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diff --git a/monte-carlo.tex b/monte-carlo.tex index 370ac0f..b24fa9d 100644 --- a/monte-carlo.tex +++ b/monte-carlo.tex @@ -89,7 +89,7 @@ extra degree of freedom, allowing the method to be used in a subcategory of interesting fields \cite{alexandrowicz_swendsen-wang_1989, wang_clusters_1989, ray_metastability_1990}. Static fields have also been applied by including a separate metropolis or heat bath update step after cluster formation -\cite{destri_swendsen-wang_1992, lauwers_critical_1989}, and other categories +\cite{destri_swendsen-wang_1992, lauwers_critical_1989, ala-nissila_numerical_1994}, and 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 @@ -192,9 +192,12 @@ in the following way. \begin{enumerate} \item Pick a random site $m_0$ and add it to the stack. - \item Pick a random rotation $r\in R$ of order two from the set of such - rotations with a probability distribution of the form - $P_{m_0}(r)=f(\J(m_0,r\cdot m_0))$. + \item Select a rotation $r\in R$ of order two. For detailed balance it is + sufficient to ensure that probability distribution $f_{m_0}(r\mid \set s)$ the + rotation is sampled from depend only on $Z(s,r\cdot s)$ for every site, + since these numbers are invariant under a cluster flip, e.g., $Z(s',r\cdot + s')=Z(r\cdot s,r\cdot(r\cdot s))=Z(r\cdot s,(rr)\cdot s)=Z(r\cdot + s,s)=Z(s,r\cdot s)$. \item While the stack isn't empty, \begin{enumerate} \item pop site $m$ from the stack. @@ -204,7 +207,7 @@ in the following way. \item For every $j$ such that $\{m,j\}\in E$, add site $j$ to the stack with probability \[ - p_r(s_m,s_j)=\min\{0,1-e^{\beta(\J(r\cdot s_m,s_j)-\J(s_m,s_j))}\}. + p_r(s_m,s_j\mid q)=\min\{0,1-qe^{\beta(\J(r\cdot s_m,s_j)-\J(s_m,s_j))}\}. \] \item Take $s_m\mapsto r\cdot s_m$. \end{enumerate} @@ -225,12 +228,12 @@ C\subset E$ is related to the probability of the reverse process $P(\set{s'}\to\ \[ \begin{aligned} \frac{P(\set s\to\set{s'})}{P(\set{s'}\to\set s)} - &=\frac{P_{m_0}(r)}{P_{m_0}(r^{-1})}\prod_{\{i,j\}\in + &=\frac{f_{\set s}(r\mid m_0)}{f_{\set s'}(r^{-1}\mid m_0)}\prod_{\{i,j\}\in C}\frac{p_r(s_i,s_j)}{p_{r^{-1}}(s_i',s_j')}\prod_{\{i,j\}\in\partial C}\frac{1-p_r(s_i,s_j)}{1-p_{r^{-1}}(s'_i,s'_j)}\\ &=\prod_{\{i,j\}\in\partial C}e^{\beta(\J(r\cdot s_i,s_j)-\J(s_i,s_j))} - =\frac{p_r(s_i,s_j)}{p_{r}(s_i,s_j)}\frac{e^{-\beta\H(\set + =\frac{e^{-\beta\H(\set s)}}{e^{-\beta\H(\set{s'})}}, \end{aligned} \] @@ -340,6 +343,7 @@ representation, this form of the Hamiltonian might be considered the `ghost transformation' representation. \section{Examples} +\label{sec:examples} Several specific examples from Table~\ref{table:models} are described in the following. @@ -357,7 +361,10 @@ wang_clusters_1989, ray_metastability_1990}. The algorithm has been implemented by one of the authors in an existing interactive Ising simulator at \texttt{https://mattbierbaum.github.io/ising.js} \cite{bierbaum_ising.js_nodate}. -\subsection{The $\mathrm O(n)$ model} In the $\mathrm O(n)$ model spins are described by vectors on the +\subsection{The $\mathrm O(n)$ model} +\label{sec:examples:on} + +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 @@ -394,6 +401,11 @@ of large $L$ the expected square angle between neighbors being \[ \avg{\theta^2}\simeq\frac{(n-1)T}{D+H/2} \] +We shall in in the numeric experiments of Section \ref{sec:performance} that +this choice ameliorates the problem but probably is not the best. A more +subtle technique---for instance by matching statistics of the effective +Gaussian model that results in these circumstances to the cluster +statistics---may result in better performance. \subsection{The Potts model} In the $q$-state Potts model spins are described by elements of $\{1,\ldots,q\}$. Its symmetry group is the symmetric group @@ -428,6 +440,7 @@ of the system. A variant of the algorithm has been applied without a field \cite{evertz_stochastic_1991}. \section{Performance} +\label{sec:performance} 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 @@ -454,24 +467,19 @@ 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 +$B(s)=hs/\beta$ using standard methods \cite{ossola_dynamic_2004}. 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} - \caption{Collapse of the correlation time $\tau$ of the 2D square lattice - 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}. The solid black line shows a plot of - $(hL^{\beta\delta/\nu})^{-z\nu/\beta\delta}$. +\begin{figure*} + \include{fig_correlation-times} + \caption{ + Correlation times $\tau$ scaled by the average cluster size as a function + of external field for various models at various system sizes. } \label{fig:correlation_time-collapse} -\end{figure} +\end{figure*} 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 @@ -538,10 +546,50 @@ conjecture. \include{fig_generator-times} \end{figure} +\section{Applying Nonlinear Fields to the xy Model} + +This far our numeric work has quantified the performance of existing +techniques. Here, we apply our general framework in a new way: +harmonic perturbations to the low-temperature {\sc xy}, or \twodee O(2), +model. We consider fields of the form $B_n(s)=h_n\cos(n\theta(s))$, where +$\theta$ is the angle made between $s$ and the $x$-axis, say. Corrections of +these types are expected to appear in realistic models of systems na\"ively +expected to exhibit Kosterlitz--Thouless critical behavior due to the presence +of the lattice or substrate. Whether these fields are relevant or irrelevant +in the renormalization group sense determines whether those systems spoil or +admit that critical behaviour. Among many fascinating +\cite{jose_renormalization_1977, kankaala_theory_1993, +ala-nissila_numerical_1994, dierker_consequences_1986, selinger_theory_1988} +results that emerge from systems with one or more of these fields applied, +it is predicted that $h_4$ is relevant while $h_6$ is not at some +sufficiently high temperatures below the Kosterlitz--Thouless point +\cite{jose_renormalization_1977}. + \begin{figure} \include{fig_harmonic-susceptibilities} + \caption{Susceptibilities as a function of system size for a \twodee O(2) + model at $T=0.7$ and with (top) fourfold symmetric and (bottom) eightfold + symmetric perturbing fields. Different field strengths are shown in + different colors. + } + \label{fig:harmonic-susceptibilities} \end{figure} +We made a basic investigation of this result using our algorithm. Since we ran +the algorithm at fairly high fields we did not choose reflections though the +origin uniformly. Instead, we choose the planes of reflection first by +rotating our starting spin by $2\pi m/n$ degrees for $m$ uniformly taken from +$1,\ldots,n$ and generating a normal to the plane from that direction as +described in Section \ref{sec:examples:on}. The resulting susceptibilities as +a function of system size are shown for various field strengths in +Fig.~\ref{fig:harmonic-susceptibilities}. In the fourfold case, for each field +strength there is a system size at which the divergence in the susceptibility +is cut off, while for the sixfold case we measured no such cutoff, even up to +strong fields. This confirms the expected result, that even in a strong field +the sixfold perturbations preserve the critical behavior. + +\section{Conclusions} + 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 @@ -560,7 +608,7 @@ perturbations on spin models can be tested numerically \cite{jose_renormalization_1977, blankschtein_fluctuation-induced_1982, bruce_coupled_1975, manuel_carmona_$n$-component_2000}. -<<<<<<< HEAD + \appendix \section{$\mathrm O(n)$ model at high field} @@ -629,9 +677,6 @@ whence \end{align} \section{Calculating autocorrelation time} -\begin{figure*} - \include{fig_correlation-times} -\end{figure*} \begin{acknowledgments} This work was supported by NSF grant NSF DMR-1719490. |