Non-Abelian fermionization and fractional quantum Hall transitions

Abstract

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 $\ensuremath{\nu}\ensuremath{\approx}2.3$ and that $\ensuremath{\nu}$ 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 $\mathrm{U}({N}_{c})$ gauge group coupled to ${N}_{f}=1$ fermion. We study one class of theories in a controlled limit where ${N}_{f}\ensuremath{\gg}{N}_{c}$ and calculate $\ensuremath{\nu}$ to leading nontrivial order in the absence of disorder. Although these theories do not yield an anomalously large exponent $\ensuremath{\nu}$ within the large ${N}_{f}\ensuremath{\gg}{N}_{c}$ 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)].