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Diffstat (limited to 'monte-carlo.tex')
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diff --git a/monte-carlo.tex b/monte-carlo.tex index f88a053..a305aa8 100644 --- a/monte-carlo.tex +++ b/monte-carlo.tex @@ -130,11 +130,14 @@ 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, 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 +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 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 +correlation time near the critical point of several models suggests that the +`ghost' 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 @@ -390,9 +393,9 @@ 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}. +just using the `ghost spin' representation. This algorithm or algorithms +based on the same decomposition of the Hamiltonian have been applied +by several researchers \cite{alexandrowicz_swendsen-wang_1989, wang_clusters_1989, ray_metastability_1990}. \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 |