From e7220fadb6b3775e55afd9a07831e90136cfaf24 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 7 Nov 2017 10:43:35 -0500 Subject: many changes, including added references --- monte-carlo.tex | 48 ++++++++++++++++++++++++++++++++++++++++-------- 1 file changed, 40 insertions(+), 8 deletions(-) (limited to 'monte-carlo.tex') diff --git a/monte-carlo.tex b/monte-carlo.tex index fe2a315..2490b31 100644 --- a/monte-carlo.tex +++ b/monte-carlo.tex @@ -82,7 +82,7 @@ \begin{document} -\title{An efficient cluster algorithm for the Ising model in an external field} +\title{Efficiently sampling Ising states in an external field} \author{Jaron Kent-Dobias} \author{James P.~Sethna} \affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} @@ -90,15 +90,43 @@ \date\today \begin{abstract} - An abstract. + We introduce an extension of the Wolff algorithm that preforms efficiently + in an external magnetic field. Near the Ising critical point, the + correlation time of our algorithm has a conventional scaling form that + reduces to that of the Wolff algorithm at zero field. As an application, we + directly measure scaling functions of observables in the metastable state of + the 2D Ising model. \end{abstract} \maketitle -\section{Introduction} - The Ising model is a simple model of a magnet comprised of locally interacting -spins. +spins. Like most large thermal systems, computation of its properties cannot +be carried out explicitly and is preformed using Monte Carlo methods. Near its +continuous phase transition, divergent correlation length leads to divergent +correlation time in any locally-updating algorithm, hampering computation. +At zero external field, this was largely alleviated by cluster algorithms, +like the Wolff algorithm, whose dynamics are nonlocal and each step flips +groups of spins whose size diverges with the correlation length. However, the +Wolff algorithm only works at zero field. We describe an extension of this +algorithm that works in arbitrary external field while preserving the Wolff +algorithm's small dynamic exponent. + +The Wolff algorithm works by first choosing a random spin and adding it to an +empty cluster. Every neighbor of that spin pointed in the same direction as +the spin is added to the cluster with probability $1-e^{-2\beta J}$, where +$\beta=1/T$ and $J$ is the coupling between sites. This process is iterated +again for neighbors of every spin added to the cluster. When all sites +surrounding the cluster have been exhausted, the cluster is flipped. Our +algorithm is a simple extension of this. An extra spin is introduced (often +referred to as a ``ghost spin'') that couples to all others with coupling $H$. +The traditional Wolff algorithm is then preformed on this larger lattice exactly as described above, +with the extra spin treated no differently from any others. Observables in the +original system can be exactly estimated on the new one using a simple +mapping. As an application, we use our algorithm to measure critical scaling functions +of the 2D Ising model in its metastable phase. + +\section{Introduction} Consider an undirected graph $G=(V,E)$ describing a system of interacting spins. The set of vertices $V=\{1,\ldots,N\}$ enumerates the sites of the network, and the @@ -147,7 +175,8 @@ $S^n$ according to the Boltzmann distribution $e^{-\beta\H(s)}$, so that averages of observables made using their samples asymptotically approach the true expected value. -The Metropolis--Hastings algorithm is very popular for systems in statistical +The Metropolis--Hastings algorithm +\cite{metropolis1953equation,hastings1970monte} is very popular for systems in statistical physics. A starting state $s\in S^n$ is randomly perturbed to the state $s'$, usually by flipping one spin. The change in energy $\Delta\H=\H(s')-\H(s)$ due to the perturbation is then computed. If the change is negative the perturbed @@ -177,7 +206,7 @@ take many perturbations to move between in configuration space. \end{algorithm} \end{figure} -The Wolff algorithm solves many of these problems, but only at zero external +The Wolff algorithm \cite{wolff1989collective} solves many of these problems, but only at zero external field, $H=0$. This algorithm solves the problem of critical slowing-down by flipping carefully-constructed clusters of spins at once in a way that samples high-correlated states quickly while also always accepting prospective states. @@ -196,6 +225,9 @@ cluster acceptance or rejection across the spins in the cluster as they are added. In this version, the cluster is abandoned with probability $1-e^{-\beta H}$ every time a spin is added to it. +$z=0.29(1)$ \cite{wolff1989comparison,liu2014dynamic} $z=0.35(1)$ for +Swendsen--Wang \cite{swendsen1987nonuniversal} + \begin{figure} \begin{algorithm}[H] \begin{algorithmic} @@ -491,7 +523,7 @@ a given average if the associated state is in the reduced space of interest. Thanks! \end{acknowledgments} -%\bibliography{monte-carlo} +\bibliography{monte-carlo} \end{document} -- cgit v1.2.3-54-g00ecf