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-rw-r--r--data/cluster-size/cluster-size_2vector3d.dat7
-rw-r--r--data/cluster-size/cluster-size_3vector3d.dat7
-rw-r--r--fig_clusters_ising2d.eps42
-rw-r--r--fig_correlation_collapse-hL.tex6
-rw-r--r--figs/fig_correlation_collapse-hL.gplot2
-rw-r--r--monte-carlo.pdfbin211006 -> 189179 bytes
-rw-r--r--monte-carlo.tex504
7 files changed, 261 insertions, 307 deletions
diff --git a/data/cluster-size/cluster-size_2vector3d.dat b/data/cluster-size/cluster-size_2vector3d.dat
index b1d07b7..23f0d4f 100644
--- a/data/cluster-size/cluster-size_2vector3d.dat
+++ b/data/cluster-size/cluster-size_2vector3d.dat
@@ -29,6 +29,7 @@
8 2.20167 2511.88999999999987 501.841463414634 4.319170260792832
8 2.20167 5011.86999999999989 507.2258064516132 2.933977464671744
8 2.20167 10000. 511.98755186721996 0.012448132782592
+
16 2.20167 0.00001 243.1411744110428 2.431011634765824
16 2.20167 0.0000199526 252.5322712418263 2.520912407564288
16 2.20167 0.0000398107 253.76404465212622 2.537209270288384
@@ -60,6 +61,7 @@
16 2.20167 2511.88999999999987 3997.2430278884435 39.557926898192385
16 2.20167 5011.86999999999989 4058.5870445344112 22.549504065769472
16 2.20167 10000. 4080.846774193549 13.085977988919296
+
32 2.20167 0.00001 986.1265014644081 9.859696529113087
32 2.20167 0.0000199526 984.3604448964771 9.837040297869311
32 2.20167 0.0000398107 1006.0389259506484 10.0341567143936
@@ -91,6 +93,7 @@
32 2.20167 2511.88999999999987 32104.675889328064 287.3561132376064
32 2.20167 5011.86999999999989 32237.24899598392 234.80085736269413
32 2.20167 10000. 32511.387096774182 150.84190567831962
+
64 2.20167 0.00001 4064.1647201144015 40.60314739749683
64 2.20167 0.0000199526 3831.507650574287 38.19021513785344
64 2.20167 0.0000398107 4188.66782299095 41.814975126175746
@@ -122,6 +125,7 @@
64 2.20167 2511.88999999999987 257573.15748031496 2009.5231733897626
64 2.20167 5011.86999999999989 256347.9367588931 2264.411607160324
64 2.20167 10000. 261751.60245901643 261.52246707172145
+
128 2.20167 0.00001 17218.50733331795 171.69256298001204
128 2.20167 0.0000199526 18378.8810154216 182.93576751172813
128 2.20167 0.0000398107 21719.71231105522 216.67560846000129
@@ -153,10 +157,11 @@
128 2.20167 2511.88999999999987 2.0513495555555564e6 17998.436177113254
128 2.20167 5011.86999999999989 2.0773365813008118e6 12176.020058098631
128 2.20167 10000. 2.0781455720000006e6 11877.88158764109
+
256 2.20167 0.00001 99796.95495047965 993.0432408086118
256 2.20167 0.0000199526 118960.46114502712 1183.6311268166205
256 2.20167 0.0000398107 148297.9400047447 1477.7606331841905
256 2.20167 0.0000794328 194551.36132918936 1937.9314215910113
256 2.20167 0.000158489 251457.49189555127 2511.3250818634547
256 2.20167 0.000316228 324895.6469723896 3238.5258641571186
-256 2.20167 0.000630957 433035.5322706618 4330.322589835591 \ No newline at end of file
+256 2.20167 0.000630957 433035.5322706618 4330.322589835591
diff --git a/data/cluster-size/cluster-size_3vector3d.dat b/data/cluster-size/cluster-size_3vector3d.dat
index bcbdc89..30c9f63 100644
--- a/data/cluster-size/cluster-size_3vector3d.dat
+++ b/data/cluster-size/cluster-size_3vector3d.dat
@@ -29,6 +29,7 @@
8 1.44325 2511.88999999999987 497.873517786561 4.809981675440128
8 1.44325 5011.86999999999989 504.82995951416984 3.678634721823232
8 1.44325 10000. 507.7768595041316 2.980009383246336
+
16 1.44325 0.00001 216.68961952935527 2.166391254450176
16 1.44325 0.0000199526 219.48383212166758 2.192636421758976
16 1.44325 0.0000398107 219.69445132802866 2.195624944164864
@@ -60,6 +61,7 @@
16 1.44325 2511.88999999999987 3962.0758620689653 38.27452665491456
16 1.44325 5011.86999999999989 4046.096 25.382901526335488
16 1.44325 10000. 4072.6788617886186 17.727087666458623
+
32 1.44325 0.00001 826.8353329757553 8.262531302096896
32 1.44325 0.0000199526 837.3603080731689 8.368123500691455
32 1.44325 0.0000398107 806.5711674789724 8.05936187572224
@@ -91,6 +93,7 @@
32 1.44325 2511.88999999999987 31823.757462686564 309.3486080652411
32 1.44325 5011.86999999999989 32246.916996047406 235.3584664114299
32 1.44325 10000. 32426.70967741933 195.48239847224116
+
64 1.44325 0.00001 3300.994286090191 32.9843773669376
64 1.44325 0.0000199526 3377.702937935479 33.7408356099031
64 1.44325 0.0000398107 3339.950045898211 33.335401827794946
@@ -122,6 +125,7 @@
64 1.44325 2511.88999999999987 258340.48031496076 1803.419695280816
64 1.44325 5011.86999999999989 260464.34136546188 1137.2822096961863
64 1.44325 10000. 258819.8306451612 1828.5416650933535
+
128 1.44325 0.00001 12447.981724311225 124.36513455760998
128 1.44325 0.0000199526 12928.512209922817 128.95709454925824
128 1.44325 0.0000398107 15446.982934044607 154.19837102476492
@@ -153,9 +157,10 @@
128 1.44325 2511.88999999999987 2.0259372646048097e6 19879.654187178392
128 1.44325 5011.86999999999989 2.0418392758620698e6 19846.025820302737
128 1.44325 10000. 2.0769803904382477e6 9952.510969695437
+
256 1.44325 0.00001 60161.45099983107 599.3263746074542
256 1.44325 0.0000199526 77901.23797637496 777.8785821259203
256 1.44325 0.0000398107 101219.99686797833 1009.7149891387064
256 1.44325 0.0000794328 136260.0666018779 1358.5048457738978
256 1.44325 0.000158489 180213.0812666957 1798.52251791727
-256 1.44325 0.000316228 244606.69981441813 2439.4315712125995 \ No newline at end of file
+256 1.44325 0.000316228 244606.69981441813 2439.4315712125995
diff --git a/fig_clusters_ising2d.eps b/fig_clusters_ising2d.eps
index 829e12c..df82215 100644
--- a/fig_clusters_ising2d.eps
+++ b/fig_clusters_ising2d.eps
@@ -1,7 +1,7 @@
%!PS-Adobe-2.0 EPSF-2.0
%%Title: fig_clusters_ising2d.tex
%%Creator: gnuplot 5.2 patchlevel 2
-%%CreationDate: Thu Apr 26 20:22:42 2018
+%%CreationDate: Mon Apr 30 17:44:57 2018
%%DocumentFonts:
%%BoundingBox: 50 50 542 252
%%EndComments
@@ -441,7 +441,7 @@ SDict begin [
/Creator (gnuplot 5.2 patchlevel 2)
% /Producer (gnuplot)
% /Keywords ()
- /CreationDate (Thu Apr 26 20:22:42 2018)
+ /CreationDate (Mon Apr 30 17:44:57 2018)
/DOCINFO pdfmark
end
} ifelse
@@ -6376,8 +6376,7 @@ LCb setrgbcolor
59 2 V
60 2 V
59 2 V
-.0323 g 9497 3093 M
-7868 2598 L
+.0323 g 7868 2598 M
59 9 V
59 1 V
59 -7 V
@@ -6408,8 +6407,7 @@ LCb setrgbcolor
59 1 V
59 4 V
59 1 V
-.0968 g 9644 3261 M
-8015 2607 L
+0 g .0968 g 8015 2607 M
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@@ -6439,8 +6437,7 @@ LCb setrgbcolor
59 4 V
59 -3 V
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-.2258 g 9705 3204 M
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+0 g .2258 g 8163 2620 M
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@@ -6468,8 +6465,7 @@ LCb setrgbcolor
59 7 V
59 5 V
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-.4839 g 9705 3008 M
-8310 2639 L
+0 g .4839 g 8310 2639 M
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@@ -6494,8 +6490,7 @@ LCb setrgbcolor
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-1 g 9705 2887 M
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+0 g 1 g 8458 2732 M
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@@ -8170,7 +8165,7 @@ LCb setrgbcolor
59 0 V
59 0 V
60 0 V
-.0968 g 1394 823 M
+0 g .0968 g 1394 823 M
59 -8 V
59 -6 V
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@@ -8201,7 +8196,7 @@ LCb setrgbcolor
60 0 V
59 0 V
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-.2258 g 1541 828 M
+0 g .2258 g 1541 828 M
60 -11 V
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@@ -10260,7 +10255,7 @@ LCb setrgbcolor
59 -15 V
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60 -5 V
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59 71 V
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@@ -10291,7 +10286,7 @@ LCb setrgbcolor
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59 -14 V
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@@ -11948,8 +11943,7 @@ LCb setrgbcolor
59 -3 V
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+.0323 g 7904 803 M
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@@ -12038,8 +12030,7 @@ LCb setrgbcolor
59 10 V
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-.4839 g 9705 1129 M
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+.1219 g .4839 g 8346 796 M
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@@ -12063,8 +12054,7 @@ LCb setrgbcolor
59 27 V
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+1 g 8494 844 M
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diff --git a/fig_correlation_collapse-hL.tex b/fig_correlation_collapse-hL.tex
index bb0ee8b..f70bc37 100644
--- a/fig_correlation_collapse-hL.tex
+++ b/fig_correlation_collapse-hL.tex
@@ -8,7 +8,7 @@
{\GNUPLOTspecial{"
%!PS-Adobe-2.0 EPSF-2.0
%%Creator: gnuplot 5.2 patchlevel 2
-%%CreationDate: Thu Apr 26 20:22:42 2018
+%%CreationDate: Mon Apr 30 17:46:22 2018
%%DocumentFonts:
%%BoundingBox: 0 0 246 151
%%EndComments
@@ -448,7 +448,7 @@ SDict begin [
/Creator (gnuplot 5.2 patchlevel 2)
% /Producer (gnuplot)
% /Keywords ()
- /CreationDate (Thu Apr 26 20:22:42 2018)
+ /CreationDate (Mon Apr 30 17:46:22 2018)
/DOCINFO pdfmark
end
} ifelse
@@ -2335,7 +2335,7 @@ grestore
end
showpage
}}%
- \put(2709,140){\makebox(0,0){\strut{}$hL^{-\beta\delta/\nu}$}}%
+ \put(2709,140){\makebox(0,0){\strut{}$hL^{\beta\delta/\nu}$}}%
\put(360,1738){%
\special{ps: gsave currentpoint currentpoint translate
630 rotate neg exch neg exch translate}%
diff --git a/figs/fig_correlation_collapse-hL.gplot b/figs/fig_correlation_collapse-hL.gplot
index bfda33f..1c3be1e 100644
--- a/figs/fig_correlation_collapse-hL.gplot
+++ b/figs/fig_correlation_collapse-hL.gplot
@@ -15,7 +15,7 @@ data = "data/correlation.dat"
set logscale xy
set xrange [0.0005:200000]
set ylabel offset 1.5,0 '$\tau L^{-z}$'
-set xlabel '$hL^{-\beta\delta/\nu}$'
+set xlabel '$hL^{\beta\delta/\nu}$'
set format x '$10^{%T}$'
set yrange [0.08:1.1]
set nokey
diff --git a/monte-carlo.pdf b/monte-carlo.pdf
index a8dcab7..c80b3c2 100644
--- a/monte-carlo.pdf
+++ b/monte-carlo.pdf
Binary files differ
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}