Edge states protected by bulk topology of photonic crystals show robustness against short-range disorder, making robust information transfer possible. Here, topological photonic crystals under long-range deformations are investigated. The vertices of each regular hexagon in a honeycomb crystalline structure are shifted randomly to establish a deformed system. By increasing the degree of random deformations, a transition from an ordered system to an amorphous system are investigated, where the close of the topological band gap is clearly shown. We further present comprehensive investigations into excitation methods of the proposed deformed system. Due to the lack of strict periodicity, excitation of topological edge modes becomes difficult. We discuss chiral and linearly polarized sources as two different methods respectively. It is found that chiral sources are sensitive and rely on the ordered lattice. Even a weak long-range deformation can bring fluctuations to transmission. We further designed and fabricated metal-dielectric-metal sandwichlike samples working in the microwave band. Using a linearly polarized source, we detected the existence of topological transport in the deformed system. This work investigates excitation and robustness of bulk topology against long-range deformations and may open the way for exploiting topological properties of materials with a deformed lattice.

Multi-Weyl semimetals feature band crossings with the dispersion that is, in general, linear in only one direction, and as a consequence their band structure is characterized by the monopole charge $n$ which can be greater than one. We show that a single screw dislocation defect oriented in the direction connecting the nodal points, which acts as an effective pseudomagnetic flux tube, can serve as a direct probe of the monopole charge $n\ge 1$ characterizing the bulk band structure of a multi-Weyl semimetal. To this end, as a proof of principle, we propose a rather simple mesoscopic setup in which the monopole charge leaves a direct imprint on the conductance measured in the plane perpendicular to the dislocation. In particular, the ratio of the positions of the neighboring maxima in the conductance as a function of the gate voltage can serve to deduce the monopole charge, while the value of the effective pseudomagnetic flux can be extracted from the position of a conductance maximum. We expect that these findings will prompt further studies on the role of multiple dislocations, as well as other topological lattice defects, such as grain boundaries and disclinations, in topological nodal materials.

Topology plays an increasing role in physics beyond the realm of topological insulators in condensed mater. From geophysical fluids to active matter, acoustics or photonics, a growing family of systems presents topologically protected chiral edge modes. The number of such modes should coincide with the bulk topological invariant (e.g., Chern number) defined for a sample without boundary, in agreement with the bulk-edge correspondence. However, this is not always the case when dealing with continuous media where there is no small scale cutoff. The number of edge modes actually depends on the boundary condition, even when the bulk is properly regularized, showing an apparent paradox where the bulk-edge correspondence is violated. In this paper, we solve this paradox by showing that the anomaly is due to ghost edge modes hidden in the asymptotic part of the spectrum, which have a signature at finite frequency both in the local density of states and in a channel geometry. We provide a general formalism based on scattering theory to detect all edge modes properly, so that the bulk-edge correspondence is restored in a broader sense, implying in particular that chiral edge modes are not necessarily topological, and conversely. Our approach is illustrated through the odd-viscous shallow-water model and the massive Dirac Hamiltonian.

We numerically investigate the properties of the quasihole excitations above the bosonic fractional Chern insulator state at filling $\nu =1/2$, in the specific case of the Harper-Hofstadter Hamiltonian with hard-core interactions. For this purpose, we employ a tree tensor network technique, which allows us to study systems with up to $N=18$ particles on a $16\times 16$ lattice and experiencing an additional harmonic confinement. First, we observe the quantization of the quasihole charge at fractional values and its robustness against the shape and strength of the impurity potentials used to create and localize such excitations. Then, we numerically characterize quasihole anyonic statistics by applying a discretized version of the relation connecting the statistics of quasiholes in the lowest Landau level to the depletions they create in the density profile [E. Macaluso et al., Phys. Rev. Lett. 123, 266801 (2019)]. Our results give a direct proof of the anyonic statistics for quasiholes of fractional Chern insulators, starting from a realistic Hamiltonian. Moreover, they provide strong indications that this property can be experimentally probed through local density measurements, making our scheme readily applicable in state-of-the-art experiments with ultracold atoms and superconducting qubits.

Topological pumping and duality transformations are paradigmatic concepts in condensed matter and statistical mechanics. In this paper, we extend the concept of topological pumping of particles to topological pumping of quantum correlations. We propose a scheme to find pumping protocols for highly correlated states by mapping them to uncorrelated ones. We show that one way to achieve this is to use dualities, because they are nonlocal transformations that preserve the topological properties of the system. By using them, we demonstrate that topological pumping of kinks and clusterlike excitations can be realized. We find that the entanglement of these highly correlated excitations is strongly modified during the pumping process and the interactions enhance the robustness against disorder. Our paper paves the way to explore topological pumping beyond the notion of particles and opens an avenue to investigate the relation between correlations and topology.

Fragile topology is firstly demonstrated in acoustic crystals and then a realistic scheme is proposed to manipulate the transport of acoustic topological edge states (ATESs), i.e., by coupling them with side acoustic cavities. We find that single-mode cavities can completely flip the ATES pseudospin to form a perfect reflection, as long as their resonant frequencies fall into the topological band gap. The perfect reflection of the ATESs is protected by the fragile topology, which is proved by the one-dimensional topological waveguide-cavity transport theory. This fragile topologically protected perfect reflection is immune to the conventional defects (such as bending and disorder) and provides a realistic paradigm for manipulating the ATES transport. As examples, two potential applications, i.e., distance sensors and acoustic switches, are proposed based on the perfect reflection.

The integer quantum Hall (QH) effects characterized by topologically quantized and nondissipative transport are caused by an electrically insulating incompressible phase that prevents backscattering between chiral metallic channels. We probed the incompressible area susceptible to the breakdown of topological protection using a scanning-gate technique incorporating nonequilibrium transport. The obtained pattern revealed the filling-factor- ($\nu$-) dependent evolution of the microscopic incompressible structures located along the edge and in the bulk region. We found that these specific structures, respectively, attributed to the incompressible edge strip and bulk localization, show good agreement in terms of $\nu$-dependent evolution with a calculation of the equilibrium QH incompressible phases, indicating the robustness of the QH incompressible phases under the nonequilibrium condition. Further, we found that the $\nu$ dependency of the incompressible patterns is, in turn, destroyed by a large imposed current during the deep QH effect breakdown. These results demonstrate the ability of our method to image the microscopic transport properties of a topological two-dimensional system.

We construct and study an intriguing model of one-dimensional interacting fermion wire, which is enforced to be in a topological phase by the spin-orbit locking structure in the interactions, regardless of the dimerization direction of the Peierls distortion, in sharp contrast to a conventional wisdom. Thorough analyses based upon the bosonization, the renormalization group technique, and the mean-field theory have been made. The novelty of being a pair-hopping fermionic wire lies in that symmetry breaking and the formation of (symmetry-protected) topological configurations are not independent, enriching our understanding of the interplay between topological phase and symmetry breaking. An experimentally feasible scheme is proposed for realizing the model with cold atoms and available techniques.

We propose an efficient scheme for simulating the topological phases of matter based on silicon-vacancy (SiV) center arrays in phononic crystals. This phononic band-gap structure allows for long-range spin-spin interactions with a tunable profile. Under a particular periodic microwave driving, the band-gap-mediated spin-spin interaction can be further designed with the form of the Su-Schrieffer-Heeger (SSH) Hamiltonian. In momentum space, we investigate the topological characters of the SSH model and show that the topological nontrivial phase can be obtained through modulating the periodic driving fields. Furthermore, we explore the zero-energy topological edge states at the boundary of the color center arrays and study the robust quantum information transfer via the topological edge states. This setup provides a scalable and promising platform for studying topological quantum physics and quantum information processing with color centers and phononic crystals.

Weyl semimetals are materials with topologically nontrivial band structures both in the bulk and on the surface, hosting chiral nodes which are sinks and sources of Berry curvature. Weyl semimetals have been predicted and recently measured to exhibit large nonlinear optical responses. This discovery, along with their high mobilities, makes Weyl semimetals relevant to a broad spectrum of applications in optoelectronic, nanophotonic, and quantum optical devices. Although there is growing interest in understanding and characterizing the linear and nonlinear behaviors of Weyl semimetals, an ab initio calculation of the linear optical and optoelectronic responses at finite temperature remains largely unexplored. Here, we specifically address the temperature dependence of the linear optical response in type-I Weyl semimetals TaAs and NbAs. We evaluate, from first principles, the scattering lifetimes due to electron-phonon and electron-electron interactions and incorporate these lifetimes in evaluating an experimentally relevant frequency-, polarization-, and temperature-dependent complex dielectric function for each semimetal. From these calculations, we present linear optical conductivity predictions which agree well where experiment exists (for TaAs) and guide the way for future measurements of type-I Weyl semimetals. Importantly, we also examine the optical conductivity's dependence on the chemical potential, a crucial physical parameter which can be controlled experimentally and can elucidate the role of the Weyl nodes in optoelectronic response. Throughout this paper, we present design principles for Weyl optoelectronic devices that use photogenerated carriers in type-I Weyl semimetals.

Symmetry indicators have proven to be extremely helpful in identifying topological crystalline insulators using symmetry-group representations of their Bloch states. An extension of this approach to superconducting systems requires defining an appropriate atomic limit for Bogoliubov-de-Gennes Hamiltonians. Here, we introduce such a notion of atomic limit and derive a ${\mathbb{Z}}_{2^d}$-valued symmetry indicator for inversion-symmetric superconductors in $d$ dimensions. This indicator allows for a refined topological classification including higher-order phases for systems in the superconducting symmetry classes D and DIII. We further elucidate their bulk-boundary correspondence using Dirac surface theories. Requiring only the normal-state band structure and the superconducting order-parameter symmetry as input, this indicator is well suited for a search of topological superconductors using first-principles calculations.

Nonreciprocal non-Hermitian systems provide an unconventional localization mechanism of topological zero modes via the non-Hermitian skin effect. While fundamental theoretical characterizations of this effect involve the biorthogonal system of right and left eigenmodes, the recent demonstration of this effect for a zero mode in a robotic metamaterial (Ghatak et al., arXiv:1907.11619) is based on the direct experimental observation of the conventional right eigenvectors. Here I show that such nonreciprocal mechanical metamaterials reveal their underlying biorthogonality in the directly observable response of the system to external excitation. Applied to the experiment, this nonreciprocal response theory predicts that the zero-mode skin effect coincides with an extended phase where the system is highly sensitive to physical perturbations, leading to a diverging response in the limit of a large system.

Amorphous systems have rapidly gained attention as promising platforms for topological matter. In this work, we establish a scaling theory of amorphous topological phase transitions driven by the density of lattice points in two dimensions. By carrying out a finite-size scaling analysis of topological invariants averaged over discrete and continuum random geometries, we discover critical properties of Chern and ${\mathbb{Z}}_2$ glass transitions. Even for short-range hopping models, the Chern glass phase may persist down to the fundamental lower bound given by the classical percolation threshold. While the topological indices accurately satisfy the postulated one-parameter scaling, they do not generally flow to the closest integer value in the thermodynamic limit. Furthermore, the value of the critical exponent describing the diverging localization length varies continuously along the phase boundary and is not fixed by the symmetry class of the Hamiltonian. We conclude that the critical behavior of amorphous topological systems exhibit characteristic features not observed in disordered systems, motivating a wealth of interesting research directions.

We study Josephson junctions based on second-order topological superconductors (SOTSs) which can be realized in quantum spin Hall insulators with a large inverted gap in proximity to unconventional superconductors. We find that tuning the chemical potential in the superconductor strongly modifies the induced pairing of the helical edge states, resulting in topological phase transitions. In a corresponding Josephson junction, a $0-\pi$ transition is realized by tuning the chemical potentials in the superconducting leads. This striking feature is stable in junctions with respect to different sizes, doping the normal region, and the presence of disorder. Our transport results can serve as novel experimental signatures of SOTSs. Moreover, the $0-\pi$ transition constitutes a fully electric way to create or annihilate Majorana bound states in the junction without any magnetic manipulation.

When two lasers are applied to a noncentrosymmetric material, it can generate light at the difference of the incoming frequencies $\mathrm{\Delta }\omega$, a phenomenon known as difference frequency generation (DFG), well characterized in semiconductors. In this work, we derive a general expression for DFG in metals, which we use to show that the DFG in chiral topological semimetals under circular polarized light is quantized in units of $e^3/h^2$ and independent of material parameters, including the scattering time $\tau$, when $\mathrm{\Delta }\omega \gg {\tau }^{-1}$. In this regime, DFG provides a simpler alternative to measure a quantized response in metals compared to previous proposals based on single frequency experiments. Our general derivation unmasks, in addition, a free-carrier contribution to the circular DFG beyond the semiclassical one. This contribution can be written as a Fermi surface integral, features strong frequency dependence, and oscillates with a $\pi /2$ shift with respect to the quantized contribution. We make predictions for the circular DFG of chiral and nonchiral materials using generic effective models, and ab initio calculations for TaAs and RhSi. Our work provides a complete picture of the DFG in the length gauge approach, in the clean, noninteracting limit, and highlights a plausible experiment to measure topologically quantized photocurrents in metals.

We propose the ${\mathbb{Z}}_Q$ Berry phase as a topological invariant for higher-order symmetry-protected topological (HOSPT) phases for two- and three-dimensional systems. It is topologically stable for electron-electron interactions assuming the gap remains open. As a concrete example, we show that the Berry phase is quantized in ${\mathbb{Z}}_4$ and characterizes the HOSPT phase of the extended Benalcazar-Bernevig-Hughes (BBH) model, which contains the next-nearest-neighbor hopping and the intersite Coulomb interactions. In addition, we introduce the ${\mathbb{Z}}_4$ Berry phase for the spin-model analog of the BBH model. Furthermore, we demonstrate the Berry phase is quantized in ${\mathbb{Z}}_4$ for the three-dimensional version of the BBH model. We also confirm the bulk-corner correspondence between the ${\mathbb{Z}}_4$ Berry phase and the corner states in the HOSPT phases.

We show that homogeneous lattice gauge theories can realize nonequilibrium quantum phases with long-range spatiotemporal order protected by gauge invariance instead of disorder. We study a kicked ${\mathbb{Z}}_2$-Higgs gauge theory and find that it breaks the discrete temporal symmetry by a period doubling. In a limit solvable by Jordan-Wigner analysis we extensively study the time-crystal properties for large systems and further find that the spatiotemporal order is robust under the addition of a solvability-breaking perturbation preserving the ${\mathbb{Z}}_2$ gauge symmetry. The protecting mechanism for the nonequilibrium order relies on the Hilbert space structure of lattice gauge theories, so that our results can be directly extended to other models with discrete gauge symmetries.

Higher-order topological (HOT) phases feature boundary (such as corner and hinge) modes of codimension $d_c>1$. We here identify an antiunitary operator that ensures the spectral symmetry of a two-dimensional HOT insulator and the existence of cornered localized states ($d_c=2$) at precise zero energy. Such an antiunitary symmetry allows us to construct a generalized HOT insulator that continues to host corner modes even in the presence of a weak anomalous Hall insulator and spin-orbital density-wave orderings, and is characterized by a quantized quadrupolar moment $Q_{xy}=0.5$. Similar conclusions can be drawn for the time-reversal symmetry breaking HOT $p+id$ superconductor and the corner localized Majorana zero modes survive even in the presence of weak Zeeman coupling and $s$-wave pairing. Such HOT insulators also serve as the building blocks of three-dimensional second-order Weyl semimetals, supporting one-dimensional hinge modes.

Higher-order topological (HOT) states, hosting topologically protected modes on lower-dimensional boundaries, such as hinges and corners, have recently extended the realm of the static topological phases. Here we demonstrate the possibility of realizing a two-dimensional Floquet second-order topological insulator, featuring corner-localized zero quasienergy modes and characterized by quantized Floquet qudrupolar moment $Q_{xy}^{\mathrm{Flq}}=0.5$, by periodically kicking a quantum spin Hall insulator (QSHI) with a discrete fourfold ($C_4$) and time-reversal ($\mathcal{T}$) symmetry breaking Dirac mass perturbation. Furthermore, we show that $Q_{xy}^{\mathrm{Flq}}$ becomes independent of the choice of origin as the system approaches the thermodynamic limit. We also analyze the dynamics of a corner mode after a sudden quench, when the $C_4$ and $\mathcal{T}$ symmetry breaking perturbation is switched off, and find that the corresponding survival probability displays periodic appearances of complete, partial and no revival for long time, encoding the signature of corner modes in a QSHI. Our protocol is sufficiently general to explore the territory of dynamical HOT phases in insulators and gapless systems.

Majorana fermions are predicted to arise at the ends of nanowire devices which combine superconductivity, strong spin-orbit coupling, and an external magnetic field. By manipulating networks of these devices with suitable gating, it has been suggested that braiding operations may be performed which act as logic operations, suitable for quantum computation. However, the unavoidable misalignment of the magnetic field in any realistic device geometry has raised questions about the feasibility of such braiding. In this paper, we numerically simulate braiding operations in devices with Y-junction and tuning-fork geometries using an experimentally motivated nanowire model. We study how the static and dynamical features vary with geometric parameters and identify parameter choices that optimize the probability of a successful braid. Notably, we find that there is an optimal Y-junction half-angle (about ${20}^{\circ }$ for our parameter values), which balances two competing mechanisms that reduce the energy gap to excitations. In addition, we find that a tuning-fork geometry has significant advantages over a Y-junction geometry, as it substantially reduces the effect of dynamical phase oscillations that complicate the braiding process. Our results suggest that performing a successful braid is in principle possible with such devices and is within experimental reach.