From Luttinger liquid to non-Abelian quantum Hall states

We formulate a theory of non-Abelian fractional quantum Hall states by considering an anisotropic system consisting of coupled, interacting one-dimensional wires. We show that Abelian bosonization provides a simple framework for characterizing the Moore-Read state, as well as the more general Read-Rezayi sequence. This coupled wire construction provides a solvable Hamiltonian formulated in terms of electronic degrees of freedom, and provides a direct route to characterizing the quasiparticles and edge states in terms of conformal field theory. This construction leads to a simple interpretation of the coset construction of conformal field theory, which is a powerful method for describing non-Abelian states. In the present context, the coset construction arises when the original chiral modes are fractionalized into coset sectors, and the different sectors acquire energy gaps due to coupling in “different directions.” The coupled wire construction can also can be used to describe anisotropic lattice systems, and provides a starting point for models of fractional and non-Abelian Chern insulators. This paper also includes an extended introduction to the coupled wire construction for Abelian quantum Hall states, which was introduced earlier.

Figure 1

(a) An array of coupled wires in a perpendicular magnetic field. (b) Energy bands for noninteracting electron in a magnetic field, which shifts the momentum of the wires. Landau level gaps, as well as the integer quantum Hall effect edge states, are apparent. (c) At special fractional filling factors, there exist momentum-conserving correlated tunneling processes that lead to fractional quantum Hall states. The process shown describes the Laughlin state at filling $\nu =\frac{1}{3}$.

Figure 2

(a) Schematic representation of the correlated tunneling processes in (2.27) that lead to the Laughlin state. Vertical arrows represent $\varphi$ and circular arrows represent $\theta$. After the change of variables (2.28), the model describes strips of $\nu =1/m$ quantum Hall fluid coupled by tunneling electrons between their edge states.

Figure 3

(a) Schematic of tunneling processes in (2.45) that lead to second-level Abelian hierarchy states. (b) After the transformation (2.47), the model describes coupled strips of $\nu =2n/(m_0+m_1)$ quantum Hall state coupled by tunneling electrons between the two channels of edge states.

Figure 4

(a) A two-component coupled wire model, viewed as a bilayer. (b) Schematic of tunneling processes in (3.3)–(3.5) that lead to the Moore-Read state for bosons at filling $\nu =1$. (b) After the transformations (3.6), (3.7), and (3.12), the model describes coupled strips of $\nu =1$ Moore-Read state coupled by tunneling electrons between the edge states, which are characterized by a $c=1$ chiral charge mode and a $c=\frac{1}{2}$ chiral Majorana fermion mode.

Figure 5

The upper panels are schematic diagrams depicting the coupling of the chiral edge modes. The solid lines represent the charge modes ${\tilde{\varphi }}_{j,\rho }^{R/L}$. The dashed lines represent the chiral Majorana modes ${\xi }_{j,\sigma }^{R/L}$ and ${\eta }_{j,\sigma }^{R/L}$. Solid (dashed) arcs represent stronger (weaker) coupling. (a) Describes the Moore-Read (MR) state, with $c=\frac{3}{2}$ edge states. (b) Describes a strong pairing (SP) state with $c=1$ edge states, and (c) describes a bilayer (B) state with $c=2$ edge states. The lower panels are ternary phase diagrams as a function of $t$, $u$, and $v$ in the cases where (d) $t_{ab}=t$, independent of $a$ and $b$ and (e) $t_{ab}=t{\delta }_{ab}$. The dashed line in (d) is the solvable line, where the decoupling of the chiral modes is perfect.

Figure 6

Coupling of edge states at the decoupled point described by the dashed line in Fig. 5. The right- and left-moving $U\left(1\right)$ charge modes and the $SU{\left(2\right)}_2/U\left(1\right)$ Majorana fermion ($Z_2$ parafermion) modes are coupled on neighboring wires, while the right- and left-moving $SU{\left(2\right)}_1\times SU{\left(2\right)}_1/SU{\left(2\right)}_2$ modes are coupled on the same wire. This pattern leaves $U\left(1\right)$ and $SU{\left(2\right)}_2/U\left(1\right)$ chiral modes at the edge. This provides a concrete interpretation for the coset construction for the Moore-Read state.

Figure 7

(a) Coupled wire construction with a staggered magnetic field. For $\delta b=\overline{b}$, the theory is equivalent to the two-component boson system shown in Fig. 4. (b) Schematic diagram showing the coupling of the edge states, similar to Fig. 4.

Figure 8

Along the solvable line $u_{ab}=v_{ab}$, the right- and left-moving $U\left(1\right)$ charge modes and the $SU{\left(2\right)}_k/U\left(1\right)$ $Z_k$ parafermion modes are coupled only on neighboring wires, while the right- and left-moving ${\left[SU{\left(2\right)}_1\right]}^k/SU{\left(2\right)}_k$ modes are coupled only on the same wire. This pattern leaves $U\left(1\right)$ and $SU{\left(2\right)}_k/U\left(1\right)$ chiral modes at the edge. This provides a concrete interpretation for the coset construction for the Read-Rezayi state.

Figure 9

Lattice of minima of the periodic potential $-{\sum }_{ab}cos\sqrt{2}{\vec{d}}_{ab}·{\vec{\theta }}_{\sigma }$ as a function of ${\vec{\theta }}_{\sigma }$. The shaded region shows the set of distinct values of ${\vec{\theta }}_{\sigma }$, which is compactified to a torus due to the vortices created by $expi\sqrt{2}{\vec{d}}_{ab}·{\vec{\phi }}_{\sigma }$. There are thus three distinct, but equivalent, minima, labeled A, B, and C.

Figure 10

Schematic diagram for the generalized Read-Rezayi state at filling $\nu =k/(2+kq)$. Groups of $k$ wires are coupled to one another by ${\mathcal{O}}_{j+1/2,ab}^t$, which require a specific staggered magnetic field for momentum conservation. Wires within a group are coupled by ${\mathcal{O}}_{j,ab}^{u,v}$, which are independent of the field. Representative examples of ${\mathcal{O}}_{j+1/2,ab}^t$ and ${\mathcal{O}}_{j,ab}^{u,v}$ are shown.