arXiv:cond-mat/9909230

The two-dimensional bond-diluted quantum Heisenberg model at the classical percolation threshold

Anders W. Sandvik

Published Sep 15, 1999• 0 citations

Abstract

The two-dimensional antiferromagnetic S=1/2 Heisenberg model with random bond dilution is studied using quantum Monte Carlo simulation at the percolation threshold (50% of the bonds removed). Finite-size scaling of the staggered structure factor averaged over the largest connected clusters of sites on L*L lattices shows that long-range order exists within the percolating fractal clusters in the thermodynamic limit. This implies that the order-disorder transition driven by bond-dilution occurs exactly at the percolation threshold and that the exponents are classical. This result should apply also to the site-diluted system.

DOI: 10.48550/arxiv.cond-mat/9909230

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arXiv:cond-mat/9909230

The two-dimensional bond-diluted quantum Heisenberg model at the classical percolation threshold

Anders W. Sandvik

Published Sep 15, 1999• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.cond-mat/9909230

Related papers

Classical percolation transition in the diluted two-dimensional S=12 Heisenberg antiferromagnet

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The two-dimensional antiferromagnetic $S=1/2$ Heisenberg model with random site dilution is studied using quantum Monte Carlo simulations. Ground-state properties of the largest connected cluster on $L\ifmmode\times\else\texttimes\fi{}L$ lattices, with L up to 64, are calculated at the classical percolation threshold. In addition, clusters with a fixed number ${N}_{c}$ of spins on an infinite lattice at the percolation density are studied for ${N}_{c}$ up to 1024. The disorder averaged sublattice magnetization per spin extrapolates to the same nonzero infinite-size value for both types of clusters. Hence, the percolating clusters, which are fractal with dimensionality $d=91/48,$ have antiferromagnetic long-range order. This implies that the order-disorder transition driven by site dilution occurs exactly at the percolation threshold and that the exponents are classical. The same conclusion is reached for the bond-diluted system. The full sublattice magnetization versus site dilution curve is obtained in terms of a decomposition into a classical geometrical factor and a factor containing all the effects of quantum fluctuations. The spin stiffness is shown to obey the same scaling as the conductivity of a random resistor network.The two-dimensional antiferromagnetic $S=1/2$ Heisenberg model with random site dilution is studied using quantum Monte Carlo simulations. Ground-state properties of the largest connected cluster on $L\ifmmode\times\else\texttimes\fi{}L$ lattices, with L up to 64, are calculated at the classical percolation threshold. In addition, clusters with a fixed number ${N}_{c}$ of spins on an infinite lattice at the percolation density are studied for ${N}_{c}$ up to 1024. The disorder averaged sublattice magnetization per spin extrapolates to the same nonzero infinite-size value for both types of clusters. Hence, the percolating clusters, which are fractal with dimensionality $d=91/48,$ have antiferromagnetic long-range order. This implies that the order-disorder transition driven by site dilution occurs exactly at the percolation threshold and that the exponents are classical. The same conclusion is reached for the bond-diluted system. The full sublattice magnetization versus site dilution curve is obtained in terms of a decomposition into a classical geometrical factor and a factor containing all the effects of quantum fluctuations. The spin stiffness is shown to obey the same scaling as the conductivity of a random resistor network.

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The two-dimensional antiferromagnetic

S=1/2

Heisenberg model with random site dilution is studied using quantum Monte Carlo simulations. Ground-state properties of the largest connected cluster on $L\ifmmode\times\else\texttimes\fi{}L$ lattices, with L up to 64, are calculated at the classical percolation threshold. In addition, clusters with a fixed number

{N}_{c}

of spins on an infinite lattice at the percolation density are studied for

{N}_{c}

up to 1024. The disorder averaged sublattice magnetization per spin extrapolates to the same nonzero infinite-size value for both types of clusters. Hence, the percolating clusters, which are fractal with dimensionality

d=91/48,

have antiferromagnetic long-range order. This implies that the order-disorder transition driven by site dilution occurs exactly at the percolation threshold and that the exponents are classical. The same conclusion is reached for the bond-diluted system. The full sublattice magnetization versus site dilution curve is obtained in terms of a decomposition into a classical geometrical factor and a factor containing all the effects of quantum fluctuations. The spin stiffness is shown to obey the same scaling as the conductivity of a random resistor network.

O(2)

quantum rotor and the Heisenberg antiferromagnet. We study two analytic descriptions of the

O(2)

quantum rotor near the percolation threshold. First a spin-wave expansion is shown to predict long-ranged order, but there are statistically rare points on the cluster that violate the standard assumptions of spin-wave theory. A real-space renormalization group (RSRG) approach is then used to understand how these rare points modify ordering of the

O(2)

rotor. A new class of fixed points of the RSRG equations for disordered one-dimensional bosons is identified and shown to support the existence of long-range order on the percolation backbone in two dimensions. These results are relevant to experiments on bosons in optical lattices and superconducting arrays, and also (qualitatively) for the diluted Heisenberg antiferromagnet ${\mathrm{La}}_{2}{(\mathrm{Zn},\mathrm{Mg})}_{x}{\mathrm{Cu}}_{1\ensuremath{-}x}{\mathrm{O}}_{4}$.