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arXiv:0805.2075 · Physical Review B

Quantum order by disorder in a spin-one frustrated magnet on the kagome lattice

Sergei V. Isakov, Yong Baek Kim

Published Mar 9, 2009• 4 citations
arXivHTMLPDF

Abstract

We study the $XXZ$ 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.

DOI
10.1103/physrevb.79.094408

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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.
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 $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ 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.
We report on a systematic study of two-dimensional, periodic, frustrated Ising models with quantum dynamics introduced via a transverse magnetic field. The systems studied are the triangular and kagom\'e lattice antiferromagnets, fully frustrated models on the square and hexagonal (honeycomb) lattices, a planar analog of the pyrochlore antiferromagnet, a pentagonal lattice antiferromagnet, as well as two quasi-one-dimensional lattices that have considerable pedagogical value. All of these exhibit a macroscopic degeneracy at T=0T=0T=0 in the absence of the transverse field, which enters as a singular perturbation. We analyze these systems with a combination of a variational method at weak fields, a perturbative Landau-Ginzburg-Wilson approach from large fields, as well as quantum Monte Carlo simulations utilizing a cluster algorithm. Our results include instances of quantum order arising from classical criticality (triangular lattice) or classical disorder (pentagonal and probably hexagonal) as well as notable instances of quantum disorder arising from classical disorder (kagom\'e). We also discuss the effect of finite temperature, as well as the interplay between longitudinal and transverse fields---in the kagom\'e problem the latter gives rise to a nontrivial phase diagram with bond-ordered and bond-critical phases in addition to the disordered phase. We also note connections to quantum-dimer models and thereby to the physics of Heisenberg antiferromagnets in short-ranged resonating valence-bond phases that have been invoked in discussions of high-temperature superconductivity.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
We study the ground-state phase diagram of the quantum spin-1/2\text{spin-}1/2spin-1/2 Heisenberg model on the kagome lattice with first- (J1<0)({J}_{1}<0)(J1​<0), second- (J2<0)({J}_{2}<0)(J2​<0), and third-neighbor interactions (Jd>0)({J}_{d}>0)(Jd​>0) by means of analytical low-energy field theory and numerical density-matrix renormalization group (DMRG) studies. The results offer a consistent picture of the Jd-dominant{J}_{d}\text{-dominant}Jd​-dominant regime in terms of three sets of spin chains weakly coupled by the ferromagnetic interchain interactions J1,2{J}_{1,2}J1,2​. When either J1{J}_{1}J1​ or J2{J}_{2}J2​ is much stronger than the other one, the model is found to support one of two cuboctohedral phases, cuboc1, and cuboc2. These cuboc states host noncoplanar long-ranged magnetic order and possess finite scalar spin chirality. However, in the compensated regime ${J}_{1}\ensuremath{\simeq}{J}_{2}$, a valence bond crystal phase emerges between the two cuboc phases. We find excellent agreement between an analytical theory based on coupled spin chains and unbiased DMRG calculations, including at a very detailed level of comparison of the structure of the valence bond crystal state. To our knowledge, this is the first such comprehensive understanding of a highly frustrated two-dimensional quantum antiferromagnet. We find no evidence of either the one-dimensional gapless spin liquid or the chiral spin liquids, which were previously suggested by parton mean-field theories.
In the classical Ising model on the kagome lattice, there are macroscopically degenerate classical states with exponentially decayed spin correlation. As the singular quantum perturbation is introduced by applying the transverse magnetic field, the quantum ground state remains disordered, dubbed by ``disorder by disorder.'' Here, we compute the temperature dependence of the spin susceptibility and the specific heat for different strengths of the field to understand the effect of the singular quantum perturbation in this system. The relevance to the ZnCu3(OH)6Cl2\mathrm{Zn}{\mathrm{Cu}}_{3}{(\mathrm{O}\mathrm{H})}_{6}{\mathrm{Cl}}_{2}ZnCu3​(OH)6​Cl2​ will be also commented.
We present a large-SSS study of a quantum spin ice Hamiltonian, introduced by Huang et al. [Phys. Rev. Lett. 112, 167203 (2014)], on the kagome lattice. This model involves a competition between the frustrating Ising term of classical kagome ice, a Zeeman magnetic field hhh, and a nearest-neighbor transverse spin-flip term ${S}_{i}^{x}{S}_{j}^{x}\ensuremath{-}{S}_{i}^{y}{S}_{j}^{y}$. Recent quantum Monte Carlo (QMC) simulations by Carrasquilla et al. [Nat. Commun. 6, 7421 (2015)], uncovered lobes of a disordered phase for large Ising interaction and $h\ensuremath{\ne}0$---a putative quantum spin liquid phase. Here, we examine the nature of this model using large-SSS expansion. We show that the ground state properties generally have the same trends with those observed in QMC simulations. In particular, the large-SSS ground state phase diagram captures the existence of the disordered lobes.