Spectrum and diffusion for a class of tight-binding models on hypercubes

Abstract

We propose a class of exactly solvable anisotropic tight-binding models on an infinite-dimensional hypercube. The energy spectrum is analytically computed and is shown to be fractal and/or absolutely continuous according to the value hopping parameters. In both cases, the spectral and diffusion exponents are derived. The main result is that, even if the spectrum is absolutely continuous, the diffusion exponent for the wave packet may be anything between 0 and 1 depending upon the class of models.PACS numbers: 05.60.+w, 73.20.At The interplay between the energy spectrum and the diffusion process for quantum systems is still an open question. Indeed, recent studies have shown that the spreading of a wavepacket is determined by several exponents that only depend on the nature of the spectrum (absolutely continuous, singular continuous, pure point, or any mixture) [1][2][3]. In particular, for quasiperiodic systems, such as quasicrystals (QC), it has been shown that an anomalous behaviour of the wavepacket spreading occurs when the spectrum is singular continuous [4] (see below). In such systems, scaling laws occur at many levels. We will concentrate upon two classes of scaling exponents due to their importance for electronic and transport properties. The first one concerns spectral exponents, defined aswhere N is the local density of states (LDOS). An absolutely continuous spectrum in some interval I of energy implies α(E) = 1 for E ∈ I and E in the spectrum. A pure point spectrum in I, namely the LDOS is a sum of Dirac peaks in I, implies α(E) = 0 on the part of the spectrum contained in I. Finally, if 0 < α(E) < 1 for E in some part of the spectrum implies singular continuous spectrum there. This is actually what happens for 1D QC [5,6] such as the Fibonacci chain. Such singular continuous spectrum occurs also for 1D chains with a potential given by a substitution sequence [7][8][9][10]. The problem of 2D electrons on a square lattice under magnetic field that can be mapped onto a 1D Harper equation [11] also display a singular continuous spectrum [12] for incommensurate fluxes, and the spectral exponents have been computed numerically [13]. A number of non rigorous results suggesting singular spectra were obtained for models on higher dimensional QC such as the labyrinth model [14], the octagonal tiling [15], in a regime of small hopping. At last let us also mention recent studies on Jacobi matrices associated to iterated functions systems (IFS) [17,18] generalizing Julia sets [19,20] leading to the calculation of the spectral exponents. The next class concerns diffusion exponents, given by L E (t) ∼ t β(E) as t → ∞ where L E (t) characterizes the spreading of a typical wave packet of energy E after time t [21] (see eq. (16) for a definition of the exponents). Ballistic motion such as the electronic motion in a perfect crystal, corresponds to β = 1. Strong localization is defined by demanding that sup t>0 L E (t) < ∞ implying β(E) = 0. The weak localization regime appearing in weakly disordered metals genera...

Dynamics of continuous-time quantum walks in restricted geometries

“…[16,39] the Appendix we prove that on infinite d-dimensional hypercubic lattices both the average chemical displacement defined in Sec. 2.1 and the Euclidean displacement depend linearly on time and that this kind of behaviour survives, at short times, also for finite lattices.…”

“…Finally, we stress that analytical results on the average displacement performed by a quantum particle on discrete structures are rather sparse (see e.g. [16,39]); in the Appendix we prove that on infinite d-dimensional hypercubic lattices both the average chemical displacement defined in Sec. 2.1 and the Euclidean displacement depend linearly on time and that this kind of behaviour survives, at short times, also for finite lattices.…”

“…The value of the 'diffusion exponent' β allows us to distinguish between normal (β = 1) and anomalous (β = 1) diffusion [15]. As for quantum transport, it is possible to introduce analogous exponents, characterizing the temporal spreading of a wave packet [16]. However, even when they take place over the same structure, quantum and classical walks can exhibit dramatically different behaviours.…”