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Acta Physica Polonica A

Heisenberg Antiferromagnet on a 1/7-Depleted Triangular Lattice

J. Schulenburg, Johannes Richter, Donald D. Betts

Published May 1, 2000• 20 citations

Abstract

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.

DOI: 10.12693/aphyspola.97.971

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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 the spin-

1∕2

Heisenberg model in a lattice that interpolates between the triangular and the kagome lattices. The exchange interaction along the bonds of the kagome lattice is

J

, and the one along the bonds connecting kagome and nonkagome sites is ${J}^{\ensuremath{'}}$, so that ${J}^{\ensuremath{'}}=J$ corresponds to the triangular limit and ${J}^{\ensuremath{'}}=0$ to the kagome one. We use variational and exact diagonalization techniques. We analyze the behavior of the order parameter for the antiferromagnetic phase of the triangular lattice, the spin gap, and the structure of the spin excitations as functions of ${J}^{\ensuremath{'}}∕J$. Our results indicate that the antiferromagnetic order is not affected by the reduction of ${J}^{\ensuremath{'}}$ down to ${J}^{\ensuremath{'}}∕J\ensuremath{\simeq}0.2$. Below this value, antiferromagnetic correlations grow weaker, a description of the ground state in terms of a N\'eel phase renormalized by quantum fluctuations becomes inadequate, and the finite-size spectra develop features that are not compatible with antiferromagnetic ordering. However, this phase does not appear to be connected to the kagome phase as well, as the low-energy spectra do not evolve with continuity for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}0$ to the kagome limit. In particular, for any nonzero value of ${J}^{\ensuremath{'}}$, the latter interaction sets the energy scale for the low-lying spin excitations, and a gapless triplet spectrum, destabilizing the kagome phase, is expected.

We study the ground-state phase diagrams and properties of spin-1/2 Heisenberg models on the diamond-like-decorated square and triangular lattices. The diamond-like-decorated square (triangular) lattice is a lattice in which the bonds of a square (triangular) lattice are replaced with diamond units. The diamond unit has two types of antiferromagnetic exchange interactions, and the ratio λ of the strength of the diagonal bond to that of the other four edges determines the ground-state properties. In particular, the macroscopically degenerated tetramer-dimer states, which are equivalent to the dimer covering states of the original lattices, are stabilized for λc < λ < 2, where the value of λc depends on the lattices. To determine the phase diagrams and boundaries λc, we employ the modified spin wave (MSW) method and the quantum Monte Carlo (QMC) method to estimate the ground-state energies of the ferrimagnetic states for λ < λc, where we can consider the mixed spin-1 and spin-1/2 Lieb-lattice and triangular Lieb-lattice Heisenberg antiferromagnets instead, and obtain λc(square)=0.974 and λc(triangular)=0.988. We also calculate the long-range order (LRO) parameters using the MSW and QMC methods and find the scaling relations where the spin reductions of each sublattice are inversely proportional to the number of sublattice sites. We prove these scaling relations by applying an infinitesimal uniform magnetic field. Furthermore, by examining the calculation process in the MSW, we clarify the mathematical structure behind the scaling relations for the sublattice LROs.

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.