Research
Multifractality at the metalinsulator 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 disorderdriven localisationdelocalisation 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 localisationdelocalisation transition and characterise the insulating and metallic phases, providing useful information about the critical parameters and wavefunction correlations. "Multifractal analysis with the probability density function at the threedimensional Anderson transition" A. Rodriguez, L. J. Vasquez, R. A. Römer. Phys. Rev. Lett. 102, 1064064 (2009) 
Multifractal electronic state at the threedimensional Anderson transition 
Spectral properties of lowdimensional disordered systems
Due to technological advances it is possible to control the dimensionality of confinement of electronic carriers within manufactured micro and nanostructures: twodimensional 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 lowdimensional 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 1D 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 nondifferentiable 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 1D 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. "Onedimensional models of disordered quantum wires: general formalism" A. Rodriguez. J. Phys. A 39, 1430314327 (2006) 
Apparent fractal behaviour of the DOS for an infinite 1D disordered system 
Transport properties of quantum wires with correlated disorder
According to the Scaling Theory of localisation, in 1D systems all electronic states must be localised in the presence of disorder. It was then believed for a long time that onedimensional disordered systems could not exhibit complex features like a metalinsulator transition. However further research has shown that a large variety of different situations can occur in 1D. In particular the existence of statistical correlations in the disorder can give rise to the emergence of extended states. If the correlations are longrange a band of effectively extended states appear in the system leading to a metalinsulator transition. Shortrange 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 shortrange correlated disorder on the electronic transmission in finite samples is nonnegligible due to the existence of states with localisation lengths larger than the system size. We study a natural model of binary correlations in different 1D quantum wires which induces a noticeable improvement on the transmission efficiency and whose effects seem to be universal independently of the potential model considered. "Onedimensional quantum wires with PöschlTeller potentials" A. Rodriguez, J. M. Cerveró. Phys. Rev. B 74, 10420120 (2006) 
Transmission efficiency in correlation space for a 1D disordered wire. 
NonHermitian Hamiltonians and imaginary potentials
The inelastic scattering processes occurring in mesoscopic samples as a consequence of a nonzero 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 phasebreaking 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 onedimensional 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 PTsymmetric potentials, i.e. those invariant under the combined action of parity (P) and timereversal (T) symmetry. "Absorption in atomic wires" J. M. Cerveró, A. Rodriguez. Phys. Rev. A 70, 05270513 (2004) 
