It is conventional wisdom that the helical edge states of a quantum spin Hall (QSH) insulator are particularly stable due to the topological protection of time-reversal symmetry. Here, we report experimental observation of an edge-dependent quantum (pseudo)spin Hall effect by employing two Kekulé electric circuits with molecule-zigzag and partially bearded edges, where the chirality of the circulating current in the unit cell mimics the electron spin. We observe a helicity flipping of the topological in-gap modes emerging in opposite parameter regions for the two edge geometries. Experimental findings are interpreted in terms of the mirror winding number defined in the unit cell, the choice of which exclusively depends on the edge shape. Our work offers a deeper understanding of the boundary effect on the QSH phase and paves the way for studying the spin-dependent topological physics in electric circuits.

The topological invariants of band insulators are usually assumed to depend only on the connectivity between orbitals and not on their intracell position (orbital embedding), which is a separate piece of information in the tight-binding description. For example, in two dimensions, the orbital embedding is known to change the Berry curvature but not the Chern number. Here, we consider one-dimensional inversion-symmetric insulators classified by a ${\mathbb{Z}}_2$ topological invariant $\vartheta =0$ or $\pi$, related to the Zak phase, and show that $\vartheta$ crucially depends on orbital embedding. We study three two-band models with bond, site, or mixed inversion: the Su-Schrieffer-Heeger model (SSH), the charge density wave model (CDW), and the Shockley model. The SSH (resp. CDW) model is found to have a unique phase with $\vartheta =0$ (resp. $\pi$). However, the Shockley model features a topological phase transition between $\vartheta =0$ and $\pi$. The key difference is whether the two orbitals per unit cell are at the same or different positions.

We present a methodology to address, from first principles, charge-spin interconversion in two-dimensional materials with spin-orbit coupling. Our study relies on an implementation of density functional theory based quantum transport formalism adapted to such purpose. We show how an analysis of the $k$-resolved spin polarization gives the necessary insight to understand the different charge-spin interconversion mechanisms. We have tested it in the simplest scenario of isolated graphene in a perpendicular electric field where effective tight-binding models are available to compare with. Our results show that the flow of an unpolarized current across a single layer of graphene produces, as expected, a spin separation perpendicular to the current for two of the three spin components (out-of-plane and longitudinal), which is the signature of the spin Hall effect. Additionally, it also yields an overall spin accumulation for the third spin component (perpendicular to the current), which is the signature of the Rashba-Edelstein effect. Even in this simple example, our results reveal an unexpected competition between the Rashba and the intrinsic spin-orbit coupling. Remarkably, the sign of the accumulated spin density does not depend on the electron or hole nature of the injected current for realistic values of the Rashba coupling.

We consider subgap bands induced in a two-dimensional superconductor by a densely packed chain of magnetic moments with ferromagnetic or spiral alignments. We show that by contrast with sparsely packed chains a consistent description requires that all wavelengths are taken into account for the scattering at the magnetic moments. The resulting subgap states are a composition of Yu-Shiba-Rusinov-type states and magnetic scattering states, whose mixture becomes especially important to understand the nature and dimensional renormalization of gap closures for spiral magnetic alignments under increasing scattering strength, particularly as the spiral becomes commensurate with the Fermi wavelength. The results are fully analytic in the form of Green's functions and provide the tools for further analysis of the properties of the subgap states.

We investigate the topological classification of the subgap bands induced in a two-dimensional superconductor by a densely packed chain of magnetic moments with ferromagnetic or spiral alignments. The wave functions for these bands are composites of Yu-Shiba-Rusinov-type states and magnetic scattering states and have a significant spatial extension away from the magnetic moments. We show that this spatial structure prohibits a straightforward extraction of a Hamiltonian useful for the topological classification. To address the latter correctly, we construct a family of spatially varying topological Hamiltonians for the subgap bands adapted for the broken translational symmetry caused by the chain. The spatial dependence in particular captures the transition to the topologically trivial bulk phase when moving away from the chain by showing how this, necessarily discontinuous, transition can be understood from an alignment of zeros with poles of Green's functions. Through the latter, the topological Hamiltonians reflect a characteristic found otherwise primarily in strongly interacting systems.

We study theoretically AB-stacked honeycomb bilayers driven by light in resonance with an infrared phonon within a tight-binding description. We characterize the phonon properties of honeycomb bilayers with group theory and construct an electronic time-dependent tight-binding model for the system following photoexcitation in resonance with an infrared phonon. We adopt an atomically adiabatic approximation [V. Mohanty and E. J. Heller, Proc. Natl. Acad. Sci. 116, 18316 (2019)] to describe classically vibrating nuclei, but obtain the Floquet quasienergy spectrum associated with the time-dependent model exactly. We introduce a general scheme to disentangle the complex low-frequency Floquet spectrum to elucidate the relevant Floquet bands. As a prototypical example, we consider bilayer graphene. We find that light in the low-frequency regime can induce a band gap in the quasienergy spectrum in the vicinity of the $K$ points even if it is linearly polarized, in contrast with the expectations within the Born-Oppenheimer approximation and the high-frequency regime. Finally, we analyze the diabaticity of the driven electron and driven phonon processes and find contrasting effects on the autocorrelation functions at the same driving frequency: Driven phonons preserve the character of the initial state while driven electrons exhibit strong deviations within a few drive cycles. The procedure outlined here can be applied to other materials to describe the combined effects of low-frequency light on phonons and electrons.

The transition from insulator to conductor can be realized in some materials but requires modification of both the arrangement of atoms and their electronic configurations. This is often achieved by doping. Here we reveal a different mechanism the lattice may adopt to induce such a transition. Experiments showed the surprising finding that limited exposure to subband-gap light caused a permanent transition from an insulator state to a conductor state in the insulating oxide ${\mathrm{Ga}}_2{\mathrm{O}}_3$, with a nine orders of magnitude increase in electronic conduction. Furthermore, annealing up to 400 °C did not suppress or decrease the induced conductivity. Photoexcitation by light-induced modification in the charge state of defects and subsequent lattice distortion around them was suggested to be the underlying mechanism behind this transition. Density functional theory calculations confirmed that modifying the charge state of defects leads to redistribution of the localized electrons and massive structural distortion in the surrounding lattice, causing large shifts in the density of states and introducing new states with shallower energy levels. Both experimental and theoretical results revealed the introduction of stable shallow energy levels, explaining the mechanism behind the transition from an insulator to a conductor state by light. We suggest that this mechanism may occur in other wide band-gap metal oxides leading to drastic modification in their electronic properties.

The phonon Boltzmann transport equation (BTE) has been widely utilized to study thermal transport in solids. While for a number of materials the exact solution to the BTE has been obtained for a uniform heat flow, problems arising in micro/nanoscale heat transport have been analyzed within the relaxation time approximation (RTA). Since the RTA breaks down at temperatures low compared to the Debye temperature, this approximation prevents the study of an important class of high Debye temperature materials such as diamond, graphite, graphene, and some other two-dimensional materials. We present a full scattering matrix formalism that goes beyond the RTA approximation and obtain a Green's function solution for the linearized BTE, which leads to an explicit expression for the phonon distribution and temperature field produced by an arbitrary spatiotemporal distribution of heat sources in an unbounded medium. The presented formalism is capable of describing a wide range of phenomena, from heat dissipation by nanoscale hot spots to the propagation of second sound waves. We provide numerical results for graphene for a spatially sinusoidal heating profile and discuss the importance of using the full scattering matrix compared to the RTA.

The generation of spin-entangled electrons is an important prerequisite for future solid-state quantum technologies. Cooper pairs in a superconductor can be split into separate electrons in a spin-singlet state, however, detecting their entanglement remains an open experimental challenge. Proposals to detect the entanglement by violating a Bell inequality typically require a large number of current cross-correlation measurements, and not all entangled states can be detected in this way. Here, we instead formulate an entanglement witness that can detect the spin-entanglement using only three cross-correlation measurements of the currents in the outputs of a Cooper pair splitter. We identify the optimal measurement settings for witnessing the entanglement, and we illustrate the use of our entanglement witness with a realistic model of a Cooper pair splitter for which we evaluate the cross-correlations of the output currents. Specifically, we find that the entanglement of the spins can be detected even with a moderate level of decoherence. Our work thereby paves the way for an experimental detection of the entanglement produced by Cooper pair splitters.

We discuss the optical forces exerted on parity-time $\left(\mathcal{PT}\right)$ symmetric heterostructures under normal incidence of a single and two counterpropagating plane waves. The underlying strategy is through generalized parametric space, stemming from consideration of the $\mathcal{PT}$-symmetry condition and Lorentz reciprocity theorem. In such a generalized parametric space, we are able to not only exhaustively indicate various $\mathcal{PT}$ phases and extraordinary wave phenomena but also deduce the directionality and magnitudes of optical forces. We find that when the system is illuminated by a normally incident wave, it can exhibit the symmetric pushing effect in the exact symmetry phase, unidirectional null, and bidirectional null forces (BNF) at the exceptional point, and pulling-pushing flipped forces in the broken symmetry phase, with BNF found at the pushing-pulling turning point. In two counterpropagating plane wave interferences, the magnitudes as well as the directionality of the resultant optical force can be tuned by a relative phase of incident waves. More interestingly, we observe that a null force independent of the relative phase occurred in a specific region of the broken symmetry phase and exceptional point. In addition, we offer several $\mathcal{PT}$-symmetric heterostructures to support our findings. Our results may benefit applications in $\mathcal{PT}$ optomechanics and force rectifiers.

We propose two mechanisms to realize the second-order topological insulator (SOTI) state in spinless hexagonal lattices, viz., chemical modification and anti-Kekulé/Kekulé distortion of hexagonal lattices. Correspondingly, we construct two models and demonstrate the nontrivial band topology of the SOTI state characterized by the second Stiefel-Whitney class $w_2$ in the presence of inversion symmetry $\left(P\right)$ and time-reversal symmetry $\left(T\right)$. Based on the two mechanisms and using first-principles calculations and symmetry analysis, we predict three categories of real light element material candidates, i.e., hydrogenated and halogenated two-dimensional (2D) hexagonal group-IV materials $XY$ $(X=\text{C}, \mathrm{Si}, \mathrm{Ge}, \mathrm{Sn},$ $Y=\text{H}, \mathrm{F}, \mathrm{Cl})$, 2D hexagonal group-V materials (blue phosphorene, blue arsenene, and black phosphorene, black arsenene), and the recent experimentally synthesized anti-Kekulé/Kekulé order graphenes and the counterparts of silicene/germanene/stanene. We explicitly demonstrate the nontrivial topological invariants and the existence of the protected corner states with a fractional charge for these candidates with a giant bulk band gap (up to 3.5 eV), which could facilitate the experimental verification by scanning tunneling microscopy. Our approaches and proposed abundant real material candidates will greatly enrich 2D SOTIs and promote their intriguing physics research.

The charge current in a topological insulator (TI) will induce a spin accumulation (Edelstein effect or EE), from which the spin current will be generated. Inversely, the spin current injection into the TI will induce a charge current called the inverse Edelstein effect (IEE). Some experimental and theoretical works have been done for the understanding of either EE or IEE. However, little experimental work incorporating both processes in the same TI sample has been done. In this work, we propose a phenomenological model to understand EE and IEE in the TI-based system. Based on this model, efficiencies of EE and IEE can be directly derived, which is consistent with previous theoretical work based on Boltzmann transport theory and obeying the energy conservation law. We also measure EE and IEE efficiencies experimentally in a TI/Ru/CoFeB system by spin-torque ferromagnetic resonance and spin pumping, respectively. The experimental results are consistent with our model, which proves that the spin-charge conversion in TI can be understood in the framework of (I)EE instead of (inverse) spin Hall effect. By combining theories and experiments, we find that enhancing interfacial transparency is crucial for enhancing EE efficiency, and avoiding metallic contact is crucial for enhancing IEE efficiency.

Muons implanted into ionic fluorides often lead to a so-called F–$\mu$–F state, in which the time evolution of the muon spin contains information about the geometry and nature of the muon site. Nuclei more distant from the muon than the two nearest-neighbor fluorine ions result in decoherence of the F–$\mu$–F system, and this can yield additional quantitative information about the state of the muon. We demonstrate how this idea can be applied to the determination of muon sites within the ionic fluorides $\alpha \text{−}{\mathrm{PbF}}_2$ and ${\mathrm{YF}}_3$, which contain fluoride ions in different crystallographic environments. Our results can be used to distinguish between different crystal phases and provide strong evidence for the existence of anion Frenkel defects in $\alpha \text{−}{\mathrm{PbF}}_2$.

${\mathrm{ErTe}}_3$ is a model system used to explore thermal transport in a layered charge density wave (CDW) material. We present thermal diffusivity, resistivity, and specific-heat data: There is a sharp decrease in thermal conductivity parallel and perpendicular to the primary CDW at the CDW transition temperature. Yet, the resistivity changes more gradually. Using the Wiedemann-Franz law well above and below $T_c$ a consistent description of the thermal transport applies with essentially independent electron and phonon contributions. In the critical regime, no such description is possible; the observed behavior corresponds to a strongly coupled electron-phonon critical “soup.”

We propose novel triple-$\mathit{q}$ multipole orders as possible candidates for the two distinct low-temperature symmetry broken phases in quadrupolar system ${\mathrm{PrV}}_2{\mathrm{Al}}_{20}$. An analysis of the experiment under [111] magnetic fields indicates that the ferro octupole moments in the lower temperature phase arise from the antiferro octupole interactions. We demonstrate that the triple-$\mathit{q}$ multipole orders can solve this seemingly inconsistent issue. Anisotropies of quadrupole moments stabilize a triple-$\mathit{q}$ order, which further leads to the second transition to a coexisting phase with triple-$\mathit{q}$ octupole moments. The cubic invariant of quadrupole moments formed by the triple-$\mathit{q}$ components and the characteristic couplings with the octupole moments in their free energy play important roles. We analyze a multipolar exchange model by mean-field approximation and discuss the temperature and magnetic field phase diagrams. Many of the microscopic results, such as the number of phases and the magnitudes of critical fields in the phase diagrams, are qualitatively consistent with the experiments in ${\mathrm{PrV}}_2{\mathrm{Al}}_{20}$.

Nonlinear photoionization of dielectrics and semiconductors is widely treated in the framework of the Keldysh theory whose validity is limited to photon energies that are small compared to the band gap and relatively low laser intensities. The time-dependent density functional theory (TDDFT) simulations, which are free of these limitations, enable one to gain insight into nonequilibrium dynamics of the electronic structure. Here we apply TDDFT to investigate the photoionization of silicon crystal by ultrashort laser pulses in a wide range of laser wavelengths and intensities and compare the results with predictions of the Keldysh theory. Photoionization rates derived from the simulations considerably exceed the data obtained with the Keldysh theory within the validity range of the latter. Possible reasons for the discrepancy are discussed and we provide fundamental data on the photoionization rates beyond the limits of the Keldysh theory. By investigating the features of the Stark shift as a function of photon energy and laser field strength, a manifestation of the transient Wannier-Stark ladder states is revealed, which become blurred with increasing laser field strength. Finally, it is shown that the TDDFT simulations can potentially provide reliable data on the electron damping time that is of high importance for large-scale modeling.

Two-dimensional topological insulators (2DTIs), which host the quantum spin Hall (QSH) effect, are one of the key materials in next-generation spintronic devices. To date, experimental evidence of the QSH effect has only been observed in a few materials, and thus, the search for new 2DTIs is at the forefront of physical and materials science. Here, we report experimental evidence of a 2DTI in the van der Waals material ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$. First-principles calculations show that each monolayer of ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ is a 2DTI with weak interlayer interactions. Combined transport, angle-resolved photoemission spectroscopy, and scanning tunneling microscopy measurements confirm the existence of a band gap at the Fermi level and topological edge states inside the gap. These results demonstrate that ${\mathrm{Ta}}_2{\mathrm{Pd}}_3{\mathrm{Te}}_5$ is a promising material for fabricating spintronic devices based on the QSH effect.

The topological classification of fermion systems in mixed states is a long-standing quest. For Gaussian states, reminiscent of noninteracting unitary fermions, some progress has been made. While the topological quantization of certain observables such as the Hall conductivity is lost for mixed states, directly observable many-body correlators exist which preserve the quantized nature and naturally connect to known topological invariants in the ground state. For systems that break time-reversal (TR) symmetry, the ensemble geometric phase was identified as such an observable which can be used to define a Chern number in $(1+1)$ and two dimensions. Here we propose a corresponding ${\mathbb{Z}}_2$ topological invariant for systems with TR symmetry. We show that this mixed-state invariant is identical to well-known ${\mathbb{Z}}_2$ invariants for the ground state of the so-called fictitious Hamiltonian, which for thermal states is just the ground state of the system Hamiltonian itself. We illustrate our findings for finite-temperature states of a paradigmatic ${\mathbb{Z}}_2$ topological insulator, the Kane-Mele model.

We report the quantum calibration of a magnetic force microscope (MFM) by measuring the two-dimensional magnetic stray-field distribution of the MFM tip using a single nitrogen vacancy (NV) center in diamond. From the measured stray-field distribution and the mechanical properties of the cantilever a calibration function is derived allowing to convert MFM images to quantum calibrated stray-field maps. This approach overcomes limitations of prior MFM calibration schemes and allows quantum calibrated nanoscale stray-field measurements in a field range inaccessible to scanning NV magnetometry. Quantum calibrated measurements of a stray-field reference sample allow its use as a transfer standard, opening the road towards fast and easily accessible quantum traceable calibrations of virtually any MFM.