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arXiv:0712.2269 · Physical Review B

Ordered phases of the anisotropic kagome-lattice antiferromagnet in a magnetic field

Edwin Miles Stoudenmire, Leon Balents

Published May 8, 2008• 15 citations
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Abstract

The antiferromagnetic Heisenberg model on an anisotropic kagome lattice may be a good minimal model for real magnetic systems as well as a limit from which the isotropic case can be better understood. We therefore study the nearest-neighbor Heisenberg antiferromagnet on an anisotropic kagome lattice in a magnetic field. Such a system should be well described by weakly interacting spin chains, and we motivate a general form for the interaction by symmetry considerations and by perturbatively projecting out the interchain spins. In the spin-$1∕2$ case, we find that the system exhibits a quantum phase transition from a ferrimagnetic ordered state to an $XY$ ordered state as the field is increased. Finally, we discuss the appearance of magnetization plateaux in the ferrimagnetic phase.

DOI
10.1103/physrevb.77.174414

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The antiferromagnetic Heisenberg model on an anisotropic kagome lattice may be a good minimal model for real magnetic systems as well as a limit from which the isotropic case can be better understood. We therefore study the nearest-neighbor Heisenberg antiferromagnet on an anisotropic kagome lattice in a magnetic field. Such a system should be well described by weakly interacting spin chains, and we motivate a general form for the interaction by symmetry considerations and by perturbatively projecting out the interchain spins. In the spin-1∕21∕21∕2 case, we find that the system exhibits a quantum phase transition from a ferrimagnetic ordered state to an XYXYXY ordered state as the field is increased. Finally, we discuss the appearance of magnetization plateaux in the ferrimagnetic phase.
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.
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 ${\ensuremath{\kappa}}_{{c}_{2}}=1.82\ifmmode\pm\else\textpm\fi{}0.03$.
Magnetization plateaus in quantum spin systems emerges in two-dimensional frustrated systems such as kagome lattice. The spin-1/2 antiferromagnetic Heisenberg model on square-kagome lattice is also appropriate for the study of magnetization plateau. Motivated by recent experimental finding of such a square-kagome lattice with nonequivalent three bonds, we investigate the phase diagrams and magnetization plateaus of the lattice by the exact diagonalization method. In addition to previously reported the 1/3 and 2/3 plateaus in the model with two equivalent bonds, we find a new 2/3 plateau whose magnetic structure is characterized by spontaneously broken four-fold rotational symmetry. The plateau appears only in the case of three nonequivalent bonds. We propose the possibility of finding plateaus including the new one.
The phase diagram of the XXZXXZXXZ spin-1 quantum magnet on the kagome lattice is studied for all cases where the zzz-component nearest-neighbor spin interaction Jz{J}_{z}Jz​ is antiferromagnetic. Besides Jz{J}_{z}Jz​ and the nearest-neighbor in-plane spin interaction Jxy{J}_{xy}Jxy​, the system is also parametrized by an on-site anisotropic term D(Sz)2D{({S}^{z})}^{2}D(Sz)2. In the zero magnetic field case, the six previously introduced phases, found using various methods, are the nondegenerate gapped photon phase, which breaks no space symmetry or spin symmetry; the sixfold degenerate phase with plaquette order, which breaks both time-reversal symmetry and translational symmetry; the ``superfluid'' (ferromagnetic) phase with an in-plane global U(1) symmetry broken when Jxy<0{J}_{xy}<0Jxy​<0; the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order when Jxy>0{J}_{xy}>0Jxy​>0; the nematic phase when D<0D<0D<0 and large; and a phase with resonating dimers on each hexagon. We obtain all of these phases and partial information about their quantum phase transitions in a single framework by studying condensation of defects in the sixfold plaquette phases. The transition between nematic phase and the sixfold degenerate plaquette phase is potentially an unconventional second-order critical point. In the case of a nonzero magnetic field along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$, another ordered phase with translation symmetry broken is opened up in the nematic phase. Due to the breaking of time-reversal symmetry by the field, a supersolid phase emerges between the sixfold plaquette order and the superfluid phase. This phase diagram might be accessible in nickel compounds, organic compound $m\text{\ensuremath{-}}\mathrm{MPYNN}∙\mathrm{B}{\mathrm{F}}_{4}$, or optical lattices of atoms with three degenerate states on every site.
We investigate the anisotropic effect of magnetization plateaus in the antiferromagnetic Heisenberg model on a kagome strip chain. The kagome strip chain Heisenberg model, composed of a hexagonal net of triangles forming five-site unit cells, exhibits four magnetization plateaus in the presence of an applied magnetic field. Using numerical density matrix renormalization group method, we find that the magnetization plateaus are stable against anisotropic interactions in the same direction of the applied magnetic field, but the plateaus become much smaller with anisotropic interactions in other directions. We further show the anisotropic effect of the magnon excitations of the 0.6 plateau state using linear spin wave theory. The magnon bandwidth remains small when tuning the anisotropic interactions along the field where the magnetization plateau is stable, while the band becomes more dispersive with anisotropic interactions perpendicular to the field. In addition, upon tuning down the interaction strength for the two lower legs below a critical value, the Hamiltonian of the kagome strip chain is dominated by two separate spin chains. This can be used to determine the effective lattice structure in materials with strong distortions. Our results enhance the theoretical understanding of the anisotropic effect and the nature of magnetization plateaus in kagome strip chain materials, which can contribute to the design and manipulation of kagome materials with tailored properties.
We consider the S = 1 / 2 antiferromagnetic Heisenberg model on a frustrated kagome-lattice bilayer with strong nearest-neighbor interlayer coupling and examine its low-temperature magnetothermodynamics using a mapping onto a rhombi gas on the kagome lattice. Besides, we use finite-size numerics to illustrate the validity of the classical lattice-gas description. Among our findings there are (i) the absence of an order-disorder phase transition and (ii) the sensitivity of the specific heat at low temperatures to the shape of the system just below the saturation magnetic field even in the thermodynamic limit.