Fractionalized quantum excitations in correlated two-dimensional topological phases

1D topological systems for next-generation electronics

Topological phases play a novel and fundamental role in matter and display extraordinary robustness to smooth changes in material parameters or disorder. A crossover between topological material and quantum information may lead to inherent fault-tolerant quantum simulations and quantum computing. Quantum features may be preserved by being encoded among topological structures of physical evolution systems. This requires stimulation, manipulation, and observation of topological phenomena at the single quantum particle level, which has not, however, yet been realized. It is asked whether the quantum features of single photons can be preserved in topological structures. The boundary states are experimentally observed at the genuine single-photon level and the performance of the topological phase is demonstrated to protect the quantum features against diffusion-induced decoherence in coupled waveguides and noise decoherence from the ambient environment. Compatibility between macroscopic topological states and microscopic single photons in the ambient environment is thus confirmed, leading to a new avenue to "quantum topological photonics" and providing more new possibilities for quantum materials and quantum technologies.

<p indent="0mm">The design of metamaterials plays an essential role in effectively controlling acoustic waves, which has led to many fascinating effects, such as acoustic cloaking, negative refraction and extraordinary transmission. Inspired by topological states discovered in condensed matter physics, a new perspective has emerged in considering the band topology of acoustic metamaterials, i.e., topological acoustics. The hallmark of topological states is bulk-boundary correspondence, lower dimensional boundary states protected by bulk topology. Recently, researchers have found that the bulk topology can also correspond to boundary states with two and three dimensions lower than the bulk, such as one-dimensional hinge states or zero-dimensional corner states in three-dimensional systems. This kind of topological state is the so-called higher-order topology. To date, various higher-order topological states have been successfully achieved in both two- and three-dimensional acoustic systems. However, these previous studies focused mainly on periodic systems. The quasi-periodic systems, which break the translational symmetry due to unique rotational symmetries, are rarely studied, especially in experiments. In this work, we experimentally investigate the higher-order topological states in acoustic two-dimensional Penrose quasicrystals possessing five-fold rotational symmetry. The Penrose patterns are constructed using inflation and deflation methods, starting with a decagon as the initial pattern. Since the centers of rotation are different for odd and even iterations, we choose the third and fourth iterations as two typical patterns. After placing uniform acoustic scatters on the vertices of the patterns, two acoustic quasicrystals are created, named G3 (Generation-3) and G4 (Generation-4). The size is controlled by the radius of the acoustic rods (<italic>r</italic>) and the length (<italic>l</italic>). Due to the long-range order and self-similarity characteristics of quasicrystals, the wavefunctions tend to be distributed in a repeating local patch. As the contrast ratio <italic>r</italic>/<italic>l</italic> increases, they also tend to split in the energy spectrum, making higher-order topological states possible. Here, we fix <italic>l</italic><sc>=10 cm</sc> and <italic>r</italic>/<italic>l</italic>=0.15, and focus on the states of frequency window ranging from 3 to <sc>5 kHz.</sc> These states can be divided into three series according to their different sound pressure distributions in two different local patches, i.e., <italic>d_xy</italic>- and <italic>d</italic>_{<italic>x</italic>2-<italic>y</italic>2}-like field distributions in local patch A and <italic>p_x</italic>-like in local patch B. For models G3 and G4 associated with their pentagon versions G3P and G4P, we find localized states at five corners of G3, G4, and G4P, and also appearing at the middle of the edges in G3P. Their topological properties are further analyzed by calculating the fractional corner charge density through the local density of states, which indicates that those in G3P are trivial while others are topological. Both trivial and topological localized states are experimentally observed via response spectra measurement. G3 and G4 models have similar topological properties, possessing topological corner states protected by five-fold rotational symmetry. Because of the different rotation centers, if the boundary conditions were changed (cut into regular pentagons), the topological corner states of G3P could disappear, while they still exist in G4P for the non-vanishing local patch A at the corners. Our results can be conveniently extended to G5 and G6 or even more complicated quasicrystal models, enriching acoustic topological states in quasi-periodic systems. Such unique five-fold rotational symmetry-protected topological corner states may also contribute to developing novel acoustic functional devices, facilitating acoustic wave localization and energy harvesting.

Topological surface states are a class of novel electronic states that are of potential interest in quantum computing or spintronic applications. Unlike conventional two-dimensional electron states, these surface states are expected to be immune to localization and to overcome barriers caused by material imperfection. Previous experiments have demonstrated that topological surface states do not backscatter between equal and opposite momentum states, owing to their chiral spin texture. However, so far there is no evidence that these states in fact transmit through naturally occurring surface defects. Here we use a scanning tunnelling microscope to measure the transmission and reflection probabilities of topological surface states of antimony through naturally occurring crystalline steps separating atomic terraces. In contrast to non-topological surface states of common metals (copper, silver and gold), which are either reflected or absorbed by atomic steps, we show that topological surface states of antimony penetrate such barriers with high probability. This demonstration of the extended nature of antimony's topological surface states suggests that such states may be useful for high current transmission even in the presence of atomic-scale irregularities-an electronic feature sought to efficiently interconnect nanoscale devices.

Topological materials with exotic quantum properties are promising candidates for quantum spin electronics. Different classes of topological materials, including Weyl semimetal, topological superconductor, topological insulator and Axion insulator, etc., can be connected to each other via quantum phase transition. For example, it is believed that a trivial band insulator can be twisted into topological phase by increasing spin-orbital coupling or changing the parameters of crystal lattice. With the results of LDA calculation and measurement by angle-resolved photoemission spectroscopy (ARPES), we demonstrate in this work that the electronic structure of SrSn2As2 single crystal has the texture of band inversion near the critical point. The results indicate the possibility of realizing topological quantum phase transition in SrSn2As2 single crystal and obtaining different exotic quantum states.

Dynamical classification of topological quantum phases

Interaction induced topological states remain one of the fascinating phases in condensed matter physics. The exciton condensate has recently sparked renewed interest due to the discovery of new candidate materials and its driving force to realize exotic topological states. In this work, we explore the exciton orders induced high-order topology in the bilayer quantum spin Hall insulators and find that the topological excitonic corner states can be realized by tuning the gate and magnetic field. When an in-plane Zeeman field is applied to the system, two or four excitonic boundary-obstructed corner states emerge in the bilayer system for distinct possible $s$-wave excitonic pairings. Besides, we also find a two-dimensional excitonic Weyl nodal phase, which supports flat band edge states connecting the bulk Weyl nodes.

Topological electronic materials exhibit many novel physical properties, such as low dissipation transport and high carrier mobility. These extraordinary properties originate from their non-trivial topological electronic structures in momentum space. In recent years, topological phase transitions based on topological electronic materials have gradually become one of the hot topics in condensed matter physics. Using first-principles calculations, we explore the topological phase transitions driven by in-plane strain in ternary pyrochlore oxide Tl&lt;sub&gt;2&lt;/sub&gt;Ta&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt;. Firstly, we analyze the atomic-orbital-resolved band structure and find that the O (p&lt;sub&gt;&lt;i&gt;x&lt;/i&gt;&lt;/sub&gt;+p&lt;sub&gt;&lt;i&gt;y&lt;/i&gt;&lt;/sub&gt;) and p&lt;sub&gt;&lt;i&gt;z&lt;/i&gt;&lt;/sub&gt; orbitals of the system near the Fermi level have band inversion, indicating the emergence of topological phase transitions in the system. Then the tight-binding models are constructed to calculate the &lt;i&gt;Z&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt; topological invariants, which can determine the topologically non-trivial feature of the system. Finally, topological properties such as surface states and a three-dimensional Dirac cone are studied. It is found that Tl&lt;sub&gt;2&lt;/sub&gt;Ta&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; without strain is a semimetal with a quadratic band touching point at Fermi level, while the in-plane strain can drive the topological phase transition via breaking crystalline symmetries. When the system is under the –1% in-plane compression strain and without considering the spin orbit coupling (SOC), the application of strain results in two triply degenerate nodal points formed in the –&lt;i&gt;Z&lt;/i&gt; to &lt;i&gt;Γ&lt;/i&gt; direction and &lt;i&gt;Γ&lt;/i&gt; to &lt;i&gt;Z&lt;/i&gt; direction, respectively. When the SOC is included, there are two fourfold degenerate Dirac points on the –&lt;i&gt;Z&lt;/i&gt; to &lt;i&gt;Γ&lt;/i&gt; path and &lt;i&gt;Γ&lt;/i&gt; to &lt;i&gt;Z&lt;/i&gt; path&lt;i&gt;,&lt;/i&gt; respectively. Thus, the –1% in-plane compression strain makes the system transit from the quadratic contact point semimetal to a Dirac semimetal. When 1% in-plane expansion strain is applied and the SOC is neglected, there exists one band intersection along &lt;i&gt;Y→&lt;/i&gt;&lt;i&gt;Γ&lt;/i&gt;. When the SOC is taken into consideration, the gap is opened. Therefore, the 1% in-plane expansion strain drives Tl&lt;sub&gt;2&lt;/sub&gt;Ta&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; into a strong topological insulator. In addition, the system is also expected to have strong correlation effect and superconductivity due to the possible flat band. This work can guide the study of topological phase transitions in three-dimensional materials and provide a good material platform for the design of low-dissipation electronic devices.

Quantum phase transition is one of the most interesting aspects in quantum many-body systems. Recently, geometric quantum discord has been introduced to signature the critical behavior of various quantum systems. However, it is well-known that topological quantum phase transition can not be described by the conventional Landau's symmetry breaking theory, and thus it is unknown that whether previous study can be applicable in this case. Here, we study the topological quantum phase transition in Kitaev's 1D p-wave spinless quantum wire model in terms of its ground state geometric quantum discord. The derivative of geometric quantum discord is nonanalytic at the critical point, in both zero temperature and finite temperature cases. The scaling behavior and the universality are verified numerically. Therefore, our results clearly show that all the key ingredients of the topological phase transition can be captured by the nearest neighbor and long-range geometric quantum discord.

We demonstrate a new class of nanostructured topological materials that exhibit a topological quantum phase arising from nanoscale structural motifs. Based on first-principles calculations, we show that an array of bismuth nanotubes (Bi-NTs), a superlattice of Bi-NTs with periodicity in the order of tube diameter, behaves as a nanostructured two-dimensional (2D) quantum spin Hall (QSH) insulator, as confirmed from the calculated band topology and 1D helical edge states. The underpinning mechanism of the QSH phase in the Bi-NT array is revealed to be inversion of bonding-antibonding levels of molecular orbitals of constituent nanostructural elements in place of atomic-orbital band inversion in conventional QSH insulators. The quantized edge conductance of the QSH phase in a Bi-NT array can be more easily isolated from bulk contributions and their properties can be highly tuned by tube size, representing distinctive advantages of nanostructured topological phases. Our finding opens a new avenue for topological materials by extending topological phases into nanomaterials with molecular-orbital-band inversion.

This is an introductory review of the physics of topological quantum matter with cold atoms. Topological quantum phases, originally discovered and investigated in condensed matter physics, have recently been explored in a range of different systems, which produced both fascinating physics findings and exciting opportunities for applications. Among the physical systems that have been considered to realize and probe these intriguing phases, ultracold atoms become promising platforms due to their high flexibility and controllability. Quantum simulation of topological phases with cold atomic gases is a rapidly evolving field, and recent theoretical and experimental developments reveal that some toy models originally proposed in condensed matter physics have been realized with this artificial quantum system. The purpose of this article is to introduce these developments. The article begins with a tutorial review of topological invariants and the methods to control parameters in the Hamiltonians of neutral atoms. Next, topological quantum phases in optical lattices are introduced in some detail, especially several celebrated models, such as the Su-Schrieffer-Heeger model, the Hofstadter-Harper model, the Haldane model and the Kane-Mele model. The theoretical proposals and experimental implementations of these models are discussed. Notably, many of these models cannot be directly realized in conventional solid-state experiments. The newly developed methods for probing the intrinsic properties of the topological phases in cold atom systems are also reviewed. Finally, some topological phases with cold atoms in the continuum and in the presence of interactions are discussed, and an outlook on future work is given.

Networks of rigid bars connected by joints, termed linkages, provide a minimal framework to design robotic arms and mechanical metamaterials built of folding components. Here, we investigate a chain-like linkage that, according to linear elasticity, behaves like a topological mechanical insulator whose zero-energy modes are localized at the edge. Simple experiments we performed using prototypes of the chain vividly illustrate how the soft motion, initially localized at the edge, can in fact propagate unobstructed all of the way to the opposite end. Using real prototypes, simulations, and analytical models, we demonstrate that the chain is a mechanical conductor, whose carriers are nonlinear solitary waves, not captured within linear elasticity. Indeed, the linkage prototype can be regarded as the simplest example of a topological metamaterial whose protected mechanical excitations are solitons, moving domain walls between distinct topological mechanical phases. More practically, we have built a topologically protected mechanism that can perform basic tasks such as transporting a mechanical state from one location to another. Our work paves the way toward adopting the principle of topological robustness in the design of robots assembled from activated linkages as well as in the fabrication of complex molecular nanostructures.

Dimensional evolution between one- ($1D$) and two-dimensional ($2D$) topological phases is investigated systematically. The crossover from a $2D$ topological insulator to its $1D$ limit shows oscillating behavior between a $1D$ ordinary insulator and a $1D$ topological insulator. By constructing a $2D$ topological system from a $1D$ topological insulator, it is shown that there exist possibly weak topological phases in $2D$ time-reversal invariant band insulators, one of which can be realized in anisotropic systems. The topological invariant of the phase is $Z_{2}=0$. However the edge states may appear along specific boundaries. It can be interpreted as arranged $1D$ topological phases, and have symmetry-protecting nature as the corresponding $1D$ topological phase. Robust edge states can exist under specific conditions. These results provide further understanding on $2D$ time-reversal invariant insulators, and can be realized experimentally.

The discovery of the topological states has become a key topic in condensed matter physics with the focus evolving from the Dirac or Weyl points to high-dimension topological states of the nodal lines and nodal surfaces. For a topological material to manifest its quantum properties and become useful in applications, the topological states need to be genuine and clean so that they lie close to the Fermi level without other trivial bands existing at the Fermi level. While a number of high-dimension topological materials are predicted, only a few of them have been synthesized and confirmed and the genuine and clean ones are especially scarce. Here we report the realization of the genuine clean multiple high-dimension topological states in NaAlSi. By performing high-resolution angle-resolved photoemission measurements and band structure calculations, we have observed two sets of nodal surfaces and the formation of two homocentric nodal ring states in NaAlSi. The observed nodal rings are distinct in that the inner one is a type-{\uppercase\expandafter{\romannumeral1}} nodal ring while the outer one is a type-{\uppercase\expandafter{\romannumeral1}} nodal ring embedded with four type-{\uppercase\expandafter{\romannumeral3}} nodal points. All the bands involved in the nodal rings lie very close to the Fermi level with no other trivial bands coexisting at the Fermi level. These observations make NaAlSi a desirable topological material to explore for novel quantum states and exotic properties.

In topological materials, shielding of bulk and surface states by crystalline symmetries has provided hitherto unknown access to electronic states in condensed matter physics. Interestingly, photoexcited carriers relax on an ultrafast timescales, demonstrating large transient mobility that can be harnessed for the development of ultrafast optoelectronic devices. In addition, these devices are much more effective than topologically trivial systems because topological states are resilient to the corresponding symmetry‐invariant perturbations. By using optical pump probe measurements, the relaxation dynamics of a topologically nontrivial chiral single crystal, PtAl, is systematically described. Based on the experimental data on transient reflectivity and electronic structures, it is found that the carrier relaxation process involves both acoustic and optical phonons with oscillation frequencies of 0.06 and 2.94 THz, respectively, in picosecond time scale. PtAl with a space group ofP213 allows only one non‐zero susceptibility element, i.e.,d14, in second harmonic generation (SHG) with a large value of 468(1) pm V–1, which is significantly higher than that observed in standard GaAs(111) and ZnTe(110) crystals. The intensity dependence of the SHG signal in PtAl reveals a non‐perturbative origin. The present study on PtAl provides deeper insight into topological states that will be useful for ultrafast optoelectronic devices.