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Journal of the Physical Society of Japan

Extension of the Method of Exact Diagonalization of Quantum Spin Models to Finite Face Centred Cubic Lattices and Estimation of theT=0Properties of theS=1/2XYFerromagnet on the Infinite fcc Lattice

G.E. Stewart, Donald D. Betts, James Flynn

Published Oct 15, 1997• 6 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.66.3231

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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%.
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
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 well-known method of exact diagonalization on finite lattices to estimate the properties of quantum spin systems on the square lattice was greatly enhanced recently by the introduction of additional finite lattices based on nonsquare parallelogram tiles. In this paper an analogous approach is taken for quantum spin systems on the simple cubic lattice; finite simple cubic lattices are formed from space-filling nearly cubic parallelepipeds on which triply periodic boundary conditions are imposed. We found 78 useful finite simple cubic lattices of N vertices, 7 ≤ N ≤ 27, of which 20 are bipartite. Our finite-lattice estimates of the energy and spontaneous magnetization of the of the S = ‰XY ferromagnet at T = 0 agree to better than one part in a thousand with the best spin wave, variational, and series expansion estimates. However, the agreement of the finite lattice estimates of the properties of the Heisenberg antiferromagnet is considerably worse. Resume : Nous connaissons bien la mOthode de diagonalisation sur rOseau carrO fini de spins quantiques. Elle a OtO notablement amOliorOe rOcemment par liintroduction de rOseaux en parallOlogrammes non orthogonaux. Nous proposons une approche similaire pour le problme des rOseaux simples cubiques de spins quantiques. Nous formons des rOseaux simples cubiques ‡ partir de parallOlipipdes presque cubiques qui remplissent liespace et au quels nous imposons des conditions limites cycliques triples. Nous avons trouvO 78 rOseaux cubiques simples utilisables avec N vertex, 7 ≤ N ≤ 27, dont 20 sont bipartites. Cette approche sur rOseau nous permet diOvaluer liOnergie et la magnOtisation spontanOe de S = ‰XY ferromagnOtisme ‡ T = 0 avec une prOcision meilleure que un pour mille par rapport aux meilleures Ovaluations par les mOthodes diondes de spin, variationnelle ou diexpansion en sOrie. Par contre, cette mOthode niest pas en accord avec liantiferromagnOtisme diHeisenberg. (Traduit par la rOdaction)
The spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ model on an infinite square lattice at zero temperature has been studied and is presented here. Our methods using finite square lattices have been the exact diagonalization and quantum Monte Carlo methods. The physical properties estimated per lattice vertex include ground-state and first-excited-state energy, staggered magnetization, and susceptibility in the parallel and perpendicular directions, spin stiffness, and spin-wave velocity. In the XXZ\mathrm{XXZ}XXZ model range from the Ising model to the Heisenberg model our estimates of physical properties compare very well with estimates by several other methods published in several other papers. In the XXZ\mathrm{XXZ}XXZ range between the Heisenberg and XY\mathrm{XY}XY models there is very little in the literature to compare with our finite lattice estimates of physical properties.
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.