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Multifractality at the metal-insulator transition

The fluctuations and correlations of wave amplitudes are of primary importance for the understanding of many classical and quantum systems. This is arguably most pronounced in the physics of Anderson localisation. At the critical point of the disorder-driven localisation-delocalisation transition the electronic states are neither extended nor localised but reveal large fluctuations in the wavefunction amplitudes at all length scales. The eigenstates at this phase transition show a multifractal behaviour. Multifractality implies that different ranges of values of the wavefunction amplitudes scale with the linear size of the system according to different fractal dimensions. The ensemble of all the different fractal dimensions that characterise the wavefunctions is known as the singularity or multifractal spectrum, which can be obtained from the probability density function for the wavefunction amplitudes. Multifractal analysis can then be used to study the localisation-delocalisation transition and characterise the insulating and metallic phases, providing useful information about the critical parameters and wavefunction correlations.

(PDF Document) "Multifractal analysis with the probability density function at the three-dimensional Anderson transition" A. Rodriguez, L. J. Vasquez, R. A. Römer. Phys. Rev. Lett. 102, 106406-4 (2009)

Multifractal electronic state at the three-dimensional Anderson transition

Spectral properties of low-dimensional disordered systems

Due to technological advances it is possible to control the dimensionality of confinement of electronic carriers within manufactured micro- and nano-structures: two-dimensional electron gases (2DEG), quantum wires within which the carriers can only move in one dimension, or quantum dots where all degrees of freedom of the carriers are quantised. To study the electronic properties of these low-dimensional systems one important element to be considered is the presence of disorder. Disorder can strongly alter the properties of the system, giving rise to insulating behaviours due to electronic localisation. In 1-D systems, the distribution of states (DOS) in the thermodynamic limit exhibits some unexpected behaviour. In contrast to the smooth and differentiable DOS corresponding to a periodic system, the DOS when disorder appears manifests itself as a non-differentiable curve that fluctuates strongly within some energy intervals, inside which the electronic states have always finite localisation lengths. Preliminary statistical analysis suggests that the DOS for an infinite 1-D disordered structure could behave as a fractal. This exotic feature is apparently related to the nature of the distributions that define the disorder in the system: fractality seemingly disappears whenever a configurational parameter of the system is governed by a continuous probability distribution.

(PDF Document) "One-dimensional models of disordered quantum wires: general formalism" A. Rodriguez. J. Phys. A 39, 14303-14327 (2006)

Fractal DOS

Apparent fractal behaviour of the DOS for an infinite 1-D disordered system

Transport properties of quantum wires with correlated disorder

According to the Scaling Theory of localisation, in 1-D systems all electronic states must be localised in the presence of disorder. It was then believed for a long time that one-dimensional disordered systems could not exhibit complex features like a metal-insulator transition. However further research has shown that a large variety of different situations can occur in 1-D. In particular the existence of statistical correlations in the disorder can give rise to the emergence of extended states. If the correlations are long-range a band of effectively extended states appear in the system leading to a metal-insulator transition. Short-range correlations can generate isolated extended states, resonances of the transmission, but their effect disappears in the thermodynamic limit, i.e. for an infinite system. Nevertheless the effect of short-range correlated disorder on the electronic transmission in finite samples is non-negligible due to the existence of states with localisation lengths larger than the system size. We study a natural model of binary correlations in different 1-D quantum wires which induces a noticeable improvement on the transmission efficiency and whose effects seem to be universal  independently of the potential model considered.

(PDF Document) "One-dimensional quantum wires with Pöschl-Teller potentials" A. Rodriguez, J. M. Cerveró. Phys. Rev. B 74, 104201-20 (2006)

transmission efficiency

Transmission efficiency in correlation space for a 1-D disordered wire.

Non-Hermitian Hamiltonians and imaginary potentials

The inelastic scattering processes occurring in mesoscopic samples as a consequence of a non-zero temperature can noticeably change the coherent transport fingerprints of these structures. The worsening of electronic transmission due to such effects is expected but in some situations the competition between the phase-breaking mechanisms and the quantum coherent interferences can improve conductance in certain energetic regimes. This is the case, for example, of disordered structures. Inelastic processes can be modelled by small absorptions which in turn can be described by extending the nature of the quantum potentials to the complex domain. In the context of disordered systems we consider imaginary potentials in one-dimensional quantum wires to model electronic absorption and to try to describe decoherence mechanisms that could eventually lead to delocalisation of the electronic states. From a more general perspective we are also interested in PT-symmetric potentials, i.e. those invariant under the combined action of parity (P) and time-reversal (T) symmetry.

(PDF Document) "Absorption in atomic wires" J. M. Cerveró, A. Rodriguez. Phys. Rev. A 70, 052705-13 (2004)

scattering diagram scarf

Scattering diagram for the complex Scarf potential as function of its real (V1) and imaginary (V2) amplitudes. (See reference on the left)