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Canadian Journal of Physics

Variational study on the ground state of the spin XY magnet

Masuo Suzuki, Seiji Miyashita

Published Jul 1, 1978• 57 citations
Publisher

Abstract

An approximate wave function of the ground state of the spin [Formula: see text] XY magnet is derived using a variational method. This wave function yields estimates of the ground state energy and long-range order which agree very well with the results obtained by Betts and Oitmaa by a finite lattice method.
DOI
10.1139/p78-121

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Journal of the Physical Society of Japan•Jun 1, 1977•1 citation
The ground-state properties of the spin-(1/2 Heisenberg antiferromagnet on a square lattice are studied by using a simple variational wave function that interpolates continuously between the N\'eel state and short-range resonating-valence-bond states. Exact calculations of the variational energy for small systems show that the state with the lowest energy has long-range antiferromagnetic order. The staggered magnetization in this state is approximately 70% of its maximum possible value. The variational estimate of the ground-state energy is substantially lower than the value obtained for the nearest-neighbor resonating-valence-bond wave function.
We present here a new approach to determine an accurate variational wavefunction for general quantum antiferromagnets, completely defined by the requirement to reproduce the simple and well known spin-wave expansion. By this wavefunction, it is possible to obtain the correct behavior of the long distance correlation functions for the one dimensional S=1/2S=1/2S=1/2 antiferromagnet, i.e. {\em a system without long range order}. The qualitative difference between the integer and half integer case is also easily understood with this variational approach. Finally we present numerical results for the 2D XY model, showing that the present wavefunction has an overlap $> 0.99 $ to the exact ground state of the S=1/2S=1/2S=1/2 model for finite system up to 6×66 \times 66×6 clusters.
Numerical studies by Betts and Oitmaa have led those authors to conjecture that in the XY\mathrm{XY}XY model the ground-state magnetization Mz{M}_{z}Mz​ is zero. This is a model of spins on a lattice with interactions $\ensuremath{-}J({{S}_{n}}^{x}{{S}_{m}}^{x}+{{S}_{n}}^{y}{{S}_{m}}^{y})$, that can also describe a hard-sphere boson fluid. In the present note I prove the conjecture for arbitrary spins S=12,1,32,…\mathcal{S}=\frac{1}{2}, 1, \frac{3}{2}, \dots{}S=21​,1,23​,… for positive JJJ, and treat various generalizations.
On the basis of the method of equation of motion, the ground states of the quantum spin systems with a long-range order are studied. The equations of motion for the spin operators are linearized by introduction of the effective field, which is the order parameter itself. The linearized equations of motion and the use of approximate commutation relations for spin operators lead to the boson representation of the systems. It is emphasized that in the ground state with the long-range order along the x -axis, the relationship ( S j y ) 2 +( S j z ) 2 =1/2 should be retained in the average. By use of the above constraint for spin fluctuations, the values of the long-range order and ground-state energy are calculated within the boson approximation. The results agree well with ones obtained by different methods.