Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1805.04019 · Physical review. E

Cluster representations and the Wolff algorithm in arbitrary external fields

Jaron Kent-Dobias, James P. Sethna

Published Dec 7, 2018• 42 citations
arXivHTMLPDF

Abstract

We introduce a natural way to extend celebrated spin-cluster Monte Carlo algorithms for fast thermal lattice simulations at criticality, such as the Wolff algorithm, to systems in arbitrary fields, be they linear magnetic vector fields or nonlinear anisotropic ones. By generalizing the ``ghost spin'' representation to one with a ``ghost transformation,'' global invariance to spin symmetry transformations is restored at the cost of an extra degree of freedom which lives in the space of symmetry transformations. The ordinary cluster-building process can then be run on the representation. We show that this extension preserves the scaling of accelerated dynamics in the absence of a field for Ising, Potts, and $\mathrm{O}(n)$ models and demonstrate the method's use in modeling the presence of novel nonlinear fields. We also provide a c++ library for the method's convenient implementation for arbitrary models.

DOI
10.1103/physreve.98.063306

Related papers

Accelerated Monte Carlo by embedded cluster dynamics

Richard C. Brower, N. A. Gross, Kevin J. M. Moriarty

We present an overview of the new methods for embedding Ising spins in continuous fields to achieve accelerated cluster Monte Carlo algorithms. The methods of Brower and Tamayo and Wolff are summarized and variations are suggested for the O(N) models based on multiple embedded Z2 spin components and/or correlated projections. Topological features are discussed for the XY model and numerical simulations presented for d=2, d=3 and mean field theory lattices.

Journal of Computational Physics•Jul 1, 1991•3 citations

Anisotropic Scaling and Generalized Conformal Invariance at Lifshitz Points

Michel Pleimling, Malte Henkel

The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.18(2)$, $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.238(5)$, and $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.

arXiv:hep-th/0103194•Physical Review Letters•Aug 31, 2001•49 citations

Parallelized traveling cluster approximation to study numerically spin-fermion models on large lattices

Anamitra Mukherjee, Niravkumar D. Patel, Chris Bishop +1 more

Lattice spin-fermion models are important to study correlated systems where quantum dynamics allows for a separation between slow and fast degrees of freedom. The fast degrees of freedom are treated quantum mechanically while the slow variables, generically referred to as the "spins," are treated classically. At present, exact diagonalization coupled with classical Monte Carlo (ED + MC) is extensively used to solve numerically a general class of lattice spin-fermion problems. In this common setup, the classical variables (spins) are treated via the standard MC method while the fermion problem is solved by exact diagonalization. The "traveling cluster approximation" (TCA) is a real space variant of the ED + MC method that allows to solve spin-fermion problems on lattice sizes with up to 10(3) sites. In this publication, we present a novel reorganization of the TCA algorithm in a manner that can be efficiently parallelized. This allows us to solve generic spin-fermion models easily on 10(4) lattice sites and with some effort on 10(5) lattice sites, representing the record lattice sizes studied for this family of models.

arXiv:1502.07028•Physical Review E•Jun 8, 2015•24 citations

Transforming generalized Ising models into Boltzmann machines

Nobuyuki Yoshioka, Yutaka Akagi, Hosho Katsura

We find an exact mapping from the generalized Ising models with many-spin interactions to equivalent Boltzmann machines, i.e., the models with only two-spin interactions between physical and auxiliary binary variables accompanied by local external fields. More precisely, the appropriate combination of the algebraic transformations, namely the star-triangle and decoration-iteration transformations, allows one to express the model in terms of fewer-spin interactions at the expense of the degrees of freedom. Furthermore, the benefit of the mapping in Monte Carlo simulations is discussed. In particular, we demonstrate that the application of the method in conjunction with the Swendsen-Wang algorithm drastically reduces the critical slowing down in a model with two- and three-spin interactions on the Kagomé lattice.

arXiv:1812.05269•Physical review. E•Mar 8, 2019•17 citations

Single-cluster dynamics for the random-cluster model

Youjin Deng, Xiaofeng Qian, Henk W. J. Blöte

We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the q-state Potts model to noninteger values q>1. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full-cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer q, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents z(exp)=0.07 (1), 0.521 (7), and 1.007 (9) for q=2, 3, and 4, respectively. For noninteger q, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.

arXiv:0907.1711•Physical Review E•Sep 30, 2009•5 citations

Single-cluster-update Monte Carlo method for the random anisotropy model

U. Rößler

A Wolff-type cluster Monte Carlo algorithm for random magnetic models is presented. The algorithm is demonstrated to reduce significantly the critical slowing down for planar random anisotropy models with weak anisotropy strength. Dynamic exponents $z\ensuremath{\lesssim}1.0$ of best cluster algorithms are estimated for models with ratio of anisotropy to exchange constant $D/J=1.0$ on cubic lattices in three dimensions. For these models, critical exponents are derived from a finite-size scaling analysis.

Physical review. B, Condensed matter•Jun 1, 1999•9 citations
We introduce a natural way to extend celebrated spin-cluster Monte Carlo algorithms for fast thermal lattice simulations at criticality, such as the Wolff algorithm, to systems in arbitrary fields, be they linear magnetic vector fields or nonlinear anisotropic ones. By generalizing the ghost spin'' representation to one with a ghost transformation,'' global invariance to spin symmetry transformations is restored at the cost of an extra degree of freedom which lives in the space of symmetry transformations. The ordinary cluster-building process can then be run on the representation. We show that this extension preserves the scaling of accelerated dynamics in the absence of a field for Ising, Potts, and O(n)\mathrm{O}(n)O(n) models and demonstrate the method's use in modeling the presence of novel nonlinear fields. We also provide a c++ library for the method's convenient implementation for arbitrary models.
The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.18(2)$, $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.238(5)$, and $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.
Lattice spin-fermion models are important to study correlated systems where quantum dynamics allows for a separation between slow and fast degrees of freedom. The fast degrees of freedom are treated quantum mechanically while the slow variables, generically referred to as the "spins," are treated classically. At present, exact diagonalization coupled with classical Monte Carlo (ED + MC) is extensively used to solve numerically a general class of lattice spin-fermion problems. In this common setup, the classical variables (spins) are treated via the standard MC method while the fermion problem is solved by exact diagonalization. The "traveling cluster approximation" (TCA) is a real space variant of the ED + MC method that allows to solve spin-fermion problems on lattice sizes with up to 10(3) sites. In this publication, we present a novel reorganization of the TCA algorithm in a manner that can be efficiently parallelized. This allows us to solve generic spin-fermion models easily on 10(4) lattice sites and with some effort on 10(5) lattice sites, representing the record lattice sizes studied for this family of models.
We find an exact mapping from the generalized Ising models with many-spin interactions to equivalent Boltzmann machines, i.e., the models with only two-spin interactions between physical and auxiliary binary variables accompanied by local external fields. More precisely, the appropriate combination of the algebraic transformations, namely the star-triangle and decoration-iteration transformations, allows one to express the model in terms of fewer-spin interactions at the expense of the degrees of freedom. Furthermore, the benefit of the mapping in Monte Carlo simulations is discussed. In particular, we demonstrate that the application of the method in conjunction with the Swendsen-Wang algorithm drastically reduces the critical slowing down in a model with two- and three-spin interactions on the Kagomé lattice.
We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the q-state Potts model to noninteger values q>1. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full-cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer q, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents z(exp)=0.07 (1), 0.521 (7), and 1.007 (9) for q=2, 3, and 4, respectively. For noninteger q, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.
A Wolff-type cluster Monte Carlo algorithm for random magnetic models is presented. The algorithm is demonstrated to reduce significantly the critical slowing down for planar random anisotropy models with weak anisotropy strength. Dynamic exponents $z\ensuremath{\lesssim}1.0$ of best cluster algorithms are estimated for models with ratio of anisotropy to exchange constant D/J=1.0D/J=1.0D/J=1.0 on cubic lattices in three dimensions. For these models, critical exponents are derived from a finite-size scaling analysis.