Solvable models for neutral modes in fractional quantum Hall edges

We describe solvable models that capture how impurity scattering in certain fractional quantum Hall edges can give rise to a neutral mode—i.e., an edge mode that does not carry electric charge. These models consist of two counterpropagating chiral Luttinger liquids together with a collection of discrete impurity scatterers. Our main result is an exact solution of these models in the limit of infinitely strong impurity scattering. From this solution, we explicitly derive the existence of a neutral mode and we determine all of its microscopic properties including its velocity. We also study the stability of the neutral mode and show that it survives at finite but sufficiently strong scattering. Our results are applicable to a family of Abelian fractional quantum Hall states of which the $\nu =2/3$ state is the most prominent example.

Figure 1

Toy model for a FQH edge with impurity scattering: two counterpropagating chiral Luttinger liquids with parameters $k_1$ and $k_2$ together with a regular lattice of impurity scatterers with spacing $\ell$.

Figure 2

Phonon band structure of the toy model in the limit $U\to \infty$, for the case $k_1=1,k_2=3$, and $v=\ell =1$. The zeros of the $n=0$ band correspond to low-energy phonon modes: $k=0$ corresponds to a right-moving charge mode, while $k=\pi /\ell$ corresponds to a left-moving neutral mode.

Figure 3

Generalized impurity lattice model with three impurities per unit cell. The unit cell has length $\ell$ while the spacings between the impurities are ${\ell }_1,{\ell }_2$, and ${\ell }_3$.

Figure 4

A schematic figure illustrating the difference between band structures with an odd and even number of impurities per unit cell. In the odd case (a), the phonon bands have zeros at $k=0$ and $k=\pi /\ell$. In the even case (b), the phonon bands have both zeros at $k=0$.

Figure 5

Random impurity model: two counterpropagating chiral Luttinger liquids in a circular geometry of length $L$, together with $M$ randomly positioned impurity scatterers.