Multifractality and Conformal Invariance at 2D Metal-Insulator Transition in the Spin-Orbit Symmetry Class

Abstract

We study the multifractality (MF) of critical wave functions at boundaries and corners at the metal-insulator transition (MIT) for noninteracting electrons in the two-dimensional (2D) spinorbit (symplectic) universality class. We find that the MF exponents near a boundary are different from those in the bulk. The exponents at a corner are found to be directly related to those at a straight boundary through a relation arising from conformal invariance. This provides direct numerical evidence for conformal invariance at the 2D spin-orbit MIT. The presence of boundaries modifies the MF of the whole sample even in the thermodynamic limit. Anderson metal-insulator transitions (MITs), i.e., localization-delocalization transitions of noninteracting electrons, are continuous phase transitions driven by disorder. At the transition, wave functions (WFs) are neither localized nor simply extended, but are complicated scale invariant fractals exhibiting multifractal behavior characterized by a continuous set of scaling exponents [1,2,3,4]. In two dimensions (2D) the universal critical properties at a host of conventional phase transitions are known to be described by conformal field theories (CFTs) [5]. It is natural to expect that disorder-averaged observables at a localization transition in 2D are also governed by a CFT. If so, then conformal symmetry should impose severe constraints on averages of local quantities, including moments of WF amplitudes.In a recent Letter Subramaniam et al. [6] extended the notion of multifractality (MF) to the boundaries of the sample ("surface MF"), and showed that near boundaries critical WFs are characterized by MF exponents that are different from those in the bulk. Moreover, it was predicted that the MF of the entire system depends crucially on the presence or absence of boundaries, even in the thermodynamic limit. In this Letter we study surface MF at the MIT in 2D for non-interacting electrons with spin-orbit scattering (symplectic universality class) [7], and extend the surface MF analysis to boundaries with corners ("corner MF"). Conformal symmetry, if present, would lead (following [8]) to a simple exact prediction relating corner and surface MF exponents. Here, we show numerically that at the 2D spin-orbit MIT this prediction is indeed valid, thereby providing direct evidence for the presence of conformal symmetry at this MIT. We also confirm the dependence of the MF of the whole system on the presence of boundaries, as predicted in [6].We begin by introducing corner and surface MF [6, 9] for a rhombus [Fig.

Multifractal analysis of the metal-insulator transition in the three-dimensional Anderson model. I. Symmetry relation under typical averaging

“…Numerical calculations have since then supported this symmetry in f (α) in the one-dimensional power-law random-banded-matrix model 27 and the twodimensional Anderson transition in the spin-orbit symmetry class. 16,28 In the present work we numerically verify that this symmetry in the singularity spectrum also holds in the three-dimensional (3D) Anderson model. In order to address this hypothesis with sufficient accuracy, we have considered the box-and system-size scaling of the typical average of P q in computing the f (α).…”

Multifractal analysis of the metal-insulator transition in the three-dimensional Anderson model. II. Symmetry relation under ensemble averaging

“…So far the symmetry has been numerically found in different critical models below three dimensions. 16,19 In a previous work the role of the symmetry law in the MIT for the 3D Anderson model was thoroughly studied by the authors using the typical average of the scaling law for the gIPR. 20 This has been usually regarded as the preferred way to perform MFA.…”

Boundary Multifractality at the Integer Quantum Hall Plateau Transition: Implications for the Critical Theory

“…It is natural to expect that effective (field) theories describing IQH plateau transitions should generally also possess conformal symmetry (cf. [4]). …”

“…(in some interval [4] around q ¼ 1=2). Equivalently, the MF wave functions can be characterized by the so-called singularity spectra f x ð x Þ related to Á x q by a Legendre transform:…”