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arXiv:2011.03666 · 中国科学通报：英文版

Observation of topological phase with critical localization in a quasi-periodic lattice

Teng Xiao, Dizhou Xie, Zhaoli Dong, Tao Chen, Wei Yi, Bo Yan

Published Jul 21, 2021• 82 citations
arXivHTMLPDF
DOI
10.1016/j.scib.2021.07.025

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The topological phase transitions and localization properties in a 1D p-wave superconductor under Fibonacci quasi-periodic potential modulation are investigated in this work. By calculating the Z2 topological invariant, the topological phase diagram of the system is determined numerically. It is found that the system can transition from a topologically trivial phase to a topological Anderson superconductor phase through the Fibonacci quasi-periodic modulation. Moreover, under certain parameters, the system undergoes multiple topological Anderson superconductor phase transitions, accompanied by the emergence of zero-energy modes. However, in the case of strong disorder, the topological Anderson superconductor phase is destroyed, indicating that the topological Anderson superconductor phase can be induced only within a finite range of parameters. Furthermore, by calculating and analyzing the fractal dimension and the mean inverse participation ratio (MIPR) order parameter, the localization properties of the system are analyzed. The results show that regardless of how the disorder intensity increases, the fractal dimension values of most eigenstates always remain within a range of 0–1. Subsequently, the variations in the fractal dimensions of all eigenstates for different system sizes are studied. The results show that the fractal dimension values of most eigenstates are away from 0 and 1. These results indicate that the wavefunction in the bulk of the topological Anderson superconductor phase induced by Fibonacci quasi-periodic potential is a critical state wavefunction, with the system overall being in a critical phase. The stability of the critical phase is confirmed by scale behavior of MIPR as shown in figure. It differs from the traditional topological Anderson superconductor phase induced by random disorder or AA-type quasi-periodic disorder. The results provide new insights into and references for studying topological phase transitions and localization transitions in 1D p-wave superconductors.
Topological state characterization in interacting quasi-periodic systems remains challenging because conventional band-topology approaches become invalid in the presence of strong many-body correlations and broken translational symmetry. This study develops a generalized many-body Zak phase framework for investigating topological phase transitions in Bose–Einstein condensates confined within quasi-periodic potential wells. By introducing a quasimomentum manifold derived from quasi-periodic phase symmetry and constructing gauge-continuous ground-state families, topological invariants are directly extracted from many-body wavefunctions without relying on Bloch bands, Brillouin zones, or bulk energy gaps. The proposed approach is integrated with density matrix renormalization group calculations to establish a unified characterization of superfluid, localized, and critical regimes. Results show that a stable topologically nontrivial phase emerges at U/J = 3.2 and ∆/J = 2.8, where the momentum distribution exhibits symmetric peaks at k = ±π/2 with an amplitude of 1.87. Open-boundary analysis reveals strongly localized edge modes with approximately 90% density concentration near system boundaries, while the entanglement spectrum displays a symmetry-protected doubly degenerate structure. The results demonstrate that interaction and quasi-periodicity jointly stabilize nontrivial topological states and associated wave-localization phenomena. The proposed framework provides a rigorous methodology for analyzing topological phase evolution, geometric-phase transitions, and wave propagation characteristics in strongly correlated quasi-periodic systems.
Abstract The disordered quantum world hosts three fundamental types of states, being extended, localized, and critical. Among various quantum phases formed by their interplay, the most nontrivial is the tripartite phase, wherein all three types of states coexist in the spectrum and are separated by anomalous mobility edges. Here, we report the realization of this exotic phase in a quasi-periodically driven orbital optical lattice with ultracold atoms. The optical lattice with a quasiperiodic Floquet modulation which couples the sss and ppp orbitals is realized in experiment and shown to host the tripartite phase based on exact theory. We establish a novel two-stage protocol to achieve a controlled preparation and detection of the tripartite phase, with all three types of quantum states being observed. The characteristic exponents of these states are determined from expansion dynamics, showing their distinct universal transport properties. Our study paves the way for a comprehensive exploration of the unconventional critical phenomena and localization physics with ultracold atoms.