Chern-Simons theory of magnetization plateaus of the spin-$\frac{1}{2}$ quantum XXZ Heisenberg model on the kagome lattice

Frustrated spin systems on kagome lattices have long been considered to be a promising candidate for realizing exotic spin-liquid phases. Recently, there has been a lot of renewed interest in these systems with the discovery of materials such as volborthite and herbertsmithite that have kagomelike structures. In the presence of an external magnetic field, these frustrated systems can give rise to magnetization plateaus of which the plateau at $m=\frac{1}{3}$ is considered to be the most prominent. Here, we study the problem of the antiferromagnetic spin-$\frac{1}{2}$ quantum XXZ Heisenberg model on a kagome lattice by using a Jordan-Wigner transformation that maps the spins onto a problem of fermions coupled to a Chern-Simons gauge field. This mapping relies on being able to define a consistent Chern-Simons term on the lattice. Such a lattice Chern-Simons term had previously only been written for the square lattice and was used to successfully study the unfrustrated Heisenberg antiferromagnet on the square lattice. At a mean-field level, these ideas have also been applied to frustrated systems by ignoring the details of the Chern-Simons term. However, fluctuations are generally strong in these models and are expected to affect the mean-field physics. Using a recently developed method to rigorously extend the Chern-Simons term to the frustrated kagome lattice, we can now formalize the Jordan-Wigner transformation on the kagome lattice. We then discuss the possible phases that can arise at the mean-field level from this mapping and focus specifically on the case of $\frac{1}{3}$ filling ($m=\frac{1}{3}$ plateau) and analyze the effects of fluctuations in our theory. We show that in the regime of $\text{XY}$ anisotropy, the ground state at the $\frac{1}{3}$ plateau is equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction $\frac{1}{2}$ and that at the $\frac{5}{9}$ plateau it is equivalent to the first bosonic Jain daughter state at filling fraction $\frac{2}{3}$.

Figure 1

Kagome lattice unit cell with three sites attached to the fluxes in their corresponding plaquettes.

Figure 2

The field $A_1$ has nontrivial commutation relations with only these six links. This is the “locality” condition that is imposed.

Figure 3

Hofstadter spectrum for the $\text{XY}$ model as a function of sublattice density $\langle n\rangle$. The numbers shown are the Chern numbers of the respective filled bands. The magnetization plateaus at $\frac{1}{3},\frac{5}{9},\frac{2}{3}$ correspond to the three vertical jumps shown in this figure, respectively, at densities $\langle n\rangle =\frac{1}{3},\frac{2}{9},\frac{1}{6}$.

Figure 4

The mean-field spectrum at $\frac{1}{3}$ filling for $\lambda =0$ and $0.6$. At $\frac{1}{3}$ filling, only the bottom band is filled. The spectrum is gapped for all values of $\lambda$ except for $\lambda \approx 0.6$ where the bottom band touches the middle band and the Chern numbers of the corresponding bands switch.