Localization effects in quantum percolation

Abstract

We present a detailed study of the quantum site percolation problem on simple cubic lattices, thereby focussing on the statistics of the local density of states and the spatial structure of the single particle wavefunctions. Using the Kernel Polynomial Method we refine previous studies of the metal-insulator transition and demonstrate the non-monotonic energy dependence of the quantum percolation threshold. Remarkably, the data indicates a "fragmentation" of the spectrum into extended and localised states. In addition, the observation of a chequerboard-like structure of the wavefunctions at the band centre can be interpreted as anomalous localisation.PACS numbers: 71.23. An, 71.30.+h, 05.60.Gg, 72.15.Rn Disordered structures attracted continuing interest over the last decades, and besides the Anderson localisation problem 1 quantum percolation 2,3 is one of the classical subjects of this field. Current applications concern e.g. transport properties of doped semiconductors 4 and granular metals 5 , metal-insulator transition in twodimensional n-GaAs heterostructures 6 , wave propagation through binary inhomogeneous media 7 , superconductorinsulator and (integer) quantum Hall transitions 8,9 , or the dynamics of atomic Fermi-Bose mixtures 10 . Another important example is the metal-insulator transition in perovskite manganite films and the related colossal magnetoresistance effect, which in the meantime are believed to be inherently percolative. 11 In disordered solids the percolation problem is characterised by the interplay of pure classical and quantum effects. Besides the question of finding a percolating path of "accessible" sites through a given lattice the quantum nature of the electrons imposes further restrictions on the existence of extended states and, consequently, of a finite DC-conductivity. As a particularly simple model describing this situation we consider a tight-binding oneelectron Hamiltonian,on a simple cubic lattice with N = L 3 sites and random on-site energies ǫ i drawn from the bimodal distributionThe two energies ǫ A and ǫ B could, for instance, represent the potential landscape of a binary alloy A p B 1−p , where each site is occupied by an A or B atom with probability p or 1−p, respectively. In the limit ∆ = (ǫ B −ǫ A ) → ∞ the wavefunction of the A sub-band vanishes identically on the B-sites, making them completely inaccessible for the quantum particles. We then arrive at a situation where non-interacting electrons move on a random ensemble of lattice points, which, depending on p, may span the entire lattice or not. The corresponding Hamiltonian readswhere the summation extends over nearest-neighbour Asites only and, without loss of generality, ǫ A is chosen to be zero. Within the classical percolation scenario the percolation threshold p c is defined by the occurrence of an infinite cluster A ∞ of adjacent A sites. For the simple cubic lattice this site-percolation threshold is p c = 0.3117. 12 In the quantum case, the multiple scattering of the particles at the irregular boundari...

Mott-Hubbard and Anderson metal-insulator transitions in correlated lattice fermions with binary disorder

“…6) and are identified either as cluster resonances 61,62 or as "anomalous" localized states. 63 The resonance states are similar bound eigenstates but with a finite life-time. They appear because of special geometrical configurations of the impurity atoms.…”

“…Panel (a) shows the behavior of an extended state, panel (b) presents the behavior of an Anderson localized state, and panel (c) shows the behavior of an "anomalous" localized state with its typical bimodal structure. 63 …”

“…The "anomalous" localized states are in fact extended states over the whole lattice but they are insulating and do not contribute to the dc conductivity. 63 On a bipartite lattice these states have small wave function amplitudes on one sublattice and large amplitudes on the other sublattice. The typical η behaviors of the PDFs for given frequencies are shown in Fig.…”

Study of off-diagonal disorder using the typical medium dynamical cluster approximation

“…5 a comparison of the average and typical DOS calculated with the DCA and the TMDCA (N c = 4 3 ) as compared with the kernel polynomial method (KPM). [47][48][49][50] In the KPM analysis, instead of diagonalizing the Hamiltonian directly, the local DOS is expressed in term of an infinite series of Chebyshev polynomials. In practice, the truncated series leads to Gibbs oscillations.…”

Hölder mean applied to Anderson localization

“…It is plausible that both MITs could be detected by knowing the local density of states (LDOS), as it discriminates between a metal and an insulator, which is driven by correlations and disorder [4]. In many recent works [5,6,7], the change of the whole LDOS distribution function at the Anderson transition were explored for the Bethe and for the simple cubic lattices, where many other effects, like interaction, percolation and binary alloy disorder, were included.…”