The recently discovered second-order photonic topological insulators (SPTIs) are characterized by gapped edge states and robust corner states, and they provide novel approaches to the traditional ways to manipulate light. In a general case, the overlapped band gap of nontrivial and trivial photonic crystals composing SPTIs is narrow, which barely allows for the production of strongly localized states. Here, we introduce an intelligent numerical approach for the inverse design of large classes of SPTIs with great flexibility for controlling the properties of topological edge and corner states. In the optimized designs, the overlapped band gap of the nontrivial and trivial photonic crystals substantially exceeds that of the existing SPTIs, and it enables the existence of highly localized corner and edge states with nearly flat dispersion. We design several structures supporting both topological edge and corner states at the desired frequencies. Through programming newly created SPTIs, we suggest a strategy for routing topological edge and corner states. Our findings pave the way for the development of integrated photonic devices with topological protection and innovative functionalities.

We investigate the order of the topological quantum phase transition in a two-dimensional quadrupolar topological insulator within a thermodynamic approach. Using numerical methods, we separate the bulk, edge, and corner contributions to the grand potential and detect different phase transitions in the topological phase diagram. The transitions from the quadrupolar to the trivial or to the dipolar phase are well captured by the thermodynamic potential. On the other hand, we have to resort to a grand potential based on the Wannier bands to describe the transition from the trivial to the dipolar phase. The critical exponents and the order of the phase transitions are determined and discussed in the light of the Josephson hyperscaling relation.

In this Rapid Communication we introduce a thermal magnetotorsional effect (TME) as a topological response in magnetic Weyl semimetals. We predict that magnetization gradients perpendicular to the Weyl node separation give rise to temperature gradients depending only on the local positions of the Weyl nodes. The TME is a consequence of magnetization-induced effective torsional spacetime geometry and the finite-temperature Nieh-Yan anomaly. Similarly to the anomalous Hall effect and chiral anomaly, the TME has a universal material-independent form. We predict that the TME can be observed in the magnetic Weyl semimetal ${\mathrm{EuCd}}_2{\mathrm{As}}_2$.

A dc (e.g., electric) field with commensurate lattice direction turns a single-particle band structure in $d=3$ dimensions into an infinite set of equally spaced irreducible $(d-1)=2$-dimensional Wannier-Stark (WS) band structures that are spatially localized along the field direction. Particle transport is expected to be suppressed once the WS bands are gapped in energy. The topological character of the irreducible band structure leads to one-dimensional sets of boundary states which fill the energy gaps. As a result, eigenmodes are smoothly connected in energy and space and yield anomalous particle transport throughout the ladder. The number of chiral boundary modes can be tuned by the dc field strength and manifests through the distribution of dissipated energy and spatial motion, and the temperature dependence of angular momentum carried by particles.

Illumination of circularly polarized light is an alternative to electrical methods for spin injection. Spins are injected at the point of light illumination, and then they diffuse radially due to the in-plane gradient of the spin density. This diffusing spin current is converted into a circular charge current by the inverse spin Hall effect (ISHE) and is usually not detectable. However, by shining a light on an asymmetric part of the sample, such as near the edges, we detected this current as a helicity-dependent component in the measured photocurrent. This is not due to the circular photogalvanic effect because it occurs even at normal incidence of the light. A finite-element-method simulation shows that the ISHE-induced circular charge current generates an electric dipole at the edge of the sample, causing the measured charge current. The asymmetric light illumination shown here is a simple way to control the output of the ISHE, which may lead to novel spintronic devices.

Topological invariants play a key role in the characterization of topological states. Because of the existence of exceptional points, it is a great challenge to detect topological invariants in non-Hermitian systems. We put forward a dynamic winding number, the winding of realistic observables in long-time average, for exploring band topology in both Hermitian and non-Hermitian two-band models via a unified approach. We build a concrete relation between dynamic winding numbers and conventional topological invariants. In one dimension, the dynamic winding number directly gives the conventional winding number. In two dimensions, the Chern number is related to the weighted sum of all the dynamic winding numbers of phase singularity points. This work opens a new avenue to measure topological invariants via time-averaged spin textures without requesting any prior knowledge of the system topology.

We introduce a theoretical framework for the concept of three-dimensional (3D) twistronics by developing a generalized Bloch band theory for 3D layered systems with a constant twist angle $\theta$ between successive layers. Our theory employs a nonsymmorphic symmetry that enables a precise definition of an effective out-of-plane crystal momentum, and also captures the in-plane moiré pattern formed between neighboring twisted layers. To demonstrate topological physics that can be achieved through 3D twistronics, we present two examples. In the first example of chiral twisted graphite, Weyl nodes arise because of inversion-symmetry breaking, with $\theta$-tuned transitions between type-I and type-II Weyl fermions, as well as magic angles at which the in-plane velocity vanishes. In the second example of a twisted Weyl semimetal, the twist in the lattice structure induces a chiral gauge field $\mathcal{A}$ that has a vortex-antivortex lattice configuration. Line modes bound to the vortex cores of the $\mathcal{A}$ field give rise to 3D Weyl physics in the moiré scale. We also discuss possible experimental realizations of 3D twistronics.

Topological phenomena in non-Hermitian systems have recently become a subject of great interest in the photonics and condensed-matter communities. In particular, the possibility of observing topologically protected edge states in non-Hermitian lattices has sparked an intensive search for systems where this kind of states are sustained. Here we present a study of the emergence of topological edge states in two-dimensional Haldane lattices exhibiting balanced gain and loss. In line with recent studies on other Chern insulator models, we show that edge states can be observed in the so-called broken $\mathcal{P}\mathcal{T}$-symmetric phase, that is, when the spectrum of the gain-loss-balanced system's Hamiltonian is not entirely real. More importantly, we find that such topologically protected edge states emerge irrespective of the lattice boundaries, namely, zigzag, bearded, or armchair.

We study a class of topological charge density wave states exhibiting monopole harmonic symmetries. The density wave ordering is equivalent to pairing in the particle-hole channel due to Fermi surface nesting under interactions. When electron and hole Fermi surfaces carry different Chern numbers, the particle-hole pairing exhibits a nontrivial Berry phase inherited from band-structure topology independent of the concrete density wave ordering mechanism. The associated density wave gap functions become nodal, and the net nodal vorticity is determined by the monopole charge of the pairing Berry phase. The gap function nodes become zero-energy Weyl nodes of the bulk spectra of quasiparticle excitations. These states can occur in doped Weyl semimetals with nested electron and hole Fermi surfaces enclosing Weyl nodes of the same chirality in the weak-coupling regime. Topologically nontrivial low-energy Fermi arc surface states appear in the density wave ordering state as a consequence of the emergent zero-energy Weyl nodes.

Unsupervised machine learning is a cornerstone of artificial intelligence as it provides algorithms capable of learning tasks, such as classification of data, without explicit human assistance. We present an unsupervised deep learning protocol for finding topological indices of quantum systems. The core of the proposed scheme is a “topological data augmentation” procedure that uses seed objects to generate ensembles of topologically equivalent data. Such data, assigned with dummy labels, can then be used to train a neural network classifier for sorting arbitrary objects into topological equivalence classes. Importantly, we also show how to retrieve the local quantities corresponding to the learned topological indices from the intermediate outputs of the trained network. Our protocol is explicitly illustrated on two-band insulators in one and two dimensions, characterized by a winding number and a Chern number respectively. Using the augmentation technique also in the classification step, to classify a family of topologically equivalent objects instead of a single object, we can achieve accuracy arbitrarily close to $100\%$ even for indices outside the training regime. Apart from the method's applicability to topological classification, it also provides a new perspective on data augmentation in supervised machine learning, where given sufficient mathematical structure the set of category-preserving deformations can be rigorously defined.

The hitherto-studied topological nature in Ce monopnictides ($\mathrm{Ce}Pn$) has been discussed based solely on their $p\text{−}d$ band inversion, despite the existence of $f$ electrons. Interestingly, however, we have found that CeN, the lightest $\mathrm{Ce}Pn$, has nontrivial $Z_2$ topology originating from $f\text{−}d$ band inversion. Through density-functional theory and dynamical mean-field theory calculations, we have demonstrated that, among the correlated $f$-electron systems of $\mathrm{Ce}Pn$, the lightest CeN is a coherent narrow $f$-band system having an $f\text{−}d$ band-inverted topological Kondo insulator nature, whereas the heaviest CeBi is a strongly localized $f$-electron system having a conventional $p\text{−}d$ band-inverted topological insulator nature. For CeN, topological surface states (TSSs) are identified clearly on three different (001), (110), and (111) surfaces. For CeBi, however, TSSs are identified only on the (111) surface. Furthermore, intriguing topological-crystalline insulator-type TSSs are identified on the (110) surfaces of both CeN and CeBi.

We consider a setup consisting of two coupled sheets of bilayer graphene in the regime of strong spin-orbit interaction, where electrostatic confinement is used to create an array of effective quantum wires. We show that for suitable interwire couplings the system supports a topological insulator phase exhibiting Kramers partners of gapless helical edge states, while the additional presence of a small in-plane magnetic field and weak proximity-induced superconductivity leads to the emergence of zero-energy Majorana corner states at all four corners of a rectangular sample, indicating the transition to a second-order topological superconducting phase. The presence of strong electron-electron interactions is shown to promote the above phases to their exotic fractional counterparts. In particular, we find that the system supports a fractional topological insulator phase exhibiting fractionally charged gapless edge states and a fractional second-order topological superconducting phase exhibiting zero-energy ${\mathbb{Z}}_{2m}$ parafermion corner states, where $m$ is an odd integer determined by the position of the chemical potential.

We identify the possibility of realizing higher order topological (HOT) phases in noncrystalline or amorphous materials. Starting from two- and three-dimensional crystalline HOT insulators, accommodating topological corner states, we gradually enhance structural randomness in the system. Within a parameter regime, as long as amorphousness is confined by an outer crystalline boundary, the system continues to host corner states, yielding amorphous HOT insulators. However, as structural disorder percolates to the edges, corner states start to dissolve into amorphous bulk, and ultimately the system becomes a trivial insulator when amorphousness plagues the entire system. These outcomes are further substantiated by computing the quadrupolar (octupolar) moment in two (three) dimensions. Therefore, HOT phases can be realized in amorphous solids, when wrapped by a thin (lithographically grown, for example) crystalline layer. Our findings suggest that crystalline topological phases can be realized even in the absence of local crystalline symmetry.

Magnetic hopfions are topologically protected three-dimensional solitons that are constituted by a tube which exhibits a topologically nontrivial spin texture in the cross-section profile and is closed to a torus. Here we show that the hopfion's locally uncompensated emergent field leads to a topological Hall signature, although the topological Hall effect vanishes on the global level. The topological Hall signature is switchable by magnetic fields or electric currents and occurs independently of the anomalous and conventional Hall effects. It can therefore be exploited to electrically detect hopfions in experiments and even to distinguish them from other textures like skyrmion tubes. Furthermore, it can potentially be utilized in spintronic devices. Exemplarily, we propose a hopfion-based racetrack data storage device and simulate the electrical detection of the hopfions as carriers of information.

We classify subsystem symmetry-protected topological (SSPT) phases in $3+1$ dimensions $(3+1\text{D}$) protected by planar subsystem symmetries: short-range entangled phases which are dual to long-range entangled Abelian fracton topological orders via a generalized “gauging” duality. We distinguish between weak SSPTs, which can be constructed by stacking $2+1\text{D}$ SPTs, and strong SSPTs, which cannot. We identify signatures of strong phases, and show by explicit construction that such phases exist. A classification of strong phases is presented for an arbitrary finite Abelian group. Finally, we show that fracton orders realizable via $p$-string condensation are dual to weak SSPTs, while those dual to strong SSPTs exhibit statistical interactions prohibiting such a realization.

We introduce three numerical methods for characterizing the topological phases of three-dimensional multiband Hubbard models based on twisted boundary conditions, Wilson loops, as well as the local topological marker. We focus on the half-filled, three-dimensional time-reversal-invariant Hofstadter model with finite spin-orbit coupling. Besides the weak and strong topological insulator phases we find a nodal line semimetal in the parameter regime between the two three-dimensional topological insulator phases. Using dynamical mean-field theory combined with the topological Hamiltonian approach we find stabilization of these three-dimensional topological states due to the Hubbard interaction. We study surface states which exhibit an asymmetry between left and right surfaces originating from the broken parity symmetry of the system. Our results set the stage for further research on inhomogeneous three-dimensional topological systems, proximity effects, topological Mott insulators, nontrivially linked nodal line semimetals, and circuit-based quantum simulators.

In the area of non-Hermitian physics, there has been increasing research interest in photonics. Recently, this interest has expanded to topological systems in which symmetry and topology intertwine with non-Hermiticity, giving rise to many intriguing physical effects. One of the major tasks in exploring topological systems is unveiling the bulk-boundary correspondence in the presence of non-Hermiticity. Several proposals have been put forward in this vein, including non-Bloch bulk-boundary correspondence and the non-Hermitian skin effect. However, its practical realization has remained elusive thus far. In this paper, we demonstrate a feasible design of a one-dimensional non-Hermitian Su-Schrieffer-Heeger model based on photonic coupled resonant optical waveguides (CROWs). We show that non-Hermitian asymmetric coupling can be realized by the judicious design of optical gain and loss elements into unidirectional coupling link rings. The phase transition points of a technically achievable CROW open chain are different from those of the periodic boundary, thus revealing the non-Bloch bulk-boundary correspondence. Moreover, the field distribution is found to be exponentially localized at the ends of an open-boundary chain, which demonstrates the non-Hermitian skin effect. Our results underpin the fundamental importance as well as potential applications in various optical devices such as optical couplers, beam splitters, lasers, optical trapping, etc.

We study electron scattering from a closed magnetic structure embedded in the top surface of a topological insulator. Outside the structure there is a uniform layer of ferromagnetic insulators, leading to a positive effective mass for the Dirac electrons. The mass inside the structure can be engineered to be negative, leading to a skyrmion structure. The geometric shape of the structure can be circular or deformed, leading to integrable or chaotic dynamics, respectively, in the classical limit. For a circular structure, the relativistic quantum scattering characteristics can be calculated analytically. For a deformed structure, we develop an efficient numerical method, the multiple-multipole method, to solve the scattering wave functions. We find that, for scattering from a skyrmion, an anomalous Hall effect, as characterized by strong skew scattering, can arise, which is robust against structural deformation due to the emergence of resonant modes. In the short- (long-)wavelength regime, the resonant modes manifest themselves as confined vortices (excited edge states). The origin of the resonant states is the spin phase factor of massive Dirac electrons at the skyrmion boundary. Further, in the short-wavelength regime, for a circular skyrmion, a large number of angular momentum channels contribute to the resonant modes. In this regime, in principle, classical dynamics is relevant, but we find that geometric deformations, even those as severe as leading to fully developed chaos, have little effect on the resonant modes. The vortex structure of the resonant states makes it possible to electrically charge the skyrmion, rendering feasible the electrical manipulation of its motion. In the long-wavelength regime, only the lowest angular momentum channels contribute to the resonant modes, making the skew scattering sharply directional. These phenomena can be exploited for applications in generating dynamic skyrmions for information storage and in Hall devices.

Semimetals exhibiting nodal lines or nodal surfaces represent a novel class of topological states of matter. While conventional Weyl semimetals exhibit momentum-space Dirac monopoles, these more exotic semimetals can feature unusual topological defects that are analogous to extended monopoles. In this work, we describe a scheme by which nodal rings and nodal spheres can be realized in synthetic quantum matter through well-defined periodic-driving protocols. As a central result of our work, we characterize these nodal defects through the quantum metric, which is a gauge-invariant quantity associated with the geometry of quantum states. In the case of nodal rings, where the Berry curvature and conventional topological responses are absent, we show that the quantum metric provides an observable signature for these extended topological defects. Besides, we demonstrate that quantum-metric measurements could be exploited to directly detect the topological charge associated with a nodal sphere. We discuss possible experimental implementations of Floquet nodal defects in few-level atomic systems, paving the way for the exploration of Floquet-induced extended monopoles in quantum matter.

Topological protection of transmission has been demonstrated in the edge channel in topological insulators in the face of bent paths and on-site randomness in the structure. Microwave measurements and couple-mode theory of a topological medium possessing time-reversal symmetry with a cavity adjacent to the edge channel show that spin is not conserved in scattering from the defect. The transmission time is negative near resonance with the defect.