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Physical review. B, Condensed matter

Accelerated convergence in exact-diagonalization studies

M. A. Novotny, José A. Riera, P. W. Leung, Elbio R. Dagotto

Published Sep 1, 1993• 0 citations

Abstract

A simple method is proposed and tested for obtaining accelerated convergence of quantum systems on small lattices with N sites. The main idea is to perform exact diagonalizations with some added irrelevant parameter, and use this parameter to accelerate the convergence to the infinite-lattice limit. In this paper different boundary conditions are used to improve the convergence for the Heisenberg model. In particular, we find that the application of the method to the d=1 antiferromagnetic Heisenberg model changes the rate of convergence of the ground-state energy per site, \ensuremath{\Vert}${\mathit{E}}_{0}$(N)-${\mathit{E}}_{0}$(\ensuremath{\infty})\ensuremath{\Vert}\ensuremath{\sim}${\mathit{N}}^{\mathrm{\ensuremath{-}}\mathit{x}}$, to x\ensuremath{\approxeq}4 from the value x=2, which is found using only periodic boundary conditions.

DOI: 10.1103/physrevb.48.6255

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A simple method is proposed and tested for obtaining accelerated convergence of quantum systems on small lattices with N sites. The main idea is to perform exact diagonalizations with some added irrelevant parameter, and use this parameter to accelerate the convergence to the infinite-lattice limit. In this paper different boundary conditions are used to improve the convergence for the Heisenberg model. In particular, we find that the application of the method to the d=1 antiferromagnetic Heisenberg model changes the rate of convergence of the ground-state energy per site, \ensuremath{\Vert}

{\mathit{E}}_{0}

(N)-

{\mathit{E}}_{0}

(\ensuremath{\infty})\ensuremath{\Vert}\ensuremath{\sim}

{\mathit{N}}^{\mathrm{\ensuremath{-}}\mathit{x}}

, to x\ensuremath{\approxeq}4 from the value x=2, which is found using only periodic boundary conditions.

We present results of exact diagonalization of the isotropic antiferromagnetic S=1 Heisenberg chain by the Lanczos method, for finite rings of up to N=22 sites. The Haldane gap G(N) and the ground-state energy per site e(N) converge, with increasing N, faster than a power law. By VBS and Shanks transformations, the extrapolated values are G(\ensuremath{\infty})=0.41049(2) and e(\ensuremath{\infty})=-1.401485(2). The spin-spin correlation function is well fit by exp(-r/\ensuremath{\xi})/ \ensuremath{\surd}r with \ensuremath{\xi}=6.2.

Disordered quantum antiferromagnets in two-dimensional compounds have been a focus of interest in the last years due to their exotic properties. However, with very few exceptions, the ground states of the corresponding Hamiltonians are notoriously difficult to simulate making their characterization and detection very elusive, both theoretically and experimentally. Here we propose a method to signal quantum disordered antiferromagnets by doing exact diagonalization in small lattices using random boundary conditions and averaging the observables of interest over the different disorder realizations. We apply our method to study the Heisenberg spin-1/2 model in an anisotropic triangular lattice. In this model, the competition between frustration and quantum fluctuations might lead to some spin-liquid phases as predicted from different methods ranging from spin-wave mean-field theory to 2D-DMRG or PEPS. Our method accurately reproduces the ordered phases expected of the model and signals quantum disordered phases by the presence of a large number of quasidegenerate ground states together with an undefined local order parameter. The method presents a weak dependence on finite-size effects.

The one-dimensional Heisenberg antiferromagnet of S = 1 spins is studied by numericaldiagonalization method based on the Lanczos algorithm. The Haldane gap of this system isnumerically estimated from finite-size clusters under the twisted boundary condition up tothe 30-site system. The finite-size results are analyzed by a convergence-acceleration technique; the multistep procedure of the convergence acceleration is successfully carried out.The present estimate for the gap magnitude is more precisely than the previously reportedvalue from a Lanczos-diagonalization study. It also agrees well with the result reported ina study based on the density matrix renormalization group calculation. The present studyclarifies that Lanczos-diagonalization calculations can sufficiently contribute for preciselyestimating an energy gap although this method can treat only small clusters.

The ground-state properties of the periodic Anderson model on one-dimensional chains from three to six sites are studied by exact diagonalization. An appropriate average over boundary conditions is found to be necessary in order to eliminate finite-size oscillations. In the weak-coupling limit, the results for the short-range correlations suggest a smooth transition from a ferromagnetic state at low electron concentration to an antiferromagnetic state at half-filling, rather than a sudden jump at a critical concentration.