Spinful Aubry-André model in a magnetic field: Delocalization facilitated by a weak spin-orbit coupling

Abstract

We have incorporated spin-orbit coupling into the Aubry-André model of tight-binding electron motion in the presence of periodic potential with a period incommensurate with lattice constant. This model is known to exhibit an insulator-metal transition upon increasing the hopping amplitude. Without external magnetic field, spin-orbit coupling leads to a simple renormalization of the hopping amplitude. However, when the degeneracy of the on-site energies is lifted by an external magnetic field, the interplay of Zeeman splitting and spin-orbit coupling has a strong effect on the localization length. We studied this interplay numerically by calculating the energy dependence of the Lyapunov exponent in the insulating regime. Numerical results can be unambiguously interpreted in the language of the phase-space trajectories. As a first step, we have explained the plateau in the energy dependence of the localization length in the original Aubry-André model. Our main finding is that a very weak spin-orbit coupling leads to delocalization of states with energies smaller than the Zeeman shift. The origin of the effect is the spin-orbit-induced opening of new transport channels. We have also found that restructuring of the phase-space trajectories, which takes place at certain energies in the insulating regime, causes a singularity in the energy dependence of the localization length. PACS numbers: 73.50.-h, 75.47.-m arXiv:1801.00305v1 [cond-mat.mes-hall]

The Aubry–André model as a hobbyhorse for understanding the localization phenomenon

“…Finally, in this paragraph we want to briefly summarize some of the reported predictions of the Aubry-André model. These include spin-orbit coupling effects [41], closed expressions for the energy separating localized and non-localized states [42] and coexistence of localized and extended states in interacting quasiperiodic systems [43] among others. At the many-body level, localization of the ground state is established rigorously in the weakly interacting regime for both repulsive and attractive interactions [44] and many-body localization versus thermalization and onset of equilibrium [45], which can have implications for quantum devices and quantum computation.…”

Analytical solution of open crystalline linear 1D tight-binding models

“…Concerning possible applications of this method, we highlight some of them: (i) as the groundwork of future studies in commensurate Aubry-André/Harper models [12][13][14][15][16][17][18][19][20][21][22][23], (ii) in the topological characterization of ISSH m models, such as the ISSH 3 [15,18,[44][45][46][47][48], ISSH 4 [41,[49][50][51][52][53] and ISSH 6 [50,54] models, (iii) in studies on quantum state transfer across more complex ISSH m models [55][56][57][58], and (iv) in simplifying the calculation of expectation values of arbitrary operators or interacting matrix elements in many-body problems built on these models [10].…”

“…, m and T defined in (56). In order to express W i n in terms of Chebyshev polynomials U n , we compute W i 1 = T N −1 χ i 1 and W i 2 = T N −2 χ i 2 to find, using the same inductive reasoning followed in (21)(22)(23),…”

“…However, recent progress was made regarding more general crystalline 1D models (or higher-dimensional models with OBC along a single direction) [8][9][10][11] . Here, we detail on how to find the analytical solutions of an entire family of 1D linear models under OBC, namely the extensively studied family of (commensurate) Aubry-André/Harper (AAH) models [12][13][14][15][16][17][18][19][20][21][22][23] , and accordingly general expressions for finding all the eigenstates and corresponding energy spectra are provided here in an easily algorithmizable format.…”

Localisation determines the optimal noise rate for quantum transport

“…However, in general the long-range interactions in 1D systems prevent full Anderson localisation [28,29], and recent work has shown that homogeneous long-range coupling [30] or coupling to cavities [31] can significantly alter 1D responses to disorder in ways beyond the scope of this paper. Recent years have also seen broad interest in the transient effects of dephasing on quantum diffusion, such as stochastic resonance, and many-body localisation, especially focussed on the quasiperiodic Aubry-André model [12,[32][33][34][35][36][37], as well as quantum chaotic systems Coloured areas show ± one standard deviation, each point is averaged from 100 configurations of disorder. An ordered ( ) and disordered ( ) point are highlighted, and their eigenspectra shown in the centre and right panels, respectively.…”