Multifractal properties of critical eigenstates in two-dimensional systems with symplectic symmetry

Schweitzer, L.

doi:10.1088/0953-8984/7/23/001

Cited by 20 publications

(28 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(7) for the bulk which is the consequence of the symmetry relation [12]. Incidentally, the value of the typical bulk exponent α b 0 agrees with earlier calculations [17,18] but not with [19]. Furthermore, our α…”

supporting

confidence: 86%

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

et al. 2007

View full text Add to dashboard Cite

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.

show abstract

supporting

confidence: 86%

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

et al. 2007

View full text Add to dashboard Cite

show abstract

“…According to (34) we want to fit the scaling function f to the logarithms of the RLLs, which depend on the n M system widths {M 1 , . .…”

Section: Methods Of Evaluation a Fit Procedures For The Scaling Fmentioning

confidence: 99%

Network model for a two-dimensional disordered electron system with spin-orbit scattering

1998

View full text Add to dashboard Cite

We introduce a network model to describe two-dimensional disordered electron systems with spinorbit scattering. The network model is defined by a discrete unitary time evolution operator. We establish by numerical transfer matrix calculations that the model exhibits a localization-delocalization transition. We determine the corresponding phase diagram in the parameter space of disorder scattering strength and spin-orbit scattering strength. Near the critical point we determine by statistical analysis a one-parameter scaling function and the critical exponent of the localization length to be ν = 2.51 ± 0.18. Based on a conformal mapping we also calculate the scaling exponent of the typical local density of states α0 = 2.174 ± 0.003.

show abstract

“…where the sum ′ is restricted to the eigenstates Ψ (i,j) with eigenvalues E (1,i) 35 Clearly, Eq. (6.2) becomes Eq.…”

Section: Quantum Diffusion For the Labyrinth Tilingmentioning

confidence: 99%

Energy spectra, wave functions, and quantum diffusion for quasiperiodic systems

et al. 2000

View full text Add to dashboard Cite

We study energy spectra, eigenstates and quantum diffusion for one-and two-dimensional quasiperiodic tight-binding models. As our one-dimensional model system we choose the silver mean or "octonacci" chain. The twodimensional labyrinth tiling, which is related to the octagonal tiling, is derived from a product of two octonacci chains. This makes it possible to treat rather large systems numerically. For the octonacci chain, one finds singular continuous energy spectra and critical eigenstates which is the typical behaviour for one-dimensional Schrödinger operators based on substitution sequences. The energy spectra for the labyrinth tiling can, depending on the strength of the quasiperiodic modulation, be either band-like or fractallike. However, the eigenstates are multifractal. The temporal spreading of a wavepacket is described in terms of the autocorrelation function C(t) and the mean square displacement d(t). In all cases, we observe power laws C(t) ∼ t −δ and d(t) ∼ t β . For the octonacci chain, 0 < δ < 1, whereas for the labyrinth tiling a crossover is observed from δ = 1 to 0 < δ < 1 with increasing modulation strength. Corresponding to the multifractal eigenstates, we obtain anomalous diffusion with 0 < β < 1 for both systems. Moreover, we find that the behaviour of C(t) and d(t) is independent of the shape and the location of the initial wavepacket. We use our results to check several relations between the diffusion exponent β and the fractal dimensions of energy spectra and eigenstates that were proposed in the literature.

show abstract

Multifractal properties of critical eigenstates in two-dimensional systems with symplectic symmetry

Cited by 20 publications

References 27 publications

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

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

Network model for a two-dimensional disordered electron system with spin-orbit scattering

Energy spectra, wave functions, and quantum diffusion for quasiperiodic systems

Contact Info

Product

Resources

About