Classification of symmetric polynomials of infinite variables: Construction of Abelian and non-Abelian quantum Hall states

The classification of complex wave functions of infinite variables is an important problem since it is related to the classification of possible quantum states of matter. In this paper, we propose a way to classify symmetric polynomials of infinite variables using the pattern of zeros of the polynomials. Such a classification leads to a construction of a class of simple non-Abelian quantum Hall states which are closely related to parafermion conformal field theories.

Figure 1

$W_{a,bc}$ obtained by moving $z_1^{\left(a\right)}$ along a large loop around $z_1^{\left(b\right)}$ and $z_1^{\left(c\right)}$ counts the total numbers of zeros of $f\left(z_1^{\left(a\right)}\right)$ in the loop. The crosses mark the zeros of $f\left(z_1^{\left(a\right)}\right)$ not at $z_1^{\left(b\right)}$ and $z_1^{\left(c\right)}$.

Figure 2

Pattern of zeros of $f\left(z_1^{\left(a\right)},z_2^{\left(b\right)},z_3^{\left(c\right)}\right)$ (when viewed as a function of $z_1^{\left(a\right)}$): (a) $D_{a,b+c}-D_b-D_c=1$, (b) $D_{a,b+c}-D_b-D_c=2$, and (c) $a=b=c$ and $D_{a,a+a}-D_a-D_a=2$. The dashed lines form two equilateral triangles. The zeros that are not located at any variable are marked by crosses.

Figure 3

The graph that represents $f_{C_7}$. Each line between the $i\text{th}$ dot and the $j\text{th}$ dot represent a factor $1/\left(z_i-z_j\right)$.

Figure 4

The graphs that represent $f_{C_8/Z_2}$. Each line between the $i\text{th}$ dot and the $j\text{th}$ dot represents a factor $1/\left(z_i-z_j\right)$. After the antisymmetrization, $f_{C_8/Z_2}$ from graph (a) gives rise to a nonzero antisymmetric function, while $f_{C_8/Z_2}$ from graph (b) gives rise to vanishing antisymmetric function.

Figure 5

The graph that represents $f_{C_4/Z_2}$. Each line between the $i\text{th}$ dot and the $j\text{th}$ dot represent a factor $1/\left(z_i-z_j\right)$.

Figure 6

The graphs that represent $f_{C_6/Z_2}$. Each line between the $i\text{th}$ dot and the $j\text{th}$ dot represents a factor $1/\left(z_i-z_j\right)$. After the antisymmetrization, $f_{C_6/Z_2}$ from graph (a) gives rise to a nonzero antisymmetric function, while $f_{C_6/Z_2}$ from graph (b) vanishes.