Non-Abelian fermionization and fractional quantum Hall transitions

There has been a recent surge of interest in dualities relating theories of Chern-Simons gauge fields coupled to either bosons or fermions within the condensed matter community, particularly in the context of topological insulators and the half-filled Landau level. Here, we study the application of one such duality to the long-standing problem of quantum Hall interplateaux transitions. The key motivating experimental observations are the anomalously large value of the correlation length exponent $\nu \approx 2.3$ and that $\nu$ is observed to be superuniversal, i.e., the same in the vicinity of distinct critical points [Sondhi et al., Rev. Mod. Phys. 69, 315 (1997)]. Duality motivates effective descriptions for a fractional quantum Hall plateau transition involving a Chern-Simons field with $\mathrm{U}\left(N_c\right)$ gauge group coupled to $N_f=1$ fermion. We study one class of theories in a controlled limit where $N_f\gg N_c$ and calculate $\nu$ to leading nontrivial order in the absence of disorder. Although these theories do not yield an anomalously large exponent $\nu$ within the large $N_f\gg N_c$ expansion, they do offer a new parameter space of theories that is apparently different from prior works involving Abelian Chern-Simons gauge fields [Wen and Wu, Phys. Rev. Lett. 70, 1501 (1993); Chen et al., Phys. Rev. B 48, 13749 (1993)].

Figure 1

A schematic plot of parameter space for Chern-Simons theories with bosonic and fermionic matter. Note that the orientation of the $y$ axis is inverted between the bosonic and fermionic cubes. The double arrows indicate a duality between the connected points. The pink points refer to free theories and the yellow points to “infinitely coupled” theories. Previous works have studied the large color and large flavor theories both in the fermionic and bosonic cases, labeled in orange and blue [19, 20, 34, 54, 55]. The red dot corresponds to our physical theory, while our calculation in the $N_f\gg N_c$ expansion is done in the green region. All calculations give $\nu =1$ at leading order [19, 20, 34, 54, 55], while experiments give $\nu \approx 2.3$ [1].

Figure 2

A plot of the anomalous dimension correction to $\nu$ to $\mathcal{O}\left(\alpha \right)$ in the original bosonic parameters. The $y$ axis is in units of $\alpha$. It is positive for $1.29<N_f/k^B<4.50$. The parameter $\lambda$ used in Eq. (10) is related to $N_f/k^B$ by ${\lambda }^{-1}=-k^B/N_f+1/2$.