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Journal of the Physical Society of Japan

Ground State of Quantum Spin Systems on the Triangular Lattice

Hidetoshi Nishimori, Hiizu Nakanishi

Published Jul 15, 1989• 48 citations
Publisher

Abstract

The ground states of quantum spin systems with spin 1 / 2 are investigated on the triangular lattice by numerically diagonalizing finite size systems ( N ≦27). In the ferromagnetic and antiferromagnetic X Y models the correlation functions are found to decay by power laws. The antiferromagnetic Heisenberg model is likely to be paramagnetic in the ground state. Thus the classical long-range order is destroyed by quantum effects in all of the models we studied.

DOI
10.1143/jpsj.58.2607

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The ground states of quantum spin systems with spin 1 / 2 are investigated on the triangular lattice by numerically diagonalizing finite size systems ( N ≦27). In the ferromagnetic and antiferromagnetic X Y models the correlation functions are found to decay by power laws. The antiferromagnetic Heisenberg model is likely to be paramagnetic in the ground state. Thus the classical long-range order is destroyed by quantum effects in all of the models we studied.
The spin-1/2 antiferromagnetic Heisenberg and X Y models are investigated on the triangular lattice by numerically diagonalizing finite size systems (up to 21 spins). The ground state energies of both models are found to be close to other theoretical predictions except those of Marland and Betts who performed similar analysis to ours. We have numerically constructed a resonating valence bond state of Anderson. It has a relatively large projection on the ground state of the finite Heisenberg model, although the absolute value of the projection decreases linearly with the system size. The trial function proposed by Miyashita for the X Y model does not have an appreciable projection on the numerically exact state. We present evidence for the absence of a sublattice long range order in the Heisenberg model and the existence of the same quantity in the X Y model.
Based on the triangular lattice and its depletions there are three simple frustrated antiferromagnetic Heisenberg models in two dimensions.The first two, the triangular and kagοmé lattices, have been examined in the recent past.The triangular lattice seems to have a łong range order whereas the kagomé does not show the long range order.But these results are still con-' troversial.This work is concentrated on a third type of this lattice family in order to improve the understanding of the connection between the long range order and coordination number in low dimensional systems.Bets has described the geometric properties of this lattice.It has a coordination number 5, which lies precisely between coordination numbers 6 and 4 of the other two lattices.The low-lying spectra and the correlation functions of finite lattices have been examined to discuss the possibility of a long range ordered ground state in the I/7-depleted triangular lattice.The low-lying spectrum is generated by an exact diagonalization, and the tower of states behavior points to a long range ordered ground state.
We study, on the basis of the general entangled-plaquette variational ansatz, the ground-state (GS) properties of the spin- antiferromagnetic Heisenberg model on the triangular lattice. Our numerical estimates are in good agreement with available exact results and comparable, for large system sizes, to those computed via the best alternative numerical approaches, or by means of variational schemes based on specific (i.e. incorporating problem-dependent terms) trial wave functions (WFs). The extrapolation to the thermodynamic limit of our results for lattices comprising up to N =324 spins yields an upper bound of the GS energy per site (in units of the exchange coupling) of − 0.5458(2) [−0.4074(1) for the XX model], while the estimated infinite-lattice order parameter is 0.3178(5) (i.e. ∼64% of the classical value).