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Canadian Journal of Physics

Improved finite-lattice method for estimating the zero-temperature properties of two-dimensional lattice models

Donald D. Betts, Sayuri Masui, N. Vats, G.E. Stewart

Published Jan 1, 1996• 25 citations
Publisher

Abstract

The well-known finite-lattice method for the calculation of the properties of quantum spin systems on a two-dimensional lattice at zero temperature was introduced in 1978. The method has now been greatly improved for the square lattice by including finite lattices based on parallelogram tiles as well as the familiar finite lattices based on square tiles. Dozens of these new finite lattices have been tested and graded using the [Formula: see text] ferromagnet. In the process new and improved estimates have been obtained for the XY model's ground-state energy per spin, ε0 = −0.549 36(30) and spontaneous magnetization per spin, m = 0.4349(10). Other properties such as near-neighbour, zero-temperature spin–spin correlations, which appear not to have been calculated previously, have been estimated to high precision. Applications of the improved finite-lattice method to other models can readily be carried out.

DOI
10.1139/p96-010

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The well-known finite-lattice method for the calculation of the properties of quantum spin systems on a two-dimensional lattice at zero temperature was introduced in 1978. The method has now been greatly improved for the square lattice by including finite lattices based on parallelogram tiles as well as the familiar finite lattices based on square tiles. Dozens of these new finite lattices have been tested and graded using the [Formula: see text] ferromagnet. In the process new and improved estimates have been obtained for the XY model's ground-state energy per spin, ε0 = −0.549 36(30) and spontaneous magnetization per spin, m = 0.4349(10). Other properties such as near-neighbour, zero-temperature spin–spin correlations, which appear not to have been calculated previously, have been estimated to high precision. Applications of the improved finite-lattice method to other models can readily be carried out.
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.
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 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].