One-dimensional quasicrystals with power-law hopping (1808.03585v2)

Abstract: One-dimensional quasi-periodic systems with power-law hopping, $1/r^a$, differ from both the standard Aubry-Azbel-Harper (AAH) model and from power-law systems with uncorrelated disorder. Whereas in the AAH model all single-particle states undergo a transition from ergodic to localized at a critical quasi-disorder strength, short-range power-law hops with $a>1$ can result in mobility edges. Interestingly, there is no localization for long-range hops with $a\leq 1$, in contrast to the case of uncorrelated disorder. Systems with long-range hops are rather characterized by ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but non ergodic) states. We show that both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.

One-dimensional Quasicrystals with Power-law Hopping: An Insightful Analysis

The paper of quasicrystals bridges the understanding between disordered and periodic systems. These unique structures yield fascinating localization phenomena, particularly when embedded in one-dimensional (1D) regimes with variations in hopping properties. The paper "One-dimensional quasicrystals with power-law hopping" by X. Deng et al., investigates a generalized Aubry-Azbel-Harper (GAAH) model that incorporates power-law hopping, providing novel insights into the behavior of single-particle states (SPS) in quasicrystals.

Overview of the Research

The authors explore 1D quasicrystals where particles exhibit a power-law hopping decay, characterized by a spatial distance parameter $1/r^a$. This model diverges from traditional setups such as the standard Aubry-Azbel-Harper (AAH) model. Unlike the AAH model, characterized by a uniform transition of all SPS from ergodic to localized states at a critical quasi-disorder strength, the GAAH model introduces mobility edges when $a > 1$. This model, therefore, provides a nuanced understanding of SPS behavior as it transitions between ergodic and multifractal states—especially for long-range hops where $a \leq 1$, where no localization occurs.

Key Findings and Implications

Several critical findings highlight the novel transitions and implications of power-law hopping in quasi-periodic systems:

1. Mobility Edges in Short-range Hops: The paper identifies a threshold for $a > 1$, where mobility edges demarcate ergodic and localized states. The system reveals a hierarchy of regimes, $P_s$, characterized by varying fractions of ergodic SPS. These findings propose a pathway for understanding mobility edge phenomena in shallow lattices and systems with next-to-nearest neighbor hopping.
2. Ergodic-to-Multifractal Transitions in Long-range Hops: For $a \leq 1$, while localization is nonexistent, the authors report transitions from ergodic to multifractal phases. These transitions indicate the emergence of extended non-ergodic states, detailed by critical fractal dimensions $D_2$.
3. Experimental Viability: The authors highlight the potential for experimental validation through expansion dynamics, which can manifest both mobility and ergodic-to-multifractal transitions. Such experiments are particularly feasible with systems such as ultracold atoms, polar molecules, and laser-driven ions.

Practical and Theoretical Implications

The practical implications extend to the manipulation and optimization of material properties, where understanding non-ergodic behavior could influence the design of novel electronic and photonic systems. From a theoretical perspective, the research advances the foundational knowledge of quasi-periodic systems, addressing the nature of localization and multifractality in complex multifaceted settings beyond uncorrelated disorder models.

Speculative Future Directions

In exploring future developments, the research sets the stage for further examination of many-body localization phenomena influenced by power-law interactions. Of particular interest is the potential identification of non-ergodic metallic phases, a domain ripe for deeper explorations within condensed matter physics and quantum systems design.

In summary, this paper makes significant strides in characterizing the diverse behavior of SPS in 1D quasicrystals with power-law hopping. By unraveling new ergodic-to-multifractal transitions and mobility edge scenarios, the authors provide a robust framework for advancing both the theoretical and experimental facets of quasicrystal paper.