© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:cond-mat/9910477 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Optimized energy calculation in lattice systems with long-range interactions

M. Krech, Erik Luijten

Published Feb 1, 2000• 21 citations

Abstract

We discuss an efficient approach to the calculation of the internal energy in numerical simulations of spin systems with long-range interactions. Although, since the introduction of the Luijten-Blote algorithm, Monte Carlo simulations of these systems no longer pose a fundamental problem, the energy calculation is still an O(N2) problem for systems of size N. We show how this can be reduced to an O(N log N) problem, with a break-even point that is already reached for very small systems. This allows the study of a variety of, until now hardly accessible, physical aspects of these systems. In particular, we combine the optimized energy calculation with histogram interpolation methods to investigate the specific heat of the Ising model and the first-order regime of the three-state Potts model with long-range interactions.

DOI: 10.1103/physreve.61.2058

Related papers

MONTE CARLO METHOD FOR SPIN MODELS WITH LONG-RANGE INTERACTIONS

Erik Luijten, Henk W. J. Blöte

We introduce a Monte Carlo method for the simulation of spin models with ferromagnetic long-range interactions in which the amount of time per spin-flip operation is independent of the system size, in spite of the fact that the interactions between each spin and all other spins are taken into account. We work out two algorithms for the q-state Potts model and discuss the generalization to systems with other interactions and to O (n) models. We illustrate the method with a simulation of the mean-field Ising model, for which we have also analytically calculated the leading finite-size correction to the dimensionless amplitude ratio <m2>2/<m4> at the critical temperature.

International Journal of Modern Physics C•Jun 1, 1995•156 citations

Studies of 3D-Ising model with long-range interaction

Boris Kryzhanovsky, Olga G. Maksimova, Leonid B. Litinskii +1 more

We use the Monte-Carlo simulations to study thermodynamic properties of Ising models on cubic lattices accounting for interaction with next-nearest neighbors. The results of simulations are in good agreement with an analytic formula that defines the critical inverse temperature as a function of the interaction parameter. The behaviors of the internal energy and the heat capacity indicate that at the critical point the heat capacity has an infinite singularity.

International Conference on Information Technology and Nanotechnology (ITNT)•Sep 20, 2021•1 citation

First-order transition in the one-dimensional three-state Potts model with long-range interactions

Zvonko Glumac, Katarina Uzelac

The first-order phase transition in the three-state Potts model with long-range interactions decaying as ${1/r}^{1+\ensuremath{\sigma}}$ has been examined by numerical simulations using the recently proposed Luijten-Bl\"ote algorithm. By applying scaling arguments to the interface free energy, the Binder's fourth-order cumulant, and the specific heat maximum, the change in the character of the transition through variation of parameter \ensuremath{\sigma} was studied.

arXiv:cond-mat/9807417•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1998•18 citations

Quantum Monte Carlo Simulation of the Spin 1/2XXZModel on the Square Lattice

Yutaka Okabe, Macoto Kikuchi

We systematically study the spin 1/2 quantum XXZ model on the square lattice using a quantum Monte Carlo method. The temperature dependence of the energy, specific heat and order parameter is calculated for the systems of sizes up to 16×16. Investigating the size dependence carefully, we exarnine the ground-state energy and the existence of the long-range order.

Journal of the Physical Society of Japan•Dec 15, 1988•107 citations

Local-states method for the calculation of free energies in Monte Carlo simulations of lattice models

A. G. Schlijper, A. R. D. van Bergen, Berend Smit

We present and demonstrate an accurate, reliable, and computationally cheap method for the calculation of free energies in Monte Carlo simulations of lattice models. Even in the critical region it yields good results with comparatively short simulation runs. The method combines upper and lower bounds on the thermodynamic limit entropy density to yield not only an accurate estimate of the free energy but a bound on the possible error as well. The method is demonstrated on the two- and three-dimensional Ising models and the three-dimensional, three-states Potts model.

Physical Review A•Jan 1, 1990•21 citations

Minimal energy ensemble Monte Carlo algorithm for the partition function of fermions coupled to classical fields

Przemysław R. Grzybowski, Łukasz Czekaj, Mariusz M. Nogala +2 more

Models of noninteracting fermions coupled to auxiliary classical fields are relevant to the understanding of a wide variety of problems in many-body physics, e.g., the description of manganites, diluted magnetic semiconductors, or strongly interacting electrons on lattices. We present a flat-histogram Monte Carlo algorithm that simulates a statistical ensemble that allows one to directly acquire the partition function at all temperatures for such systems. The defining feature of the algorithm is that it utilizes the complete thermodynamic information from the full energy spectrum of noninteracting fermions available during sampling of the configuration space of the classical fields. We benchmark the method for the classical Ising and Potts models in two dimensions, as well as the Falicov-Kimball model describing itinerant electrons interacting with heavy ions.

arXiv:1508.01439•Physical review. E•Jun 20, 2016•0 citations

We discuss an efficient approach to the calculation of the internal energy in numerical simulations of spin systems with long-range interactions. Although, since the introduction of the Luijten-Blote algorithm, Monte Carlo simulations of these systems no longer pose a fundamental problem, the energy calculation is still an O(N2) problem for systems of size N. We show how this can be reduced to an O(N log N) problem, with a break-even point that is already reached for very small systems. This allows the study of a variety of, until now hardly accessible, physical aspects of these systems. In particular, we combine the optimized energy calculation with histogram interpolation methods to investigate the specific heat of the Ising model and the first-order regime of the three-state Potts model with long-range interactions.

We introduce a Monte Carlo method for the simulation of spin models with ferromagnetic long-range interactions in which the amount of time per spin-flip operation is independent of the system size, in spite of the fact that the interactions between each spin and all other spins are taken into account. We work out two algorithms for the q-state Potts model and discuss the generalization to systems with other interactions and to O (n) models. We illustrate the method with a simulation of the mean-field Ising model, for which we have also analytically calculated the leading finite-size correction to the dimensionless amplitude ratio <m2>2/<m4> at the critical temperature.

We present and demonstrate an accurate, reliable, and computationally cheap method for the calculation of free energies in Monte Carlo simulations of lattice models. Even in the critical region it yields good results with comparatively short simulation runs. The method combines upper and lower bounds on the thermodynamic limit entropy density to yield not only an accurate estimate of the free energy but a bound on the possible error as well. The method is demonstrated on the two- and three-dimensional Ising models and the three-dimensional, three-states Potts model.

Models of noninteracting fermions coupled to auxiliary classical fields are relevant to the understanding of a wide variety of problems in many-body physics, e.g., the description of manganites, diluted magnetic semiconductors, or strongly interacting electrons on lattices. We present a flat-histogram Monte Carlo algorithm that simulates a statistical ensemble that allows one to directly acquire the partition function at all temperatures for such systems. The defining feature of the algorithm is that it utilizes the complete thermodynamic information from the full energy spectrum of noninteracting fermions available during sampling of the configuration space of the classical fields. We benchmark the method for the classical Ising and Potts models in two dimensions, as well as the Falicov-Kimball model describing itinerant electrons interacting with heavy ions.