccc}
@@ -193,42 +195,35 @@ well for these cases, but we will drop the additional index notation for clarity
   \label{table:models}
 \end{table*}
 
-The goal of statistical mechanics as applied to these systems is to compute
-expectation values of observables $A:X^N\to\R$. Assuming the ergodic
-hypothesis holds (for systems with broken-symmetry states, it does not), the
-expected value $\avg A$ of an observable $A$ is its average over every state
-$\vec s$
-in the configuration space $X^N$ weighted by the probability $p(\vec s)$ of
-that state appearing, or
+The goal of statistical mechanics is to compute expectation values of
+observables $A:X^N\to\R$. Assuming the ergodic hypothesis holds (for systems
+with broken-symmetry states, it does not), the expected value $\avg A$ of an
+observable $A$ is its average over every state $\vec s$ in the configuration
+space $X^N$ weighted by the Boltzmann probability of that state appearing, or
 \[
 \avg A
   =\frac{\int_{X^N}A(\vec s)e^{-\beta\H(\vec s)}\,\dd\mu(\vec s)}
-  {\int_{X^N}e^{-\beta\H(\vec s)}\,\dd\mu(\vec s)}
+  {\int_{X^N}e^{-\beta\H(\vec s)}\,\dd\mu(\vec s)},
 \]
 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 $X^N$. These values are estimated by Monte
-Carlo techniques by constructing a finite sequence of states $\{\vec
-s_1,\ldots,\vec s_M\}$ such that
+$\mu(Y)=\mu(Y_1)\cdots\mu(Y_N)$ is the simple extension of the measure on $X$
+to a measure on $X^N$. These values are estimated using Monte Carlo techniques
+by constructing a finite sequence of states $\{\vec s_1,\ldots,\vec s_M\}$
+such that
 \[
-  \avg A\simeq\frac1M\sum_{i=1}^MA(\vec s_i)
+  \avg A\simeq\frac1M\sum_{i=1}^MA(\vec s_i).
 \]
 Sufficient conditions for this average to converge to $\avg A$ as $M\to\infty$
 are that the process that selects $\vec s_{i+1}$ given the previous states be
 Markovian (only depends on $\vec s_i$), ergodic (any state can be accessed),
 and obey detailed balance (the ratio of probabilities that $\vec s'$ follows
-  $\vec s$ and vice versa is equal to the ratio of weights for $\vec s$ and
+$\vec s$ and vice versa is equal to the ratio of weights for $\vec s$ and
 $\vec s'$ 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{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
-nonzero, in fact \emph{arbitrary}, external fields.
-
-The Wolff algorithm proceeds in the following way.
+While any of several related cluster algorithms can be described for this
+system, we will focus on the Wolff algorithm \cite{wolff_collective_1989}. In
+the absence of an external field, e.g., B(s)=0, the Wolff algorithm proceeds
+in the following way.
 \begin{enumerate}
   \item Pick a random site and a random rotation $r\in R$ of order two, and add the site to
     a stack.
@@ -249,98 +244,88 @@ The Wolff algorithm proceeds in the following way.
 \end{enumerate}
 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  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
+must satisfy ergodicity and detailed balance. Ergodicity is satisfied since we
+have ensured that the subset of elements in $R$ that are order two acts
+transitively on $K$, e.g., for any $s,t\in X$ there exists $r\in R$ such that
+$r\cdot s=t$. Since there is a nonzero probability that only one spin is
+rotated and that spin can be rotated into any state, ergodicity follows. The
+probability $P(\vec s\to\vec s')$ that the configuration $\vec s$ is brought
+to $\vec s'$ by the flipping 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}
 \[
-  \begin{aligned}
   \frac{P(\vec s\to\vec s')}{P(\vec s'\to\vec s)}
-  &=\prod_{\{i,j\}\in
+  =\prod_{\{i,j\}\in
   C}\frac{p_r(s_i,s_j)}{p_{r^{-1}}(s_i',s_j')}\prod_{\{i,j\}\in\partial
   C}\frac{1-p_r(s_i,s_j)}{1-p_{r^{-1}}(s'_i,s'_j)}
-  =\prod_{\{i,j\}\in
-  C}\frac{p_r(s_i,s_j)}{p_{r}(r\cdot s_i,r\cdot s_j)}\prod_{\{i,j\}\in\partial
-  C}\frac{1-p_r(s_i,s_j)}{1-p_{r^{-1}}(r\cdot s_i,s_j)}
-  \\
-  &=\prod_{\{i,j\}\in
-  C}\frac{p_r(s_i,s_j)}{p_{r}(s_i,s_j)}\prod_{\{i,j\}\in\partial
+  =\prod_{\{i,j\}\in\partial
   C}e^{\beta(Z(r\cdot s_i,s_j)-Z(s_i,s_j))}
-  =\frac{e^{-\beta\H(\vec s)}}{e^{-\beta\H(\vec s')}}
-\end{aligned}
+  =\frac{p_r(s_i,s_j)}{p_{r}(s_i,s_j)}\frac{e^{-\beta\H(\vec
+  s)}}{e^{-\beta\H(\vec s')}},
 \]
 \end{widetext}
-whence detailed balance is satisfied. Ergodicity is satisfied since we have
-ensured that the subset of elements in $R$ that are order two acts
-transitively on $K$, e.g., for any $s,t\in X$ there exists $r\in R$ such that
-$r\cdot s=t$. Since there is a nonzero probability that only one spin is
-rotated and that spin can be rotated into any state, ergodicity follows.
-
-The function of the algorithm described above depends on the fact that the
-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 $\tilde V=\{0,1,\ldots,N\}$
-adds a new `ghost' site $0$ which is connected by
+whence detailed balance is also satisfied. 
+
+This algorithm relies on the fact that the coupling $Z$ depends only on
+relative orientation of the spins---global reorientations do not affect the
+Hamiltonian. The external field $B$ breaks this symmetry. However, it can be
+restored. Define a new graph $\tilde G=(\tilde V,\tilde E)$, where $\tilde
+V=\{0,1,\ldots,N\}$ adds the new `ghost' site $0$ which is connected by
 \[
   \tilde E=E\cup\big\{\{0,i\}\mid i\in V\big\}
 \]
-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
+to all other sites.  Instead of assigning the ghost site a spin whose value
+comes from $X$, we assign it values in the symmetry group $s_0\in R$, so that
+the configuration space of the new model is $R\times X^N$. We introduce the
+Hamiltonian $\tilde\H:R\times X^N\to\R$ defined by
 \[
 \begin{aligned}
   \tilde\H(s_0,\vec s)
   &=-\!\!\!\!\sum_{\{i,j\}\in E}\!\!\!\!Z(s_i,s_j)
   -\sum_{i\in V}B(s_0^{-1}\cdot s_i)\\
-  &=-\!\!\!\!\sum_{\{i,j\}\in\tilde E}\!\!\!\!\tilde Z(s_i,s_j)
+  &=-\!\!\!\!\sum_{\{i,j\}\in\tilde E}\!\!\!\!\tilde Z(s_i,s_j),
 \end{aligned}
 \]
-where the new coupling $\tilde Z:(R\cup X)\times(R\cup X)\to\R$ is defined for $s,t\in
-R\cup X$ by
+where the new coupling $\tilde Z:(R\cup X)\times(R\cup X)\to\R$ is defined for
+$s,t\in R\cup X$ by
 \[
   \tilde Z(s,t) =
   \begin{cases}
     Z(s,t) & \text{if $s,t\in X$} \\
     B(s^{-1}\cdot t) & \text{if $s\in R$} \\
-    B(t^{-1}\cdot s) & \text{if $t\in R$} 
+    B(t^{-1}\cdot s) & \text{if $t\in R$}.
   \end{cases}
   \label{eq:new.z}
 \]
-Note that this modified coupling is invariant under the action of group
-elements: for any $r,s_0\in R$ and $s\in X$,
+The modified coupling is invariant under the action of group elements: for any
+$r,s_0\in R$ and $s\in X$,
 \[
 \begin{aligned}
   \tilde Z(rs_0,r\cdot s)
   &=B((rs_0)^{-1}\cdot (r\cdot s))\\
-  &=B((s_0^{-1}r^{-1})\cdot(r\cdot s))\\
-  &=B((s_0^{-1}r^{-1}r)\cdot s)\\
   &=B(s_0^{-1}\cdot s)
   =\tilde Z(s_0,s)
 \end{aligned}
 \]
-The invariance $\tilde Z$ to rotations given other arguments follows from the
-invariance properties of $Z$.
-
-We have produced a system that incorporates the field function $B$ whose
-Hamiltonian is invariant to global rotations, but how does it relate to our
-previous system, whose properties we actually want to measure? If $A:X^N\to\R$
-is an observable of the original system, one can construct an observable
-$\tilde A:R\times X^N\to\R$ of the new system defined by
+The invariance of $\tilde Z$ to rotations given other arguments follows from
+the invariance properties of $Z$.
+
+We have produced a system incorporating the field function $B$ whose
+Hamiltonian is invariant under global rotations, but how does it relate to our
+old system, whose properties we actually want to measure? If $A:X^N\to\R$ is
+an observable of the original system, we construct an observable $\tilde
+A:R\times X^N\to\R$ of the new system defined by
 \[
   \tilde A(s_0,\vec s)=A(s_0^{-1}\cdot\vec s)
 \]
 whose expectation value in the new system equals that of the original
-observable in the old system. First, note that $\tilde\H(1,\vec s)=\H(\vec s)$. Since
-the Hamiltonian is invarient under global rotations, it follows that for any
-$g\in R$, $\tilde\H(g,g\cdot\vec s)=\tilde\H(g^{-1}g,g^{-1}g\cdot\vec
-s)=\tilde\H(1,\vec s)=\H(\vec s)$.
-Using the invariance properties of the measure on $X$ and introducing a
-measure $\rho$ on $R$, it follows that
+observable in the old system. First, note that $\tilde\H(1,\vec s)=\H(\vec
+s)$. Since the Hamiltonian is invariant under global rotations, it follows
+that for any $g\in R$, $\tilde\H(g,g\cdot\vec s)=\H(\vec s)$.  Using the
+invariance properties of the measure on $X$ and introducing a measure $\rho$
+on $R$, it follows that
 \[
 \begin{aligned}
   \avg{\tilde A}
@@ -372,154 +357,125 @@ measure $\rho$ on $R$, it follows that
 }{\int_{X^N}e^{-\beta\H(\vec s')}\dd\mu(\vec
     s')
   }
-  =\avg A
+  =\avg A.
 \end{aligned}
 \]
-To summarize, spin systems in a field may be treated in the following way.
+Using this equivalence, spin systems in a field may be treated in the
+following way.
 \begin{enumerate}
   \item Add a site to your lattice adjacent to every other site.
-  \item Initialize a ``spin'' at that site that is a representation of a
+  \item Initialize a `spin' at that site whose value is a representation of a
     member of the symmetry group of your ordinary spins.
   \item Carry out the ordinary Wolff cluster-flip procedure on this new
     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. In contrast with the simpler ghost spin
-representation, this form of the Hamiltonian mya be considered the ``ghost
-transformation'' representation.
+representation, this form of the Hamiltonian might be considered the `ghost
+transformation' representation.
+
 
 \section{Examples}
 
 \subsection{The Ising Model}
 
-In the Ising model, spins are drawn from the set $\{1,-1\}$. The symmetry
-group of this model is $C_2$, the cyclic group on two elements, which can be
-conveniently represented by the multiplicative group with elements $\{1,-1\}$,
-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. The ghost spin version of the algorithm has
-been applied by several researchers previously
-\cite{wang_clusters_1989,ray_metastability_1990,destri_swendsen-wang_1992,lauwers_critical_1989}
+In the Ising model spins are drawn from the set $\{1,-1\}$. Its symmetry group
+is $C_2$, the cyclic group on two elements, which can be conveniently
+represented by a multiplicative group with elements $\{1,-1\}$, exactly the
+same as the spins themselves. The only nontrivial element is of order two.
+Since the symmetry group and the spins are described by the same elements,
+performing the algorithm on the Ising model in a field is fully described by
+just using the `ghost spin' representation.  This algorithm has been applied
+by several researchers \cite{wang_clusters_1989, ray_metastability_1990,
+destri_swendsen-wang_1992, lauwers_critical_1989}.
 
 \subsection{The $\mathrm O(n)$ Model}
 
-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
-about some hyperplane through the origin and $\pi$ rotations about any axis
-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. The ghost-spin version of the algorithm
-has been used to apply a simple vector field by previous researchers
-\cite{dimitrovic_finite-size_1991}.
+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
+$\pi$ rotations about any axis through the origin. Since the former generate
+the entire group, reflections alone suffice to provide ergodicity. The `ghost
+spin' version of the algorithm has been used to apply a simple vector field to
+the $\mathrm O(3)$ model \cite{dimitrovic_finite-size_1991}. The method is
+quickly generalized to spins whose symmetry groups other compact Lie groups.
 
 \subsection{The Potts \& Clock Models}
 
-In both the $q$-state Potts and clock models, spins are described by
-$\Z/q\Z$, the set of integers modulo $q$. The symmetry group of this model is the dihedral group
-$D_q=\{r_0,\ldots,r_{q-1},s_0,\ldots,s_{q-1}\}$, the group of symmetries of a
-regular $q$-gon. The element $r_n$ represents a rotation of the polygon by
-$2\pi n/q$, and the element $s_n$ represents a reflection composed with a
-rotation $r_n$. The group acts on the spins by permutation: $r_n\cdot
-m={n+m}\pmod q$
-and $s_n\cdot m={-(n+m)}\pmod q$. Intuitively, this can be thought of
-as the natural action of the group on the vertices of a regular polygon that have
-been numbered $0$ through $q-1$. The elements of $D_q$ that are of order 2 are
-all reflections and $r_{q/2}$ if $q$ is even, though the former can generate
-the latter. While the reflections do not necessarily generate the entire group, for any
-$n,m\in\Z/q\Z$ there
-exists a
-reflection that takes $n\to m$, ensuring
-ergodicity. The elements of the dihedral group can be stored simply as an
-integer and a boolean that represents whether the element is a pure rotation or a
-reflection. The principle difference between the Potts and clock models is
-that, in the latter case, the form of the coupling $Z$ allows a geometric
-interpretation as being two-dimensional vectors fixed with even spacing along
-the unit circle.
-
-
-\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
+In both the $q$-state Potts and clock models spins are described by elements
+of $\Z/q\Z$, the set of integers modulo $q$. Its symmetry group is the
+dihedral group $D_q=\{r_0,\ldots,r_{q-1},s_0,\ldots,s_{q-1}\}$, the group of
+symmetries of a regular $q$-gon. The element $r_n$ represents a rotation by
+$2\pi n/q$, and the element $s_n$ represents a reflection composed with the
+rotation $r_n$. The group acts on spins by permutation: $r_n\cdot m={n+m}\pmod
+q$ and $s_n\cdot m={-(n+m)}\pmod q$. This is the natural action of the group
+on the vertices of a regular polygon that have been numbered $0$ through
+$q-1$. The elements of $D_q$ of order 2 are all reflections and $r_{q/2}$ if
+$q$ is even, though the former can generate the latter. While reflections do
+not necessarily generate the entire group, their action on $\Z/q\Z$ is
+transitive.
+
+\subsection{Roughening Models}
+
+Though not often thought of as a spin model, roughening of surfaces can be
+described in this framework. Spins are described by integers $\Z$ and their
 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
-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
-$Z(i,j)=(i-j)^2$, though it may also be any function of the absolute
-difference $|i-j|$.
-Because random 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.
-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. A variant of the algorithm has been
-applied without a field before \cite{evertz_stochastic_1991}.
-
-
-%\begin{figure}
-%  \centering
-%  \input{fig_correlation}
-%  \caption{The autocorrelation time $\tau$ of the internal energy $\H$ for a $n=32\times32$ square-lattice Ising
-%  model simulated by the three nonzero field methods detailed in this paper.
-%  The top plot is for simulations at high temperature, $T=5.80225$, the middle
-%plot is for simulations at the critical temperature, $T=2.26919$, and the
-%bottom plot is for simulations at }
-%  \label{fig:correlation}
-%\end{figure}
+i\in\Z\}$, whose action on the spin $j\in\Z$ is given by $r_i\cdot j=i+j$ and
+$s_i\cdot j=-i-j$. The elements of order two are reflections $s_i$, whose
+action on $\Z$ is transitive. The coupling can be any function of the absolute
+difference $|i-j|$.  Because random choice of reflection will almost always
+result in energy changes so large that the whole system is flipped, it is
+better to select random reflections about integers close to the average state
+of the system.  A variant of the algorithm has been applied without a field
+\cite{evertz_stochastic_1991}.
 
-\section{Dynamic scaling}
 
-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 
+\section{Dynamic scaling}
 
+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
+considered successful if it likewise extends the efficiency of the Wolff
+algorithm into that domain.
 
-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
+At a critical point, correlation time $\tau$ scales with system size
+$L=N^{-D}$ as $\tau\sim L^z$. Cluster algorithms are celebrated for their
+small dynamic exponents $z$. In the vicinity of an ordinary critical point,
+the renormalization group predicts scaling behavior for the correlation time
+as a function of temperature $t$ and field $h$ of the form
 \[
-  \tau=t^{-z\nu}\mathcal T(ht^{-\beta\delta},Lt^\nu)
-  =h^{-z\nu/\beta\delta}\mathcal T'(ht^{-\beta\delta},Lt^\nu).
+  \tau=h^{-z\nu/\beta\delta}\mathcal T(ht^{-\beta\delta},hL^{\beta\delta/\nu}).
 \]
-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 $B(s)=hs/\beta$ 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.
+If a given dynamics for a system at zero field results in scaling like $L^z$,
+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
+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}
-  \input{fig_correlation-temp}
   \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
-    $ht^{-\beta\delta}$.
+    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}.
   }
   \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
+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
+without is to analyze the distribution of cluster sizes. The success of the
+algorithm at zero field is related to the fact that the 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 Swendsen--Wang decomposition of the system scales
 consistently near the critical point if it has the form
 \[
   P_{\text{SW}}(s)=s^{-\tau}f(ts^\sigma,th^{-1/\beta\delta},tL^{1/\nu}).
@@ -531,15 +487,14 @@ proportional to their size, or
   \begin{aligned}
     \avg{s_{\text{\sc 1c}}}&=\sum_ssP_{\text{\sc
     1c}}(s)=\sum_ss\frac sNP_{\text{SW}}(s)\\
-    &=t^{-\gamma}g(th^{-1/\beta\delta},tL^{1/\nu})\\
-    &=L^{\gamma/\nu}\mathcal G(ht^{-\beta\delta},hL^{\beta\delta/\nu})
+    &=L^{\gamma/\nu}g(ht^{-\beta\delta},hL^{\beta\delta/\nu}).
   \end{aligned}
 \]
 
-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)
+For the Ising model, an additional scaling relation can be written. Since 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}}}
@@ -554,36 +509,35 @@ 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}. 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.
+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}. 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.
 
 \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.
+    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 $g(0,x)\propto x^{2/\delta}$ 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.
+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
+spin, but a ghost transformation. We provided evidence that algorithmic
+extensions deriving from this method are the natural way to extend cluster
+methods in the presence of a field, in the sense that they appear to reproduce
+the scaling of dynamic properties 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
@@ -591,8 +545,8 @@ 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{jose_renormalization_1977}
-\cite{blankschtein_fluctuation-induced_1982,bruce_coupled_1975,manuel_carmona_$n$-component_2000}.
+\cite{jose_renormalization_1977, blankschtein_fluctuation-induced_1982,
+bruce_coupled_1975, manuel_carmona_$n$-component_2000}.
 
 \begin{acknowledgments}
 \end{acknowledgments}