Quartic multifractality and finite-size corrections at the spin quantum Hall transition

Abstract

The spin quantum Hall (or class C) transition represents one of the few localization-delocalization transitions for which some of the critical exponents are known exactly. Not known, however, is the multifractal spectrum, ${\tau }_q$, which describes the system-size scaling of inverse participation ratios $P_q$, i.e., the $q$-moments of critical wavefunction amplitudes. We here report simulations based on the class C Chalker-Coddington network and demonstrate that ${\tau }_q$ is (essentially) a quartic polynomial in $q$. Analytical results fix all prefactors except the quartic curvature that we obtain as $\gamma =(2.22\pm 0.15)\cdot 10^{-3}$. In order to achieve the necessary accuracy in the presence of sizable corrections to scaling, we have analyzed the evolution with system size of the entire $P_q$-distribution function. As it turns out, in a sizable window of $q$-values this distribution function exhibits a (single-parameter) scaling collapse already in the pre-asymptotic regime, where finite-size corrections are not negligible. This observation motivates us to propose a novel approach for extracting ${\tau }_q$ based on concepts borrowed from the Kolmogorov-Smirnov test of mathematical statistics. We believe that our work provides the conceptual means for high-precision investigations of multifractal spectra also near other localization-delocalization transitions of current interest, especially the integer (class A) quantum Hall effect.

arXiv comment16 pages, 32 figures

References in corpus (22)

• Anderson Transitions
• The spin quantum Hall effect in unconventional superconductors
• Multifractality and critical fluctuations at the Anderson transition
• Exact exponents for the spin quantum Hall transition
• Exact relations between multifractal exponents at the Anderson transition
• Scattering, reflection and impedance of waves in chaotic and disordered systems with absorption
• Quantum Hall plateau transitions in disordered superconductors
• Multifractality at the quantum Hall transition: Beyond the parabolic paradigm
• Quantum and classical localisation, the spin quantum Hall effect and generalisations
• Finite Size Effects and Irrelevant Corrections to Scaling near the Integer Quantum Hall Transition
• Statistics of Impedance, Local Density of States, and Reflection in Quantum Chaotic Systems with Absorption
• Universal Multifractality in Quantum Hall Systems with Long-Range Disorder Potential
• The integer quantum Hall plateau transition is a current algebra after all
• Multifractality and Conformal Invariance at 2D Metal-Insulator Transition in the Spin-Orbit Symmetry Class
• Classification and symmetry properties of scaling dimensions at Anderson transitions
• Wave function statistics and multifractality at the spin quantum Hall transition
• Symmetries of multifractal spectra and field theories of Anderson localization
• Multifractality at the spin quantum Hall transition
• Gaussian free fields at the integer quantum Hall plateau transition
• Spectrum-wide quantum criticality at the surface of class AIII topological phases: An "energy stack" of integer quantum Hall plateau transitions
• Modeling Disordered Quantum Systems with Dynamical Networks
• Statistics of Conductances and Subleading Corrections to Scaling near the Integer Quantum Hall Plateau Transition

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• Generalized multifractality at spin quantum Hall transition
• Topological quantum criticality of the disordered Chern insulator
• Conformal Invariance and Multifractality at Anderson Transitions in Arbitrary Dimensions
• Generalized multifractality in the spin quantum Hall symmetry class with interaction
• Generalized surface multifractality in 2D disordered systems
• Instanton analysis for the spin quantum Hall symmetry class: Non-perturbative corrections to physical observables and generalized multifractal spectrum
• Quantum Hall criticality in an amorphous Chern insulator
• Boundary multifractality in the spin quantum Hall symmetry class with interaction
• Generalized multifractality at the spin quantum Hall transition: Percolation mapping and pure-scaling observables