Experimental indications have been reported suggesting that laser pulses shining on materials with relativistic dispersion can produce currents that survive long after the illumination has died out. Such residual currents, i.e., remnant currents, have applications in petahertz logical gates. The remnant currents' strength strongly depends on the pulse shape. We develop an analytical formula that allows one to optimize the pulse shape for remnant current production; we predict remnant currents exceeding the values observed so far by up to five orders of magnitude. This can be achieved by using single-cycle pulses instead of the previously employed multicycle pulses. In fact, remnant currents can be almost as strong as the peak current under irradiation.

$\mathrm{InBi}\left(001\right)$ is formed epitaxially on $\mathrm{InAs}\left(111\right)$-A by depositing Bi onto an In-rich surface. Angle-resolved photoemission measurements reveal topological electronic surface states, close to the $\overline{\mathrm{M}}$ high symmetry point. This demonstrates a heteroepitaxial system entirely in the III–V family with topological electronic properties. InBi shows coexistence of Bi and In surface terminations, in contradiction with other III–V materials. For the Bi termination, the study gives a consistent physical picture of the topological surface electronic structure of $\mathrm{InBi}\left(001\right)$ terminated by a Bi bilayer rather than a surface formed by splitting to a Bi monolayer termination. Theoretical calculations based on relativistic density functional theory and the one-step model of photoemission clarify the relationship between the $\mathrm{InBi}\left(001\right)$ surface termination and the topological surface states, supporting a predominant role of the Bi bilayer termination. Furthermore, a tight-binding model based on this Bi bilayer termination with only Bi–Bi hopping terms, and no Bi–In interaction, gives a deeper insight into the spin texture.

We investigate the topological phases and corresponding edge states in silicene with fixed intrinsic and extrinsic Rashba spin-orbit couplings. The results show that silicene can exhibit four distinct topological phases, such as two types of valley-polarized quantum anomalous Hall effect (QAHE) and two types of QAHE that can be effectively controlled by the exchange field and electric field. In addition, these topological phases only correspond to the valley edge states distributed at the boundary of silicene. In particular, when the off-resonant circularly polarized (ORCP) light is further considered, silicene can host richer topological phases that are characterized by the Chern numbers from $C=-3$ to $C=3$, where the corresponding mixed edge states—where the valley and nonvalley edge states could coexist—can remarkably appear. More interestingly, the special higher Chern numbers $C=\pm 3$ determine two valley edge states and one nonvalley edge state, with the right-moving or left-moving direction. These peculiar edge states are attributed to the ORCP light destroying the collapse of the Dirac cones that the valley edge states bridge. Additionally, silicene can experience multiple edge-state transitions for identical Chern numbers $C=\pm 2$ between the valley edge state, nonvalley edge state, and two types of mixed edge state, under the ORCP light. This work may provide a platform and point of view for investigating other forms of topological edge state and is beneficial for designing multifunctional topological devices.

We have investigated the time-resolved low-energy charge dynamics in counterpropagating edge channels formed in the quantum Hall regime of an electron-hole bilayer system in band-inverted InAs/InGaSb composite quantum wells. In the studied magnetic field range, the system remains in the topological phase where an equal number of counterpropagating electronlike and holelike edge channels exist in the bulk energy gap, analogous to the helical edge channels in a quantum spin Hall insulator. When the Fermi level was set in the conduction band, the injected charge pulses propagated unidirectionally as chiral edge magnetoplasmon (EMP) wave packets. In contrast, in the nonchiral regime with the Fermi level set in the band gap, the injected charge pulses propagated identically in both directions with significantly reduced velocity. In addition, as the channel length increased, we observed a sharp crossover from EMP transport to diffusive transport, manifested as pronounced waveform broadening and dissipation. Simulations using a distributed capacitance model including interchannel tunneling well reproduced the observed behavior in both chiral and nonchiral regimes, revealing that interchannel charge transfer between counterpropagating channels plays a crucial role. The slow EMP transport in the nonchiral regime and its crossover to diffusive transport are consequences of the strong interchannel interaction and interchannel tunneling, respectively. Our results will thus provide insights into charge dynamics and scattering mechanism in analogous one-dimensional systems, including a quantum spin Hall insulator with helical edge modes and its expected Tomonaga-Luttinger liquid behavior.

Exploration of topology protected by a certain symmetry is central in condensed matter physics. A recent idea of subsymmetry protected (SSP) topology—remains of a broken symmetry can still protect specific topological boundary states—has been developed and demonstrated in a photonic system [Nat. Phys. 19, 992 (2023)], but not in topological superconductors. Here, we extend this idea further by applying subsymmetry protecting perturbation (SSPP) to one-dimensional topological insulating and superconducting systems using the Su-Schrieffer-Hegger (SSH) and Kitaev models. Using the tight-binding and low-energy effective theory, we show that the SSP boundary states retain topological properties while the SSPP results in the asymmetry of boundary states. For the SSH model, an SSP zero-energy edge state localized on one edge possesses quantized polarization. In contrast, the other edge state is perturbed to have nonzero energy, and its polarization is not quantized. For topological superconductors, SSP zero-energy Majorana boundary states for spinful Kitaev models emerge on only one edge, contrary to the conventional belief that Majorana fermions emerge at opposite edges. Our findings can be used as a platform to expand our understanding of topological materials as they broaden our understanding of the symmetry in a topological system and a method to engineer Majorana fermions.

There is extensive current interest in electronic topology in correlated settings. In strongly correlated systems, contours of Green's function zeros may develop in frequency-momentum space, and their role in correlated topology has increasingly been recognized. However, whether and how the zeros contribute to electronic properties is a matter of uncertainty. Here we address the issue in an exactly solvable model for a Mott insulator. We show that the Green's function zeros contribute to several physically measurable correlation functions in a way that does not run into inconsistencies. In particular, the physical properties remain robust to chemical potential variations up to the Mott gap, as it should be based on general considerations. Our work sets the stage for further understandings of the rich interplay among topology, symmetry, and strong correlations.

The mixed-valence compound ${\mathrm{SmB}}_6$ with partially filled samarium $4f$ flat bands hybridizing with $5d$ conduction bands is a paramount example of a correlated topological heavy-fermion system. In this study, we revisit the topology of ${\mathrm{SmB}}_6$ within the formalism of topological quantum chemistry and symmetry-based indicators. Based on this approach, we reach a detailed classification of the material's topology, and we reformulate the origin of topology in terms of band representations. We corroborate these aspects by constructing a minimal TB model that is consistent with all the symmetries of the system and is able to reproduce the topology of the material. We finally unveil the previously overlooked formation of multiple topological gaps, and we discuss its implications for experiments.

The recent observation of zero-biased photocurrent on the surface of topological insulators allows to identify spin-orbit-coupled two-dimensional Dirac cones as ideal platforms for the manipulation of the tilt of Dirac cone. We show that the in-plane effective magnetic field $\tilde{B}$ implements a moving frame transformation on the topological insulators' helical surface states. As a result, photo-excited electrons on the surface undergo a Galilean boost proportional to the effective in-plane magnetic field $\tilde{B}$. The boost velocity is transversely proportional to $\tilde{B}$. This explains why the experimentally observed photocurrent depends linearly on $\tilde{B}$. Our theory, while consistent with the observation that at leading order the effect does not depend on the polarization of the incident radiation, at next leading order in $\tilde{B}$ predicts a polarization dependence in both parallel and transverse directions to the polarization. We also predict two induced Fermi-surface effects that can serve as further confirmation of our moving frame theory. Based on the estimated value $\zeta \approx 0.34$ of the tilt parameter for a magnetic field of $\tilde{B}\sim 3\mathrm{T}$, our geometric picture qualifies the surface Dirac cone of magnetic topological insulators as an accessible platform for the synthesis and experimental investigation of strong analog gravitational phenomena.

Identification of a nontrivial ${\mathbb{Z}}_2$ index in a spinful two-dimensional insulator indicates the presence of an odd, quantized (pseudo)spin-resolved Chern number, ${\mathcal{C}}_s=({\mathcal{C}}_{\uparrow }-{\mathcal{C}}_{\downarrow })/2$. However, the statement is not biconditional. An odd spin-Chern number can survive when the familiar ${\mathbb{Z}}_2$ index vanishes. Identification of solid-state systems hosting an odd, quantized ${\mathcal{C}}_s$ and trivial ${\mathbb{Z}}_2$ index is a pressing issue due to the potential for such insulators to admit band gaps optimal for experiments and quantum devices. Nevertheless, they have proven elusive due to the computational expense associated with their discovery. In this work, a neural network capable of identifying the spin-Chern number is developed and used to identify the first solid-state systems hosting a trivial ${\mathbb{Z}}_2$ index and odd ${\mathcal{C}}_s$. We demonstrate the potential of one such system, ${\mathrm{Ti}}_2{\mathrm{CO}}_2$, to support Majorana corner modes via the superconducting proximity effect.

We establish the quantum mechanics of composite fermions based on the dipole picture initially proposed by Read. It comprises three complimentary components: a wave equation for determining the wave functions of a composite fermion in ideal fractional quantum Hall states and when subjected to external perturbations, a wave-function Ansatz for mapping a many-body wave function of composite fermions to a physical wave function of electrons, and a microscopic approach for determining the effective Hamiltonian of the composite fermion. The wave equation resembles the ordinary Schrödinger equation but has drift velocity corrections that are not present in the Halperin-Lee-Read theory. The wave-function Ansatz constructs a physical wave function of electrons by projecting a state of composite fermions onto a half-filled bosonic Laughlin state of vortices. Remarkably, Jain's wave-function Ansatz can be reinterpreted as the new Ansatz in an alternative wave-function representation of composite fermions. The dipole picture and the effective Hamiltonian can be derived from the microscopic model of interacting electrons confined in a Landau level, with all parameters determined. In this framework, we can construct the physical wave function of a fractional quantum Hall state deductively by solving the wave equation and applying the wave-function Ansatz, based on the effective Hamiltonian derived from first principles, rather than relying on intuition or educated guesses. For ideal fractional quantum Hall states in the lowest Landau level, the approach reproduces the well-established results of the standard theory of composite fermions. We further demonstrate that the reformulated theory of composite fermions can be easily generalized for flat Chern bands.

We propose a new platform to detect signatures of the presence of Majorana bound states (MBSs) in the Fraunhofer pattern of Josephson junctions featuring quantum spin Hall edge states on the normal part and Majorana bound states at the NS interfaces. We use a tight-binding model to demonstrate a drastic change in the periodicity of the Fraunhofer pattern when comparing trivial and nontrivial regimes. We explain these results in terms of the presence of additional parallel-spin electron-hole reflections, which due to the spin-momentum locking, occur as cross Andreev reflections, accumulating a different magnetic flux and yielding a change in the Fraunhofer periodicity. We show that this detection scheme exhibits some advantages compared to previous ones as it is robust against disorder and finite temperature and works in equilibrium. Furthermore, we introduce a scattering model that captures the main results of the microscopic calculations with MBSs and extend our discussion to the main differences found using accidental zero-energy Andreev bound states.

A delicate tension complicates the relationship between the topological magnetoelectric effect (TME) in three-dimensional (3D) ${\mathbb{Z}}_2$ topological insulators (TIs) and time-reversal symmetry (TRS). TRS underlies a particular ${\mathbb{Z}}_2$ topological classification of the electronic ground state of crystalline band insulators and the associated quantization of the magnetoelectric response coefficient calculated using bulk linear-response theory but, according to standard symmetry arguments, simultaneously forbids a nonzero magnetoelectric coefficient in any physical finite-size system. This contrast between theories of magnetoelectric response in formal bulk models and in real finite-sized materials originates from the distinct approaches required to introduce notions of (electronic) polarization and orbital magnetization in these fundamentally different environments. In this work we argue for a modified interpretation of the bulk linear-response calculations in nonmagnetic ${\mathbb{Z}}_2$ TIs that is more plainly consistent with TRS and use this interpretation to discuss the effect's observation—still absent over a decade after its prediction. Our analysis is reinforced by microscopic bulk and thin-film calculations carried out using a simplified but still realistic effective model for the well established ${\mathrm{V}}_2{\mathrm{VI}}_3 [\text{V}=\text{(Sb,}\text{Bi)}$ and $\text{VI}=\text{(Se,}\text{Te)}]$ family of nonmagnetic ${\mathbb{Z}}_2$ TIs. When a uniform dc magnetic field is included in this model, the anomalous $n=0$ Landau levels (LLs) play the central role, both in thin films and in bulk. In the former case, only the $n=0$ LL eigenfunctions can support a dipole moment, which vanishes if there are no magnetic surface dopants and is quantized in the thick-film limit if magnetic dopants at the top and bottom surfaces have opposite orientation. In the latter case, the Hamiltonian projected into the $n=0$ LL subspace is a one-dimensional Su-Schrieffer-Heeger model with ground-state polarization that is quantized in accordance with the bulk linear-response coefficient calculated for (a lattice regularization of) the starting 3D model. Motivated by analytical results, we conjecture a type of microscopic bulk-boundary correspondence: a bulk insulator with (generalized) TRS supports a magnetoelectric coefficient that is purely itinerant (which is generically related to the geometry of the ground state) if and only if magnetic surface dopants are required for the TME to manifest in finite samples thereof. We conclude that in nonmagnetic ${\mathbb{Z}}_2$ TIs the TME is activated by magnetic surface dopants, that the charge-density response to a uniform dc magnetic field is localized at the surface and specified by the configuration of those dopants, and that the TME is qualitatively less robust against disorder than the integer quantum Hall effect.

Anomalous topological phases, where edge states coexist with topologically trivial Chern bands, can only appear in periodically driven lattices. When the driving is smooth and continuous, the bulk-edge correspondence is guaranteed by the existence of a bulk invariant known as the winding number. However, in lattices subject to periodic discrete step walks the existence of edge states does not only depend on bulk invariants but also on the boundary. This is a consequence of the absence of an intrinsic time dependence or micromotion in discrete step walks. We report the observation of edge states and a simultaneous measurement of the bulk invariants in anomalous topological phases in a two-dimensional synthetic photonic lattice subject to discrete step walks. The lattice is implemented using time multiplexing of light pulses in two coupled fibre rings, in which one of the dimensions displays real-space dynamics and the other one is parametric. The presence of edge states is inherent to the periodic driving and depends on the properties of the boundary in the implemented two-band model with zero Chern number. We provide a suitable expression for the topological invariants whose calculation does not rely on micromotion dynamics.

We present a direct comparison between the stroboscopic and nonstroboscopic effective approaches for ultracold atoms in shaken honeycomb lattices, focusing specifically on the optimal driving introduced by A. Verdeny and F. Mintert [Phys. Rev. A 92, 063615 (2015)]. In the fast-driving regime, we compare the effective nonstroboscopic Hamiltonian derived through a perturbative expansion with a nonperturbative calculation of the stroboscopic Floquet Hamiltonian, obtained through a direct numerical approach. We find that the leading off-diagonal tunneling coefficients, which define the undriven model, retain their isotropic nature under the effect of the driving, and their values are independent of the computational approach used. This consistency ensures the robustness of certain topological properties, such as the position of Dirac points, as expected. Conversely, the next-to-leading diagonal tunneling coefficients, generated dynamically, can vary across different approaches without affecting the system topology. This suggests that the topological phase diagram is most conveniently represented in terms of the physical parameters characterizing the driving and directly accessible in experiments, rather than in terms of the tunneling parameters.

Topological insulators have been studied intensively over the last decades. Earlier research focused on Hermitian Hamiltonians, but recently, peculiar and interesting properties were found by introducing non-Hermiticity. In this work, we apply a quantum geometric approach to various Hermitian and non-Hermitian versions of the Su-Schrieffer-Heeger (SSH) model. We find that this method allows one to correctly identify different topological phases and topological phase transitions for all SSH models, but only when using the metric tensor containing both left and right eigenvectors. Whereas the quantum geometry of Hermitian systems is Riemannian, introducing non-Hermiticity leads to pseudo-Riemannian and complex geometries, thus significantly generalizing from the quantum geometries studied thus far. One remarkable example of this is the mathematical agreement between topological phase transition curves and lightlike paths in general relativity, suggesting a possibility of simulating space-time in non-Hermitian systems. We find that the metric in non-Hermitian phases degenerates in such a way that it effectively reduces the dimensionality of the quantum geometry by one. This implies that within linear response theory, one can perturb the system by a particular change of parameters while maintaining a zero excitation rate.

Topological phase transitions go beyond Ginzburg and Landau's paradigm of spontaneous symmetry breaking and occur without an associated local order parameter. Instead, such transitions can be characterized by the emergence of nonlocal order parameters, which require measurements on extensively many particles simultaneously—an impossible venture in real materials. On the other hand, quantum simulators have demonstrated such measurements, making them prime candidates for experimental confirmation of nonlocal topological order. Here, building upon the recent advances in preparing few-particle fractional Chern insulators using ultracold atoms and photons, we propose a realistic scheme for detecting the hidden off-diagonal long-range order (HODLRO) characterizing Laughlin states. Furthermore, we demonstrate the existence of this hidden order in fractional Chern insulators, specifically for the $\nu =1/2$-Laughlin state in the isotropic Hofstadter-Bose-Hubbard model. This is achieved by large-scale numerical density matrix renormalization group (DMRG) simulations based on matrix product states, for which we formulate an efficient sampling procedure providing direct access to HODLRO in close analogy to the proposed experimental scheme. We confirm the characteristic power-law scaling of HODLRO, with an exponent $1/\nu =2$, and show that its detection requires only a few thousand snapshots. This makes our scheme realistically achievable with current technology and paves the way for further analysis of nonlocal topological orders, e.g., in topological states with non-Abelian anyonic excitations.

We investigate the transport and energy spectrum properties of a three-dimensional high-order topological structure formed by stacking two-dimensional square diatomic Chern insulator lattices. Electron-hole symmetry and the energy spectrum degeneracy at individual points in the semimetallic phase are proven to be due to chiral and antiunitary symmetries in the periodic system. Additionally, we explore the influence of boundary conditions in a slab system with varying surface atom connectivity, and we demonstrate analytically the presence of zero-energy surface states in specific configurations. Moreover, we describe the emergence of two chiral hinge states driven by a perpendicular phase in the nanowire geometry. Next, the quantum Hall resistance is computed in the cross-configuration of a four-lead device. In this paper, we demonstrate that the trajectories of hinge states, determined by the number of layers in parallelepiped finite structure, give rise to fractional Hall plateaus.

Symmetry is one of the cornerstones of modern physics and has profound implications in different areas. In symmetry-protected topological systems, symmetries are responsible for protecting surface states, which are at the heart of the fascinating properties exhibited by these materials. When the symmetry protecting the edge mode is broken, the topological phase becomes trivial. By engineering losses that break the symmetry protecting a topological Hermitian phase, we show that a new genuinely non-Hermitian symmetry emerges, which protects and selects one of the boundary modes: the topological monomode. Moreover, the topology of the non-Hermitian system can be characterized by an effective Hermitian Hamiltonian in a higher dimension. To corroborate the theory, we experimentally investigated the non-Hermitian one- and two-dimensional SSH models using photonic lattices and observed dynamically generated monomodes in both cases. We classify the systems in terms of the (non-Hermitian) symmetries that are present and calculate the corresponding topological invariants.