Chern-Simons theory of the magnetization plateaus of the spin-1/2 quantum XXZ Heisenberg model on Kagome Lattice

TL;DR

This paper models the magnetization plateaus in a Kagome lattice spin system using a Chern-Simons approach, revealing fractional quantum Hall states at specific magnetization levels.

Contribution

It introduces a rigorous lattice Chern-Simons formulation for the Kagome lattice and applies it to analyze magnetization plateaus in the XXZ Heisenberg model.

Findings

At the 1/3 plateau, the ground state resembles a bosonic fractional quantum Hall Laughlin state.

At the 5/9 plateau, the state is akin to a bosonic Jain state at 2/3 filling.

The approach connects frustrated magnetism with topological quantum states.

Abstract

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 Kagome like 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-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. 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 $XY$ anisotropy the ground state at the $1/3$ plateau is equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction $1/2$ and that at the $5/9$ plateau it is equivalent to the first bosonic Jain daughter state at filling fraction $2/3$.