Conformal field theory of the integer quantum Hall plateau transition
Martin R. Zirnbauer
TL;DR
This work proposes a conformal field theory for the integer quantum Hall plateau transition by formulating an ${f A}_1|{f A}_1$ nonlinear sigma model on the Riemannian symmetric superspace ${f X}_{{f A}_1|{f A}_1}$ with a Wess-Zumino term. The theory is engineered to have Euclidean signature, a partition function $Z=1$ (hence central charge $c=0$), BRST invariance, and an enlarged chiral symmetry ${ m G}_{L} imes{ m G}_{R}$; a truly marginal coupling $f$ remains, but is fixed by matching the short-distance classical conductance near absorbing boundaries to the universal diffusion result. The marginal coupling yields a predicted typical point-contact conductance exponent $X_t=2/ ext{π}$, in agreement with numerical simulations, and the quantum conductance moments are governed by a continuum of ${ m PSL}(2|2)$ representations with scaling dimensions $oldsymbol{\Delta}_{oldsymbol{\lambda}}=f^2(oldsymbol{\lambda}^2+1)$. The formulation connects to BRST/topological mechanisms, AdS$_3$-related sigma-model structures, and provides a nonperturbative CFT framework for the plateau transition with clear experimental and numerical consistency, while leaving open the precise localization-length exponent and further nonperturbative tests.
Abstract
A solution to the long-standing problem of identifying the conformal field theory governing the transition between quantized Hall plateaus of a disordered noninteracting 2d electron gas, is proposed. The theory is a nonlinear sigma model with a Wess-Zumino-Novikov-Witten term, and fields taking values in a Riemannian symmetric superspace based on H^3 x S^3. Essentially the same conformal field theory appeared in very recent work on string propagation in AdS_3 backgrounds. We explain how the proposed theory manages to obey a number of tight constraints, two of which are constancy of the partition function and noncriticality of the local density of states. An unexpected feature is the existence of a truly marginal deformation, restricting the extent to which universality can hold in critical quantum Hall systems. The marginal coupling is fixed by matching the short-distance singularity of the conductance between two interior contacts to the classical conductivity sigma_xx = 1/2 of the Chalker-Coddington network model. For this value, perturbation theory predicts a critical exponent 2/pi for the typical point-contact conductance, in agreement with numerical simulations. The irrational exponent is tolerated by the fact that the symmetry algebra of the field theory is Virasoro but not affine Lie algebraic.