Generalized multifractality at the spin quantum Hall transition: Percolation mapping and pure-scaling observables

TL;DR

This paper develops a percolation mapping and constructs pure-scaling observables to analyze generalized multifractality at the spin quantum Hall transition, providing exact exponents and challenging existing conformal invariance models.

Contribution

It introduces a percolation mapping for certain observables and constructs positive pure-scaling eigenfunction observables, enabling precise numerical and analytical analysis of multifractality.

Findings

Exact analytical exponents for generalized multifractality observables.

Numerical determination of exponents up to order q=5.

Generalized parabolicity does not hold, ruling out certain conformal models.

Abstract

This work extends the analysis of the generalized multifractality of critical eigenstates at the spin quantum Hall transition in two-dimensional disordered superconductors [J. F. Karcher et al, Annals of Physics, 435, 168584 (2021)]. A mapping to classical percolation is developed for a certain set of generalized-multifractality observables. In this way, exact analytical results for the corresponding exponents are obtained. Furthermore, a general construction of positive pure-scaling eigenfunction observables is presented, which permits a very efficient numerical determination of scaling exponents. In particular, all exponents corresponding to polynomial pure-scaling observables up to the order $q=5$ are found numerically. For the observables for which the percolation mapping is derived, analytical and numerical results are in perfect agreement with each other. The analytical and numerical results unambiguously demonstrate that the generalized parabolicity (i.e., proportionality to eigenvalues of the quadratic Casimir operator) does not hold for the spectrum of generalized-multifractality exponents. This excludes Wess-Zumino-Novikov-Witten models, and, more generally, any theories with local conformal invariance, as candidates for the fixed-point theory of the spin quantum Hall transition. The observable construction developed in this work paves a way to investigation of generalized multifractality at Anderson-localization critical points of various symmetry classes.