The chirality of polar skyrmion bubbles is critical to their topological identities and potential applications. Although previous phase-field models could simulate polar skyrmion bubbles, the Bloch-type feature key to the chirality is not produced. Here, we introduce an extended phase-field model with biquadratic coupling between polarization and its gradient. Taking the ${\mathrm{SrTiO}}_3$/${\mathrm{PbTiO}}_3$/${\mathrm{SrTiO}}_3$ heterostructure as an example, it is found that with the increase of the coupling strength, a transition occurs from achiral to chiral skyrmion bubbles, and ultimately to mixed-chiral bubbles with zero net chirality. However, the chirality of the skyrmion bubbles is random and external electric field and temperature could not result in a uniform chirality. Notably, the homochiral vortex tubes would transform into the skyrmion bubbles with uniform chirality under the electric field, while achiral precursors lead to random chirality. These results provided a theoretical framework and practical strategies for chirality engineering in polar skyrmions.

Moiré superlattice systems offer versatile platforms for studying strongly correlated electrons, where both isotropic Mott insulator and highly anisotropic Luttinger liquid regimes have been experimentally realized. However, the physics interpolating between these limits remains largely elusive, which motivates the pursuit of a moiré platform with in situ tunable hopping anisotropy. Here, we demonstrate that twisted rhombic homobilayers with band extrema at the $Y$ valley can realize a single-band triangular lattice Hubbard model with tunable hopping anisotropy via displacement field. We employ the density matrix renormalization group method to study this model with interaction $U=10t$ and anisotropy $0.5\le t^{\prime }/t\le 1.5$ at half filling, revealing a rich ground state phase diagram, including a chiral spin liquid phase, nonmagnetic phases, and a Néel antiferromagnetic phase. Our work establishes a moiré platform with in situ tunable hopping anisotropy in a single-band triangular Hubbard model, opening avenues for more systematic studies of the interplay between anisotropy and electronic correlations.

High-quality single crystals of ${\mathrm{Yb}}_3{\mathrm{Sc}}_2{\mathrm{Ga}}_3{\mathrm{O}}_{12}$ with a hyperkagome spin lattice were grown by using the floating-zone method. The magnetic susceptibility, specific heat and thermal conductivity were measured down to very low temperatures $(<100 \mathrm{mK})$ to characterize the magnetic properties and probe the nature of ground state. The experimental results demonstrate that the ground state is spin disordered one with effective 1/2 spin and antiferromagnetic coupling, pointing to a possible quantum spin liquid, although the clear characteristic behaviors of quantum spin liquid were not observed in these results. The specific heat data display broad peaklike feature that is likely caused by the short-range spin correlations. The ultralow-temperature thermal conductivity exhibits rather strong magnetic field dependence and zero residual term $({\kappa }_0/T)$, which indicates the scattering between phonons and magnetic excitations. In addition, both the specific heat and thermal conductivity data display some field-induced magnetic transitions.

The macroscopic theory of anyon condensation, rooted in the categorical structure of topological excitations, provides a complete classification of gapped boundaries in topologically ordered systems, where distinct boundaries correspond to the condensation of different Lagrangian algebras. However, an intrinsic and direct understanding of anyon condensation in lattice models, grounded in the framework of Lagrangian algebras, remains undeveloped. In this paper, we propose a systematic framework for constructing all gapped boundaries of Kitaev's quantum double models directly from the data of Lagrangian algebras. Central to our approach is the observation that bulk interactions in the quantum double models admit two complementary interpretations: the anyon-creating picture and the anyon-probing picture. Generalizing this insight to the boundary, we derive the consistency condition for boundary ribbon operators that respect the mathematical axiomatic structure of Lagrangian algebras. Solving these conditions yields explicit expressions for the local boundary interactions required to realize gapped boundaries. We also provide three families of solutions that cover a broad range of cases. Our construction provides a microscopic characterization of the bulk-to-boundary anyon condensation dynamics via the action of ribbon operators. Moreover, all these boundary terms are supported within a common effective Hilbert space, making further studies on pure boundary phase transitions natural and convenient. Given the broad applicability of anyon condensation theory, we believe that our approach can be generalized to planar topological codes, extended string-net models, or higher-dimensional topologically ordered systems.

A precision measurement of the photoionization of pure sodium and potassium nanoparticles isolated in a beam enabled an accurate determination of their work functions as a function of temperature. In addition to resolving and quantifying the initial gradual decrease of the work function with temperature, which is associated with thermal expansion, the experiment revealed that the work function then undergoes a distinct drop both in magnitude and in slope that signifies the onset of nanoparticle melting. This establishes that a structural phase transition can be detected via a high-resolution measurement of the photoemission threshold. The melting temperature of nanoparticles with diameters of 7–9 nm is reduced by nearly 100 K relative to the bulk value. This suppression aligns with predictions from the Gibbs-Thomson equation which describes finite-size phase transitions.

Photonic crystals (PCs) with localized optical cavity modes arising from topological domain-wall line defects are simulated for optical biosensing by numerical solution of Maxwell's equations. These consist of a square lattice of square silicon blocks with a significant photonic band gap (PBG). Optical transmission through the PBG at specific frequencies occurs by defect-mediated optical tunneling. Biofluid flows perpendicular to light propagation, through a channel containing the PC, defined by silica sidewalls and an underlying silica substrate. Replacing the silicon blocks with thin silicon strips throughout the domain-wall region, analyte binding coincides with regions of maximal field intensity. As a result, the sensitivity is improved by almost 16 times higher than the previous designs. We analyze optical mode hybridization of two nearby domain walls and its close relation to the transmission levels and correlations in frequency shifts of nearby optical resonances in response to analyte bindings. We illustrate three high-sensitivity chips each with three domain-wall defects, all of which can distinguish three analyte bindings and their combinations completely in a single spectroscopic measurement. In a photonic crystal, consisting of silicon squares embedded in a water background and a $5\mu \text{m}$ lattice spacing, the biosensor sensitivity to a thin analyte binding layer is nearly 3000 nm/RIU and to the overall background biofluid is over 8000 nm/RIU.

Recent experiments on multilayer graphene systems have rekindled interest in electronic crystal phases in two dimensions—but now for phases enriched by nontrivial quantum geometry. In this work, we introduce a simple continuum model with tunable Berry curvature distribution and total flux, enabling a systematic study of crystallization in geometrically nontrivial bands. In the noninteracting limit, the addition of a $C_6$-symmetric periodic potential yields a rich phase diagram, for which we provide several analytical insights. Notably, we derive a general formula for the Chern number in the weak-potential regime that is broadly applicable to single-band projected models. Removing the periodic potential and treating Coulomb interactions self-consistently at the Hartree–Fock level, the resulting phase diagrams host a variety of crystalline states, including anomalous Hall crystals, halo Wigner crystals in which localized electrons spontaneously acquire orbital angular momentum leading to depleted electron occupation at the zone center, and a halo anomalous Hall crystal that combines these properties with a finite Chern number. We identify why these phases are energetically favorable through analytical and energetic considerations. Our results provide insight into the interplay between crystallization and band geometry, while also offering a simple toy model amenable to numerical methods beyond mean-field.

The absence of inversion symmetry in chiral tellurium (Te) creates unique spin textures within its electron waves. However, understanding textured optical waves within Te remains a challenge due to the semiclassical limitations of the long-wavelength approximation. To unveil deep microscopic optical eigenwaves in Te, we develop a spin-resolved deep-microscopic optical band structure, analogous to its electronic counterpart. We demonstrate that the degeneracies in this optical band structure are lifted by the twisted lattice of Te, which induces optical gyrotropy. Our theory shows excellent agreement with experimental optical gyrotropy measurements. At the lattice level, we reveal that the chirality of Te manifests as deep-microscopic spin texture within the bulk optical eigenwaves. Our framework uncovers the finite-momentum origin of optical activity and provides a microscopic basis to quantify light-matter interactions in chiral crystalline materials.

Altermagnetism is a collinear magnetic order in which opposite spin species are exchanged under a real-space rotation. Hence, the search for physical realizations has focused on crystalline solids with specific rotational symmetry. Here, we show that altermagnetism can also emerge in noncrystalline systems, such as amorphous solids, despite the lack of global rotational symmetries. We construct a minimal Hamiltonian with two directional orbitals per site on an amorphous lattice with interactions that are invariant under spin rotation. Altermagnetism then arises due to spontaneous symmetry breaking in the spin and orbital degrees of freedom around each atom, displaying a common point group symmetry. This form of altermagnetism exhibits anisotropic spin transport and spin spectral functions, both experimentally measurable. Our mechanism generalizes to any lattice and any altermagnetic order, opening the search for altermagnetic phenomena to noncrystalline systems.

Flat band (FB) systems provide ideal playgrounds for studying correlated physics, whereas multi-orbital characteristics in real materials are distinguished from most simple FB models. Here, we propose a systematic and versatile symmetry-based framework for FB constructions in tight-binding (TB) models based on symmetric compact localized states (CLSs), integrating lattice and orbital degrees of freedom. The essential perspective is to capture destructive interference, which is universal for FBs, by reinterpreting the Hamiltonian as a linear mapping and searching for symmetric CLSs in the kernel. To illustrate the versatility of our framework, we construct three representative FB models: one in two dimensions (2D) and the rest in three dimensions (3D). All of them lack special lattice structures and incorporate high orbitals. Further, we unveil 3D FBs exhibiting exotic nodal-line band touchings. For a comprehensive understanding, we derive a concise criterion for determining all band touchings and demonstrate their symmetry-protected nature. Our work offers a systematic approach to construct FBs across diverse lattice systems and opens new avenues for understanding and engineering FB systems, with potential implications for correlated quantum phenomena and exotic phases of matter.

Artificial Kitaev chains based on arrays of quantum dots are promising platforms for realizing Majorana bound states (MBSs). In a two-site Kitaev chain, it is possible to find these non-Abelian zero-energy excitations at certain points in parameter space (sweet spots). These states, commonly referred to as poor man's Majorana bound states (PMMs), are challenging to find and stabilize experimentally. In this work, we investigate the evolution of the sweet spots as we increase the number of sites of the Kitaev chain. To this end, we use the Bogoliubov-de Gennes representation to study the excitations of the system, and the scattering matrix and Green's functions formalisms to calculate the zero-bias conductance. Our results show that the sweet spots evolve into a region that grows bigger and becomes gradually more protected as the number of sites $N$ increases. Because of the protection of the MBSs, we refer to this region as a topological island. We obtain similar results by considering a realistic spinful model with finite magnetic fields in a chain of normal-superconducting quantum dots. For long chains, $N\ge 20$, we show the emergence of strictly zero-energy plateaus robust against disorder. We demonstrate that the topological islands can be observed by performing conductance measurements via a quantum dot side-coupled to the Kitaev chain. Our results indicate that the conductance $G$ through the dot can be directly associated with the non-Abelian statistics of the MBSs. Finally, our work shows that the fine-tuning required to create and detect PMMs in a two-site Kitaev chain is significantly relaxed as the length of the chain increases and details how PMMs evolve into MBSs. Our results are consistent with experimental reports for two- and three-site chains.

EuAuSb is a magnetic Weyl semimetal candidate that exhibits in-plane ferromagnetic ordering and a helical magnetic order across the plane at temperatures below 3.9 K. These magnetic characteristics can drive an interaction-induced reconstruction of the Fermi surface. Here, we report a rare instance of Fermi surface splitting and collapsing driven by exchange coupling, as revealed by strongly temperature-dependent quantum oscillation frequencies in EuAuSb. Shubnikov–de Haas measurements reveal two distinct quantum oscillation frequencies ($F_{\alpha }=80\mathrm{T}$ and $F_{\beta }=165\mathrm{T}$) at 2 K. As the temperature increases toward 40 K, the oscillation peaks at these two frequencies gradually merge into a single peak centered around 128 T. This behavior signifies the collapse of the spin-split Fermi surfaces, driven by the weakening of short-range magnetic correlations and the exchange coupling between conduction electrons and local magnetic moments as temperature increases. Our results are further supported by density functional theory calculations, angle-resolved photoemission spectroscopy, and magnetization measurements.

We studied the displacement of magnetic domains under temperature gradients in perpendicularly magnetized $\mathrm{Ta}/[\mathrm{Pt}/{\mathrm{Co}}_{68}{\mathrm{B}}_{32}/{\mathrm{Ir}]}_{\times 10}/\mathrm{Pt}$ multilayer tracks with microfabricated Pt heaters and thermometers by magnetic force microscopy. Subtracting out the effects of the Oersted field from the heating current reveals the pure temperature gradient driven motion, which is always toward the heater. The higher the thermal gradient along the track is (owing to the proximity to the heater or larger heater currents), the greater the observed displacements of the domains are, up to a velocity of around 1 nm/s in a temperature gradient of 20 $\mathrm{K}/\mu \mathrm{m}$. This velocity lies in the creep regime. Quantitative estimates of the strength of different driving mechanisms for the effect that have been proposed theoretically show that entropic forces dominate over those arising from the spin Seebeck and spin-dependent Seebeck effects in driving the domain motion.

Characterizing magnetic fluctuations is one of the keys to understanding the origin of superconductivity in the spin-triplet superconductor ${\mathrm{UTe}}_2$, which exhibits two superconducting (SC) phases (SC1 and SC2) under pressure: SC1 where the superconducting transition temperature of $T_{\mathrm{c}}$ decreases with pressure, whereas $T_{\mathrm{c}}$ of SC2 rises with pressure. Previously, D. Ambika et al. [Phys. Rev. B 105, L220403 (2022)] reported the possible coexistence of ferromagnetic (FM) and antiferromagnetic (AFM) spin fluctuations in ${\mathrm{UTe}}_2$ under pressure from their nuclear magnetic resonance (NMR) measurements. To delve into the relationship between the magnetic fluctuations and the two SC phases, we have carried out detailed ${}^{125}\mathrm{Te}$ NMR measurements on a single crystal of ${\mathrm{UTe}}_2$ with $T_{\mathrm{c}}=1.6\mathrm{K}$ at various pressures ranging from 0 to 2.05 GPa. By comparing the temperature $T$ dependence of nuclear spin-lattice relaxation rates divided by temperature $1/T_{1}T$ with that of the Knight shift $K$ for magnetic fields along the $a$, $b$, and $c$ directions, we evidence the enhancement of AFM spin fluctuations with increasing pressure. Based on the results, we suggest that FM spin fluctuations are more favorable for SC1 and AFM spin fluctuations are crucial for SC2. Our findings will inspire further study on this material to understand the peculiar SC phases in detail.

Kagome materials with charge density waves (CDWs) are fascinating quantum systems, offering an ideal platform to explore intertwined orders and to uncover novel mechanisms behind CDW formation. Chemical models have been developed and applied to predict CDW in $A{M}_6{X}_6$-type kagome materials, such as the rattling chain model based on ${\mathrm{ScV}}_6{\mathrm{Sn}}_6$ and the magnetic energy-saving model based on FeGe. In this study, we successfully synthesized ${\mathrm{Ti}}_{0.85}{\mathrm{Fe}}_6{\mathrm{Ge}}_6$ single crystals using the vapor transport method. As predicted by the rattling chain model, these crystals are expected to exhibit kagome CDW behavior. Magnetization measurements indicate that ${\mathrm{Ti}}_{0.85}{\mathrm{Fe}}_6{\mathrm{Ge}}_6$ is an easy-axis antiferromagnet with $T_{\mathrm{N}}=$ 488 K, and transport measurements reveal metallic behavior primarily driven by electron-type carriers. However, no clear signatures of a CDW were observed in ${\mathrm{Ti}}_{0.85}{\mathrm{Fe}}_6{\mathrm{Ge}}_6$. Density functional theory calculations demonstrate a markedly distinct electronic structure compared to related compounds: instead of a carrier-doping-induced rigid shift, the density of states shifted away from the Fermi level. Consistent with our structural investigations, the absence of a CDW and the unusual band structure can be attributed to the bonding characteristic within ${\mathrm{Ti}}_{0.85}{\mathrm{Fe}}_6{\mathrm{Ge}}_6$. The strong covalent bonds of Ti-Ge1b, along with the solid Ge1b-Ge1b dimers, prevent the Ti-Ge1b-Ge1b-Ti chain from rattling. The presence of an Fe-Fe antibonding state at the Fermi level enhances the spin polarization and depletes the electronic density around the Fermi level. Our results suggest that both the ionic radius and the bonding characteristics of the filler atom are crucial for the formation of CDWs in kagome materials. These factors can serve as supplementary terms to the rattling chain model, providing new insights for the discovery of novel kagome CDW materials.

Narrow nanometer-sized damage trails created by swift heavy ions, so-called “ion tracks”, reflect a material's response to intense local electronic excitation. Ion tracks are commonly described as cylindrical damage zones with circular cross-sections. This assumption largely results from limitations of current characterization techniques to resolve ion track morphologies with sufficient detail. Here we present measurements of the cross-sectional morphology of ion tracks with angstrom-level precision using synchrotron-based small-angle x-ray scattering. We analyzed the track shape in single-crystalline fluorapatite, tourmaline, and synthetic $\alpha$-quartz irradiated with 185 MeV ${}^{197}\mathrm{Au}$ ions along different crystallographic directions. Our results reveal a clear anisotropy in the track cross-sections: while [0001]-oriented tracks have a largely circular cross-section, tracks along $\langle 10\overline{1}0\rangle$ show a distinct anisotropy. This anisotropy cannot be explained solely by electronic energy loss, but instead reflects the influence of the intrinsic physical properties of the crystals. The track dimensions correlate with the elastic properties in different crystallographic directions showing smaller cross-sections along directions of higher elastic stiffness. Furthermore, crystals with predominantly covalent bonding and higher thermal conductivity exhibit significantly smaller tracks. These findings highlight how anisotropic physical characteristics of single-crystal govern ion track formation, providing insight into the interplay between irradiation effects and crystal anisotropy.

Recent studies have focused on the relationship between charge density wave (CDW) collective electronic ground states and nontrivial topological states. ${\mathrm{YTe}}_3$, a nonmagnetic quasi-two-dimensional chalcogenide, has been reported to exhibit a CDW state below 334 K. Using angle-resolved photoemission spectroscopy (ARPES) and density functional theory (DFT), we establish that ${\mathrm{YTe}}_3$ is a CDW-induced Kramers nodal line (KNL) metal, a recently proposed topological state of matter. Scanning tunneling microscopy and low energy electron diffraction reveal two orthogonal domains, each with a unidirectional CDW and a similar wave vector (${\mathbf{q}}_{{}_{\mathrm{CDW}}}$). When the influence of twin domains is considered, the effective band structure (EBS) computations that utilize DFT-calculated bands using a noncentrosymmetric structure determined by x-ray crystallography, show excellent agreement with ARPES. The noncentrosymmetry of ${\mathrm{YTe}}_3$ is established by Raman spectroscopy. The Fermi surface and ARPES intensity plots show weak shadow bands displaced by ${\mathbf{q}}_{{}_{\mathrm{CDW}}}$ from the main bands. These are linked to CDW modulation, as the EBS calculation confirms. Bilayer split main and shadow bands suggest the existence of crossings, according to theory and experiment. DFT bands, including spin-orbit coupling, indicate existence of a KNL along the $\mathrm{\Sigma }$ direction from multiple crossings of bands dispersing perpendicular to it. Additionally, doubly degenerate bands are only found along the KNL at all energies, with some bands dispersing through the Fermi level.

We discuss a theoretical description of the inelastic electron tunneling spectra (IETS) of a magnetic nanosystem (an atom or a molecule) adsorbed on a solid surface measured in a scanning tunneling microscope (STM). We represent the nanosystem by means of a cluster Hubbard model, which allows us to study scenarios when the tunneling electrons sequentially interact with several magnetic centers inside the nanosystem or when the magnetic centers are made out of heavy atoms with a strong spin-orbit coupling and large orbital moments. The sequential tunneling through multiple centers is illustrated on an adatom probed by an STM tip with a nickelocene molecule attached to it. For atoms with large orbital moments, we find that the exchange interaction between the atom and the spin $\mathbf{s}$ of the tunneling electron is richer than the usually assumed Heisenberg form $\mathbf{S}·\mathbf{s}$ or $\mathbf{J}·\mathbf{s}$, where $\mathbf{S}$ and $\mathbf{J}$ are the spin and total orbital moments of the atom. For atoms in axially symmetric environments, the $\mathbf{J}·\mathbf{s}$ exchange would restrict the transitions accessible by IETS to those fulfilling $|\mathrm{\Delta }{J}_z|\le 1$, where $J_z$ is the projection of the total angular momentum to the symmetry axis, whereas we arrive at a more permissive selection rule $|\mathrm{\Delta }{J}_z|\le 2\ell +1$, where $\ell$ is the orbital momentum quantum number of the partially filled atomic shell carrying the magnetic moment.

Traditionally, the effect of the time-reversal operator during ferroelastic phase transition has been overlooked, as the transition is mechanically driven and relies solely on pure point group operations. Here, we propose nonrelativistic spin-split antiferromagnetic-ferroelastic systems, in which the existence of specific F-operators (a combination of ${180}^{\circ }$ rotation around an axis perpendicular to the spins in the spin space and spatial point group operation) can induce spin-flipping phenomena during the transition. Unlike the traditional spin modulation mechanism that depends on the coupling between ferroelastic and magnetic orders, this system enables spin reversal without any change of the magnetic order. By investigating all collinear spin groups and their subgroups, we demonstrate the symmetry constraints and systematically classify all ferroelastic species capable of inducing spin reversal in conjunction with ferroelastic phase transitions. First-principles calculations indicate that two-dimensional ${\mathrm{V}}_2\mathrm{SO}$ serves as an exemplary material for exploring this ferroelastic phase. Furthermore, we suggest that such materials may be utilized to develop innovative multiferroic ferroelastic tunnel junctions. Our findings provide new insights into the understanding of ferroelastic phase as well as new ways to realize nonrelativistic spin-split antiferromagnetic-ferroelastic devices.

The one-dimensional (1D) domain wall of 2D ${\mathbb{Z}}_2$ topological orders is studied theoretically. The Ising domain wall model is shown to have an emergent SU(2)${}_1$ conformal symmetry because of a hidden nonsymmorphic octahedral symmetry. While a weak magnetic field is an irrelevant perturbation to the bulk topological orders, it induces a domain wall transition from the Tomonaga-Luttinger liquid to a ferromagnetic order, which spontaneously breaks the anomalous ${\mathbb{Z}}_2$ symmetry and the time-reversal symmetry on the domain wall. Moreover, the gapless domain wall state also realizes a 1D topological quantum critical point between a ${\mathbb{Z}}_2^T$-symmetry-protected topological phase and a trivial phase, thus demonstrating the holographic construction of topological transitions.