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arXiv:1409.7070 · Physical Review Letters

Quantum Selection of Order in an XXZ Antiferromagnet on a Kagome Lattice

A. L. Chernyshev, Mike E. Zhitomirsky

Published Dec 3, 2014• 85 citations
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Abstract

Selection of the ground state of the kagome-lattice $XXZ$ antiferromagnet by quantum fluctuations is investigated by combining nonlinear spin-wave and real-space perturbation theories. The two methods unanimously favor $\mathbf{q}=0$ over $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ magnetic order in a wide range of the anisotropy parameter $0\ensuremath{\le}\mathrm{\ensuremath{\Delta}}\ensuremath{\lesssim}0.72$. Both approaches are also in accord on the magnitude of the quantum order-by-disorder effect generated by topologically nontrivial, looplike spin-flip processes. A tentative $S\text{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}$ phase diagram of the model is proposed.

DOI
10.1103/physrevlett.113.237202

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Selection of the ground state of the kagome-lattice XXZXXZXXZ antiferromagnet by quantum fluctuations is investigated by combining nonlinear spin-wave and real-space perturbation theories. The two methods unanimously favor q=0\mathbf{q}=0q=0 over 3\ifmmode×\else\texttimes\fi3\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}3​\ifmmode×\else\texttimes\fi3​ magnetic order in a wide range of the anisotropy parameter 0\ensuremath≤\ensuremathΔ\ensuremath≲0.720\ensuremath{\le}\mathrm{\ensuremath{\Delta}}\ensuremath{\lesssim}0.720\ensuremath≤\ensuremathΔ\ensuremath≲0.72. Both approaches are also in accord on the magnitude of the quantum order-by-disorder effect generated by topologically nontrivial, looplike spin-flip processes. A tentative S\ensuremath-\ensuremathΔS\text{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}S\ensuremath-\ensuremathΔ phase diagram of the model is proposed.
We study the competition between $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ (RT3) and q=0q=0q=0 (Q0) magnetic orders in spin-1 and $\text{spin}\ensuremath{-}\frac{3}{2}$ kagome-lattice XXZ antiferromagnets with varying XY anisotropy parameter $\mathrm{\ensuremath{\Delta}}$, using series expansion methods. The Hamiltonian is split into two parts: an H0{H}_{0}H0​ which favors the classical order in the desired pattern and an H1{H}_{1}H1​, which is treated in perturbation theory by a series expansion. We find that the ground state energy series for the RT3 and Q0 phases are identical up to sixth order in the expansion, but ultimately a selection occurs, which depends on spin and the anisotropy $\mathrm{\ensuremath{\Delta}}$. Results for ground state energy and the magnetization are presented. These results are compared with recent spin-wave theory and coupled-cluster calculations. The series results for the phase diagram are close to the predictions of spin-wave theory. For the spin-1 model at the Heisenberg point ($\mathrm{\ensuremath{\Delta}}=1$), our results are consistent with a vanishing order parameter, that is, an absence of a magnetically ordered phase. We also develop series expansions for the ground state energy of the spin-1 Heisenberg model in the trimerized phase. We find that the ground state energy in this phase is lower than those of magnetically ordered ones, supporting the existence of a spontaneously trimerized phase in this model.
We study the XXZXXZXXZ spin-one quantum magnet with single-ion anisotropy on the kagome lattice as an example where quantum fluctuations on highly degenerate classical ground states lead to various exotic quantum ground states. Previous studies have predicted several quantum phases, but different analytical approaches do not necessarily lead to the same physical picture. In this work, we use quantum Monte Carlo computations to critically examine some of the predictions made in the string-net mean-field theory and the degenerate perturbation theory combined with duality analysis and effective-field theory. It is found that the resulting phase diagram differs from some of the previous predictions. Further implications of our results to different analytical approaches are discussed.
We study general conditions in which disordered, spin liquid, and valence-bond ordered phases occur in the Kagome lattice quantum Ising antiferromagnets. Different quantum dynamical processes in the Ising model, with and without total Ising spin conserved, are analytically shown to yield all three characteristic quantum paramagnetic phases in the Kagome system. Special emphasis is given to the XXZXXZXXZ model that can be sensibly compared to the Kagome lattice Heisenberg antiferromagnet. It is explicitly demonstrated that the total-spin-conserving dynamics can yield a resonant valence bond liquid phase with very short-ranged correlations, but also a valence-bond ordered phase compatible with the one proposed to explain the seemingly gapless singlet states of the Heisenberg antiferromagnet on the Kagome lattice. Likely consequences for generic spin models are discussed. The analysis combines compact U(1) gauge theory, duality transformations, lattice-field-theoretical methods, and variational approach.
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In this paper we present several examples of order due to disorder phenomena and mainly focus on the Heisenberg antiferromagnet with nearest neighbour interactions on the Husimi cactus, a system which locally has the same topology as the Kagomé lattice. This system has a huge classical degeneracy corresponding to an extensive number of degrees of freedom. We show that unlike thermal fluctuations, quantum fluctuations partially lift this degeneracy and favour a discrete subset of classical ground states. In order to clarify the origin of these effects, we have set up a general semiclassical analysis of the order from disorder phenomenon and clearly identified the differences between classical and quantum fluctuations. This semiclassical approach also enables us to classify various situations where a selection mechanism still occurs. Moreover, once a discrete set of ground states has been preselected, our analysis suggests that tunelling processes within this set should be the dominant effect underlying the strange low-energy spectrum of Kagomé-like lattices.
We study in this paper the Heisenberg antiferromagnet with nearest neighbours interactions on the Husimi cactus, a system which has locally the same topology as the Kagomé lattice. This system has a huge classical degeneracy corresponding to an extensive number of degrees of freedom. We show that unlike thermal fluctuations, quantum fluctuations lift partially this degeneracy and favour a discrete subset of classical ground states. In order to clarify the origin of these effects, we have set up a general semi-classical analysis of the order from disorder phenomenon and clearly identified the differences between classical and quantum fluctuations. This semi-classical approach also enables us to classify various situations where a selection mechanism still occurs. Moreover, once a discrete set of ground states has been preselected, our analysis suggests that tunelling processes within this set should be the dominant effect underlying the strange low energy spectrum of Kagomé-like lattices.