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author | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2018-04-27 14:20:03 -0400 |
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committer | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2018-04-27 14:20:03 -0400 |
commit | 7062cdc1982e580054b7c1c717342e4213d34d6a (patch) | |
tree | d0864fd0d98a2d97c1ede9417b88ac8abef56d3c /monte-carlo.tex | |
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diff --git a/monte-carlo.tex b/monte-carlo.tex index 96e5f91..222c5e8 100644 --- a/monte-carlo.tex +++ b/monte-carlo.tex @@ -115,23 +115,23 @@ of doing this, approximating thermodynamic quantities by sampling the distribution of systems states. For a particular system, a Monte Carlo algorithm is better the faster it arrives at a statistically independent sample. This is typically a problem at critical points, where critical slowing -down \cite{wolff1990critical} results in power-law divergences of any dynamics. Celebrated cluster +down \cite{wolff_critical_1990} results in power-law divergences of any dynamics. Celebrated cluster algorithms largely addressed this for many spin systems in the absence of -external fields by using nonlocal updates \cite{janke1998nonlocal} whose clusters undergo a percolation -transition at the critical point of the system \cite{coniglio1980clusters} and that in relatively small -dynamic exponents \cite{wolff1989comparison,du2006dynamic,liu2014dynamic,wang1990cluster}, -including the Ising, $n$-component \cite{wolff1989collective}, and Potts -\cite{swendsen1987nonuniversal,baillie1991comparison} models. These +external fields by using nonlocal updates \cite{janke_nonlocal_1998} whose clusters undergo a percolation +transition at the critical point of the system \cite{coniglio_clusters_1980} and that in relatively small +dynamic exponents \cite{wolff_comparison_1989,du_dynamic_2006,liu_dynamic_2014,wang_cluster_1990}, +including the Ising, $\mathrm O(n)$ \cite{wolff_collective_1989}, and Potts +\cite{swendsen_nonuniversal_1987,baillie_comparison_1991} models. These algorithms rely on the natural symmetry of the systems in question under global rotations, so the general application of external fields is not trivial. Some success has been made in extending these algorithms to systems in certain external fields based on applying the ghost site representation -\cite{coniglio1989exact} of certain +\cite{coniglio_exact_1989} of certain spin systems that returns global rotation invariance to spin Hamiltonians at the cost of an extra degree of freedom, but these results only allow the application of a narrow category of fields -\cite{alexandrowicz1989swendsen,destri1992swendsen,lauwers1989critical,wang1989clusters}. +\cite{alexandrowicz_swendsen-wang_1989,destri_swendsen-wang_1992,lauwers_critical_1989,wang_clusters_1989}. We show that the scaling of correlation time near the critical point of several models suggests that this approach is a natural one, e.g., that it extends the celebrated scaling of dynamics in @@ -147,10 +147,10 @@ set of states accessible by a spin, and $R$ is the \emph{symmetry group} of $X$. The set $X$ must admit a measure $\mu$ that is invariant under the action of $R$, e.g., for any $A\subseteq X$ and $r\in R$, $\mu(r\cdot A)=\mu(A)$. This trait is shared by the counting measure on any discrete set, or by any group acting by isometries -on a Riemannian manifold, such as $O(n)$ on $S^{n-1}$ in the $n$-component -model. Finally, the subset of elements in $R$ of order two must act +on a Riemannian manifold, such as $\mathrm O(n)$ on $S^{n-1}$ in the $\mathrm O(n)$ +model \cite{caracciolo_wolff-type_1993}. Finally, the subset of elements in $R$ of order two must act transitively on $X$. This property, while apparently obscure, is shared by any -symmetric space \cite{loos1969symmetric} or by any transitive, finitely generated isometry group. In fact, all the examples listed here have spins spaces with natural +symmetric space \cite{loos_symmetric_1969} or by any transitive, finitely generated isometry group. In fact, all the examples listed here have spins spaces with natural metrics whose symmetry group is the set of isometries of the spin spaces. We put one spin at each site of the lattice described by $G$, so that the state of the entire spin system is described by elements $\vec s\in X\times\cdots\times @@ -176,7 +176,7 @@ well for these cases, but we will drop the additional index notation for clarity \hline\hline Ising & $\{-1,1\}$ & $\Z/2\Z$ & $0\cdot s\mapsto s$, $1\cdot s\mapsto -s$ & $st$ & $Hs$ \\ - $n$-component & $S^{n-1}$ & $\mathrm O(n)$ & $M\cdot s\mapsto Ms$ & $s^{\mathrm T}t$ & $H^{\mathrm T}s$\\ + $\mathrm O(n)$ & $S^{n-1}$ & $\mathrm O(n)$ & $M\cdot s\mapsto Ms$ & $s^{\mathrm T}t$ & $H^{\mathrm T}s$\\ Potts & $\mathbb Z/q\mathbb Z$ & $D_n$ & $r_m\cdot s=m+s$, $s_m\cdot s=-m-s$ & $\delta(s,t)$ & $\sum_mH_m\delta(m,s)$\\ Clock & $\mathbb Z/q\mathbb Z$ & $D_n$ & $r_m\cdot s=m+s$, $s_m\cdot @@ -189,7 +189,7 @@ well for these cases, but we will drop the additional index notation for clarity their external fields are also given. Other fields are possible, of course: for instance, some are interested in modulated fields $H\cos(2\pi k\theta(s))$ for integer $k$ and $\theta(s)$ giving the angle of $s$ to some axis applied - to $n$-component models \cite{jose1977renormalization}.} + to $\mathrm O(n)$ models \cite{jose_renormalization_1977}.} \label{table:models} \end{table*} @@ -222,7 +222,7 @@ $\vec s'$ in the ensemble). While any several related cluster algorithms can be described for this system, we will focus on the Wolff algorithm in particular -\cite{wolff1989collective}. We will first describe a generalized version of the celebrated Wolff algorithm +\cite{wolff_collective_1989}. We will first describe a generalized version of the celebrated Wolff algorithm in the standard case where $B(s)=0$. After reflecting on the technical requirements of that algorithm, we will introduce a transformation to our system and Hamiltonian that allows the same algorithm to be applied with @@ -401,11 +401,11 @@ elements, performing the algorithm on the Ising model in a field is very accurately described by simply adding an extra spin coupled to all others and running the ordinary algorithm. The ghost spin version of the algorithm has been applied by several researchers previously -\cite{wang1989clusters,ray1990metastability,destri1992swendsen,lauwers1989critical} +\cite{wang_clusters_1989,ray_metastability_1990,destri_swendsen-wang_1992,lauwers_critical_1989} -\subsection{The $n$-component Model} +\subsection{The $\mathrm O(n)$ Model} -In the $n$-component model, spins are described by vectors on the $(n-1)$-sphere, +In the $\mathrm O(n)$ model, spins are described by vectors on the $(n-1)$-sphere, so that $X=S^{n-1}$. The symmetry group of this model is $O(n)$, $n\times n$ orthogonal matrices. The symmetry group acts on the spins by matrix multiplication. The elements of $O(n)$ that are order two are reflections @@ -460,7 +460,8 @@ changes so big that the whole system is always flipped, it is better to select random reflections about integers close to the average state of the system. Continuous roughening models---where the spin states are described by real numbers and the symmetry group is $\mathrm E(1)$, the Euclidean group for -one-dimensional space---are equally well described. +one-dimensional space---are equally well described. A variant of the algorithm has been +applied without a field before \cite{evertz_stochastic_1991}. %\begin{figure} @@ -494,8 +495,8 @@ If a given dynamics for a system at zero field results in scaling like $t^{-z\nu}$, one should expect its natural extension in the presence of a field to scale like $h^{-z\nu/\beta\delta}$. We measured the autocorrelation time for the 2D square-lattice model at a variety of system sizes, -temperatures, and fields using methods here -\cite{geyer1992practical}. The resulting scaling behavior, plotted in +temperatures, and fields $B(s)=hs/\beta$ using methods here +\cite{geyer_practical_1992}. The resulting scaling behavior, plotted in Fig.~\ref{fig:correlation_time-collapse}, is indeed consistent with the zero-field scaling behavior. @@ -554,7 +555,10 @@ should go as $(hL^{\beta\delta})^{2/\delta}$ for large argument. We further conjecture that this scaling behavior should hold for other models whose critical points correspond with the percolation transition of Wolff clusters. This behavior is supported by our numeric work along the critical isotherm for various Ising, Potts, and -$\mathrm O(n)$ models, shown in Fig.~\ref{fig:cluster_scaling}. As can be +$\mathrm O(n)$ models, shown in Fig.~\ref{fig:cluster_scaling}. Fields for the +Potts and $\mathrm O(n)$ models take the form +$B(s)=(h/\beta)\sum_m\cos(2\pi(s-m)/q)$ and $B(s)=(h/\beta)[1,0,\ldots,0]s$ +respectively. As can be seen, the average cluster size collapses for each model according to the scaling hypothesis, and the large-field behavior likewise scales as we expect from the na\"ive Ising conjecture. @@ -587,7 +591,8 @@ prior methods. Instead of simply applying a spin-like field, this method allows for the application of \emph{arbitrary functions} of the spins. For instance, theoretical predictions for the effect of symmetry-breaking perturbations on spin models can be tested numerically -\cite{jose1977renormalization}. +\cite{jose_renormalization_1977} +\cite{blankschtein_fluctuation-induced_1982,bruce_coupled_1975,manuel_carmona_$n$-component_2000}. \begin{acknowledgments} \end{acknowledgments} |