Elliptic mirror of the quantum Hall effect

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, whose group structure is determined by the spin structure of the toroidal target space (an elliptic curve). Hall quantization is protected by the topology of stable holomorphic vector bundles $\mathcal{V}$ on this space, and plateau values ${\ensuremath{\sigma}}_{H}^{{}_{\ensuremath{\bigoplus}}}=\ensuremath{\mu}\ensuremath{\in}\mathbb{Q}$ of the Hall conductivity are rational because such bundles are classified by their slope $\ensuremath{\mu}(\mathcal{V})=\mathrm{deg}(\mathcal{V})/\mathrm{rk}(\mathcal{V})$, where $\mathrm{deg}(\mathcal{V})$ is the degree and $\mathrm{rk}(\mathcal{V})$ is the rank of $\mathcal{V}$. By exploiting a quantum equivalence called 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 has a critical delocalization exponent ${\ensuremath{\nu}}_{\mathrm{tor}}=18ln2/({\ensuremath{\pi}}^{2}{G}^{4})=2.6051\ensuremath{\cdots}$ ($G$ is Gauss' constant) that is in excellent agreement with the value ${\ensuremath{\nu}}_{\mathrm{num}}=2.607\ifmmode\pm\else\textpm\fi{}\phantom{\rule{0.16em}{0ex}}.004$ 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 ${\ensuremath{\nu}}_{\mathrm{exp}}=2.3\ifmmode\pm\else\textpm\fi{}0.2$. The modular model suggests how these theoretical and experimental results may be reconciled, but to determine if these theoretical models really are in the quantum Hall universality class, improved finite-size scaling experiments are urgently needed.