This study formulated the longitudinal spin and orbital currents and torques in the nonequilibrium Green's function formalism for magnetic tunnel junctions with spin-orbit coupling (SOC). The analytical results for the $y$ and $z$ components of the spin currents without SOC are consistent with those previously reported. We show that the torque induced by the SOC is transmitted through the junction, resulting in an orbital coupling between the two magnetic layers. Orbital torque deduced from the analytical expressions exhibits several unique features. Orbital mixing is crucial to the orbital torque, and the torque may remain finite even for nonmagnetic-ferromagnetic tunnel junctions. The result was interpreted in terms of long-range character of the magnetic anisotropy and nondegeneracy of orbital states. The effects of finite voltage applied to the junctions have also been clarified.

Spin-orbit torque (SOT) is a promising technique for next-generation magnetic random-access memory. Recent experiments have shown that materials with low-symmetry crystalline or magnetic structures can generate anomalous SOT with an out-of-plane component, which is crucial for switching the perpendicular magnetization of adjacent ferromagnetic (FM) layers in a field-free condition. In this study, we derive the threshold current for field-free perpendicular magnetization switching using anomalous SOT and numerically calculate the magnetic moment trajectory in an FM free layer for currents smaller and greater than the threshold. We also investigate the dependence of switching time and energy consumption on applied current, finding that the minimum energy consumption decreases with an increasing out-of-plane torque proportion. Additionally, we explore the relationships between the threshold current and anisotropy strength, out-of-plane torque proportion, FM free-layer thickness, and Gilbert damping constant. The results show a negative correlation between the threshold current and out-of-plane torque proportion, and positive correlations with the other three parameters. Finally, we demonstrate that even when the applied current is smaller than the threshold current, it can still add an effective exchange bias field $H_{\mathrm{bias}}$ on the FM free layer. The $H_{\mathrm{bias}}$ is proportional to the applied current $J_{\mathrm{SOT}}$, facilitating the determination of anomalous SOT efficiency. Our findings provide insights into the design of spintronic devices that favor field-free switching of perpendicular magnetization using anomalous SOT and offer a means of adjusting the exchange bias field to control FM layer magnetization depinning.

Odd-frequency superconductivity is an exotic superconducting state in which the symmetry of the gap function is odd in frequency. Here, we show that an inherent odd-frequency mode emerges dynamically under application of a Lorentz transformation of the anomalous Green function with the general frequency-dependent gap function. To see this, we consider a Dirac model with quartic potential and perform a mean-field analysis to obtain a relativistic Bogoliubov–de Gennes system. Solving the resulting Gor'kov equations yields expressions for relativistic normal and anomalous Green functions. The form of the relativistically invariant pairing term is chosen such that it reduces to BCS form in the nonrelativistic limit. We choose an ansatz for the gap function in a particular frame which is even frequency and analyze the effects on the anomalous Green function under a boost into a relativistic frame. The odd-frequency pairing emerges dynamically as a result of the boost. In the boosted frame, the order parameter contains terms which are both even and odd in frequency. The relativistic correction to the anomalous Green function to first order in the boost parameter is completely odd in frequency. In this paper, we provide evidence that odd-frequency pairing may form intrinsically within relativistic superconductors.

In this paper, we provide theoretical approaches to identify the influence of the quasidisorder on a two-dimensional system. We discover that in the system there is a topological phase transition accompanied by a transverse Anderson localization. The topological features are characterized by the band gap, the edge-state spectra, the transport conductance, and the Chern number. The localization transition is clearly demonstrated by the investigations of the partial inverse participation ratio, the average of level spacing ratio, and the fraction dimension. The results reveal the topological nature of the bulk delocalized states. Our work facilitates the understanding of the synchronicity between the topological phase transition and the transverse Anderson localization in two-dimensional disordered systems.

We investigate Bloch oscillations of wave packets in monolayer phosphorene with broken inversion symmetry. We find that the real-space trajectories and Berry and group velocities of Bloch electron undergo Bloch oscillations in the system. The strong dependence of Bloch dynamics on the crystal momentum is illustrated. It is shown that the spin-orbit interaction crucially affects the dynamics of the Bloch electron. We also demonstrate the dynamics in external electric and magnetic fields within the framework of Newton's equations of motion, leading to the geometric visualization of such oscillatory motion. In the presence of both applied in-plane electric and transverse magnetic fields, the system undergoes a dynamical transition from confined to deconfined state and vice versa, tuned by the relative strength of the fields.

The Kitaev model on a honeycomb lattice with bond-dependent Ising interactions offers an exactly solvable model of a quantum spin liquid (QSL) with fractionalized excitations: gapped $Z_2$ fluxes and gapless linearly dispersing Majorana fermions in the isotropic limit ($K_x=K_y=K_z$). We explore the phase diagram along two axes, an external magnetic field $h$, applied out of plane of the honeycomb, and anisotropic interactions $K_z$ larger than the other two ($K_x=K_y\equiv K$). For $K_z/K\gg 2$ and $h=0$, the matter Majorana fermions have the largest gap, and the system is described by a gapped $Z_2$ toric code. One of the central questions we address is whether the fractionalized excitations in the different phases have sharp signatures that can be detected in experiments. We show that while the response to single-spin excitations is broad, the spectral function corresponding to two-spin excitations across a bond has sharp signatures that can be attributed to specific anyons. In the toric code regime, the $\epsilon =e\times m$ fermion, formed from the bosonic Ising electric ($e$) and magnetic ($m$) charges, disperses along a specific one-dimensional direction that provides a fingerprint of fractionalization. At lower $K_z$ in the center of the Abelian phase, in a regime we dub the primordial fractionalized regime, the field generates a hybridization between the $\epsilon$ fermion and the Majorana matter fermion, resulting in a $\psi$ fermion which too has a distinct quasi-one-dimensional dispersion. All the other phases in the field-anisotropy plane are naturally obtained from this primordial soup. These highly constrained fractonlike dispersions can be observable by inelastic light and neutron scattering, thereby providing “smoking gun” signatures of fractionalization in the QSL phase. Our analysis is based on calculations of susceptibilities, topological entanglement entropy, and excitation dynamics, obtained using exact diagonalization and density matrix renormalization group, and supported by perturbation theory.

Exchanging particles on graphs, or more concretely on networks of quantum wires, has been proposed as a means to perform fault-tolerant quantum computation. This was inspired by braiding of anyons in planar systems. However, exchanges on a graph are not governed by the usual braid group but instead by a graph braid group. By imposing compatibility of graph braiding with fusion of topological charges, we obtain generalized hexagon equations. We find the usual planar anyon solutions but also more general braid actions. We illustrate this with Abelian, Fibonacci, and Ising fusion rules.

We studied the exciton properties in double layers of transition metal dichalcogenides (TMDs) with a dielectric spacer between the layers. We developed a method based on an expansion of Chebyshev polynomials to solve the Wannier equation for the exciton. Corrections to the quasiparticle bandgap due to the dielectric environment were also included via the exchange self-energy calculated within a continuum model. We systematically investigated hetero double-layer systems for TMDs with chemical compounds $\mathit{MX}{}_2$, showing the dependence of the inter- and intralayer excitons binding energies as a function of the spacer width and the dielectric constant. Moreover, we discussed how the exciton energy and its wave function, which includes the effects of the changing bandgap, depend on the geometric system setup.

Triple-fold band degeneracy (TBD) is an intriguing phase of topological semimetals which appears as the intermediate state between the four-folded degenerated Dirac points and the two-folded Weyl points. The resemblance of TBD in the photonics is interesting, because it indicates nonzero Berry curvature in momentum space and nontrivial topological property in real space. Recently, it was reported that TBD could be realized in the classical wave systems including phononic and photonic crystals. Unfortunately, those periodic structures can hardly provide a systematic strategy nor a compact layout in practical implementations. Hence, the resemblance of TBD is still a big challenge in topological photonics. In this work, we demonstrate that homogeneous gyromagnetic metamaterials (GMs) can be a good framework to stimulate the peculiar TBD. We discuss the emulation of triple degeneracy in dispersive GMs by carefully tuning the constitutive parameters. The introduction of nonlocality in the constitutive parameter of GMs provides the freedom of topology control of the GMs. Further, Fermi arc surface states are discussed and shown to be reflectionless against sharp corners. Remarkably, Fermi arc surface states, which occur at the boundary between the vacuum and the GMs, can be stretchable by tailoring the gap width of the GM's band structure. Our work provides a practical and flexible route to resemble TBD, which is of fundamental importance in designing reflectionless and unidirectional photonic topological devices.

On the basis of density functional plus dynamical mean-field theory calculations we explore the electronic structure of the $\mathrm{Tl}{\mathrm{Ni}}_2{\mathrm{Se}}_2$ superconductor, showing how the normal-state spectra are reshaped by dynamical many-particle interactions. Upon electron-doping the $3d^8$ Fermi liquid metal, hidden orbital selectivity with coexisting Mott localized, pseudogapped, and narrow Kondo-quasiparticle electronic excitations is predicted for $\mathrm{Tl}{\mathrm{Ni}}_2{\mathrm{Se}}_2$. By specifically focusing on the energy spectrum that is relevant to the mixed-valence heavy-electron normal state of unconventional superconductors, our findings have the potential to enhance our microscopic understanding of the intricate interplay between many-particle physics and electron band filling, paving the way for further advancements.

The anomalous Hall effect has considerable impact on the progress of condensed matter physics and occurs in systems with time-reversal symmetry breaking. Here we theoretically investigate the anomalous Hall effect in the nonmagnetic transition metal pentatellurides ${\mathrm{ZrTe}}_5$ and ${\mathrm{HfTe}}_5$. In the presence of Zeeman splitting and Dirac mass, there is an intrinsic anomalous Hall conductivity induced by the Berry curvature in the semiclassical treatment. In a finite magnetic field, the anomalous Hall conductivity rapidly decays to zero for constant spin splitting and vanishes for the magnetic-field-dependent Zeeman energy. A semiclassical formula is derived to depict the magnetic field dependence of the Hall conductivity, which is beneficial for experimental data analysis. Lastly, when the chemical potential is fixed in the magnetic field, a Hall conductivity plateau arises, which may account for the observed anomalous Hall effect in experiments.

Using high resolution tender x-ray resonant inelastic scattering and hard x-ray nonresonant inelastic scattering beyond the dipole limit we were able to detect electronic excitations in intermetallic ${\mathrm{UGa}}_2$ that are highly atomic in nature. Analysis of the spectral lineshape reveals that the local $5f^2$ configuration characterizes the correlated nature of this ferromagnet. The orientation and directional dependence of the spectra indicate that the ground state is made of the ${\mathrm{\Gamma }}_1$ singlet and/or ${\mathrm{\Gamma }}_6$ doublet symmetry. With the ordered moment in the $ab$ plane, we infer that the magnetism originates from the higher lying ${\mathrm{\Gamma }}_6$ doublet being mixed with the ${\mathrm{\Gamma }}_1$ singlet due to intersite exchange, qualifying ${\mathrm{UGa}}_2$ to be a true quantum magnet. The ability to observe atomic excitations is crucial to resolve the ongoing debate about the degree of localization versus itineracy in U intermetallics.

We investigate the dispersive paramagnetic excitons on the honeycomb lattice that originate from the crystalline electric field split localized $f$-electron states in the paramagnetic state due to intersite exchange. We start with a symmetry analysis of possible Ising-type singlet-singlet and $xy$-type singlet-doublet models. The former supports only symmetric intersite exchange while the latter additionally allows for antisymmetric Dzyaloshinski-Moriya exchange interactions. We calculate the closed expressions for magnetic exciton dispersion using both response function formalism and bosonic Bogoliubov approach. We do this for the most general model that shows inversion-symmetry breaking on the honeycomb lattice but also discuss interesting special cases. By calculating Berry curvatures and Chern numbers of paramagnetic excitons we show that the $xy$ model supports nontrivial topological states in a wide range of parameters. This leads to the existence of excitonic topological edge states with Dirac dispersion lying in the zone boundary gap without the presence of magnetic order.

We present Ru $L_3$-edge resonant inelastic x-ray scattering (RIXS) measurements of spin-orbit and $d\text{−}d$ excitations in exfoliated nanolayers of the Kitaev spin-liquid candidate ${\mathrm{RuCl}}_3$. Whereas the spin-orbit excitations are independent of thickness, we observe a pronounced redshift and broadening of the $d\text{−}d$ excitations in layers with thicknesses below $\sim 7$ nm. Aided by model calculations, we attribute these effects to distortions of the ${\mathrm{RuCl}}_6$ octahedra near the surface. Our study paves the way towards RIXS investigations of electronic excitations in various other two-dimensional materials and heterostructures.

Graphene structures play a fundamental role in the exploration of edge states, topological insulators, and electronic transport applications. In this paper, we propose a graphenelike moiré photonic lattice formed by superimposing two identical triangle sublattices with an antiphase and a specific twist angle and study the edge and corner states of such a lattice in linear and nonlinear regimes. The graphenelike moiré lattice is truncated into three types of edges in analogy with those of graphene. Through band-structure calculations, we find that only the zigzag edge supports edge states, while the other two edges do not exhibit such states. A truncated graphenelike moiré photonic lattice with zigzag edges is constructed by utilizing the laser-writing technique. We observe the complex edge states and corner states in the linear regime at the zigzag edge. Furthermore, by introducing nonlinearity, the optical localization occurs at the edges and corners. Our research provides a route for exploring topological edge states and nonlinear edge solitons in coherent moiré photonic lattices and brings possibilities for subsequent technological development in electronic or optoelectronic devices.

In this work we employ macrospin simulations of the magnetization dynamics obtained from the Landau-Lifshitz-Gilbert equation coupled to angular momentum transport yielded by the nonequilibrium Green's function formalism to investigate two possible origins of the back-hopping effect in magnetic tunnel junctions with one layer of a synthetic antiferromagnet as reference layer. One possible origin discussed in this work is an additional sign reversal of the dampinglike (parallel) torque component as a function of the applied voltage. The other is the destabilization and switching of the reference layer by the spin-transfer torque. We find that this destabilization gives rise to a dynamic behavior manifesting in a back-and-forth switching of the free layer, while the sign reversal of the dampinglike torque results in a second hysteresis loop.

Superconductivity has been experimentally observed in monolayer ${\mathrm{WTe}}_2$, in which the suggested in-plane field measurements are of a spin-triplet nature. Furthermore, it has been proposed that with a $p$-wave pairing, the material is a second-order topological superconductor with a pair of Majorana zero modes at the corners of a finite sample. We show that for a repulsive on-site interaction and sizable Fermi surfaces, the desired $p$-wave state arises naturally due to the Kohn-Luttinger mechanism, and indeed a finite superconducting sample hosts corner Majorana zero modes. We study the behavior of the critical temperature in response to external in-plane magnetic fields. We find an enhancement to the critical temperature that depends on the direction of the magnetic field, which can be directly verified experimentally.

Colloidal gel systems exhibit increasingly slow relaxation and ultra-long-ranged spatial correlations of the dynamics similar to other jammed materials. These cooperative dynamics point to the presence of long-ranged stress correlation in these systems, which remain largely uninvestigated in the literature. In this work, we systematically investigate the nature of stress correlations in soft colloidal gel materials in the limit of moderate to high packing fractions and strong attraction. In this regime, centrosymmetric potential description for particle interaction fails as strong attraction can lead to frictional contacts, as shown explicitly in previous experiments. Accordingly, we model the system similarly to the cohesive granular media with Langevin dynamics to incorporate the effects of rolling and sliding resistant contacts and thermal fluctuations. We show that the spatial stress correlations are long ranged with very slow spatial decay close to the gel point. Similarly to previous studies on the frictional granular matter, the full stress autocorrelation matrix is dictated by the pressure and torque autocorrelations due to mechanical balance and material isotropy constraints. Surprisingly, it is observed that the gel materials do not behave as a normal elastic solid close to the gel point as assumed loosely in the literature because the real-space pressure fluctuations decay slower than normal. Furthermore, we link the abnormal pressure fluctuations to the non-hyperuniform behavior of the system (granular matter and gel) with respect to the local packing fraction fluctuations, thus relating the deviations from the normal elastic behavior across various jammed systems under a common framework.

A magnetic bubble crystal is a two-dimensional soliton lattice consisting of multiple spin-density waves similar to a magnetic skyrmion crystal. Nevertheless, the emergence of the bubble crystal with a collinear spin texture is rare compared to that of the skyrmion crystal with a noncoplanar spin texture. Here, we theoretically report the stabilization mechanisms of the bubble crystal in tetragonal magnets. By performing numerical calculations based on an efficient steepest descent method for an effective spin model with magnetic anisotropy and multiple spin interactions in momentum space on a two-dimensional square lattice, we construct magnetic-field-temperature phase diagrams for various sets of model parameters. We find that the bubble crystal is stabilized at finite temperatures near the skyrmion crystal by an easy-axis anisotropic two-spin interaction. Through a detailed analysis, we also show that the high-harmonic wave-vector interaction and the biquadratic interaction play important roles in the stability of the bubble crystal. Our results indicate a close relationship between the bubble crystal and the skyrmion crystal in terms of the stabilization mechanisms, which suggests the possibility of the bubble crystal in the skyrmion-hosting materials by controlling the easy-axis magnetic anisotropy through external and/or chemical pressure.

Intrinsic room-temperature ferromagnetic ordering and inversion asymmetry of V-based transition metal dichalcogenide (TMD) monolayers with trigonal prismatic coordinates are highly desirable to develop spintronic devices. Motivated by facilitating the performance improvement of V-based TMDs in data-storage applications, we investigated the influence of the interface magnetic proximity effect, mediated by Zeeman-type spin-orbit interaction (SOI), on the magnetic exchange interaction and magnetic anisotropy of the 2H-${\mathrm{VSe}}_2$ monolayer. It has already been observed that strain and interlayer coupling does not have a significant effect on the magnetic anisotropy of V-based TMDs. Here we demonstrate using density functional theory (DFT) that the proximity effect of the archetypal metallic material ${\mathrm{NbSe}}_2$ at the van der Waals interface induces strong out-of-plane magnetic anisotropy in 2H-${\mathrm{VSe}}_2$. The transition of magnetization from in plane to out of plane in 2H-${\mathrm{VSe}}_2$ holds great promise toward the realization of faster and smaller magnetic bits in lower power-consuming memory devices such as random access memories. The valley polarization of the ${\mathrm{VSe}}_2$ system is significantly reversed by applying Zeeman-type SOI of ${\mathrm{NbSe}}_2$ substrate. Also, by combining the two-band $\mathit{k}·\mathit{p}$ model with high-throughput DFT calculations, the change of $\mathit{k}·\mathit{p}$ parameters of 2H-${\mathrm{VSe}}_2$ is examined due to Zeeman-type SOI induced by ${\mathrm{NbSe}}_2$.