From 3902c90c737febdd0a501a8762e2ab4e4d1a6513 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 9 May 2018 17:12:45 -0400
Subject: minor changes

---
 monte-carlo.tex | 120 ++++++++++++++++++++++++++++++--------------------------
 1 file changed, 64 insertions(+), 56 deletions(-)

(limited to 'monte-carlo.tex')

diff --git a/monte-carlo.tex b/monte-carlo.tex
index bff5ffb..8eef5b0 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -88,7 +88,7 @@
 
 \begin{document}
 
-\title{A natural extension of cluster algorithms in arbitrary symmetry-breaking fields}
+\title{Accelerating Monte Carlo: Wolff in arbitrary external fields}
 \author{Jaron Kent-Dobias}
 \author{James P.~Sethna}
 \affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA}
@@ -96,50 +96,54 @@
 \date\today
 
 \begin{abstract}
-  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
+  We introduce a natural way to extend celebrated spin-cluster Monte Carlo
+  algorithms for fast thermal lattice simulations at criticality, like Wolff, to
+  systems in arbitrary fields. The method relies on the generalization of the
+  `ghost spin' representation to one with a `ghost transformation' that
+  restores invariance to spin symmetries at the cost of an extra degree of
+  freedom. The ordinary cluster-building process can then be run 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.
+  preserves the scaling of accelerated dynamics in the absence of a field.
 \end{abstract}
 
 \maketitle
 
-Spin systems are important in the study of statistical physics and phase
+Lattice models are important in the study of statistical physics and phase
 transitions. Rarely exactly solvable, they are typically studied by
-approximation and numeric methods. Monte Carlo techniques are a common way of
+approximate and numerical 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,
+timescales. Celebrated cluster algorithms largely addressed this in the absence of symmetry-breaking 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}.
+These result in relatively small dynamic exponents for many spin
+systems \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 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
+symmetry operations on the spins. Some success has been made in extending these
+algorithms to systems in certain external fields by adding a `ghost site'
+\cite{coniglio_exact_1989} 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
+of freedom, allowing the method to be used in a subcategory of interesting
+fields \cite{alexandrowicz_swendsen-wang_1989, destri_swendsen-wang_1992,
+lauwers_critical_1989, wang_clusters_1989}. 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 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.'
+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 $V=\{1,\ldots,N\}$
+We will pose the problem in a general way, but several specific examples can
+be found in Table~\ref{table:models} for concreteness. 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
@@ -168,8 +172,9 @@ 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.
+paper (that the algorithm obeys detailed balance and ergodicity) hold equally
+well for these cases, but we will drop the additional index notation for
+clarity. Statements about efficiency may not.
 
 \begin{table*}[htpb]
   \begin{tabular}{l||ccccc}
@@ -191,7 +196,7 @@ 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 $\mathrm O(n)$ models \cite{jose_renormalization_1977}.}
+    to the $\mathrm O(2)$ model \cite{jose_renormalization_1977}.}
   \label{table:models}
 \end{table*}
 
@@ -254,18 +259,20 @@ 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{widetext}
 \[
-  \frac{P(\vec s\to\vec s')}{P(\vec s'\to\vec s)}
+  \begin{aligned}
+  &\frac{P(\vec s\to\vec s')}{P(\vec s'\to\vec s)}
   =\prod_{\{i,j\}\in
   C}\frac{p_r(s_i,s_j)}{p_{r^{-1}}(s_i',s_j')}\prod_{\{i,j\}\in\partial
-  C}\frac{1-p_r(s_i,s_j)}{1-p_{r^{-1}}(s'_i,s'_j)}
-  =\prod_{\{i,j\}\in\partial
+  C}\frac{1-p_r(s_i,s_j)}{1-p_{r^{-1}}(s'_i,s'_j)}\\
+  &\quad=\prod_{\{i,j\}\in\partial
   C}e^{\beta(Z(r\cdot s_i,s_j)-Z(s_i,s_j))}
   =\frac{p_r(s_i,s_j)}{p_{r}(s_i,s_j)}\frac{e^{-\beta\H(\vec
   s)}}{e^{-\beta\H(\vec s')}},
+\end{aligned}
 \]
-\end{widetext}
+%\end{widetext}
 whence detailed balance is also satisfied. 
 
 This algorithm relies on the fact that the coupling $Z$ depends only on
@@ -374,12 +381,10 @@ value of $\tilde A$ on the new system. In contrast with the simpler ghost spin
 representation, this form of the Hamiltonian might be considered the `ghost
 transformation' representation.
 
+Several specific examples from Table~\ref{table:models} are described in the
+following.
 
-\section{Examples}
-
-\subsection{The Ising Model}
-
-In the Ising model spins are drawn from the set $\{1,-1\}$. Its symmetry group
+\emph{The Ising model.} 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.
@@ -389,21 +394,21 @@ 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
+\emph{The $\mathrm O(n)$ model.} 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 elements
+the $\mathrm O(3)$ model \cite{dimitrovic_finite-size_1991}. Other fields of
+interest include $(n+1)$-dimensional spherical harmonics
+\cite{jose_renormalization_1977} and cubic fields
+\cite{bruce_coupled_1975,blankschtein_fluctuation-induced_1982}, which can be
+applied with the new method. The method is
+quickly generalized to spins whose symmetry groups other compact Lie groups
+
+\emph{The Potts \& clock models.} 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
@@ -414,11 +419,9 @@ 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.
+transitive and therefore the algorithm is ergodic.
 
-\subsection{Roughening Models}
-
-Though not often thought of as a spin model, roughening of surfaces can be
+\emph{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 $D_\infty=\{r_i,s_i\mid
 i\in\Z\}$, whose action on the spin $j\in\Z$ is given by $r_i\cdot j=i+j$ and
@@ -431,12 +434,17 @@ of the system.  A variant of the algorithm has been applied without a field
 \cite{evertz_stochastic_1991}.
 
 
-\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.
+algorithm into that domain. Some systems are not efficient under Wolff, and we
+don't expect this extension to help them. For instance, Ising models with
+random fields or bonds technically can be treated with Wolff
+\cite{dotsenko_cluster_1991}, but it is not efficient because the clusters
+formed do scale naturally with the correlation length
+\cite{rieger_monte_1995,redner_graphical_1998}. Other approaches, like replica methods, should
+be relied on instead
+\cite{redner_graphical_1998,chayes_graphical_1998,machta_replica-exchange_2000}.
 
 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
@@ -499,10 +507,10 @@ to the scaling functions of the magnetization and susceptibility per site by
   \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].
+    &=L^{\gamma/\nu}\big[(hL^{\beta\delta/\nu},ht^{-\beta\delta})^{-\gamma/\beta\delta}\beta \mathcal
+      Y(hL^{\beta\delta/\nu,ht^{-\beta\delta}})\\
+      &\hspace{1em}+(hL^{\beta\delta/\nu},ht^{-\beta\delta})^{2/\delta}\mathcal
+    M(hL^{\beta\delta/\nu},ht^{-\beta\delta})\big].
   \end{aligned}
 \]
 We therefore expect that, for the Ising model, $\avg{s_{\text{\sc 1c}}}$
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