Non-Abelian bosonization and modular transformation approach to superuniversality

Quantum Hall inter-plateau transitions are physical exemplars of quantum phase transitions. Near each of these transitions, the measured electrical conductivity scales with the same correlation length and dynamical critical exponents, i.e., the critical points are superuniversal. In apparent contradiction to these experiments, prior theoretical studies of quantum Hall phase transitions within the framework of Abelian Chern-Simons theory coupled to matter found correlation length exponents that depend on the value of the quantum critical Hall conductivity. Here, we use non-Abelian bosonization and modular transformations to theoretically study the phenomenon of superuniversality. Specifically, we introduce a new effective theory that has an emergent $U\left(N\right)$ gauge symmetry with any $N>1$ for a quantum phase transition between an integer quantum Hall state and an insulator. We then use modular transformations to generate from this theory effective descriptions for transitions between a large class of fractional quantum Hall states whose quasiparticle excitations have Abelian statistics. We find the correlation length and dynamical critical exponents are independent of the particular transition within a controlled 't Hooft large $N$ expansion, i.e., superuniversal! We argue that this superuniversality could survive away from this controlled large $N$ limit using recent duality conjectures.

Figure 1

Schematic zero-temperature phase diagram [16] in the space of Hall ${\rho }_{xy}$ and longitudinal resistivity ${\rho }_{xx}$. Phases are denoted by their zero-temperature complex conductivity $\sigma ={\sigma }_{xy}+i{\sigma }_{xx}$, measured in units of $e^2/h$. The blue boundary denotes the $1\to 0$ integer quantum Hall transition, while the green boundaries denote transitions we derive from the $1\to 0$ transition via modular transformations.

Figure 2

(a) One-loop fermion self-energy decomposed into $SU\left(N\right)\subset U\left(N\right)$ and $U\left(1\right)\subset U\left(N\right)$ contributions. The closed oriented loop produces the relative factor of $N$ between the second and third diagrams. (b) The leading Feynman diagrams contributing to ${\gamma }_{\overline{\psi }\psi }$ in the 't Hooft large $N$ limit. Directed lines are fermion propagators; wavy lines are $U\left(N\right)$ gauge field propagators; a double line is a $SU\left(N\right)$ gauge field propagator; a dashed line denotes a $U\left(1\right)$ gauge field propagator; insertion of $\overline{\psi }\psi$ is represented by $\otimes$.