Classical trajectories in quantum transport at the band center of bipartite lattices with or without vacancies

Abstract

Here we report on several anomalies in quantum transport at the band center of a bipartite lattice with vacancies that are surely due to its chiral symmetry, namely: no weak localization effect shows up, and, when leads have a single channel the transmission is either one or zero. We propose that these are a consequence of both the chiral symmetry and the large number of states at the band center. The probability amplitude associated to the eigenstate that gives unit transmission ressembles a classical trajectory both with or without vacancies. The large number of states allows to build up trajectories that elude the blocking vacancies explaining the absence of weak localization. PACS numbers: 73.63.Fg, 71.15.MbBipartite lattices. The possibility that qualitative differences between models with pure diagonal or non-diagonal disorder may exist, have attracted a considerable interest since Dyson's work on a phonon model in one dimension 1 . For instance it has been reported that non-diagonal disorder promotes a delocalization transition at E = 0 2,3,4 . Although this raised a controversy regarding the general statement saying that the particular type of disorder should not matter in a single parameter scaling theory 5 , in recent years a different view has emerged which ascribes that transition to the peculiar properties of bipartite lattices. These lattices are characterised by the electron-hole symmetry of the spectrum (also known as chiral symmetry) and, while diagonal disorder efficiently breaks this symmetry, pure non-diagonal disorder does not. More recently, however, it has been shown 6 that, at E = 0, standard exponential localization occurs in a system with vacancies (a defect that also preserves chirality), reopening the mentioned controversy.Here we report on some anomalies in transport properties at the band center of bipartite lattices that are indeed related to chirality, namely: i) the absence of weak localization 7,8 , and, ii) when single channels leads are connected to cavities with or without vacancies, the transmission is either zero or one. We show that these anomalies are a associated to the existence of ballistic classical trajectories which are possible due to chiral symmetry and the large number of states at E = 0.We illustrate these ideas on cavities of the square lattice with vacancies. The Hamiltonian. We consider a tight-binding Hamiltonian in L × L clusters of the square lattice with a single atomic orbital per lattice site, H = m,n ǫ m,n |m, n >< m, n| − <mn;m ′ n ′ > t m,n;m ′ n ′ |m, n >< m ′ , n ′ | (1) 0 0.2 0.4 0.6 0.8 1 Φ/Φ 0 −0.3 −0.2 −0.1 0.0 [G(Φ)−G(0)]/G 0 −0.4 −0.2 0 0.2 (a) (b) FIG. 1: Change in the conductance versus magnetic flux (both in units of their respective quanta) in: a) 78 × 78 clusters with 78 vacancies (1200 realizations were included) and leads of width W = 4 connected at opposite corners of the cavity, and b) 6 × 6 clusters with 6 vacancies (all realizations were included in the calculation) and leads of width 1 connected at opposite corners of the cavity. T...

Anomalies and Anderson localization: π-coupling of the energy bands

“…Depending on the details of the model, it has been demonstrated that this point presents anomalies in the localization length [12,14,16,22,23,26,28], divergent density of states [2,9,13], even-odd effects [2,18,19], violation of single-parameter scaling (SPS) [5,8,21,25,27] and anomalously localized states (ALS) [1]. Similar features appearing at E = 0 have also been shown to exist in higher dimensions [6,10,17,20].…”

“…In particular, it is not clear what causes the point E = 0 to be so special. It is worth noticing that the origin of the delocalization of states of the one-dimensional hopping-disordered Anderson model has already been claimed to be due to an additional (chiral) sublattice symmetry [3,6,19,18]. Additional symmetries of the Hamiltonian have also been considered as the origin of the violation of SPS at E = 0 [5,8,25].…”