Spin quantum Hall effects in a spin-1 topological paramagnet

TL;DR

The paper addresses the existence of 2D, integer-spin topological paramagnets with symmetry-protected gapless edges, extending the AKLT paradigm to higher dimensions. It develops a fermionic spin-$1$ representation with three species $f_m$ and a single-occupancy constraint $\hat{N}_f=1$, assigning Chern numbers $C_m=\pm1$ to filled bands under $U(1)_{S^z}$ symmetry. After projecting to the spin Hilbert space, a multicomponent Chern-Simons theory with a $K$-matrix yields a quantized spin Hall response $\sigma^s_{xy}=\frac{C_{+1}+C_{-1}+4C_0}{C_{+1}C_{-1}+C_0(C_{+1}+C_{-1})}$, which can be $\pm2$ for appropriate $C_m$. A concrete kagome-lattice realization is provided, including a strong-coupling derivation of a microscopic Hamiltonian $H_{kagome}$ that preserves $U(1)_{S^z}$ and lattice symmetries while breaking time reversal. This framework offers a path to realizing and engineering non-fractionalized, symmetry-protected topological phases in integer-spin magnets.

Abstract

AKLT state (or Haldane phase) in a spin-1 chain represents a large class of gapped topological paramagnets, which hosts symmetry-protected gapless excitations on the boundary. In this work we show how to realize this type of featureless spin-1 states on a generic two-dimensional lattice. These states have a gapped spectrum in the bulk but supports gapless edge states protected by spin rotational symmetry along a certain direction, and are featured by spin quantum Hall effect. Using fermion representation of integer-spins we show a concrete example of such spin-1 topological paramagnets on kagome lattice, and suggest a microscopic spin-1 Hamiltonian which may realize it.

Figures (2)

• Figure 1: (color online) Hopping Hamiltonian (\ref{['kagome:mean-field Hamiltonian']}) for $f_m$-fermion of spin-1 magnets on kagome lattice. Solid red lines are 1st nearest neighbor (NN) hopping terms with real amplitude $t_1^m$, while dashed arrows represent 2nd NN hoppings with imaginary amplitude $\space\mathrm{i}\space t_2^m\cdot C_m\nu_{{\bf r}{\bf r}^\prime}$ between site ${\bf r}$ and ${\bf r}^\prime$. Here $\nu_{{\bf r}{\bf r}^\prime}=+1$ along on the arrow direction, or $-1$ along the opposite direction. $C_m=\pm1$ is the Chern number of the lowest band of this hopping Hamiltonian. $\vec{a}_{1,2}$ denote the two Bravais lattice vectors of kagome lattice and there are three lattice sites in each unit cell.
• Figure 2: (color online) Band structure of $f_m$-fermion hopping Hamiltonian (\ref{['kagome:mean-field Hamiltonian']}) in the 1st Brillouin zone of kagome lattice. The hopping parameters are chosen as $\mu^m=t_1^m=-1$ and $t_2^m=-1/2$. The Chern number for the three bands are $\{C_m,-2C_m,C_m\}$ from bottom to top, with the choice $C_m=\pm1$.