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Canadian Journal of Physics

Exact diagonalization of the S = 1/2 XY ferromagnet on a new set of finite triangular lattices at T = 0

Donald D. Betts, K S Lee, Hua Lin

Published Mar 1, 2003• 1 citation
Publisher

Abstract

We have obtained 85 finite triangular lattices from 7 to 36 vertices. We display two very good finite lattices – 21a (well-known) and 22a (previously unknown). Over the past decade several physicists have used exact diagonalization on five tripartite triangular lattices from N = 9 to 36 to study the Heisenberg and XY antiferromagnet on the infinite triangular lattice. Nine more tripartite triangular lattices are available as shown below in the text. Our exact diagonalization of the S = 1/2 XY ferromagnetic energies and magnetization leads, by scalar equations, to the properties on the infinite lattice. We found that all but 10 of the 85 are good lattices. Finally, we obtained spin–spin correlations of two kinds, xx and zz. PACS Nos.: 75.10Jm, 05.05+q

DOI
10.1139/p03-036

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We have obtained 85 finite triangular lattices from 7 to 36 vertices. We display two very good finite lattices – 21a (well-known) and 22a (previously unknown). Over the past decade several physicists have used exact diagonalization on five tripartite triangular lattices from N = 9 to 36 to study the Heisenberg and XY antiferromagnet on the infinite triangular lattice. Nine more tripartite triangular lattices are available as shown below in the text. Our exact diagonalization of the S = 1/2 XY ferromagnetic energies and magnetization leads, by scalar equations, to the properties on the infinite lattice. We found that all but 10 of the 85 are good lattices. Finally, we obtained spin–spin correlations of two kinds, xx and zz. PACS Nos.: 75.10Jm, 05.05+q
First we introduce the upper triangular lattice form method that permits one to derive easily the complete set of finite lattices of any given number, N , of vertices for any regular infinite lattice in any number of dimensions, and we use it to obtain all finite fcc lattices of N <28. We compute the ground state energy, magnetization and spin-spin correlations of the spin one-half X Y ferromagnet on each of these lattices. Extrapolation of these data to infinite N yields T =0 estimates of the properties mentioned. Our estimates agree to a few tenths of a percent with spin wave, series expansion and variational estimates.
This paper is the third in a sequence of papers developing and applying the finite-lattice method for estimation of the zero-temperature properties of quantum spin models on infinite cubic lattices. Here we generate finite bipartite body-centred cubic (bcc) lattices of vertices. Our geometrically distinct finite lattices are defined by vectors in upper triangular lattice form. We have found that sets of two to six geometrically distinct finite bcc lattices are topologically identical, and that we thus need only one lattice of each set for our method of estimation. We have studied the spin one-half Heisenberg antiferromagnet by diagonalizing its Hamiltonian on each of the finite lattices and hence computing its ground-state properties. By extrapolation of these data we obtain estimates of the T = 0 properties on the infinite bcc lattice. Our estimate of the T = 0 energy agrees to five parts in 10 thousand with third-order spin-wave and series-expansion method estimates, while our estimate of the staggered magnetization agrees with the spin-wave estimate to within 0.25%.
This paper describes an improvement in the method of exact diagonalization of Hamiltonians of quantum spin models on finite square lattices and the statistical analysis of the data so obtained to estimate the physical properties of the models on the infinite square lattices at zero temperature. The geometry and topology of finite square lattices are described. The models studied are the spin one-half XY and Heisenberg antiferromagnets using 28 finite square lattices with up to 32 vertices. Our estimates of the energy and magnetization on each model on the infinite lattice at zero temperature compare very well with recent estimates using quantum Monte Carlo, series expansion, and spin wave estimates. Estimates of spin wave velocity and transverse susceptibilities are more scattered.PACS No.: 75.10J
The calculation of two- and four-spin correlations of the [Formula: see text] ferromagnet has been extended to an N = 21 site triangular lattice. By fitting a quadratic in 1/N to the nearest neighbour transverse pair correlations, we have estimated the ground-state energy per bond on the infinite lattice to be E0/3NJ = −0.5326 ± 0.003. We conjecture that the square of the magnetization per site vanishes very sharply as N−0.06. The nearest neighbour longitudinal correlation per bond [Formula: see text] for all two-dimensional lattices.
The calculation of two- and four-spin correlations of the [Formula: see text] antiferromagnet has been extended to an N = 21 site triangular lattice. By fitting a quadratic in 1/N to the nearest neighbour transverse spin correlations, we have estimated the ground-state energy per bond on the infinite lattice to be E0/3NJ = −0.2716 ± 0.005. The nearest neighbour longitudinal correlation is estimated to be [Formula: see text]. A short-range order parameter, the chirality, is defined and estimated for the infinite lattice. From the N dependence of the three sublattice or helical magnetization, the spin correlation is conjectured to decay algebraically as [Formula: see text].