The electromagnetic Casimir effect has a fermionic counterpart in topological insulators: Zero-point fluctuations of a massless Dirac fermion field mediate a force between magnetic scatterers. The Casimir force is insensitive to disorder that preserves the topological protection of an unpaired Dirac cone. The protection may be broken if the Dirac equation is discretized, and an exponential suppression of the Casimir effect will result if a gap opens at the Dirac point. Here, we show how this lattice artifact may be avoided, by applying a recently developed local discretization of the Euclidean action that does not suffer from the fermion-doubling obstruction of local discretizations of the Hamiltonian.

In this paper, we present an effective approach to tailor the optical properties of coupled corner states in photonic crystal (PhC) supercell arrays by symmetry breaking. Defects are strategically introduced into unit cells to break the in-plane $C_{4v}$ symmetry, and numerical simulations confirm the preservation of the band gap and two-dimensional Zak phases, in spite of the presence of defects. Notably, the size of defects is demonstrated to play a significant role in tuning the eigenfrequencies, quality ($Q$) factors, and band structures of the coupled corner states. Furthermore, these defects exhibit the ability to transform topological dark states into bright states, enabling efficient excitation by external plane waves and polarization-dependent $Q$ factors. The resonant origin of the coupled corner states is comprehensively elucidated using a multipole decomposition analysis. This study further releases the potential of the topological light-matter interaction, holding significant implications for the design and advancement of topological photonic devices with on-demand tunability and multifrequency regions for various applications such as lasing, sensing, and quantum information processing.

${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and related compounds are prototype three-dimensional topological insulators with a single Dirac cone in the surface band structure. While the topological surface states can be characterized with surface-sensitive methods, the underlying bulk energy band inversion has not been investigated in detail. Here, a study is presented that combines density-functional theory and nuclear magnetic resonance to explore the nontrivial band edge of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. It is found that the topological band inversion is not a discrete reversal of the order of the valence and conduction band at the $\mathrm{\Gamma }$ point. Rather, the bands closest to the Fermi level become well mixed and spread evenly below and above the band gap, such that the characters of the valence- and conduction-band edges become indistinguishable. Beside those bands relevant for the band inversion, i.e., Bi and Se $p_{\mathrm{z}}$, also Bi $p_{\mathrm{x}}$ and $p_{\mathrm{y}}$ states are involved. As a part of this mixture of states, the band inversion shows no edges in $k$ space.

A striking feature of three-dimensional (3D) topological insulators (TIs) is the theoretically expected topological magnetoelectric (TME) effect, which gives rise to additional terms in Maxwell's laws of electromagnetism with an universal quantized coefficient proportional to half-integer multiples of the fine-structure constant $\alpha$. In an ideal scenario one therefore expects also quantized contributions in the magnetooptical response of TIs. We review this premise by taking into account the trivial dielectric background of the TI bulk and potential host substrates, and the often present contribution of itinerant bulk carriers. We show (i) that one obtains a nonuniversal magnetooptical response whenever there is impedance mismatch between different layers and (ii) that the detectable signals due to the TME rapidly approach vanishingly small values as the impedance mismatch is detuned from zero. We demonstrate that it is methodologically impossible to deduce the existence of a TME exclusively from an optical experiment in the thin film limit of 3D TIs at high magnetic fields.

Recently, the physics of $J=3/2$ electrons on a honeycomb lattice has received attention with the suggestion of hosting an SU(8) Dirac semimetallic state. Motivated by this, in this work we provide a systematic study of the interplay of various hopping pathways and atomic spin-orbit coupling for the low-energy electrons in candidate $d^1$ transition-metal halides $M{X}_3$ $(M=\mathrm{Ti},\mathrm{Zr},\mathrm{Hf}; X=\mathrm{F},\mathrm{Cl},\mathrm{Br})$. By combining first-principles calculations and a minimal hopping Hamiltonian, we uncover the role of dominant direct metal-metal hopping on top of indirect metal-halide-metal hopping. This sets up a hierarchy of hopping pathways that centrally modify the SU(8) picture for the above materials. These hopping interactions, along with the spin-orbit coupling, lead to a plethora of exactly compensated metals instead of the SU(8) Dirac semimetal. Remarkably, the same can be understood as descendants of a topological insulator obtained by gapping out the SU(8) Dirac semimetallic phase. The resultant compensated metals have varied Fermi surface topology and are separated by Lifshitz phase transitions. We discuss the implications of the proximate Lifshitz transition, which may be accessed via strain, in the context of the relevant materials.

Non-Hermitian (NH) Hamiltonians have been shown to exhibit unique signatures, including the NH skin effect and an exponential spectral sensitivity with respect to boundary conditions. Here, we investigate as to what extent these remarkable phenomena, recently predicted and observed in a broad range of settings, may also occur in closed correlated fermionic systems that are governed by a Hermitian many-body Hamiltonian. There, an effectively NH quasiparticle description naturally arises in the Green's function formalism due to interparticle scattering that represents an inherent source of dissipation. As a concrete platform we construct an extended interacting Su-Schrieffer-Heeger (SSH) model subject to varying boundary conditions, which we analyze using exact diagonalization and nonequilibrium Green's function methods. That way, we clearly identify the presence of the aforementioned NH phenomena in the quasiparticle properties of this Hermitian model system.

Wires made of topological insulators (TI) are a promising platform for searching for Majorana bound states. These states can be probed by analyzing the fractional ac Josephson effect in Josephson junctions with the TI wire as a weak link. An axial magnetic field can be used to tune the system from trivial to topologically nontrivial. Here, we investigate the oscillations of the supercurrent in such wire Josephson junctions as a function of the axial magnetic field strength and different contact transparencies. Although the current flows on average parallel to the magnetic field we observe $h/2e, h/4e$, and even $h/8e$ periodic oscillations of the supercurrent in samples with lower contact transparencies. Corresponding tight-binding transport simulations using a Bogoliubov–de Gennes model Hamiltonian yield the supercurrent through the Josephson junctions, showing in particular the peculiar $h/4e$-periodic oscillations observed in experiments. A further semiclassical analysis based on Andreev-reflected trajectories connecting the two superconductors allows us to identify the physical origin of these oscillations. They can be related to flux-enclosing paths winding around the TI nanowire, thereby highlighting the three-dimensional character of the junction geometry compared to common planar junctions.

Topological phases have been reported on self-similar structures in the presence of a perpendicular magnetic field. Here, we present an understanding of these phases from a perspective of spectral flow and charge pumping. We study the Harper-Hofstadter model on self-similar structures constructed from the Sierpinski gasket. We numerically investigate the spectral flow and the associated charge pumping when a flux tube is inserted through the structure and the flux through the tube is varied adiabatically. We find that the nature of the spectral flow is qualitatively different from that of translationally invariant noninteracting systems with a perpendicular magnetic field. We show that the instantaneous eigenspectra can be used to understand the quantization of the charge pumped over a cycle, and hence to understand the topological character of the system. We show the correspondence between the local contributions to the Hall conductivity and the spectral flow of the edgelike states. We also show that the edgelike states can be approximated by eigenstates of the discrete angular-momentum operator, their chiral nature being a consequence of this.

Stacking-dependent physics is emerging as a fascinating research topic for two-dimensional materials. A variety of novel properties can be achieved by layer stacking according to different modes. However, most of the studies focus on the impact of electronic, superconducting, optical, and magnetic properties. In this work, we have systematically studied the stacking-dependent topological quantum states in bilayer ${\mathrm{Mn}}_2{\mathrm{Cl}}_3{\mathrm{Br}}_3$ by the first-principles electronic structure calculations. Here, ferromagnetic layer coupling is always favored in different stacking modes. Many exotic topological quantum states, such as topological nodal-ring spin-gapless semimetal state, spin-valley polarized quantum valley Hall (SVP-QVH) effect, and high Chern number quantum anomalous Hall effect can be realized in this single system. Among these states, the SVP-QVH is a new topological phase discovered for the first time, in which there are two topologically protected gapless chiral edge states that carry the same spin, localize respectively on the two polarized valleys, and propagate in opposite directions along the edges. The spin-valley polarized anomalous valley Hall effect can be further realized with electron doping or gate tuning, resulting in edge states that integrate multiple degrees of freedom of valley, spin, and charge. Our research therefore provides an idea for discovery of new topological phases and design of two-dimensional multifunctional electronic, spintronic, and topological devices.

Simultaneous manipulation of charge and spin density distributions in materials is the key element required in spintronics applications. Here we study the formation of coupled spin and charge densities arising in scattering of electrons by domains of local magnetization producing a position-dependent Zeeman field in the presence of the spin-momentum locking typical for topological insulators. Analytically and numerically calculated scattering pattern is determined by the electron energy, domain magnetization, and size. The spin-momentum locking produces strong differences with respect to the spin-diagonal scattering and leads to the scattering asymmetry with a nonzero mean scattering angle as determined by only two parameters characterizing the system. To extend the variety of possible patterns, we study scattering by diffraction gratings and propose to design them in modern nanostructures based on topological insulators to produce desired distributions of the charge and spin densities. These results can be useful for engineering of magnetic patterns for electron optics to control coupled charge and spin evolution.

Periodic driving may cause topologically protected, chiral transport along edges of a two-dimensional (2D) lattice that, without driving, would be topologically trivial. We study what happens if one adds a different on-site potential along the diagonal of such a 2D grid. In addition to the usual bulk and edge states, the system then also exhibits doublon states, analogous to two interacting particles in one dimension. A particle initially located at an edge propagates along the system's boundary. Its wave function splits when it hits the diagonal and continues propagating simultaneously along the edge and the diagonal. The strength of the diagonal potential determines the ratio between both parts. We show that for specific values of the diagonal potential, hopping onto the diagonal is prohibited so that the system effectively separates into two triangular lattices. For other values of the diagonal potential, we find a temporal delay between the two contributions traveling around and through the system. This behavior could enable the steering of topologically protected transport of light along the edges and through the bulk of laser-inscribed photonic waveguide arrays.

We study one-dimensional lattices with imaginary-valued Aubry-André-Harper (AAH) potentials. Such lattices can host edge states with purely imaginary eigenenergies, which differ from the edge states of the Hermitian AAH model and are stabilized by a non-Hermitian particle-hole symmetry. The edge states arise when the period of the imaginary potential is a multiple of four lattice constants. They are topological in origin, and can manifest on domain walls between lattices with different modulation periods and phases, as predicted by a bulk polarization invariant. Interestingly, the edge states persist and remain localized even if the gap in the real spectrum closes. These features can be used in laser arrays to select topological lasing modes under spatially extended pumping.

Recently, gate-defined Josephson junctions in magic angle twisted bilayer graphene (MATBG) were studied experimentally, and highly unconventional Fraunhofer patterns were observed. In this work, we show that an interaction-driven valley-polarized state connecting two superconducting regions of MATBG would give rise to a long-sought-after purely electric controlled ${\phi }_0$-junction in which the two superconductors acquire a finite phase difference ${\phi }_0$ in the ground state. We point out that the emergence of the ${\phi }_0$-junction stems from the valley-polarized state which breaks time-reversal symmetry, and trigonal warping effects which break intravalley inversion symmetry. Importantly, a spatially nonuniform valley-polarization order parameter at the junction can explain the key features of the observed unconventional Fraunhofer patterns. Our work explores the novel transport properties of the valley-polarized state, and we suggest that gate-defined MATBG Josephson junctions could realize the first purely electric controlled ${\phi }_0$-junctions.

To observe synchronization in large networks of classical or quantum systems demands both excellent control of the interactions between nodes and very accurate preparation of initial conditions due to the involved nonlinearities and dissipation. This limits its applicability for future devices. We demonstrate a route toward significantly enhancing the robustness of synchronized behavior in open nonlinear systems that utilizes the power of topology. In nontrivial topological lattices of quantum van der Pol oscillators, boundary synchronization emerges in the classical mean field as well as the quantum regime. In addition to its robustness against disorder and initial state perturbations, the observed dynamics is independent of the underlying topological model provided the existence of topological zero-energy modes. Our work extends the notion of topology to the general nonlinear dynamics and open quantum system realm with applications to networks where specific nodes need special protection like power grids or quantum networks.

The ground states of noninteracting fermions in one-dimension with chiral symmetry form a class of topological band insulators, described by a topological invariant that can be related to the Zak phase. Recently, a generalization of this quantity to open systems—known as the ensemble geometric phase (EGP)—has emerged as a robust way to describe topology at nonzero temperature. By using this quantity, we explore the nature of topology allowed for dissipation beyond a Lindblad description, to allow for coupling to external baths at finite temperatures. We introduce two main aspects to the theory of open-system topology. First, we discover topological phase transitions as a function of the temperature $T$, manifesting as changes in differences of the EGP accumulated over a closed loop in parameter space. We characterize the nature of these transitions and reveal that the corresponding nonequilibrium steady state can exhibit a nontrivial structure—contrary to previous studies where it was found to be in a fully mixed state. Second, we demonstrate that the EGP itself becomes quantized when key symmetries are present, allowing it to be viewed as a topological marker which can undergo equilibrium topological transitions at nonzero temperatures.

We propose a magnetic topological transistor based on ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$, in which the “on” state (quantized conductance) and the “off” state (zero conductance) can be easily switched by changing the relative direction of two adjacent electric fields (parallel vs antiparallel) applied within a two-terminal junction. We explain that the proposed magnetic topological transistor relies on a novel mechanism due to the interplay of topology, magnetism, and layer degrees of freedom in ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. Its performance depends substantially on film thickness and type of magnetic order. We show that “on” and “off” states of the transistor are robust against disorder due to the topological nature of the surface states. Our work opens an avenue for applications of layer-selective transport based on the topological van der Waals antiferromagnet ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$.

We show that the nonlinear transport of bosonic excitations in a two-dimensional honeycomb lattice of spin-orbit-coupled Rydberg atoms gives rise to disordered quantum phases which are topological and may be candidates for quantum spin liquids. As recently demonstrated by Lienhard et al. [Phys. Rev. X 10, 021031 (2020)] the spin-orbit coupling breaks time-reversal and chiral symmetries and leads to a tunable density-dependent complex hopping of spin excitations which behave as hard-core bosons. Using exact diagonalization (ED), we numerically investigate the phase diagram resulting from the competition between density-dependent and direct transport terms as well as density-density interactions. In mean-field approximation there is a phase transition from a condensate to a ${120}^{\circ }$ phase when the amplitude of the complex hopping exceeds that of the direct one. In the full model a new phase emerges close to the mean-field critical point as a result of quantum correlations induced by the density dependence of the complex hopping. We show that without density-density interactions this phase is a genuine disordered one, has large spin chirality, and is characterized by a nontrivial many-body Chern number. The Chern number is found to be robust to disorder. ED simulations of small lattices with up to 30 lattice sites give indications for a nondegenerate ground state with finite spin and collective gaps and thus for a bosonic integer quantum Hall phase, protected by $\mathrm{U}\left(1\right)$ symmetry. On the other hand, while staying finite, the many-body gap varies substantially when different twisted boundary conditions are applied, which points to a gapless phase. For very strong negative nonlinear hopping amplitudes we find another disordered regime with vanishing spin gap. This phase also has a large spin chirality and could be a gapless spin liquid but lies outside the parameter regime experimentally accessible in the Rydberg system.

We propose an alternative route to stabilize magnetic skyrmion textures which does not require Dzyaloshinkii-Moriya interactions, magnetic anisotropy, or an external Zeeman field. Instead, it solely relies on the emergence of flux in the system's ground state. We discuss scenarios that lead to a nonzero flux and identify the magnetic skyrmion ground states which become accessible in its presence. Moreover, we explore the chiral superconductors obtained for the surface states of a topological crystalline insulator when two types of magnetic skyrmion crystals coexist with a pairing gap. Our work opens perspectives for engineering topological superconductivity in a minimal fashion and promises to unearth functional topological materials and devices which may be more compatible with electrostatic control than the currently explored skyrmion-Majorana platforms.