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Canadian Journal of Physics

Zero-temperature properties of quantum spin models on a triangular lattice I: the ferromagnet

Satoshi Fujiki, Donald D. Betts

Published Aug 1, 1986• 43 citations
Publisher

Abstract

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.

DOI
10.1139/p86-154

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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].

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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

Canadian Journal of Physics•Mar 1, 2003•1 citation
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].
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.
A Pfaffian representation, of the partition function of the triangular lattice is used to derive expressions for various two, four, and six spin correlations in terms of Pfaffians. The pair correlations along a diagonal are expressed as a Toeplitz determinant whose limiting form yields the spontaneous magnetization. At the ferromagnetic critical point the correlations decay as 1/r12 with approximately radial symmetry. At the antiferromagnetic zero point the ground state is highly degenerate—it has finite entropy—and on a given sublattice the pair correlations along a row decay as ε/r12, where ε=+ε0 on the sublattice containing the origin spin and ε≃−12ε0 on the other two sublattices. Finally, the perpendicular susceptibility, χ⊥, which depends on a finite number of correlations, is calculated; its ferromagnetic behavior is similar to that of the perpendicular susceptibilities of the quadratic and honeycomb lattices, but for an antiferromagnet χ⊥ diverges as 1/T at low temperatures.
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