There has been a revival of interest in localization phenomena in quasiperiodic systems with a view to examining how they differ fundamentally from such phenomena in random systems. Motivated by this, we study transport in the quasiperiodic, one-dimentional (1d) Aubry-Andre model and its generalizations to 2d and 3d. We study the conductance of open systems, connected to leads, as well as the Thouless conductance, which measures the response of a closed system to boundary perturbations. We find that these conductances show signatures of a metal-insulator transition from an insulator, with localized states, to a metal, with extended states having (a) ballistic transport (1d), (b) superdiffusive transport (2d), or (c) diffusive transport (3d); precisely at the transition, the system displays sub-diffusive critical states. We calculate the beta function β(g) = d ln(g)/d ln (L) and show that, in 1d and 2d, single-parameter scaling is unable to describe the transition. Furthermore, the conductances show strong non-monotonic variations with L and an intricate structure of resonant peaks and subpeaks. In 1d the positions of these peaks can be related precisely to the properties of the number that characterizes the quasiperiodicity of the potential; and the L-dependence of the Thouless conductance is multifractal. We find that, as d increases, this non-monotonic dependence of g on L decreases and, in 3d, our results for β(g) are reasonably well approximated by single-parameter scaling.The single-parameter scaling theory of Abrahams, et al., [1] has played an important part in our understanding of Anderson localization and metal-insulator transitions in disordered systems, e.g., non-interacting electrons in a random potential [2]. Localization phenomena are, however, not only restricted to random systems, but also occur in other systems, the most prominent examples being systems with quasiperiodic potentials [3][4][5][6][7][8][9][10][11].Recently such quasiperiodic systems have attracted a lot of attention because of the experimental observation of many-body localization (MBL) in quasiperiodic lattices of cold atoms [12]. These have brought back into focus the need to examine the essential similarities and differences between random and quasiperiodic systems at the level of eigenstates [3][4][5][6][7][8][9][10][11], dynamics [13][14][15], and universality classes of localization-delocalization transitions [16]. It has also been argued [16] that quasiperiodic systems provide more robust realizations of Many Body Localization (MBL) than their random counterparts because the former do not have rare regions, which are locally thermal. Therefore, we may find a stable MBL phase in dimension d > 1 in a quasiperiodic system, but not in a random system, where the MBL phase may be destabilised because of such rare regions [17,18]. Non-interactingquasiperiodic systems exhibit delocalization-localization transitions even in one dimension (1d), unlike random systems in which all states are localized in dimensions d = 1 and 2 for ort...
We implement a recursive Green's function method to extract the Fock space (FS) propagator and associated self-energy across the many-body localization (MBL) transition, for one-dimensional interacting fermions in a random on-site potential. We show that the typical value of the imaginary part of the local FS self-energy, t , related to the decay rate of an initially localized state, acts as a probabilistic order parameter for the thermal to MBL phase transition and can be used to characterize critical properties of the transition as well as the multifractal nature of MBL states as a function of disorder strength W . In particular, we show that a fractal dimension D s extracted from t jumps discontinuously across the transition, from D s < 1 in the MBL phase to D s = 1 in the thermal phase. Moreover, t follows an asymmetrical finite-size scaling form across the thermal-MBL transition, where a nonergodic volume in the thermal phase diverges with a Kosterlitz-Thouless-like essential singularity at the critical point W c and controls the continuous vanishing of t as W c is approached. In contrast, a correlation length (ξ ) extracted from t exhibits a power-law divergence on approaching W c from the MBL phase.
We provide real-space and Fock-space (FS) characterizations of ergodic, nonergodic extended (NEE) and many-body localized (MBL) phases in an interacting quasiperiodic system, namely, the generalized Aubry-André-Harper model, which possesses a mobility edge in the noninteracting limit. We show that a mobility edge in the single-particle (SP) excitations survives even in the presence of interaction in the NEE phase. In contrast, all single-particle excitations get localized in the MBL phase due to the MBL proximity effect. We give complementary insights into the distinction of the NEE states from the ergodic and MBL states by computing local FS self-energies and decay length associated, respectively, with the local and the nonlocal FS propagators. Based on a finite-size scaling analysis of the typical local self-energy across the NEE to ergodic transition, we show that MBL and NEE states exhibit qualitatively similar multifractal character. However, we find that the NEE and MBL states can be distinguished in terms of the distribution of local self-energy and the decay of the nonlocal propagator in the FS, whereas the typical local FS self-energy cannot tell them apart.
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