Wave function statistics and multifractality at the spin quantum Hall transition

TL;DR

This paper investigates the multifractal properties and wave function statistics at the spin quantum Hall transition, providing analytical and numerical insights into the spectrum of exponents governing wave function and conductance scaling.

Contribution

It offers the first analytical calculation of multifractal exponents at this transition and compares them with numerical results, revealing both agreement and deviations.

Findings

Analytical values for $oldsymbol{ ext{Δ}_2=-1/4}$ and $oldsymbol{ ext{Δ}_3=-3/4}$

Numerical multifractality spectrum approximates $oldsymbol{ ext{Δ}_q oughly q(1-q)/8}$

Relations established between wave function, conductance, and Green function exponents.

Abstract

The statistical properties of wave functions at the critical point of the spin quantum Hall transition are studied. The main emphasis is put onto determination of the spectrum of multifractal exponents $\Delta_q$ governing the scaling of moments $<|\psi|^{2q}>\sim L^{-qd-\Delta_q}$ with the system size $L$ and the spatial decay of wave function correlations. Two- and three-point correlation functions are calculated analytically by means of mapping onto the classical percolation, yielding the values $\Delta_2=-1/4$ and $\Delta_3=-3/4$. The multifractality spectrum obtained from numerical simulations is given with a good accuracy by the parabolic approximation $\Delta_q\simeq q(1-q)/8$ but shows detectable deviations. We also study statistics of the two-point conductance $g$, in particular, the spectrum of exponents $X_q$ characterizing the scaling of the moments $<g^q >$. Relations between the spectra of critical exponents of wave functions ($\Delta_q$), conductances ($X_q$), and Green functions at the localization transition with a critical density of states are discussed.