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Journal of Physics A Mathematical and General

Finite-lattice extrapolation algorithms

Malte Henkel, Gunter Markus Schutz

Published Jun 7, 1988• 198 citations
Publisher

Abstract

Two algorithms for sequence extrapolation, due to van den Broeck and Schwartz (1979), and Bulirsch and Stoer (1964), are reviewed and critically compared. Applications to three-state and six-state quantum chains and to the (2+1)D Ising model show that the algorithm of Bulirsch and Stoer is superior, in particular if only very few finite-lattice data are available.

DOI
10.1088/0305-4470/21/11/019

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Two algorithms for sequence extrapolation, due to van den Broeck and Schwartz (1979), and Bulirsch and Stoer (1964), are reviewed and critically compared. Applications to three-state and six-state quantum chains and to the (2+1)D Ising model show that the algorithm of Bulirsch and Stoer is superior, in particular if only very few finite-lattice data are available.
For pt.I see ibid., vol.14, p.241-57 (1981). Two efficient methods for finding the low-lying states of Hamiltonians on finite lattices are described. The first involves constructing a finite representation of the Hamiltonian using strong-coupling eigenstates, while the second is based on the Lanczos recursion method. The methods are used to determine the mass gap of the O(2) and O(3) Heisenberg Hamiltonians in (1+1) dimensions for a sequence of finite chains. The critical behaviour of the infinite chain is then analysed by extrapolating the finite-lattice estimates using finite-size scaling. A remarkably sensitive test is developed for the presence of a phase transition. For the O(2) model data this test yields strong evidence for a phase transition with the weak-coupling phase massless, while, in the O(3) case the test supports, although more weakly, the absence of any transition.
The density of states for the three-dimensional Ising model is calculated with high precision by means of multicanonical simulations. This allows us to estimate the leading partition function zeros for lattice sizes up to L = 32. We have evaluated the critical exponent ν and the correction to scaling through an analysis of a multi-parameter fit and of the Bulirsch-Stoer (BST) extrapolation algorithm. The performance of the BST algorithm is also explored in case of the 2D Ising model, where the exact partition function zeros are known.
The Bulirsch-Stoer extrapolation algorithm was used in a statistical mechanics setting in 1984 by Henkel and Patkos. Since then it has been used numerous times in a large variety of settings to extrapolate from finite size systems to the infinite system in a large variety of situations in statistical mechanics. We investigate some of its characteristics by using it in situations where the behavior of the infinite system is known. One characteristic is the error involved with the algorithm. More importantly we investigate the dependence of the effectiveness of the algorithm on the size and number of systems used as input and find that a larger number of smaller systems results in better results than a few much larger systems.
The sequence-transformation method due to Vanden Broeck and Schwartz is used to accelerate the convergence of numerical finite-lattice data on ground-state energies to their infinite-lattice limit. Highly accurate estimates of the ground-state energy are obtained for the S=12S=\frac{1}{2}S=21​ antiferromagnetic Heisenberg and XY\mathrm{XY}XY rings. The ground-state energy of the S=1S=1S=1 antiferromagnetic Heisenberg ring is also estimated accurately.