Non-Abelian $SU(N-1)$-singlet fractional quantum Hall states from coupled wires

The construction of fractional quantum Hall (FQH) states from the two-dimensional array of quantum wires provides a useful way to control strong interactions in microscopic models and has been successfully applied to the Laughlin, Moore-Read, and Read-Rezayi states. We extend this construction to the Abelian and non-Abelian $SU(N-1)$-singlet FQH states at filling fraction $\nu =k(N-1)/[N+k(N-1\left)m\right]$ labeled by integers $k$ and $m$, which are potentially realized in multicomponent quantum Hall systems or $SU\left(N\right)$ spin systems. Utilizing the bosonization approach and conformal field theory (CFT), we show that their bulk quasiparticles and gapless edge excitations are both described by an $(N-1)$-component free-boson CFT and the $SU{\left(N\right)}_k/{\left[U\left(1\right)\right]}^{N-1}$ CFT known as the Gepner parafermion. Their generalization to different filling fractions is also proposed. In addition, we argue possible applications of these results to two kinds of lattice systems: bosons interacting via occupation-dependent correlated hoppings and an $SU\left(N\right)$ Heisenberg model.

Figure 1

The array of Luttinger liquids. The wire labeled by $j$ consists of $k$ copies of $(N-1)$-component bosons. They are put on a magnetic field with the magnetic flux $b$.

Figure 2

Weight diagram of $SU\left(3\right)$. The red solid arrows represent the weight vectors ${\mathbf{\omega }}_s$ in the fundamental representation. The blue solid arrows represent the root vectors ${\mathbf{\alpha }}_{\sigma }$.

Figure 3

Schematic view of the construction of a non-Abelian $SU\left(2\right)$-singlet FQH state for $k=2$. The left and right ${\left[SU{\left(3\right)}_1\right]}^2$ CFTs from each wire are decomposed into the two sectors ${\left[U\left(1\right)\right]}^2\times SU{\left(3\right)}_2/{\left[U\left(1\right)\right]}^2$ and ${\left[SU{\left(3\right)}_1\right]}^2/SU{\left(3\right)}_2\sim SU{\left(2\right)}_3/U\left(1\right)$. The interwire interactions ${\mathcal{O}}_{j,\sigma ,ab}^t$ open a gap in the former sector, leaving unpaired chiral gapless modes at the edges, while the intrawire interactions ${\mathcal{O}}_{j,12}^{u,s{s}^{\prime }}$ open a gap in the latter sector.

Figure 4

Minima of the cosine potential (103) are indicated by the black dots on the $({\tilde{\mathrm{\Theta }}}_1,{\tilde{\mathrm{\Theta }}}_2)$ plane scaled by $\sqrt{2}/\pi$. Different symbols enclosing the dots represent four independent minima within the compactification radii of $\tilde{\mathbf{\Theta }}$. The two vectors ${\mathbf{\alpha }}_{\uparrow ,\downarrow }$ are roots of $SU\left(3\right)$ [see Eq. (48)].

Figure 5

Weight diagram of $SU\left(4\right)$. The red solid arrows represent the weight vectors ${\mathbf{\omega }}_{\sigma }$. Combined with ${\mathbf{\omega }}_4=-{\mathbf{\omega }}_1-{\mathbf{\omega }}_2-{\mathbf{\omega }}_3$, they form the weights of the fundamental representation of $SU\left(4\right)$. The blue solid arrows represent the root vectors ${\mathbf{\alpha }}_{\sigma }$.

Figure 6

Schematic view of the construction of a non-Abelian $SU(N-1)$-singlet FQH state. The left and right ${\left[SU{\left(N\right)}_1\right]}^k$ CFTs from each wire are decomposed into the two sectors ${\left[U\left(1\right)\right]}^{N-1}\times SU{\left(N\right)}_k/{\left[U\left(1\right)\right]}^{N-1}$ and $SU{\left(N\right)}_k/{\left[U\left(1\right)\right]}^{k-1}$, which are, respectively, gapped by the interwire interactions ${\mathcal{O}}_{j,\sigma ,ab}^t$ and the intrawire interactions ${\mathcal{O}}_{j,ab}^{u,s{s}^{\prime }}$, leaving unpaired chiral gapless modes at the edges.

Figure 7

Minima of the cosine potentials (189) for $N=4$ and $k=2$, which are indicated by the spheres in the $({\tilde{\mathrm{\Theta }}}_1,{\tilde{\mathrm{\Theta }}}_2,{\tilde{\mathrm{\Theta }}}_3)$ space scaled by $\sqrt{2}/\pi$. Different colors of the spheres represent eight independent minima within the compactification radii of $\tilde{\mathbf{\Theta }}$. The three vectors ${\mathbf{\alpha }}_{1,2,3}$ are roots of $SU\left(4\right)$ [see Eq. (124)].

Figure 8

Flux configuration required for the generalized non-Abelian FQH states. For each $j$, the particle hopping from the copy 1 to 2 feels the magnetic flux $\delta b$, while the average flux over copies is set to be a uniform value $b$.

Figure 9

$SU\left(N\right)$ spin model defined on an anisotropic triangular lattice. The lattice sites are specified by the indices $j$ and $\ell$. The solid and dashed lines represent the exchange couplings $J$ and $J^{\prime }$, respectively. The circle arrows represent the three-body interactions $J_3$ on triangles.