Spatially anisotropic Heisenberg kagome antiferromagnet

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 $J^{\prime }$). Using one-dimensional bosonization, renormalization-group methods, and current algebra techniques, the ground state is determined in the limit $J^{\prime }⪡J$. We find that the dangling spins and chain spins form a spiral with $O\left(1\right)$ and $O\left(J^{\prime }/J\right)$ static moments, respectively, atop of which the chain spins exhibit a smaller $O\left[{\left(J^{\prime }/J\right)}^2\right]$ antiferromagnetically ordered component along the axis perpendicular to the spiral plane.

Figure 1

(Color online) Spatially anisotropic kagome lattice with nearest-neighbor exchange $J$ (blue solid bonds) among chain spins $\left(\mathit{S}\right)$ and $J^{\prime }$ (black dashed bonds) among chain spins and interstitial spins $\left(\mathit{s}\right)$.

Figure 2

(Color online) (a) Three-dimensional perspective view of the spin-ordering pattern. The interstitial spins (red arrows) form a coplanar spiral with an $O\left(1\right)$ local moment on the $x\text{−}y$ plane. The components of the chain spins (gray arrows) within the $x\text{−}y$ plane also form a spiral but with static moment of $O\left[J^{\prime }/J\right]$ and antiparallel to the interstitial spins. The out-of-plane components ($z$ direction) of the chain spins are nonzero but even smaller—$O\left[{\left(J^{\prime }/J\right)}^2\right]$—and ordered antiferromagnetically along and between the chains. (b) Top view of the spin-ordering pattern. The vertical component of the chain spins is indicated using $+$ (upward pointing) and $-$ (downward pointing).

Figure 3

(Color online) Exchange paths of the triangular lattice formed by the intermediate spins $\mathit{s}$ of the spatially anisotropic kagome lattice. Solid black lines denote the nearest-neighbor ferromagnetic exchange ${\mathcal{J}}_1$ (a–b and b–c bonds). Dotted black lines the nearest-neighbor antiferromagnetic exchange ${\mathcal{J}}_2$ (a–c bond). The further-neighbor interactions ${\mathcal{J}}_3$ (a–d bond) and ${\mathcal{J}}_4$ (a–e bond) for the spin ${\mathit{s}}_a$ are illustrated with red arrows. The chain bonds of the anisotropic kagome lattice are indicated in light blue (cf. Fig. 1).