The orbital angular momentum (OAM) of electron states is an essential ingredient for topological and quantum geometric quantities in solids. For example, Dirac surface states with helical spin- and orbital-angular momenta are a hallmark of a 3D topological insulator. Angle-resolved photoemission spectroscopy (ARPES) with variable circular light polarization, known as circular dichroism (CD), has been assumed to be a direct probe of OAM and, by proxy, of the Berry curvature of electronic bands in energy- and momentum-space. Indeed, topological surface states have been shown to exhibit angle-dependent CD (CDAD), and more broadly, CD is often interpreted as evidence of spin-orbit coupling. Meanwhile, it is well-established that CD originates from the photoemission matrix elements, which can have extrinsic contributions related to the experimental geometry and the inherently broken inversion symmetry at the sample surface. Therefore, it is important to broadly examine CD-ARPES to determine the scenarios in which it provides a robust probe of intrinsic material physics. We performed CD-ARPES on the canonical topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ over a wide range of incident photon energies. Not only do we observe angle-dependent CD in the surface states, as expected, but we also find CD of a similar magnitude in virtually all bulk bands. Since OAM is forbidden by inversion symmetry in the bulk, we conclude this originates from symmetry-breaking in the photoemission process. Comparison with theoretical calculations supports this view and suggests that “hidden” OAM—localized to atomic sites within each unit cell—contributes significantly. Additional effects, including inter-atomic interference and final-state resonances, are responsible for the rapid variation of the CDAD signal with photon energy.

We present a theoretical framework connecting the pseudo-Chern number in momentum space to the spectral flow index in phase space for continuous media, with specific applications to topological Langmuir-cyclotron waves (TLCWs) in magnetized plasmas at uniform finite temperatures. By deriving a rigorous correspondence between these two topological invariants, we provide a solid justification for previous studies that applied this relationship heuristically across various continuous media. For magnetized plasmas with finite-temperature effects, we confirm the existence of TLCWs through numerical computation of bulk Chern number differences and analytical calculation of the spectral flow index. These findings advance the understanding of topological phenomena in continuous media.

The use of machine learning to predict wave dynamics is a topic of growing interest, but commonly used deep-learning approaches suffer from a lack of interpretability of the trained models. Here, we present an interpretable machine learning framework for analyzing the nonlinear evolution dynamics of optical wave packets in complex wave media. We use sparse regression to reduce microscopic discrete lattice models to simpler effective continuum models, which can accurately describe the dynamics of the wave packet envelope. We apply our approach to valley-Hall domain walls in honeycomb photonic lattices of laser-written waveguides with Kerr-type nonlinearity and different boundary shapes. The reconstructed equations accurately reproduce the linear dispersion and nonlinear effects, including self-steepening and self-focusing. This scheme is proven free of the a priori limitations imposed by the underlying hierarchy of scales traditionally employed in asymptotic analytical methods. It represents a powerful interpretable machine learning technique of interest for advancing design capabilities in photonics and framing the complex interaction-driven dynamics in various topological materials.

We perform a systematic study of the signatures of fragile topology in over 50 nonmagnetic two-dimensional materials with formula ${\mathrm{AB}}_2$, belonging to space group $P\overline{3}m1$. Using group theory analysis in the framework of topological quantum chemistry, we find fragile bands near the Fermi level for all the materials studied. Since stable topological bands are also present in these systems, the interplay of both phases is discussed, showing that corner charges appear in over 80% of the materials and are linked to fragile topology. Using first-principles calculations, we predict fractionally charged corner charges protected by $C_3$ symmetry. Our work aims to broaden the scope of materials with experimentally accessible fragile bands.

Interacting fermionic chains exhibit extended regions of topological degeneracy of their ground states as a result of the presence of Majorana or parafermionic zero modes localized at the edges. In the opposite limit of infinite temperature, the corresponding nonintegrable spin chains, obtained via generalized Jordan-Wigner mapping, are known to host so-called almost strong zero modes, which are long-lived with respect to any bulk excitations. Here we study the fairly unexplored territory that bridges these two extreme cases of zero and infinite temperature. We blend two established techniques for states, the Lanczos series expansion and a tensor network ansatz, uplifting them to the level of operator algebra. This allows us to efficiently simulate large system sizes for arbitrarily long timescales and to extract the temperature-dependent decay rates. We observe that for the Kitaev-Hubbard model, the decay rate of the edge mode depends exponentially on the inverse temperature $\beta$, and on an effective energy scale ${\mathrm{\Delta }}_{\mathrm{eff}}$ that is greater than the thermodynamic gap of the system $\mathrm{\Delta }$.

The investigation of methods to control and optimize the physical properties of the intrinsic magnetic topological insulator (TI) ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ is a critical challenge for the development of functional materials for quantum technologies and spintronics. The promising approach is to substitute Mn atoms with Ge, Sn, or Pb atoms in the $\left({\mathrm{Mn}}_{1-x}{A}_x^{\mathrm{IV}}\right){\mathrm{Bi}}_2{\mathrm{Te}}_4$ ($A^{\mathrm{IV}}=\mathrm{Ge},\mathrm{Sn},\mathrm{Pb}$) solid solutions. This substitution enables manageable tuning of the system's magnetic and electronic properties. In this study, we present a detailed investigation of the electronic structure evolution in $\left({\mathrm{Mn}}_{1-x}{\mathrm{Pb}}_x\right){\mathrm{Bi}}_2{\mathrm{Te}}_4$ as a function of Pb concentration, using a variety of angle-resolved photoemission spectroscopy based techniques, including photon-energy-dependent studies, spin-resolved, and circular dichroism (CD) measurements. Special emphasis was placed on identifying experimental evidence of the theoretically predicted topological phase transitions (TPTs) in $\left({\mathrm{Mn}}_{1-x}{\mathrm{Pb}}_x\right){\mathrm{Bi}}_2{\mathrm{Te}}_4$ near $x\approx 50$%. The criteria for detecting TPT include the presence or absence of topological surface states (TSS) in the electronic structure, which can be identified by their characteristic helical spin structure in spin-resolved or CD spectra, as well as the closure of the bulk band gap. Our results show that the bulk gap in the ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ -like electronic structure decreases gradually with increasing Pb concentration up to 40%, where it almost closes. From 40% to 60%, the band gap remains unchanged, and above 80%, the ${\mathrm{PbBi}}_2{\mathrm{Te}}_4$ -like electronic structure emerges, with the bulk gap reopening. Additionally, the TSS were detected in $\left({\mathrm{Mn}}_{1-x}{\mathrm{Pb}}_x\right){\mathrm{Bi}}_2{\mathrm{Te}}_4$ samples with Pb concentrations up to at least 30% and beyond 80%, correlating with the regions where the bulk gap is open. However, no TSS were observed at $x\approx 55$%, indicating that the system is in topologically distinct phase compared to ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ or ${\mathrm{PbBi}}_2{\mathrm{Te}}_4$. At this concentration, the system may be in a semimetallic state or a trivial insulator phase with a very narrow bulk gap. The demonstrated tunability of the electronic structure in $\left({\mathrm{Mn}}_{1-x}{\mathrm{Pb}}_x\right){\mathrm{Bi}}_2{\mathrm{Te}}_4$ highlights its potential for further exploration in topological and spintronic applications.

Motivated by a recent experiment reporting the fractional quantum spin Hall effect in twisted ${\mathrm{MoTe}}_2$, we investigate microscopically the prospects of realizing exotic topologically ordered states beyond conventional quantum Hall physics. We show that a non-Abelian spin Hall insulator, a state of two copies of the non-Abelian Moore-Read state, can be stabilized at half filling of time-reversal conjugate Chern bands. We elucidate that the existence of this phase relies on the reduction of opposite-spin interactions at short distances to overcome the Ising ferromagnetism. Moreover, we demonstrate that band mixing provides a generic mechanism for this reduction to be achieved. Quite remarkably, we find that a renormalization of opposite-spin interactions at short distances as small as 15% of the moiré period is sufficient for a direct transition to a completely spin-unpolarized phase which supports the non-Abelian spin Hall insulator. Furthermore, we show that the non-Abelian spin Hall insulator can either break time-reversal symmetry or preserve it depending on the underlying topological order.

Higher-order topological phases (HOTPs) possess localized and symmetry-protected eigenmodes at corners and along hinges in two- and three-dimensional lattices. The numbers of these topological boundary modes will undergo quantized changes at the critical points between different HOTPs. In this work, we reveal unique higher-order topology induced by time-periodic driving at the critical points of topological phase transitions, which has no equilibrium counterparts and also goes beyond the description of gapped topological matter. Using an alternately coupled Creutz ladder and its Floquet-driven descendants as illustrative examples, we analytically characterize and numerically demonstrate the zero and $\pi$ corner modes that could emerge at the critical points between different Floquet HOTPs. Moreover, we propose a unified scheme of bulk-corner correspondence for both gapless and gapped Floquet HOTPs protected by chiral symmetry in two dimensions. Our work reveals the possibility of corner modes surviving topological transitions in Floquet systems and initializes the study of higher-order Floquet topology at quantum criticality.

We investigate interference between topological interfacial modes in a semiconductor photonic crystal platform with Dirac frequency dispersions, which can be exploited for interferometry switch. It is demonstrated both theoretically and experimentally that in a two-in/two-out structure with four topological waveguides, geometric phases of the two-component spinor wave functions of topological photonic modes accumulate at turning points of waveguides, which govern the interferences and split the electromagnetic energy into two output ports with relative power ratio tunable by the relative phase of inputs. We show that this photonic phenomenon is intimately related to the spin-momentum locking property of quantum spin Hall effect, and results from the symphonic contributions of three phase variables: the spinor phase and geometric phase upon design, and the global phase controlled from outside. The present findings open the door for manipulating topological interfacial modes, thus exposing a facet of topological physics. The topology-driven interference can be incorporated into other devices and is expected to have far-reaching impacts on advanced photonics, optomechanics, and phononics applications.

Due to the presence of flat Chern bands, moiré transition-metal dichalcogenide (TMD) bilayers are a platform to realize strongly correlated topological phases of fermions such as fractional Chern insulators. TMDs are also known to host long-lived excitons, which inherit the topology of the underlying Chern bands. For the particular example of ${\mathrm{MoTe}}_2/{\mathrm{WSe}}_2$ heterobilayers we perform a time-dependent Hartree-Fock calculation to identify a regime in the phase diagram where the excitons themselves form a topological flat band. This paves a way towards realizing strongly correlated states of bosons in moiré TMDs.

We study the quantum Hall effect (QHE) in the three-dimensional topological insulator HgTe, which features topological Dirac-type surface states in a bulk gap opened by strain. Despite the coexistence of multiple carrier subsystems, the system exhibits perfectly quantized Hall plateaus at high magnetic fields. Here we study the system using three different experimental techniques: Transport experiments, capacitance measurements including the quantum capacitance, and current distribution measurements using electrostatically sensitive scanning probe microscopy. Our key finding is that at sufficiently high magnetic fields, the different electronic subsystems merge into one, and the current in a quantum Hall plateau is distributed across the entire width of the Hall bar device.

We explore electrical transport through a point contact in strained HgTe, a three-dimensional topological insulator. In the absence of a magnetic field $B$, there is no quantization. However, under higher magnetic fields, we observe distinct noninteger conductance steps. Based on numerical tight-binding calculations and a phenomenological Landauer-Büttiker approach, we attribute these atypical, noninteger quantized plateaus to significant scattering effects at the point contact.

We have investigated the surface and bulk electronic structures of the superconducting type-II Dirac semimetal 1T-PdSeTe. The superconducting transition temperature $T_{\text{c}}=3.2$ K was almost twice as high as $T_{\text{c}}=1.6$ K in 1T-${\mathrm{PdTe}}_2$. Scanning transmission electron microscopy measurements showed homogeneously mixed Se and Te atoms in the chalcogen layers, consistent with the ${\mathrm{CdI}}_2$-type crystal structure. Angle-resolved photoemission spectroscopy measurements and density functional theory calculations indicated the existence of the topological surface states, and the overall band structures were similar to those of 1T-${\mathrm{PdTe}}_2$. These results suggest that the ${\mathrm{CdI}}_2$-type lattice symmetry dictates the band dispersion, regardless of atomic disorder in the chalcogen layers. As the electronic band dispersion and the local structures were persistent upon substitution, the enhancement of $T_{\text{c}}$ is likely associated with the chemical pressure. Our results provide insight into the effects of the solid solution on the surface and bulk electronic states as well as the superconducting transition temperature.

In the low-energy regime, the Su-Schrieffer-Heeger (SSH) model's key characteristics are encapsulated by a Dirac-type Hamiltonian in continuum space, i.e., localized states emerge at the boundaries. Building on this, we have developed an effective Hamiltonian to model ultracold interacting bosons on an SSH-like lattice through variational minimization under the mean-field approximation. To pinpoint the boundary states, we have developed an algorithm by generalizing the imaginary time propagator, where an initial state evolves under the squared Hamiltonian to converge to the targeted state. This algorithm has broader applicability, enabling the identification of specific eigenstates in various contexts. Furthermore, we draw a parallel to an experimentally physical setup involving a gas of ultracold bosons confined to an array of potential wells with alternating depths. By establishing the system's analogy with the SSH system, we apply our algorithm to investigate boundary states in the presence of interactions, demonstrating how these findings align with those of the continuous system.

Quantum algorithms provide a potential strategy for solving computational problems that are intractable by classical means. Computing the topological invariants of topological matter is one central problem in research on quantum materials, and a variety of numerical approaches for this purpose have been developed. However, the complexity of quantum many-body Hamiltonians makes calculations of topological invariants challenging for interacting systems. Here, we present two quantum circuits for calculating Chern numbers of two-dimensional quantum matter on quantum computers. Both circuits combine a gate-based adiabatic time evolution over the discretized Brillouin zone with particular phase estimation techniques. The first algorithm uses many qubits, and we analyze it using a tensor-network simulator of quantum circuits. The second circuit uses fewer qubits, and we implement it experimentally on a quantum computer based on superconducting qubits. Our results establish a method for computing topological invariants with quantum circuits, taking a step towards characterizing interacting topological quantum matter using quantum computers.

In quantum mechanics, supersymmetry (SUSY) posits an equivalence between two elementary degrees of freedom, bosons, and fermions defined by local rules. Here we apply it to find connections between bosonic and fermionic lattice models in the realm of condensed-matter physics and uncover a novel fivefold way topology it demands in these systems. At the single-particle level, our connections pair a bosonic and fermionic lattice model, either describing the hopping of number-conserving particles or local couplings between fermion parity-conserving particles. The pair are isospectral except for zero modes, such as flat bands, quadratic band touchings, and nexus points, whose existence is undergirded by the Witten index of the SUSY theory. We develop a unifying framework to formulate these SUSY connections in terms of general lattice graph correspondences. Notably, in this framework, the supercharge operator that generates SUSY is Hermitian and can itself be interpreted as a hopping Hamiltonian on a bipartite lattice, a feature that enables the discovery of materials or model lattices hosting the SUSY partners. To illustrate the power of SUSY, we present 16 use cases of SUSY, that span topics including frustrated magnets, Kitaev spin liquids, and topological superconductors, the majority of which turn out to provide insights into the discovery and design of flat bands and topological materials.

A generalization of the graphene honeycomb model to the case where each site in the honeycomb lattice contains a $n$-fold degenerate set of eigenstates of the $C_3$ symmetry has been recently proposed to describe several systems, including triangulene crystals and photonic lattices. These generalized honeycomb models are defined by $(n_a,n_b)$, the number of $C_3$ eigenstates in the $a$ and $b$ sites of the unit cell, resulting in $n_a+n_b$ bands. Thus, the (1,1) case gives the coventional honeycomb model that describes the two low-energy bands in graphene. Generalizations, such as (2,1), (2,2), and (3,3) display several nontrivial features, such as coexisting graphenelike Dirac cones with flat bands, both at zero and finite energy, as well as robust degeneracy points where a flat band and a parabolic band meet at the $\mathrm{\Gamma }$ point. Here we explore the edge states of this class of crystals, using as reference triangulene crystals, and we find several types of edge states absent in the conventional (1,1) honeycomb case, associated to the nontrivial features of the two-dimensional bands of the high-fold case. First, we find dispersive edge states associated to the finite-energy flat bands, that occur both at the armchair and zigzag termination. Second, in the case of noncentrosymmetric triangulene crystals that lead to a $S=1$ Dirac band, we have a bonding-antibonding pair of dispersive edge states, localized in the same edge so that their energy splitting is reduced as their localization increases, opposite to the conventional behavior of pairs of states localized in opposite edges. Third, for the (3,3) case, that hosts a gap separating a pair of flat conduction and valence bands, we find nondispersive edge states with $E=0$ in all edge terminations.

Distinct from familiar $s$-, $p$-, or $d$-wave pairings, the monopole superconducting order represents a novel class of pairing order arising from nontrivial monopole charge of the Cooper pair. In the weak-coupling regime, this order can emerge when pairing occurs between Fermi surfaces with different Chern numbers in, for example, doped Weyl semimetal systems. However, the phase of monopole pairing order is not well-defined over an entire Fermi surface, making it challenging to design experiments sensitive to both its symmetry and topology. To address this, we propose a scheme based on symmetry and topological principles to identify this elusive pairing order through a set of phase-sensitive Josephson experiments. By examining the discrepancy between global and local angular momentum of the pairing order, we can unveil the monopole charge of the pairing order, including for models with higher pair monopole charge $|q_p|=1,2$, and 3. We demonstrate the proposed probe of monopole pairing order through analytic and numerical studies of Josephson coupling in models of monopole superconductor junctions. This work opens a promising avenue to uncover the unique topological properties of monopole pairing orders and to distinguish them from known pairing orders based on spherical harmonic symmetry.

Sonic and photonic topological insulators that host topological edge states offer promising potentials for the resilient control of acoustic and electromagnetic waves, respectively. Despite the great progress on sonic or photonic topological insulators, the research of their integration, i.e., the phoxonic topological insulator, is less explored. In this work, we propose a phoxonic topological insulator that hosts acoustic and dual-polarization photonic helical edge states simultaneously. In specific, we first design a glide-symmetric phoxonic crystal with concurrent sonic and dual-polarization photonic bandgaps via the topology optimization method. Then by choosing two different unit cells from the optimized phoxonic crystal and assembling them to create a domain-wall interface, a phoxonic topological insulator that supports two pairs of gapless helical edge states within both the sonic and photonic bandgaps is constructed. Pseudospin-locked unidirectional transmissions and robust manipulations of helical edge states are demonstrated for acoustic and dual-polarization electromagnetic waves simultaneously in the proposed phoxonic topological insulator. The designed phoxonic topological insulator opens new avenues for developing topological photoacoustic devices, enabling the reliable management of both acoustic and electromagnetic waves, as well as the investigation of their interplay.