Elliptic mirror of the quantum Hall effect

TL;DR

This paper uses elliptic mirror models with modular symmetry to analyze quantum Hall effects, providing theoretical predictions that align well with experimental and numerical data, and suggesting new directions for experimental validation.

Contribution

It introduces elliptic mirror models linked via mirror symmetry to quantum Hall systems, offering a unified, topologically protected framework with precise critical exponents matching numerical results.

Findings

Critical delocalization exponent $ u_{tor} \\approx 2.605$ matches numerical data.

Scaling flow geometry agrees with experimental modular predictions.

Finite size scaling experiments can disentangle theoretical and experimental exponents.

Abstract

Toroidal sigma models of magneto-transport are analyzed, in which integer and fractional quantum Hall effects automatically are unified by a {holomorphic modular symmetry}. By exploiting a quantum equivalence called \emph{mirror symmetry}, these models are mapped to tractable mirror models (also elliptic), in which topological protection is provided by more familiar winding numbers. Phase diagrams and scaling properties of elliptic models are compared to some of the experimental and numerical data accumulated over the past three decades. The geometry of scaling flows extracted from quantum Hall experiments is in good agreement with modular predictions, including the location of many quantum critical points. One conspicuous model %(arguably the simplest and most natural one) has a critical delocalization exponent $\nu_{\rm tor} = 18 \ln 2 /(\pi^2 G^4) = 2.6051\dots$ ($G$ is Gauss' constant) that is in excellent agreement with the value $\nu_{\rm num} = 2.607\pm\,.004$ calculated in the numerical Chalker-Coddington model, suggesting that these models are in the same universality class. The real delocalization exponent may be disentangled from other scaling exponents in finite size scaling experiments, giving an experimental value $\nu_{\rm exp} = 2.3\pm 0.2$. The modular model suggests how these theoretical and experimental results may be reconciled, but in order to determine if these theoretical models really are in the quantum Hall universality class, improved finite size scaling experiments are urgently needed.