Quantum diffusion and lack of universal one-parameter scaling in one-dimensional disordered lattices with long-range coupling

More current with less particles due to power-law hopping

We reveal interesting universal transport behavior of ordered one-dimensional fermionic systems with power-law hopping. We restrict ourselves to the case where the power-law decay exponent [Formula: see text], so that the thermodynamic limit is well-defined. We explore the quantum phase-diagram of the non-interacting model in terms of the zero temperature Drude weight, which can be analytically calculated. Most interestingly, we reveal that for [Formula: see text], there is a phase where the zero temperature Drude weight diverges as filling fraction goes to zero. Thus, in this regime, counter intuitively, reducing number of particles increases transport and is maximum for a sub-extensive number of particles. Being a statement about zero-filling, this transport behavior is immune to adding number conserving interaction terms. We have explicitly checked this using two different interacting systems. We propose that measurement of persistent current due to a flux through a mesoscopic ring with power-law hopping will give an experimental signature of this phase. In persistent current, the signature of this phase survives up to a finite temperature for a finite system. At higher temperatures, a crossover is seen. The maximum persistent current shows a power-law decay at high temperatures. This is in contrast with short ranged systems, where the persistent current decays exponentially with temperature.

Resonant localized states and quantum percolation on random chains with power-law-diluted long-range couplings

We investigate the nature of one-electron eigenstates in power-law-diluted chains for which the probability of occurrence of a bond between sites separated by a distance r decays as p(r) = p/r(1+σ). Using an exact diagonalization scheme and a phenomenological finite-size scaling analysis, we determine the quantum percolation transition phase diagram in the full parameter space (p,σ). We show that the density of states displays singularities at some resonance energies associated with degenerate eigenstates localized in a pair of sites with special symmetries. This model is shown to present an intermediate phase for which there is classical percolation but no quantum percolation. Quantum percolation only takes place for σ < 0.78, a value larger than the corresponding one for the Anderson transition in long-ranged coupled chains with random diagonal disorder. The fractality of critical wavefunctions is also characterized.

Geometrical and Anderson transitions in harmonic chains with constrained long-range couplings

Low-dimensional systems with long-range couplings usually present phase transitions which are absent in the short-ranged counterpart model. In this work, we show that a harmonic chain with long-range couplings restricted by a cost function proportional to the chain length N exhibits two distinct phase transitions. In the present model, two sites at a distance r>1 are connected by a spring with probability 1/r(α) with the constraint that the total length of the non-nearest-neighbor couplings is limited to λN, where λ is a cost parameter. A geometrical phase transition is found at α=1.5 between a phase with a finite number of long-range couplings and a phase on which the number of long-range couplings is proportional to the system size. Further, the normal vibrational modes of this chain display a phase transition from delocalized to localized modes at a smaller value of α. Maximum effective disorder is reached at α=2 for which the frequency of the lowest vibrational mode exhibits a pronounced peak.

Anderson transition in low-dimensional disordered systems driven by long-range nonrandom hopping

The single-parameter scaling hypothesis predicts the absence of delocalized states for noninteracting quasiparticles in low-dimensional disordered systems. We show analytically, using a supersymmetric method combined with a renormalization group analysis, as well as numerically that extended states may occur in the one- and two-dimensional Anderson model with a nonrandom hopping falling off as some power of the distance between sites. The different size scaling of the bare level spacing and the renormalized magnitude of the disorder seen by the quasiparticles finally results in the delocalization of states at one of the band edges of the quasiparticle energy spectrum.