In this paper we explore the effects of quasiperiodicity in paradigmatic models of Chern insulators. We identify a plethora of topological phase transitions and characterize them based on spectral and localization properties. Contrary to uncorrelated disorder, gap closing and reopening topological transitions can be induced by quasiperiodicity. These can separate widely different phases, including (i) trivial and Chern insulators, both with ballistic states near the gap edges; (ii) Chern insulators with critical states around the gap edges or (iii) Chern and trivial insulators respectively with ballistic and localized gap-edge states. Transition (i) is similar to clean-limit topological transitions due to the ballistic character of the gap-edge states, but at the same time resembles (quasi)disorder driven topological Anderson insulator phenomena. On the other hand, transitions (ii) and (iii) have no clean-limit counterpart. Additionally, quasiperiodicity can also induce topological transitions into a trivial state for which the gap closes and does not reopen, a scenario that resembles more what is observed with uncorrelated disorder. However, we found that such transitions can also be non-conventional in that they can be accompanied by the emergence of intermediate metallic and critical phases where the Chern number is not quantized. Our results show that a rich variety of topological phase transitions, not previously realized experimentally nor predicted theoretically can be attained when applying quasiperiodic modulations to simple Chern insulators. Such models have previously been realized experimentally in widely different platforms, including in optical lattices and photonic or acoustic media, where quasiperiodicity effects can be incorporated. The unveiled topological phase transitions can in principle be observed experimentally with state-of-the-art techniques.
We study topological charge pumping in one-dimensional quasiperiodic systems. Since these systems lack periodicity, we cannot use the conventional approach based on the topological Chern number defined in the momentum space. Here, we develop a general formalism based on a real space picture using the so-called Bott index. We extend the Bott index that was previously used to characterize quantum Hall effects in quasiperiodic systems, and apply it to topological charge pumping in quasiperiodic systems. The Bott index allows us to systematically compute topological indices of charge pumping, rega…
We investigate localization and topological properties of a dimerized Kitaev chain with p-wave superconducting correlations and a quasiperiodically modulated chemical potential. With regard to the localization studies, we demonstrate the existence of distinct phases, such as, the extended phase, the critical (intermediate) phase, and the localized phase that arise due to the competition between the dimerization and the onsite quasiperiodic potential. Most interestingly, the critical phase comprises of two different mobility edges that are found to exist between the extended to the localized ph…
Time-periodic (Floquet) drive can give rise to novel symmetry breaking and topological phases of matter. Recently, we showed that a quintessential Floquet topological phase known as the anomalous Floquet-Anderson insulator is stable to noise on the timing of its Floquet drive. Here, we perturb the anomalous Floquet-Anderson insulator at a single incommensurate frequency, resulting in a quasiperiodic 2-tone drive. Our numerics indicate that a robust topological phase survives at weak noise with topological pumping that is more stable than the case of white noise. Within the topological phase, w…
We investigate the topology and localization of one-dimensional Hermitian and non-Hermitian Su-Schrieffer-Heeger chains with quasiperiodic hopping modulations. In the Hermitian case, phase diagrams are obtained by numerically and analytically calculating various topological and localization characters. We show the presence of topological extended, intermediate, and localized phases due to the coexistence of independent topological and localization phase transitions driven by the quasiperiodic disorder. Unlike the gapless and localized TAI phase in one-dimensional random disordered systems, we …
We derive a topological classification of the steady states of $d$-dimensional lattice models driven by $D$ incommensurate tones. Mapping to a unifying $(d+D)$-dimensional localized model in frequency space reveals anomalous localized topological phases (ALTPs) with no static analog. While the formal classification is determined by $d+D$, the observable signatures of each ALTP depend on the spatial dimension $d$. For each $d$, with $d+D=3$, we identify a quantized circulating current, and corresponding topological edge states. The edge states for a driven wire ($d=1$) function as a quantized, …
Topological Anderson insulator and reentrant topological transitions in a mosaic trimer lattice
Institute of Theoretical Physics, State Key Laboratory of Quantum Optics Technologies and Devices, Collaborative Innovation Center of Extreme Optics, Shanxi University（山西大学理论物理研究所、量子光学技术与器件国家重点实验室、极端光学协同创新中心）；Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所、凝聚态物理国家实验室）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
The effects of downfolding a Brillouin zone can open gaps and quench the kinetic energy by flattening bands. Quasiperiodic systems are extreme examples of this process, which leads to new phases and critical eigenstates. We analytically and numerically investigate these effects in a two dimensional topological insulator with a quasiperiodic potential and discover a complex phase diagram. We study the nature of the resulting eigenstate quantum phase transitions; a quasiperiodic potential can make a trivial insulator topological and induce topological insulator-to-metal phase transitions through…
We uncover the relationship of topology and disorder in a one-dimensional Su-Schrieffer-Heeger chain subjected to a slowly varying quasi-periodic modulation. By numerically calculating the disorder-averaged winding number and analytically studying the localization length of the zero modes, we obtain the topological phase diagram, which implies that the topological Anderson insulator (TAI) can be induced by a slowly varying quasi-periodic modulation. Moreover, unlike the localization properties in the TAI phase caused by random disorder, mobility edges can enter into the TAI region identified b…
A number of moiré graphene systems have nearly flat topological bands where electron motion is strongly correlated. Though microscopically these systems are only quasiperiodic, they can typically be treated as translation invariant to an excellent approximation. Here we reconsider this question for magic angle twisted bilayer graphene that is nearly aligned with a hexagonal boron nitride(h-BN) substrate. We carefully study the effect of the periodic potential induced by h-BN on the low energy physics. The combination of this potential and the moiré lattice produced by the twisted graphene gene…
We investigate the localization and topological transitions in a one-dimensional (interacting) non-Hermitian quasiperiodic lattice, which is described by a generalized Aubry-André-Harper model with irrational modulations in the off-diagonal hopping and on-site potential and with non-Hermiticities from the nonreciprocal hopping and complex potential phase. For noninteracting cases, we reveal that the nonreciprocal hopping (the complex potential phase) can enlarge the delocalization (localization) region in the phase diagrams spanned by two quasiperiodical modulation strengths. We show that the …
We study a one-dimensional $p$-wave superconductor subject to non-Hermitian quasiperiodic potentials. Although the existence of the non-Hermiticity, the Majorana zero mode is still robust against the disorder perturbation. The analytic topological phase boundary is verified by calculating the energy gap closing point and the topological invariant. Furthermore, we investigate the localized properties of this model, revealing that the topological phase transition is accompanied with the Anderson localization phase transition, and a wide critical phase emerges with amplitude increments of the non…
We construct an example of a 1$d$ quasiperiodically driven spin chain whose edge states can coherently store quantum information, protected by a combination of localization, dynamics, and topology. Unlike analogous behavior in static and periodically driven (Floquet) spin chains, this model does not rely upon microscopic symmetry protection: Instead, the edge states are protected purely by emergent dynamical symmetries. We explore the dynamical signatures of this Emergent Dynamical Symmetry-Protected Topological (EDSPT) order through exact numerics, time evolving block decimation, and analytic…
We aim to study a one-dimensional $p$-wave superconductor with quasiperiodic on-site potentials. A modified real-space-Pfaffian method is applied to calculate the topological invariants. We confirm that the Majorana zero mode is protected by the nontrivial topology the topological phase transition is accompanied by the energy gap closing and reopening. In addition, we numerically find that there are mobility edges which originate from the competition between the extended $p$-wave pairing and the localized quasi-disorder. We qualitatively analyze the influence of superconducting pairing paramet…
Emergence of non-trivial phases in interacting non-Hermitian quasiperiodic chains with power-law hopping
Department of Physics and Astronomy, National Institute of Technology, Rourkela（鲁尔基国家技术大学物理与天文系）；School of Physical Sciences, National Institute of Science Education and Research（科学教育与研究国立研究院物理科学学院）；Homi Bhabha National Institute（霍米·巴巴国立研究所）
We study the interplay of superconductivity and a wide spectrum of critical (multifractal) wave functions ("spectrum-wide quantum criticality," SWQC) in the one-dimensional Aubry-André and power-law random-banded matrix models with attractive interactions, using self-consistent BCS theory. We find that SWQC survives the incorporation of attractive interactions at the Anderson localization transition, while the pairing amplitude is maximized near this transition in both models. Our results suggest that SWQC, recently discovered in two-dimensional topological surface-state and nodal superconduct…
From Bellissard's gap labeling to atomistic quasiperiodic bilayers
Center for Advanced Systems Understanding（高级系统理解中心）；Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿-罗斯多夫亥姆霍兹中心）；Institute of Artificial Intelligence, Technische Universität Dresden（德累斯顿工业大学人工智能研究所）；Department of Physics, Yeshiva University（叶史瓦大学物理系）
We introduce a one-dimensional lattice model whose hopping amplitudes are modulated for equally spaced sites. Such mosaic lattice exhibits many interesting topological and localization phenomena that do not exist in the regular off-diagonal lattices. When the mosaic modulation is commensurate with the underlying lattice, topologically nontrivial phases with zero- and nonzero-energy edge modes are observed as we tune the modulation, where the nontrivial regimes are characterized by quantized Berry phases. If the mosaic lattice becomes incommensurate, Anderson localization will be induced purely…
In the present study, the interplay among interaction, topology, quasiperiodicity, and non-Hermiticity is studied. The hard-core bosons model on a one-dimensional lattice with asymmetry hoppings and quasiperiodic onsite potentials is selected. This model, which preserves time-reversal symmetry (TRS), will exhibit three types of phase transition: real-complex transition of eigenenergies, topological phase transition and many-body localization (MBL) phase transition. For thereal-complex transition, it is found that the imaginary parts of the eigenenergies are always suppressed by the MBL. Moreov…
We study topological superconductor in one-dimensional (1D) mosaic lattice whose on-site potentials are modulated for equally spaced sites. When the system is topologically nontrivial, Majorana zero modes appear at the two ends of the 1D lattice. By calculating energy spectra and topological invariant of the system, we find the interval of the mosaic modulation of the on-site potential, whether it is periodic, quasiperiodic, or randomly distributed, can influence the topological properties significantly. For even interval of the mosaic potential, the system will always exist in the topological…
We construct and solve a two-dimensional, chirally symmetric model of Dirac cones subjected to a quasiperiodic modulation. In real space, this is realized with a quasiperiodic hopping term. This hopping model, as we show, at the Dirac node energy has a rich phase diagram with a semimetal-to-metal phase transition at intermediate amplitude of the quasiperiodic modulation, and a transition to a phase with a diverging density of states and sub-diffusive transport when the quasiperiodic hopping is strongest. We further demonstrate that the semimetal-to-metal phase transition can be characterized b…
Non-Hermiticity significantly enriches the properties of topological models, leading to exotic features such as the non-Hermitian skin effects and non-Bloch bulk-boundary correspondence that have no counterparts in Hermitian settings. Its impact is particularly illustrating in non-Hermitian quasicrystals where the interplay between non-Hermiticity and quasiperiodicity results in the concurrence of the delocalization-localization transition, the parity-time (PT)-symmetry breaking, and the onset of the non-Hermitian skin effects. Here we experimentally simulate non-Hermitian quasicrystals using …
Nascent platforms for programmable quantum simulation offer unprecedented access to new regimes of far-from-equilibrium quantum many-body dynamics in (approximately) isolated systems. Here, achieving precise control over quantum many-body entanglement is an essential task for quantum sensing and computation. Extensive theoretical work suggests that these capabilities can enable dynamical phases and critical phenomena that exhibit topologically-robust methods to create, protect, and manipulate quantum entanglement that self-correct against large classes of errors. However, to date, experimental…
Topological metal-insulator transitions in one-dimensional non-Hermitian quasicrystals: beyond PT-symmetry
College of Physics and Optoelectronic Engineering, Ocean University of China（中国海洋大学物理与光电工程学院）；Qingdao Key Laboratory of Advanced Optoelectronics（青岛先进光电子重点实验室）；Engineering Research Center of Advanced Marine Physical Instruments and Equipment of MOE（教育部海洋高端装备工程研究中心）；Department of Physics, National University of Singapore（新加坡国立大学物理系）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Unit（马久实验室，CNRS-南法大学-苏黎世联邦理工学院-新加坡国立大学-南洋理工大学国际联合研究单位）；Center for Phononics and Thermal Energy Science, China-EU Joint Lab on Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University（同济大学物理科学与技术学院，声子学与热能科学中心，中欧纳米声子学联合实验室，上海市特殊人工微结构材料与技术重点实验室）
We study non-Hermitian Aubry-André-Harper models with p-wave pairing, where the non-Hermiticity is introduced by on-site complex quasiperiodic potentials. By analysing the $\mathcal{PT}$ symmetry breaking, winding numbers of energy spectra, localization and fractal dimensions of states, and fate of Majorana fermions, a complete phase diagram on Anderson localization and topological phase transitions is obtained. In particular, the non-Hermitian topological nature of Anderson localization phase transitions from extended to critical and then to localized phases is identified, using both analytic…
The emergence of the mobility edge (ME) has been recognized as an important characteristic of Anderson localization. The difficulty in understanding the physics of the MEs in three-dimensional (3D) systems from a microscopic image encourages the development of models in lower-dimensional systems that have exact MEs. While most of the previous studies are concerned with one-dimensional (1D) quasiperiodic systems, the analytic results that allow for an accurate understanding of two-dimensional (2D) cases are rare. In this work, we disclose an exactly solvable 2D quasicrystal model with parity-ti…
Recent progress on disorder-induced topological phases
Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of Education), Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, South China Normal University（原子与亚原子结构及量子控制重点实验室（教育部），广东省物质结构与基本相互作用卓越研究中心，华南师范大学）；Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research Institute for Physics, School of Physics, South China Normal University（广东-香港量子物质联合实验室，前沿物理研究院，物理学院，华南师范大学）；Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area (Guangdong)（粤港澳大湾区量子科学中心（广东））
We study a non-Hermitian Aubry-André-Harper model with both nonreciprocal hoppings and complex quasiperiodical potentials, which is a typical non-Hermitian quasicrystal. We introduce boundary-dependent self-dualities in this model and obtain analytical results to describe its Asymmetrical Anderson localization and topological phase transitions. We find that the Anderson localization is not necessarily in accordance with the topological phase transitions, which are characteristics of localization of states and topology of energy spectrum respectively. Furthermore, in the localized phase, single…
Non-Hermitian effects could trigger spectrum, localization and topological phase transitions in quasiperiodic lattices. We propose a non-Hermitian extension of the Maryland model, which forms a paradigm in the study of localization and quantum chaos by introducing asymmetry to its hopping amplitudes. The resulting nonreciprocal Maryland model is found to possess a real-to-complex spectrum transition at a finite amount of hopping asymmetry, through which it changes from a localized phase to a mobility edge phase. Explicit expressions of the complex energy dispersions, phase boundaries and mobil…
We introduce the one-dimensional quasireciprocal lattices where the forward hopping amplitudes between nearest neighboring sites $\{ t+t_{jR} \}$ are chosen to be a random permutation of the backward hopping $\{ t+t_{jL} \}$ or vice versa. The values of $\{ t_{jL} \}$ (or $\{t_{jR} \}$) can be periodic, quasiperiodic, or randomly distributed. We show that the Hamiltonian matrices are pseudo-Hermitian and the energy spectra are real as long as $\{ t_{jL} \}$ (or $\{t_{jR} \}$) are smaller than the threshold value. While the non-Hermitian skin effect is always absent in the eigenstates due to th…
Quasiperiodically driven fermionic systems can support topological phases not realized in equilibrium. The fermions are localized in the bulk, but support quantized energy currents at the edge. These phases were discovered through an abstract classification, and few microscopic models exist. We develop a coupled layer construction for tight-binding models of these phases in $d\in\{1,2\}$ spatial dimensions, with any number of incommensurate drive frequencies $D$. The models exhibit quantized responses associated with synthetic two- and four-dimensional quantum Hall effects in the steady state.…
