Quantum Spin Hall Effect and Spin Bott Index in a Quasicrystal Lattice

Observation of dispersive acoustic quasicrystals

Moiré quasicrystals, formed by stacking periodic structures with a relative twist between them, exhibit many exotic phenomena. Their quasiperiodicity leads to effects such as light localization-delocalization transitions, superconductivity, topological states, and quasiband dispersion. However, weak interlayer interactions, the scalar nature of acoustic fields, and longer wavelengths severely limit the demonstration of these phenomena in acoustics. Here, we report an acoustic moiré quasicrystal that not only achieves a localization-delocalization transition, but also enables wave propagation shifting from diffusion to canalization or localization as a function of the quasicrystal geometry. Unlike conventional two-dimensional materials, the designed sublattice provides tailorable anisotropy and spatial broken symmetry, allowing quasicrystal structures to exhibit reconfigurable nontrivial dispersion. Furthermore, by introducing a uniform tilt angle in the unit cells breaks the spatial symmetry of the moiré quasicrystal, resulting in partial attenuation and disappearance of the wave within the localization pattern. Our findings pave a new avenue for controlling the properties of acoustic wave patterns, and benefit potential applications in energy transfer, subwavelength wave propagation, and highly sensitive sensors.

Distinguishing local isomorphism classes in quasicrystals by high-order harmonic spectroscopy

Electron diffraction spectroscopy is a fundamental tool for investigating quasicrystal structures, which unveils the quasiperiodic long-range order. Nevertheless, it falls short in effectively distinguishing separate local isomorphism classes. This is a long outstanding problem. Here, we study the high-order harmonic generation in two-dimensional generalized Penrose quasicrystals to optically resolve different local isomorphism classes. The results reveal that: (i) harmonic spectra from different parts of a quasicrystal are identical, even though their atomic arrangements vary significantly. (ii) The harmonic yields of diverse local isomorphism classes exhibit variations, providing a way to distinguish local isomorphism classes. (iii) The rotational symmetry of harmonic yield can serve as a characteristic of quasicrystal harmonics and is consistent with the orientation order. Our results not only pave the way for confirming the experimental reproducibility of quasicrystal harmonics and identifying quasicrystal local isomorphism classes, but also shed light on comprehending electron dynamics influenced by the vertex environments.

Nematic Superconductivity and Its Critical Vestigial Phases in the Quasicrystal

We propose a general mechanism to realize nematic superconductivity (SC) and reveal its exotic vestigial phases in the quasicrystal (QC). Starting from a Penrose-Hubbard model, our microscopic studies suggest that the Kohn-Luttinger mechanism driven SC in the QC is usually gapless due to violation of Anderson's theorem, rendering that both chiral and nematic SCs are common. The nematic SC in the QC can support novel vestigial phases driven by pairing phase fluctuations above its T_{c}. Our combined renormalization group and Monte Carlo studies provide a phase diagram in which, besides the conventional charge-4e SC, two critical vestigial phases emerge, i.e., the quasinematic (QN) SC and QN metal. In the two QN phases, discrete lattice rotation symmetry is counterintuitively "quasibroken" with power-law decaying orientation correlation. They separate the phase diagram into various phases connected via Berezinskii-Kosterlitz-Thouless (BKT) transitions. These remarkable critical vestigial phases, which resemble the intermediate BKT phase in the q state (q≥5) clock model, are a consequence of the fivefold (or higher) anisotropy field brought about by the unique QC symmetry, which are absent in conventional crystalline materials.

Realization of Time-Reversal Invariant Photonic Topological Anderson Insulators

Disorder, which is ubiquitous in nature, has been extensively explored in photonics for understanding the fundamental principles of light diffusion and localization, as well as for applications in functional resonators and random lasers. Recently, the investigation of disorder in topological photonics has led to the realization of topological Anderson insulators characterized by an unexpected disorder-induced phase transition. However, the observed photonic topological Anderson insulators so far are limited to the time-reversal symmetry breaking systems. Here, we propose and realize a photonic quantum spin Hall topological Anderson insulator without breaking time-reversal symmetry. The disorder-induced topological phase transition is comprehensively confirmed through the theoretical effective Dirac Hamiltonian, numerical analysis of bulk transmission, and experimental examination of bulk and edge transmissions. We present convincing evidence for the unidirectional propagation and robust transport of helical edge modes, which are the key features of nontrivial time-reversal invariant topological Anderson insulators. Furthermore, we demonstrate disorder-induced beam steering, highlighting the potential of disorder as a new degree of freedom to manipulate light propagation in magnetic-free systems. Our work not only paves the way for observing unique topological photonic phases but also suggests potential device applications through the utilization of disorder.

Higher-order topological phases in crystalline and non-crystalline systems: a review

In recent years, higher-order topological phases have attracted great interest in various fields of physics. These phases have protected boundary states at lower-dimensional boundaries than the conventional first-order topological phases due to the higher-order bulk-boundary correspondence. In this review, we summarize current research progress on higher-order topological phases in both crystalline and non-crystalline systems. We firstly introduce prototypical models of higher-order topological phases in crystals and their topological characterizations. We then discuss effects of quenched disorder on higher-order topology and demonstrate disorder-induced higher-order topological insulators. We also review the theoretical studies on higher-order topological insulators in amorphous systems without any crystalline symmetry and higher-order topological phases in non-periodic lattices including quasicrystals, hyperbolic lattices, and fractals, which have no crystalline counterparts. We conclude the review by a summary of experimental realizations of higher-order topological phases and discussions on potential directions for future study.

Stacking Order Engineering of Two-Dimensional Materials and Device Applications

Stacking orders in 2D van der Waals (vdW) materials dictate the relative sliding (lateral displacement) and twisting (rotation) between atomically thin layers. By altering the stacking order, many new ferroic, strongly correlated and topological orderings emerge with exotic electrical, optical and magnetic properties. Thanks to the weak vdW interlayer bonding, such highly flexible and energy-efficient stacking order engineering has transformed the design of quantum properties in 2D vdW materials, unleashing the potential for miniaturized high-performance device applications in electronics, spintronics, photonics, and surface chemistry. This Review provides a comprehensive overview of stacking order engineering in 2D vdW materials and their device applications, ranging from the typical fabrication and characterization methods to the novel physical properties and the emergent slidetronics and twistronics device prototyping. The main emphasis is on the critical role of stacking orders affecting the interlayer charge transfer, orbital coupling and flat band formation for the design of innovative materials with on-demand quantum properties and surface potentials. By demonstrating a correlation between the stacking configurations and device functionality, we highlight their implications for next-generation electronic, photonic and chemical energy conversion devices. We conclude with our perspective of this exciting field including challenges and opportunities for future stacking order engineering research.

Physical Properties of an Aperiodic Monotile with Graphene-like Features, Chirality, and Zero Modes

The discovery of the Hat, an aperiodic monotile, has revealed novel mathematical aspects of aperiodic tilings. However, the physics of particles propagating in such a setting remains unexplored. In this work we study spectral and transport properties of a tight-binding model defined on the Hat. We find that (i) the spectral function displays striking similarities to that of graphene, including sixfold symmetry and Dirac-like features; (ii) unlike graphene, the monotile spectral function is chiral, differing for its two enantiomers; (iii) the spectrum has a macroscopic number of degenerate states at zero energy; (iv) when the magnetic flux per plaquette (ϕ) is half of the flux quantum, zero modes are found localized around the reflected "anti-hats"; and (v) its Hofstadter spectrum is periodic in ϕ, unlike for other quasicrystals. Our work serves as a basis to study wave and electron propagation in possible experimental realizations of the Hat, which we suggest.

Room Temperature Spin-to-Charge Conversion in Amorphous Topological Insulating Gd-Alloyed BixSe1-x/CoFeB Bilayers

Disordered topological insulator (TI) films have gained intense interest by benefiting from both the TI's exotic transport properties and the advantage of mass production by sputtering. Here, we report on the clear evidence of spin-charge conversion (SCC) in amorphous Gd-alloyed BixSe1-x (BSG)/CoFeB bilayers fabricated by sputtering, which could be related to the amorphous TI surface states. Two methods have been employed to study SCC in BSG (tBSG = 6-16 nm)/CoFeB(5 nm) bilayers with different BSG thicknesses. First, spin pumping is used to generate a spin current in CoFeB and detect SCC by the inverse Edelstein effect (IEE). The maximum SCC efficiency (SCE) is measured to be as large as 0.035 nm (IEE length λIEE) in a 6 nm thick BSG sample, which shows a strong decay when tBSG increases due to the increase of BSG surface roughness. The second method is THz time-domain spectroscopy, which reveals a small tBSG dependence of SCE, validating the occurrence of a pure interface state-related SCC. Furthermore, our angle-resolved photoemission spectroscopy data show dispersive two-dimensional surface states that cross the bulk gap until the Fermi level, strengthening the possibility of SCC due to the amorphous TI states. Our studies provide a new experimental direction toward the search for topological systems in amorphous solids.

Superconductivity and strong interactions in a tunable moiré quasicrystal

Electronic states in quasicrystals generally preclude a Bloch description1, rendering them fascinating and enigmatic. Owing to their complexity and scarcity, quasicrystals are underexplored relative to periodic and amorphous structures. Here we introduce a new type of highly tunable quasicrystal easily assembled from periodic components. By twisting three layers of graphene with two different twist angles, we form two mutually incommensurate moiré patterns. In contrast to many common atomic-scale quasicrystals2,3, the quasiperiodicity in our system is defined on moiré length scales of several nanometres. This 'moiré quasicrystal' allows us to tune the chemical potential and thus the electronic system between a periodic-like regime at low energies and a strongly quasiperiodic regime at higher energies, the latter hosting a large density of weakly dispersing states. Notably, in the quasiperiodic regime, we observe superconductivity near a flavour-symmetry-breaking phase transition4,5, the latter indicative of the important role that electronic interactions play in that regime. The prevalence of interacting phenomena in future systems with in situ tunability is not only useful for the study of quasiperiodic systems but may also provide insights into electronic ordering in related periodic moiré crystals6-12. We anticipate that extending this platform to engineer quasicrystals by varying the number of layers and twist angles, and by using different two-dimensional components, will lead to a new family of quantum materials to investigate the properties of strongly interacting quasicrystals.

Disordered topological graphs enhancing nonlinear phenomena

Complex networks play a fundamental role in understanding phenomena from the collective behavior of spins, neural networks, and power grids to the spread of diseases. Topological phenomena in such networks have recently been exploited to preserve the response of systems in the presence of disorder. We propose and demonstrate topological structurally disordered systems with a modal structure that enhances nonlinear phenomena in the topological channels by inhibiting the ultrafast leakage of energy from edge modes to bulk modes. We present the construction of the graph and show that its dynamics enhances the topologically protected photon pair generation rate by an order of magnitude. Disordered nonlinear topological graphs will enable advanced quantum interconnects, efficient nonlinear sources, and light-based information processing for artificial intelligence.

Optical absorption measurement of spin Berry curvature and spin Chern marker

In two-dimensional time-reversal symmetric topological insulators described by Dirac models, theZ2topological invariant can be described by the spin Chern number. We present a linear response theory for the spin Berry curvature that integrates to the spin Chern number, and introduce its spectral function that can be measured at finite temperature by momentum- and spin-resolved circular dichroism, which may be achieved by pump-probe type of experiments using spin- and time-resolved ARPES. As a result, the sign of the Pfaffian of theZ2invariant can be directly measured. A spin Chern number spectral function is further introduced from the optical spin current response, and is shown to be measurable from the spin-resolved opacity of two-dimensional materials under circularly polarized light, pointing to an optical measurement of theZ2invariant and a frequency sum rule. The spin Chern number expressed in real space is known to yield a spin Chern marker, and we propose that it may be measurable from spin-resolved local heating rate caused by circularly polarized light. A nonlocal spin Chern marker is further proposed to characterize the quantum criticality near topological phase transitions, and is shown to be equivalent to an overlap between spin-selected Wannier states.

Local Topological Markers in Odd Spatial Dimensions and Their Application to Amorphous Topological Matter

Local topological markers, topological invariants evaluated by local expectation values, are valuable for characterizing topological phases in materials lacking translation invariance. The Chern marker-the Chern number expressed in terms of the Fourier transformed Chern character-is an easily applicable local marker in even dimensions, but there are no analogous expressions for odd dimensions. We provide general analytic expressions for local markers for free-fermion topological states in odd dimensions protected by local symmetries: a Chiral marker, a local Z marker which in case of translation invariance is equivalent to the chiral winding number, and a Chern-Simons marker, a local Z_{2} marker characterizing all nonchiral phases in odd dimensions. We achieve this by introducing a one-parameter family P_{ϑ} of single-particle density matrices interpolating between a trivial state and the state of interest. By interpreting the parameter ϑ as an additional dimension, we calculate the Chern marker for the family P_{ϑ}. We demonstrate the practical use of these markers by characterizing the topological phases of two amorphous Hamiltonians in three dimensions: a topological superconductor (Z classification) and a topological insulator (Z_{2} classification).

The generalized method to calculate the real-space winding number for one-dimensional systems with complex multi-band-gap structure

The topological study of the complicated one-dimensional (1D) systems with multi-band-gap structures, including quasi-crystals (QCs), is very hard since the lack of effective topological invariants to describe the non-triviality of gaps. A generalized method, based on the contracted wave-function, is proposed in this work to calculate the real-space winding number for the complicated 1D systems with multi-band-gap structures. First, the effectiveness of the generalized method is demonstrated to obtain the quantized real-space winding number for the gaps and correctly predict the topological phase transition and the existing fractional charge on the edges for the periodic 4-atoms SSH model (4A-SSH model). Then, we apply the generalized method to more complicated 1D Thue-Morse (TM) systems, which is one kind of QCs. The quantized real-space winding number is obtained for two traditional gaps and two fractal gaps for the TM systems and can also correctly predict the existence of topological edge-states and fractional charge on the ends. Several new phenomena are observed, e.g. the topological phase transition and the edge-states for the gaps in multi-band-gap structures, the1/4fractional charge for the 4A-SSH model, the fluctuation of local charge and the asymmetric (but still with a quantized difference) fractional charge at the ends of TM system. The generalized method could be a powerful tool to study the topology of gaps in the complicated periodic systems or QCs.

Effective Model for Fractional Topological Corner Modes in Quasicrystals

High-order topological insulators (HOTIs), as generalized from topological crystalline insulators, are characterized with lower-dimensional metallic boundary states protected by spatial symmetries of a crystal, whose theoretical framework based on band inversion at special k points cannot be readily extended to quasicrystals because quasicrystals contain rotational symmetries that are not compatible with crystals, and momentum is no longer a good quantum number. Here, we develop a low-energy effective model underlying HOTI states in 2D quasicrystals for all possible rotational symmetries. By implementing a novel Fourier transform developed recently for quasicrystals and approximating the long-wavelength behavior by their large-scale average, we construct an effective k·p Hamiltonian to capture the band inversion at the center of a pseudo-Brillouin zone. We show that an in-plane Zeeman field can induce mass kinks at the intersection of adjacent edges of a 2D quasicrystal topological insulators and generate corner modes (CMs) with fractional charge, protected by rotational symmetries. Our model predictions are confirmed by numerical tight-binding calculations. Furthermore, when the quasicrystal is proximitized by an s-wave superconductor, Majorana CMs can also be created by tuning the field strength and chemical potential. Our work affords a generic approach to studying the low-energy physics of quasicrystals, in association with topological excitations and fractional statistics.

Photonic Z_{2} Topological Anderson Insulators

That disorder can induce nontrivial topology is a surprising discovery in topological physics. As a typical example, Chern topological Anderson insulators (TAIs) have been realized in photonic systems, where the topological phases exist without symmetry protection. In this Letter, by taking transverse magnetic and transverse electric polarizations as pseudospin degrees of freedom, we theoretically propose a scheme to realize disorder-induced symmetry-protected topological phase transitions in two-dimensional photonic crystals with a combined time-reversal, mirror, and duality symmetry T_{f}=TM_{z}D. In particular, we demonstrate that the disorder-induced symmetry-protected topological phase persists even without pseudospin conservation, thereby realizing a photonic Z_{2} TAI, in contrast to a Z-classified quantum spin Hall (QSH) TAI with decoupled spins. By formulating a new scattering approach, we show that the topology of both the QSH and Z_{2} TAIs can be manifested by the accumulated spin rotations of the reflected waves from the photonic crystals. Using a transmission structure, we also illustrate the trivialization of a disordered QSH phase with an even integer topological index caused by spin coupling.

Disorder, Low-Energy Excitations, and Topology in the Kitaev Spin Liquid

The Kitaev model is a fascinating example of an exactly solvable model displaying a spin-liquid ground state in two dimensions. However, deviations from the original Kitaev model are expected to appear in real materials. In this Letter, we investigate the fate of Kitaev's spin liquid in the presence of disorder-bond defects or vacancies-for an extended version of the model. Considering static flux backgrounds, we observe a power-law divergence in the low-energy limit of the density of states with a nonuniversal exponent. We link this power-law distribution of energy scales to weakly coupled droplets inside the bulk, in an uncanny similarity to the Griffiths phase often present in the vicinity of disordered quantum phase transitions. If time-reversal symmetry is broken, we find that power-law singularities are tied to the destruction of the topological phase of the Kitaev model in the presence of bond disorder alone. However, there is a transition from this topologically trivial phase with power-law singularities to a topologically nontrivial one for weak to moderate site dilution. Therefore, diluted Kitaev materials are potential candidates to host Kitaev's chiral spin-liquid phase.

Inducing a topological transition in graphene nanoribbon superlattices by external strain

Armchair graphene nanoribbons, when forming a superlattice, can be classified into different topological phases, with or without edge states. By means of tight-binding and classical molecular dynamics (MD) simulations, we studied the electronic and mechanical properties of some of these superlattices. MD shows that fracture in modulated superlattices is brittle, as for unmodulated ribbons, and occurs at the thinner regions, with staggered superlattices achieving a larger fracture strain than inline superlattices. We found a general mechanism to induce a topological transition with strain, related to the electronic properties of each segment of the superlattice, and by studying the sublattice polarization we were able to characterize the transition and the response of these states to the strain. For the cases studied in detail here, the topological transition occurred at ∼3-5% strain, well below the fracture strain. The topological states of the superlattice - if present - are robust to strain even close to fracture. The topological transition was characterized by means of the sublattice polarization of the states.

Structural Amorphization-Induced Topological Order

Electronic properties of crystals are inherently pertained to crystalline symmetry, so that amorphization that lowers and breaks symmetry is detrimental. One important crystalline property is electron band topology which is known to be weakened and destroyed by structural disorder. Here, we report a counterintuitive theoretical discovery that atomic structural disorder by amorphization can in fact induce electronic order of topology in an otherwise topologically trivial crystal. The resulting nontrivial topology is characterized by a nonzero spin Bott index, associated with robust topological edge states and quantized conductance. The underlying topological phase transition (TPT) from a trivial crystal to a topological amorphous is analyzed by mapping out a phase diagram in the degree of structural disorder using an effective medium theory. The atomic disorder is revealed to induce topological order by renormalizing the spectral gap toward nontriviality near the phase boundary. As a concrete example, we further show such TPT in amorphous stanane by first-principles calculations. Our findings point to possible observation of an electronic ordering transition accompanied by a structural disorder transition.

Topological Anderson Insulator in Cation-Disordered Cu2ZnSnS4

We present the first candidate for the realization of a disorder-induced Topological Anderson Insulator in a real material system. High-energy reactive mechanical alloying produces a polymorph of Cu₂ZnSnS₄ with high cation disorder. Density functional theory calculations show an inverted ordering of bands at the Brillouin zone center for this polymorph, which is in contrast to its ordered phase. Adiabatic continuity arguments establish that this disordered Cu₂ZnSnS₄ can be connected to the closely related Cu₂ZnSnSe₄, which was previously predicted to be a 3D topological insulator, while band structure calculations with a slab geometry reveal the presence of robust surface states. This evidence makes a strong case in favor of a novel topological phase. As such, the study opens up a window to understanding and potentially exploiting topological behavior in a rich class of easily-synthesized multinary, disordered compounds.

Generic Orbital Design of Higher-Order Topological Quasicrystalline Insulators with Odd Five-Fold Rotation Symmetry

In addition to crystals, topological phases in quasicrystals and disorder systems have drawn increasing attention lately. Here, we propose a generic double band-inversion mechanism underlying the higher-order topological phase in quasicrystals, that is.,"higher-order topological quasicrystalline insulator" (HOTQI), which exploits local atomic orbital and lattice symmetries. It is generally applicable to both quasicrystals and crystals with either odd-rotational (OR) or even-rotational symmetry (ERS), different from previous HOTI mechanisms whose applicability is limited by symmetry types. The HOTQI is characterized by topological corner states at the nonordinary corners of pentagonal (octagonal) samples of five-fold (eight-fold) quasicrystals, which violate the translational invariance and ordinary crystalline symmetries. The role of quasicrystalline symmetry, the robustness against symmetry breaking, and possible experimental realizations are discussed. Our findings not only provide a concrete example of HOTQIs that is incompatible with classical crystallographic symmetry but also offer useful guidance to the search of higher-order topological materials and metamaterials.

Sierpiński Structure and Electronic Topology in Bi Thin Films on InSb(111)B Surfaces

Deposition of Bi on InSb(111)B reveals a striking Sierpiński-triangle (ST)-like structure in Bi thin films. Such a fractal geometric topology is further shown to turn off the intrinsic electronic topology in a thin film. Relaxation of a huge misfit strain of about 30% to 40% between Bi adlayer and substrate is revealed to drive the ST-like island formation. A Frenkel-Kontrova model is developed to illustrate the enhanced strain relief in the ST islands offsetting the additional step energy cost. Besides a sufficiently large tensile strain, forming ST-like structures also requires larger adlayer-substrate and intra-adlayer elastic stiffnesses, and weaker intra-adlayer interatomic interactions.

Strongly Interacting Bosons in a Two-Dimensional Quasicrystal Lattice

Quasicrystals exhibit exotic properties inherited from the self-similarity of their long-range ordered, yet aperiodic, structure. The recent realization of optical quasicrystal lattices paves the way to the study of correlated Bose fluids in such structures, but the regime of strong interactions remains largely unexplored, both theoretically and experimentally. Here, we determine the quantum phase diagram of two-dimensional correlated bosons in an eightfold quasicrystal potential. Using large-scale quantum Monte Carlo calculations, we demonstrate a superfluid-to-Bose glass transition and determine the critical line. Moreover, we show that strong interactions stabilize Mott insulator phases, some of which have spontaneously broken eightfold symmetry. Our results are directly relevant to current generation experiments and, in particular, drive prospects to the observation of the still elusive Bose glass phase in two dimensions and exotic Mott phases.

Bosonic Bott Index and Disorder-Induced Topological Transitions of Magnons

We investigate the role of disorder on the various topological magnonic phases present in deformed honeycomb ferromagnets. To this end, we introduce a bosonic Bott index to characterize the topology of magnon spectra in finite, disordered systems. The consistency between the Bott index and Chern number is numerically established in the clean limit. We demonstrate that topologically protected magnon edge states are robust to moderate disorder and, as anticipated, localized in the strong regime. We predict a disorder-driven topological phase transition, a magnonic analog of the "topological Anderson insulator" in electronic systems, where the disorder is responsible for the emergence of the nontrivial topology. Combining the results for the Bott index and transport properties, we show that bulk-boundary correspondence holds for disordered topological magnons. Our results open the door for research on topological magnonics as well as other bosonic excitations in finite and disordered systems.

Layer-dependent topological phase in a two-dimensional quasicrystal and approximant

The electronic and topological properties of materials are derived from the interplay between crystalline symmetry and dimensionality. Simultaneously introducing "forbidden" symmetries via quasiperiodic ordering with low dimensionality into a material system promises the emergence of new physical phenomena. Here, we isolate a two-dimensional (2D) chalcogenide quasicrystal and approximant, and investigate their electronic and topological properties. The 2D layers of the materials with a composition close to Ta_1.6Te, derived from a layered transition metal dichalcogenide, are isolated with standard exfoliation techniques, and investigated with electron diffraction and atomic resolution scanning transmission electron microscopy. Density functional theory calculations and symmetry analysis of the large unit cell crystalline approximant of the quasicrystal, Ta₂₁Te₁₃, reveal the presence of symmetry-protected nodal crossings in the quasicrystalline and approximant phases, whose presence is tunable by layer number. Our study provides a platform for the exploration of physics in quasicrystalline, low-dimensional materials and the interconnected nature of topology, dimensionality, and symmetry in electronic systems.

Details of the topological state transition induced by gradually increased disorder in photonic Chern insulators

Using two well-defined empirical parameters, we numerically investigate the details of the disorder-induced topological state transition (TST) in photonic Chern insulators composed of two-dimensional magnetic photonic crystals (MPCs). The TST undergoes a gradual process, accompanied with some interesting phenomena as the disorder of rod positions in MPCs increases gradually. This kind of TST is determined by the competition among the topologically protected edge state, disorder-induced wave localizations and bulk states in the system. More interestingly, the disorder-induced wave localizations almost have no influence on the one-way propagation of the original photonic topological states (PTSs), and the unidirectional nature of the PTSs at the edge area can survive even when the bulk states arise at stronger disorders. Our results provide detailed demonstrations for the deep understanding of fundamental physics underlying topology and disorder and are also of practical significance in device fabrication with PTSs.

Kohn-Luttinger Mechanism Driven Exotic Topological Superconductivity on the Penrose Lattice

The Kohn-Luttinger mechanism for unconventional superconductivity (SC) driven by weak repulsive electron-electron interactions on a periodic lattice is generalized to the quasicrystal (QC) via a real-space perturbative approach. The repulsive Hubbard model on the Penrose lattice is studied as an example, on which a classification of the pairing symmetries is performed and a pairing phase diagram is obtained. Two remarkable properties of these pairing states are revealed, due to the combination of the presence of the point-group symmetry and the lack of translation symmetry on this lattice. First, the spin and spacial angular momenta of a Cooper pair is decorrelated: for each pairing symmetry, both spin-singlet and spin-triplet pairings are possible even in the weak-pairing limit. Second, the pairing states belonging to the 2D irreducible representations of the D_{5} point group can be time-reversal-symmetry-breaking topological SCs carrying spontaneous bulk super current and spontaneous vortices. These two remarkable properties are general for the SCs on all QCs, and are rare on periodic lattices. Our work starts the new area of unconventional SCs driven by repulsive interactions on the QC.

A Unified View of Topological Phase Transition in Band Theory

We develop a unified view of topological phase transitions (TPTs) in solids by revising the classical band theory with the inclusion of topology. Reevaluating the band evolution from an “atomic crystal” (a normal insulator (NI)) to a solid crystal, such as a semiconductor, we demonstrate that there exists ubiquitously an intermediate phase of topological insulator (TI), whose critical transition point displays a linear scaling between electron hopping potential and average bond length, underlined by deformation-potential theory. The validity of the scaling relation is verified in various two-dimensional (2D) lattices regardless of lattice symmetry, periodicity, and form of electron hoppings, based on a generic tight-binding model. Significantly, this linear scaling is shown to set an upper bound for the degree of structural disorder to destroy the topological order in a crystalline solid, as exemplified by formation of vacancies and thermal disorder. Our work formulates a simple framework for understanding the physical nature of TPTs with significant implications in practical applications of topological materials.

Higher-Order Topological Insulators in Quasicrystals

Current understanding of higher-order topological insulators (HOTIs) is based primarily on crystalline materials. Here, we propose that HOTIs can be realized in quasicrystals. Specifically, we show that two distinct types of second-order topological insulators (SOTIs) can be constructed on the quasicrystalline lattices (QLs) with different tiling patterns. One is derived by using a Wilson mass term to gap out the edge states of the quantum spin Hall insulator on QLs. The other is the quasicrystalline quadrupole insulator (QI) with a quantized quadrupole moment. We reveal some unusual features of the corner states (CSs) in the quasicrystalline SOTIs. We also show that the quasicrystalline QI can be simulated by a designed electrical circuit, where the CSs can be identified by measuring the impedance resonance peak. Our findings not only extend the concept of HOTIs into quasicrystals but also provide a feasible way to detect the topological property of quasicrystals in experiments.

Toward Realistic Amorphous Topological Insulators

The topological properties of materials are, until now, associated with the features of their crystalline structure, although translational symmetry is not an explicit requirement of the topological phases. Recent studies of hopping models on random lattices have demonstrated that amorphous model systems show a nontrivial topology. Using ab initio calculations, we show that two-dimensional amorphous materials can also display topological insulator properties. More specifically, we present a realistic state-of-the-art study of the electronic and transport properties of amorphous bismuthene systems, showing that these materials are topological insulators. These systems are characterized by the topological index [Formula: see text]₂ = 1 and bulk-edge duality, and their linear conductance is quantized, [Formula: see text], for Fermi energies within the topological gap. Our study opens the path to the experimental and theoretical investigation of amorphous topological insulator materials.