Quasiclassical magnetic order and its loss in a spin-$\frac{1}{2}$ Heisenberg antiferromagnet on a triangular lattice with competing bonds

Quasiclassical magnetic order and its loss in a spin- $\frac{1}{2}$ Heisenberg antiferromagnet on a triangular lattice with competing bonds

P. H. Y. Li, R. F. Bishop, and C. E. Campbell

Phys. Rev. B 91, 014426 – Published 22 January 2015

Abstract

We use the coupled cluster method (CCM) to study the zero-temperature ground-state (GS) properties of a spin- $\frac{1}{2}$ $J_{1} - J_{2}$ Heisenberg antiferromagnet on a triangular lattice with competing nearest-neighbor and next-nearest-neighbor exchange couplings $J_{1} > 0$ and $J_{2} \equiv κ J_{1} > 0$ , respectively, in the window $0 \leq κ < 1$ . The classical version of the model has a single GS phase transition at $κ^{cl} = \frac{1}{8}$ in this window from a phase with 3-sublattice antiferromagnetic (AFM) $120^{\circ}$ Néel order for $κ < κ^{cl}$ to an infinitely degenerate family of 4-sublattice AFM Néel phases for $κ > κ^{cl}$ . This classical accidental degeneracy is lifted by quantum fluctuations, which favor a 2-sublattice AFM striped phase. For the quantum model we work directly in the thermodynamic limit of an infinite number of spins, with no consequent need for any finite-size scaling analysis of our results. We perform high-order CCM calculations within a well-controlled hierarchy of approximations, which we show how to extrapolate to the exact limit. In this way we find results for the case $κ = 0$ of the spin- $\frac{1}{2}$ model for the GS energy per spin, $E / N = - 0.5521 (2) J_{1}$ , and the GS magnetic order parameter, $M = 0.198 (5)$ (in units where the classical value is $M^{cl} = \frac{1}{2}$ ), which are among the best available. For the spin- $\frac{1}{2}$ $J_{1} - J_{2}$ model we find that the classical transition at $κ = κ^{cl}$ is split into two quantum phase transitions at $κ_{1}^{c} = 0.060 (10)$ and $κ_{2}^{c} = 0.165 (5)$ . The two quasiclassical AFM states (viz., the $120^{\circ}$ Néel state and the striped state) are found to be the stable GS phases in the regime $κ < κ_{1}^{c}$ and $κ > κ_{2}^{c}$ , respectively, while in the intermediate regimes $κ_{1}^{c} < κ < κ_{2}^{c}$ the stable GS phase has no evident long-range magnetic order.

Received 22 October 2014

DOI:https://doi.org/10.1103/PhysRevB.91.014426

Authors & Affiliations

P. H. Y. Li^* and R. F. Bishop^†

School of Physics and Astronomy, Schuster Building, University of Manchester, Manchester M13 9PL, UK

C. E. Campbell

School of Physics and Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis, Minnesota 55455, USA

^*peggyhyli@gmail.com
^†raymond.bishop@manchester.ac.uk

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Vol. 91, Iss. 1 — 1 January 2015

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Images

Figure 1
The $J_{1} - J_{2}$ model on the triangular lattice with $J_{1} > 0$ and $J_{2} > 0$ , showing (a) the bonds ( $J_{1} \equiv$ black solid lines; $J_{2} \equiv$ blue dashed lines) and the Bravais lattice vectors $\hat{a}$ and $\hat{b}$ ; (b) the $120^{\circ}$ Néel antiferromagnetic (AFM) state; (c) the infinitely degenerate family of classical 4-sublattice ground states on which the spins on the same lattice are parallel to each other, with the sole constraint that the sum of four spins on different sublattices is zero; and (d) one of the three degenerate striped AFM states. For the two states shown the arrows represent the directions of the spins located on lattice sites $•$ .
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Figure 2
CCM results for the GS energy per spin, $E / N$ , as a function of the frustration parameter $κ \equiv J_{2} / J_{1}$ , for the spin- $\frac{1}{2}$ $J_{1} - J_{2}$ model on the triangular lattice (with $J_{1} \equiv + 1$ ). The left curves in each panel are based on the $120^{\circ}$ Néel AFM state as CCM model state, while the right curves are similarly based on the striped AFM state as CCM model state. (a) The $LSUB m$ results with $3 \leq m \leq 10$ , shown out to their (approximately determined) termination points. Portions of the curves with thinner lines denote the (approximately determined) unphysical regions where the magnetic order parameter takes negative values ( $M < 0$ ). (b) The corresponding $LSUB \infty (k)$ extrapolations, based on Eq. (15): the $k = 1$ curve for the $120^{\circ}$ Néel model state is based on $LSUB m$ results with $m = {5, 6, 7, 8, 9, 10}$ , while the $k = {2, 3}$ curves for both model states are based on $LSUB m$ results with $m = {4, 6, 8, 10}$ and $m = {3, 5, 7, 9}$ , respectively. The plus (+) symbols mark those points where the corresponding extrapolated solutions have vanishing magnetic order parameters, $M = 0$ [and see Fig. 3]. Those sections of the curves beyond the plus (+) symbols, shown with thinner lines, indicate unphysical regions, where $M < 0$ for these approximations (and see text for further details).
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Figure 3
CCM results for the GS magnetic order, $M$ , as a function of the frustration parameter $κ = J_{2} / J_{1}$ , for the spin- $\frac{1}{2}$ $J_{1} - J_{2}$ model on the triangular lattice (with $J_{1} > 0$ ). The left curves in each panel are based on the $120^{\circ}$ Néel AFM state as CCM model state, while the right curves are similarly based on the striped AFM state as CCM model state. (a) The $LSUB m$ results with $3 \leq m \leq 10$ , shown out to their (approximately determined) termination points. (b) The corresponding $LSUB \infty (k)$ extrapolations, based on Eq. (17): the $k = 1$ curve for the $120^{\circ}$ Néel model state is based on $LSUB m$ results with $m = {5, 6, 7, 8, 9, 10}$ , while the $k = {2, 3}$ curves for both model states are based on $LSUB m$ results with $m = {4, 6, 8, 10}$ and $m = {3, 5, 7, 9}$ , respectively. As explained in the text the extrapolation scheme of Eq. (17) is appropriate for the accurate determination of the QCPs at which $M \to 0$ . However, for zero or small dynamic frustration ( $J_{2} \approx 0$ ) the scheme of Eq. (16) is appropriate. Rather than crowd the figure with additional full curves based on Eq. (16), we show with cross ( $\times$ ) symbols the corresponding extrapolated values based on the $120^{\circ}$ Néel state, using Eq. (16), for the case $κ = 0$ only of the triangular-lattice HAFM.
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