Topological properties of Abelian and non-Abelian quantum Hall states classified using patterns of zeros

It has been shown that different Abelian and non-Abelian fractional quantum Hall states can be characterized by patterns of zeros described by sequences of integers $\left\{S_a\right\}$. In this paper, we will show how to use the data $\left\{S_a\right\}$ to calculate various topological properties of the corresponding fraction quantum Hall state, such as the number of possible quasiparticle types and their quantum numbers, as well as the actions of the quasiparticle tunneling and modular transformations on the degenerate ground states on torus.

Figure 1

As a function of $z^{\left(c\right)}$, $P_{\gamma }^{\prime }\left({\alpha }^{\left(a\right)};z^{\left(b\right)},{\alpha }^{\left(b\right)};z^{\left(c\right)},{\alpha }^{\left(c\right)};\dots \right)$ has a $D_{\gamma ;a,c}\text{th}$ order zero as $z^{\left(c\right)}\to 0$ and a $D_{b,c}\text{th}$ order zero as $z^{\left(c\right)}\to z^{\left(b\right)}$. The crosses mark the positions of other zeros that are not at $z=0$ and not at $z^{\left(b\right)}$. The total number of zeros seen by the $z^{\left(c\right)}$ in the neighborhood of $z=0$ and $z^{\left(b\right)}$ is $D_{\gamma ;a+b,c}$, which satisfies Eq. (19).

Figure 2

The $2J+1$ orbital on a sphere forms an angular-momentum $J$ representation of the $O\left(3\right)$ rotation.

Figure 3

The circular orbital wave function ${\varphi }^{\left(l\right)}\left(X^1,X^2\right)$ in the thin cylinder limit. A thick line marks the positions where ${\varphi }^{\left(l\right)}\left(X^1,X^2\right)$ is peaked.

Figure 4

(Color online) The graphic representation of an occupation distribution $\left(n_0,\dots ,n_{10}\right)=\left(2,0,1,0,2,0,0,0\right)$ for the ${\Phi }_{5/8;Z_5^{\left(2\right)}}$ state with $n=5$ and $m=8$. $l_{\text{max}}=4$ for such a distribution.

Figure 5

(Color online) If $N_{\varphi }=2J=l_{\text{max}}+N_{c}m$, the symmetric polynomial $\Phi$ of $N=N_{c}n$ variables can be put on the sphere without any defects (or quasiparticles).

Figure 6

(Color online) If $N_{\varphi }=2J=l_{\text{max}}+\left(N_c-1\right)m+n_{\varphi }$, the above occupation distribution $n_{\gamma ;l}$ (obtained by shifting the ground-state distribution in Fig. 5) describes a single quasiparticle on the sphere located at the south pole $\left(l=0\right)$. There is no quasiparticle at the north pole $\left(l=2J\right)$.

Figure 7

(a) A system described by an inverse-mass matrix $g\left(\tau \right)$. (b) A system described by an inverse-mass matrix $g\left({\tau }^{\prime }\right)$ where $\tau$ and ${\tau }^{\prime }$ are related by a modular transformation $M_T$. (c) The system in (b) is the same as the system in (a) if we make a change in coordinates. (d) The system in (c) is drawn differently.

Figure 8

The tunneling processes $A^{\left({\gamma }_1\right)}$ and $B^{\left({\gamma }_2\right)}$.

Figure 9

A general quasiparticle tunneling process.

Figure 10

(a) Two tunneling processes: $A^{\left({\gamma }_1\right)}$ and $A^{\left({\gamma }_2\right)}$. (b) The tunneling path of the above two tunneling processes can be deformed according to the fifth relation in Eq. (A21) with $i={\overline{\gamma }}_2$, $j={\gamma }_2$, $k={\gamma }_1$, $k={\overline{\gamma }}_1$, $m=0$, and $n={\gamma }_3$. (c) The fifth, the fourth, and the third relations in Eq. (A21) can reduce (b) and (c).

Figure 11

(a) The ground state ${\Phi }_{\gamma =0}$ on a torus that corresponds to the trivial quasiparticle can be represented by an empty solid torus. (b) The other ground state ${\Phi }_{\gamma }$ that corresponds to a type $\gamma$ quasiparticle can be represented by an solid torus with a loop of type $\gamma$ in the center.

Figure 12

The graphic representation of $A^{\left({\gamma }^{\prime }\right)}|\gamma ⟩$.

Figure 13

(a) The graphic representation of $A^{\left({\gamma }^{\prime }\right)}|\gamma ⟩$. (b) The graphic representation of $A^{\left({\gamma }^{\prime }\right)}|\gamma ⟩$. (c) The fifth and the fourth relations in Eq. (A21) can deformed the graph in (b) to the graph in (c). The shaded area represents the hole of the torus.

Figure 14

The amplitude of two linked local loops is a complex number $S_{{\gamma }_1{\gamma }_2}^{\text{TC}}$.