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arXiv:0807.0285 · Physical Review B

Spatially anisotropic Heisenberg kagome antiferromagnet

Andreas P. Schnyder, Oleg A. Starykh, Leon Balents

Published Nov 18, 2008• 51 citations
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Abstract

We study the quasi-one-dimensional limit of the spin-1/2 quantum Heisenberg antiferromagnet on the kagome lattice. The lattice is divided into antiferromagnetic spin chains (exchange $J$) that are weakly coupled via intermediate ``dangling'' spins (exchange ${J}^{\ensuremath{'}}$). Using one-dimensional bosonization, renormalization-group methods, and current algebra techniques, the ground state is determined in the limit ${J}^{\ensuremath{'}}⪡J$. We find that the dangling spins and chain spins form a spiral with $O(1)$ and $O({J}^{\ensuremath{'}}/J)$ static moments, respectively, atop of which the chain spins exhibit a smaller $O[{({J}^{\ensuremath{'}}/J)}^{2}]$ antiferromagnetically ordered component along the axis perpendicular to the spiral plane.

DOI
10.1103/physrevb.78.174420

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We study the quasi-one-dimensional limit of the spin-1/2 quantum Heisenberg antiferromagnet on the kagome lattice. The lattice is divided into antiferromagnetic spin chains (exchange JJJ) that are weakly coupled via intermediate ``dangling'' spins (exchange J\ensuremath′{J}^{\ensuremath{'}}J\ensuremath′). Using one-dimensional bosonization, renormalization-group methods, and current algebra techniques, the ground state is determined in the limit J\ensuremath′⪡J{J}^{\ensuremath{'}}⪡JJ\ensuremath′⪡J. We find that the dangling spins and chain spins form a spiral with O(1)O(1)O(1) and O(J\ensuremath′/J)O({J}^{\ensuremath{'}}/J)O(J\ensuremath′/J) static moments, respectively, atop of which the chain spins exhibit a smaller O[(J\ensuremath′/J)2]O[{({J}^{\ensuremath{'}}/J)}^{2}]O[(J\ensuremath′/J)2] antiferromagnetically ordered component along the axis perpendicular to the spiral plane.
We use the coupled-cluster method to study the zero-temperature properties of an extended two-dimensional Heisenberg antiferromagnet formed from spin-12\frac{1}{2}21​ moments on an infinite spatially anisotropic kagome lattice of corner-sharing isosceles triangles, with nearest-neighbor bonds only. The bonds have exchange constants J1>0{J}_{1}>0J1​>0 along two of the three lattice directions and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$ along the third. In the classical limit, the ground-state (GS) phase for $\ensuremath{\kappa}<\frac{1}{2}$ has collinear ferrimagnetic (N\'eel${}^{\ensuremath{'}}$) order where the J2{J}_{2}J2​-coupled chain spins are ferromagnetically ordered in one direction with the remaining spins aligned in the opposite direction, while for $\ensuremath{\kappa}>\frac{1}{2}$ there exists an infinite GS family of canted ferrimagnetic spin states, which are energetically degenerate. For the spin-12\frac{1}{2}21​ case, we find that quantum analogs of both these classical states continue to exist as stable GS phases in some regions of the anisotropy parameter $\ensuremath{\kappa}$, namely, for $0<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}$ for the N\'eel${}^{\ensuremath{'}}$ state and for (at least part of) the region $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}$ for the canted phase. However, they are now separated by a paramagnetic phase without either sort of magnetic order in the region ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$, which includes the isotropic kagome point $\ensuremath{\kappa}=1$ where the stable GS phase is now believed to be a topological (Z2{\mathbb{Z}}_{2}Z2​) spin liquid. Our best numerical estimates are ${\ensuremath{\kappa}}_{{c}_{1}}=0.515\ifmmode\pm\else\textpm\fi{}0.015$ and ${\ensuremath{\kappa}}_{{c}_{2}}=1.82\ifmmode\pm\else\textpm\fi{}0.03$.
We theoretically study the spatially anisotropic spin-1/2 kagome antiferromagnet with a Dzyaloshinskii-Moriya (DM) interaction using a renormalization-group analysis in the quasi-one-dimensional limit. We identify the various temperature and energy scales for ordering in the system. For a very weak DM interaction, we find a low-temperature spiral phase with the plane of the spiral selected by the DM interaction. This phase is similar to a previously identified phase in the absence of the DM interaction. However, above a critical DM interaction strength we find a transition to a phase with coexisting antiferromagnetic and dimer order, reminiscent of one-dimensional antiferromagnetic systems with a uniform DM interaction. Our results help shed light on the fate of two-dimensional systems with both strong interactions and significant spin-orbit coupling.
We study the properties of the Heisenberg antiferromagnet with spatially anisotropic nearest-neighbour exchange couplings on the kagome net, i.e. with coupling J in one lattice direction and couplings J' along the other two directions. For J/J' > 1, this model is believed to describe the magnetic properties of the mineral volborthite. In the classical limit, it exhibits two kinds of ground states: a ferrimagnetic state for J/J' 1/2. To include quantum effects self-consistently, we investigate the Sp(N) symmetric generalisation of the original SU(2) symmetric model in the large-N limit. In addition to the dependence on the anisotropy, the Sp(N) symmetric model depends on a parameter kappa that measures the importance of quantum effects. Our numerical calculations reveal that in the kappa-J/J' plane, the system shows a rich phase diagram containing a ferrimagnetic phase, an incommensurate phase, and a decoupled chain phase, the latter two with short- and long-range order. We corroborate these results by showing that the boundaries between the various phases and several other features of the Sp(N) phase diagram can be determined by analytical calculations. Finally, the application of a block-spin perturbation expansion to the trimerised version of the original spin-1/2 model leads us to suggest that in the limit of strong anisotropy, J/J' >> 1, the ground state of the original model is a collinearly ordered antiferromagnet, which is separated from the incommensurate state by a quantum phase transition.
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We investigate the quantum phases of a frustrated antiferromagnetic Heisenberg spin-1/2 model Hamiltonian on a kagom\'e-strip chain (KSC), a one-dimensional analog of the kagom\'e lattice, and construct its phase diagram in an extended exchange-parameter space. The isolated unit cell of this lattice comprises of five spin-1/2 particles, giving rise to several types of magnetic ground states in a unit cell: a spin-3/23/23/2 state as well as spin-1/21/21/2 states with and without additional degeneracies. We explore the ground-state properties of the fully connected thermodynamic system using exact diagonalization and density matrix renormalization group methods, identifying several distinct quantum phases. All but one of the phases exhibit gapless spin excitations. The exception is a dimerized spin-gapped phase that covers a large part of the phase diagram and includes the uniformly exchange coupled system. We argue that this phase can be understood by perturbing around the limit of decoupled unit cells where each unit cell has six degenerate ground states. We use degenerate perturbation theory to obtain an effective Hamiltonian, a Kugel-Khomskii model with an anisotropic spin-one orbital degree of freedom, which helps explain the origin of dimerization.