Quantum diffusion in the generalized Harper equation

Abstract

We study numerically the dynamic and spectral properties of a one-dimensional quasi-periodic system, where site energies are given by k = V cos 2πf x k with x k denoting the kth quasiperiodic lattice site. When 2πf is given by the reciprocal lattice vector G(m, n) with n and m being successive Fibonacci numbers, the variance of the wavepacket is found to grow quadratically in time, regardless of the potential strength V. For other values of f , there exists a critical value of V beyond which the growth of the wavepacket variance is bounded. In particular an anomalous diffusion takes place for 2πf corresponding to G(m, n) with generic integers m and n. The level-spacing distribution is also examined, and the corresponding exponent β is observed to decrease with V .

Phase transitions in a non-Hermitian Aubry-André-Harper model

“…On the other hand, in the localized phase V > J transport is prevented (dynamical localization), so that δ(V ) = v(V ) = 0. At the phase transition point V = V c = J, transport is intermediate between ballistic and localized; previous works have shown that transport in the lattice is nearly diffusive with an exponent δ(V c ) 0.5 [18,21,77]. This means that the phase transition is characterized by a discontinuous behavior of diffusion exponent δ(V ) near the critical point V = V c [Fig.…”

“…One-dimensional lattices with aperiodic order, i.e., displaying a long-range periodicity intermediate between ordinary periodic crystals and disordered systems, provide fascinating models to study unusual transport phenomena in a wide variety of classical and quantum systems, ranging from condensed matter systems to ultracold atoms, photonic, and acoustic systems [1][2][3][4][5][6][7]. Quasiperiodicity gives rise to a range of unusual behavior, including critical spectra, multifractal eigenstates, localization transitions at a finite modulation of the on-site potential, and mobility edges [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Typical dynamical variables that characterize single-particle transport are the largest propagation speed v of excitation in the lattice, which is bounded (to form a light cone) for short-range hopping according to the Lieb-Robinson bound [28], and the quantum diffusion exponent δ, which measures the asymptotic spreading of wave-packet variance σ 2 (t ) in the lattice according to the power law σ 2 (t ) ∼ t 2δ .…”

“…The Hamiltonian displays a Cantor-set energy spectrum with a phase transition from extended states and absolutely continuous spectrum to exponentially localized states and pure point spectrum as the amplitude V of the on-site quasiperiodic potential is increased above a critical value V c [45,46]. In terms of dynamical behavior of a wave packet, measured by the exponent δ = δ(V ) of wave spreading, the phase transition is discontinuous since δ(V ) = 1 for V < V c (ballistic transport), δ(V ) 1/2 at the critical point V = V c (almost diffusive transport), and δ(V ) = 0 in the localized phase V > V c (dynamical localization) [18,21]; see Fig. 1(a).…”

Metal-insulator phase transition in a non-Hermitian Aubry-André-Harper model

“…displaying a long-range periodicity intermediate between ordinary periodic crystals and disordered systems, provide fascinating models to study unusual transport phenomena in a wide variety of classical and quantum systems, ranging from condensed-matter systems to ultracold atoms, photonic and acoustic systems [1][2][3][4][5][6][7]. Quasiperiodicity gives rise to a range of unusual behavior including critical spectra, multifractal eigenstates, localization transitions at a finite modulation of the on-site potential, and mobility edges [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Typical dynamical variables that characterize single-particle transport are the largest propagation speed v of excitation in the lattice, which is bounded (to form a light cone) for short-range hopping according to the Lieb-Robinson bound [28], and the quantum diffusion exponent δ, which measures the asymptotic spreading of wave packet variance σ 2 (t) in the lattice according to the power law σ 2 (t) ∼ t 2δ .…”

“…The Hamiltonian displays a Cantor-set energy spectrum with a phase transition from extended states and absolutely continuous spectrum to exponentially localized states and pure point spectrum as the amplitude V of the on-site quasi-periodic potential is increased above a critical value V c [45,46]. In terms of dynamical behavior of a wave packet, measured by the exponent δ = δ(V ) of wave spreading, the phase transition is discontinuous since δ(V ) = 1 for V < V c (ballistic transport), δ(V ) 1/2 at the critical point V = V c (almost diffusive transport), and δ(V ) = 0 in the localized phase V > V c (dynamical localization) [18,21]; see Fig.1(a). However, in terms of the spreading velocity v = v(V ), defined as v ∼ σ(t)/t, the phase transition turns out to be smooth, with v(V ) continuous function of potential amplitude V and v(V ) = 0 for V ≥ V c [see Fig.1(a)].…”