Termination of typical wavefunction multifractal spectra at the Anderson metal-insulator transition: Field theory description using the functional renormalization group

Abstract

We revisit the problem of wavefunction statistics at the Anderson metal-insulator transition (MIT) of non-interacting electrons in d > 2 spatial dimensions. At the transition, the complex spatial structure of the critical wavefunctions is reflected in the non-linear behavior of the multifractal spectrum of generalized inverse participation ratios (IPRs). Beyond the crossover from narrow to broad IPR statistics, which always occurs for sufficiently large moments of the wavefunction amplitude, the spectrum obtained from a typical wavefunction associated with a particular disorder realization differs markedly from that obtained from the disorder-averaged IPRs. This phenomenon is known as the termination of the multifractal spectrum. We provide a field theoretical derivation for the termination of the typical multifractal spectrum, by combining the non-linear sigma model framework, conventionally used to access the MIT in d = 2 + epsilon dimensions, with a functional renormalization group (FRG) technique. The FRG method deployed here was originally pioneered to study the properties of the two-dimensional (2D) random phase XY model [D. Carpentier and P. Le Doussal, Nucl. Phys. B 588, 565 (2000)]. The same method was used to demonstrate the termination of the multifractal spectrum in the very special problem of 2D Dirac fermions subject to a random Abelian vector potential. Our result shows that the typical multifractal wavefunction spectrum and its termination can be obtained at a generic Anderson localization transition in d > 2, within the standard field theoretical framework of the non-linear sigma model, when combined with the FRG.

22 pages, 4 figures

References in corpus (20)

• Anderson Transitions
• The Quantum Hall Transition in Real Space: From Localized to Extended States
• Particle-hole symmetric localization in two dimensions
• Multifractality of wavefunctions at the quantum Hall transition revisited
• Multifractality at the quantum Hall transition: Beyond the parabolic paradigm
• Multifractal analysis with the probability density function at the three-dimensional Anderson transition
• Multifractal analysis of the metal-insulator transition in the 3D Anderson model I: Symmetry relation under typical averaging
• Dimensionality dependence of the wave function statistics at the Anderson transition
• Multifractal analysis of the metal-insulator transition in the 3D Anderson model II: Symmetry relation under ensemble averaging
• Density of states for the $\pi$-flux state with bipartite real random hopping only: A weak disorder approach
• Boundary multifractality at the integer quantum Hall plateau transition: implications for the critical theory
• A Classification of random Dirac fermions
• Induced vs Spontaneous Breakdown of S-matrix Unitarity: Probability of No Return in Quantum Chaotic and Disordered Systems
• Magnetic properties of Sr_2Fe_1+xMo_1-xO_6 with -1 <= x <= 0.25
• Surface criticality and multifractality at localization transitions
• Corner Multifractality for Reflex Angles and Conformal Invariance at 2D Anderson Metal-Insulator Transition with Spin-Orbit Scattering
• Anomalous dimensions of operators without derivatives in the non-linear sigma-model for disordered bipartite lattices
• Random hopping fermions on bipartite lattices: Density of states, inverse participation ratios, and their correlations in a strong disorder regime
• Strong disorder effects of a Dirac fermion with a random vector field
• Field Theory for the Global Density of States Distribution Function in Disordered Conductors

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