We report multiangle reflectivity measurements in the extreme-ultraviolet (XUV) range for mono- and bilayer ${\mathrm{MoS}}_2$ on a ${\mathrm{Si}}_3{\mathrm{N}}_4$ substrate. Using a single-sheet 2D conductivity model, we extract the complex optical response of the ${\mathrm{MoS}}_2$ bilayer between 25 and 90 eV and derive an effective refractive index by introducing a thickness equal to the interlayer spacing. The ${\mathrm{MoS}}_2$ monolayer response is consistently reproduced either by halving the 2D conductivity or the effective thickness, indicating a robust scaling with layer number. Conversely from what was previously observed around the energy gap, the resulting optical constants display a broad resonance at the Mo ${\mathrm{N}}_{2,3}$ edge with no clear core-exciton peaks below the absorption edge despite the reduced dimensionality. First-principles calculations reproduce the experimental results and show that local-field (Hartree) effects dominate the XUV response, while screened-exchange (SEX) contributions remain weak and mainly induce spectral shifts. Our analysis demonstrates that excitonic effects play a minor role in the XUV optical response of atomically thin ${\mathrm{MoS}}_2$, highlighting key differences with respect to the visible and infrared regimes, and calling for a reassessment of the use of Mo-based transition metal dichalcogenides in attosecond spectroscopy and XUV excitonics.

The time-reversal symmetry ($\mathcal{T}$) breaking is a signature of ferromagnetism, giving rise to such phenomena as the anomalous Hall effect (AHE) and orbital magnetism. Nevertheless, $\mathcal{T}$ can also be broken in certain classes of antiferromagnets, such as weak ferromagnets or altermagnets, which remain invariant under spatial inversion. In light of this similarity with ferromagnetism, it is tempting to ask whether such an unconventional antiferromagnetic (AFM) state can be represented as the simplest ferromagnetic one, i.e., within the minimal unit cell containing only one magnetic site. We show that such a representation is possible due to a special form of the spin-orbit (SO) interaction in an antipolar lattice hosting this AFM state. The inversion symmetry constrains the form of the SO interaction, which becomes invariant under the symmetry operation $\left\{\mathcal{S}\right|\mathbf{t}\}$, combining the ${180}^{\circ }$ rotation of spins ($\mathcal{S}$) with the lattice shift $\mathbf{t}$, connecting two antiferromagnetically coupled sublattices. This is the fundamental symmetry property of centrosymmetric antiferromagnets, which justifies the use of the generalized Bloch theorem and transformation to the local coordinate frame with one magnetic site per cell. It naturally explains the emergence of AHE and net orbital magnetization, and provides transparent expressions for these properties in terms of the electron hoppings and SO interaction operating between AFM sublattices, as well as the orthorhombic strain, controlling the piezomagnetic response. The idea is illustrated on a number of examples including a two-dimensional square lattice, monoclinic ${\mathrm{VF}}_4$ and ${\mathrm{CuF}}_2$, and ${\mathrm{RuO}}_2$-type materials with the rutile structure, using for these purposes realistic models derived from first-principles calculations.

In this paper, we predict that the magnetoelectric response of two-dimensional (2D) antiferromagnets contains a contribution determined by the topological properties of the ground state. The quasitopological magnetoelectric response, encoded in the spin magnetoelectric polarizability and the closely related spin multipolization, occurs when the electronic structure of the antiferromagnetic insulator is described by massive 2D Dirac fermions and is therefore native to 2D, unlike the topological magnetoelectric effect of three-dimensional topological insulators. To demonstrate the topology-induced contribution to the (spin) magnetoelectric polarizability, we compute the magnetoelectric polarizability microscopically for two distinct minimal lattice models: a spin-orbit-coupled Néel antiferromagnet and a spin-orbit-free noncollinear antiferromagnet with double-$Q$ spin order. We show that the topological origin of the revealed magnetoelectric effect can be traced back to the electromagnetic response of topological semimetals in two dimensions, and hence is ultimately governed by a strong topological invariant in one dimension. Given this dimensional hierarchy, we further consider two minimal lattice models in one dimension, both one-dimensional variants of the 2D lattice models, and show that the magnetoelectric polarizability exhibits a clear signature of nontrivial crystalline topology. Possible material realizations are discussed.

Recent experiments on trilayer transition-metal dichalcogenide heterostructures have revealed the rich behavior of dipolar excitons. Motivated by these experimental observations, we investigate the collective dynamics of planar quantum dipoles whose orientations fluctuate because of charge tunneling between the outer layers. Using large-scale quantum Monte Carlo simulations, we map out the low-temperature phase diagram as a function of experimentally tunable parameters. We uncover a diverse landscape of phases driven by dipolar correlations. Under strong dipole fluctuations, a quadrupolar superfluid emerges. Suppressing charge tunneling nucleates a droplet state stabilized by the attractive interaction between antiparallel dipoles. At high exciton densities, the system gives way to a partially fragmented condensate, characterized by a finite interaction-driven occupation of the two dipolar states alongside a dominant quadrupolar superfluid component. Furthermore, exploring large effective masses that suppress zero-point motion, we find a staggered dipolar crystal at high densities. Our detailed study of the dependence of exciton energy shifts on an external electric field provides qualitative theoretical guidance for interpreting existing data, underscoring the crucial role of the antiparallel dipolar configuration and paving the way for future experimental explorations of quantum phases of trilayer excitons.

A phase transition is an example of a “topological defect” in the space of parameters of a quantum or classical many-body system. In this paper, we consider phase diagram topological defects of higher codimension. These have the property that equilibrium states undergo some kind of nontrivial winding as one moves around the defect. We show that such topological defects exist even in classical statistical mechanical systems, and describe their general structure in this context. We then introduce the term “diabolical critical point” (DCP), which is a higher-codimension analog of a continuous phase transition, with the proximate phases of matter replaced by the nontrivial winding of the proximate equilibrium states. We propose conditions under which a system can have a stable DCP. We also discuss some examples of stable DCPs in $(1+1)$-dimensional quantum systems.

In the nonrelativistic limit, helimagnetic order is always associated with odd-parity magnetism. That is, for single-particle states, the expectation value of the electronic spin is odd in crystal momentum, which implies direct control of the spin by means of electric fields. However, the theoretical description of helimagnets is hindered by the fact that the spiral pitch may require large supercells or even be incommensurate with the lattice. In this article, we show that such issues may be remedied by use of the generalized Bloch theorem. It allows one to describe (by models or first principles) the system in terms of the primitive unit cell, from which all relevant properties can be obtained by downfolding in reciprocal space. We exemplify the procedure using ${\mathrm{MnI}}_2$ and ${\mathrm{NiI}}_2$, which are known type II multiferroics having spiral order, and the helimagnetic metal ${\mathrm{MnTe}}_2$. We analyze how the magnitude of spin splitting depends on orbital composition of bands and we show that spin splitting is maximized for states having large odd-orbital ($p$-type) character. It is straightforward to generalize the framework to handle response functions for helimagnets using only the primitive unit cell, and the present downfolding procedure thus strongly facilitates theoretical progress in the field.

The generation and control of pure spin Hall currents are central to low-dissipation spintronics. However, effectively manipulating spin Hall currents via nonvolatile means remains challenging, severely hindering their application in spintronics. Here, we show that the intrinsic nonlinear spin Hall effect, driven by a quantum metric dipole, enables a pure spin Hall current in ferroelectrics because the transverse net charge flow is prohibited by time-reversal or crystalline symmetries. We further demonstrate that ferroelectric polarization provides an all-electric means to reverse the nonlinear pure spin Hall current. By systematically enumerating all gray point groups and performing first-principles calculations, we identify five groups enabling such reversibility as well as a series of two-dimensional ferroelectrics, including $\alpha \text{−}\mathrm{Bi}$, SnSe, SnTe, ${\mathrm{NbOI}}_2$, and $\alpha \text{−}{\mathrm{In}}_2{\mathrm{Se}}_3$. Importantly, the pure spin Hall currents in these candidates are exclusively polarized along the out-of-plane direction and exhibit a uniform spin texture during transport, thereby maintaining a long spin lifetime of carriers. Our findings reveal a mechanism for a ferroelectric switchable pure spin Hall current, opening a route toward low-dissipation spintronics in ferroelectrics.

Two-dimensional honeycomb lattice ferromagnet chromium tri-iodide (${\mathrm{CrI}}_3$) has attracted tremendous interest because it retains ferromagnetism down to the monolayer limit and hosts intriguing topological magnons. As a prototypical van der Waals magnet, ${\mathrm{CrI}}_3$ provides an ideal platform for exploring the interplay between reduced dimensionality, magnetic order, and nontrivial spin excitations. Here, using inelastic neutron scattering together with improved sample quality, we uncover the magnon winding feature around the $K$-point of the hexagonal Brillouin zone, a key signature of Dirac magnons. In addition, we find that the magnon energy follows a $T^2$-renormalization behavior at elevated temperatures, consistent with magnon-magnon interactions. These results provide previously missing information on the magnon spectrum of ${\mathrm{CrI}}_3$ and further consolidate the topological nature of its spin excitations.

The phase diagram of the two-dimensional quantum $J_1\text{–}{J}_3$ Heisenberg antiferromagnet on a square lattice is a long-standing open problem. Despite recent advances in numerical techniques for quantum spin models, a detailed analytical theory is still lacking. We address this problem using a semiclassical approach based on a continuum nonlinear sigma model effective field theory, applying the nonperturbative large-$N$ technique to map out the phase diagram and determine the magnetic correlations. We show that previously overlooked interactions are crucial for stabilizing a vestigial nematic phase, both at finite and zero temperature. Our results reveal that the spontaneous breaking of global symmetries in the $J_1\text{–}{J}_3$ model is controlled by the strength of infrared quantum fluctuations which are enhanced by proximity to the classical Lifshitz point.

Ruthenium trihalides $\mathrm{Ru}{X}_3$ ($X=\text{Cl,}\text{Br,}\text{I}$) provide a tunable platform for Kitaev magnetism in two-dimensional van der Waals materials. Despite their similar crystal structures and zigzag antiferromagnetic order, ${\mathrm{RuBr}}_3$ exhibits a higher Néel temperature ($T_N$) than ${\mathrm{RuCl}}_3$, suggesting their distinct proximity to the Kitaev quantum spin-liquid phase. Using Ru $L_3$-edge resonant x-ray scattering, we show that, while the long-range zigzag order in ${\mathrm{RuBr}}_3$ disappears at $T_N$, the zigzag correlations that persist well above $T_N$ show a pronounced spectral weight redistribution along the interlayer direction. These results suggest that the enhanced interlayer magnetic interactions driven by the extended Br $4p$ orbitals stabilize three-dimensional zigzag correlations in ${\mathrm{RuBr}}_3$.

We investigate the restoration of topological protection at armchair interfaces in quantum valley Hall systems by introducing an adiabatic geometry. In conventional armchair boundaries, strong intervalley mixing between K and ${\mathrm{K}}^{\prime }$ valleys opens a gap in the interface modes and degrades topological protection. Here, we show that a smoothly varying mass domain wall suppresses intervalley mixing and closes the inner band gap of the interface modes, leading to the recovery of topological protection over a wide frequency range within the bulk band gap. Numerical transport simulations demonstrate enhanced transmission efficiency and strong robustness against structural defects. We further implement the adiabatic topological geometry as microelectromechanical system-based phononic waveguides, and experimentally verify robust wave propagation through sharply bent (${90}^{\circ }, {120}^{\circ }$, and ${150}^{\circ }$) interfaces. Suppressing valley mixing via adiabaticity provides a versatile design framework for topological waveguides and enables robust topological characteristics across a wide range of wave phenomena.

In a structure with two tunnel-coupled quantum wells of different widths, the spin Kerr effect resulting from pulsed optical pumping at the narrow-well exciton resonance reveals the wide-well electron magnetization dynamics. Our analysis shows that the effect is driven by electron exchange between narrow-well excitons and spin-polarized electrons in the wide well. A theoretical model of the spin Kerr effect has been developed accounting for the interwell electron spin exchange. In the studied double-well structure with CdTe and ${\mathrm{Cd}}_{0.98}{\mathrm{Mn}}_{0.02}\mathrm{Te}$ quantum wells separated by a ${\mathrm{Cd}}_{0.88}{\mathrm{Mg}}_{0.12}\mathrm{Te}$ barrier with a thickness of 1.6 nm (5 monolayers), the model accurately describes the experimental results and allows us to estimate the interwell electron exchange constant as ${\delta }_e\approx 1.1\times {10}^{-15}\text{eV}{\text{cm}}^2$.

We use a direct phase-resolved optical technique to study the coherence of spin waves (SWs) that are driven by surface acoustic waves (SAWs) via resonant magnetoelastic coupling. For this, we employ a piezoelectric lithium tantalate (${\mathrm{LiTaO}}_3$) substrate, equipped with micropatterned interdigital transducers for SAW excitation, which interact with SWs in a 5-nm-thin and 20-µm-wide ${\mathrm{Co}}_{40}{\mathrm{Fe}}_{40}{\mathrm{B}}_{20}$ waveguide. We detect the SAW and the SW using a phase-locked microfocused optical polarization detection experiment and use the characteristic polarization dependence to separate the SAW and SW signals. Our measurements directly image the resonant and coherent excitation of the SW by the SAW.

Hexagonal chromium diboride ${\mathrm{CrB}}_2$ is a metallic frustrated antiferromagnet with a Néel temperature $T_N\sim$ 88 K. In ${\mathrm{CrB}}_2$, Cr $3d$ electrons not only give rise to localized magnetic moments but also contribute to metallic conduction. We perform ultrasound velocity measurements on a single crystal of hexagonal ${\mathrm{CrB}}_2$ to determine its elastic properties. The temperature dependence of the $ab$-plane shear elastic modulus exhibits Curie-type softening upon cooling from $\sim 120\mathrm{K}$ down to $T_N$. This behavior is interpreted as a precursor to a symmetry-lowering lattice distortion at $T_N$, indicating that magnetic frustration is relieved via transverse magnetoelastic coupling. In addition, the $a$-axis and $c$-axis compressive elastic moduli show unusual softness and their suppression upon cooling, which are naturally explained by Fermi-surface nesting and its suppression upon cooling. The present results suggest that, in ${\mathrm{CrB}}_2$, longitudinal magnetoelastic coupling suppresses Fermi-surface nesting and enhances frustrated exchange interactions, while transverse magnetoelastic coupling plays a key role in relieving the frustration.

We perform a renormalized mean-field study of the two-dimensional repulsive Hubbard model, focusing on the intricate interplay and possible coexistence of magnetic, charge, and superconducting orders. We improve on conventional mean-field theory by utilizing a renormalization group framework that captures high-energy fluctuations. This method generates effective magnetic and $d$-wave pairing interactions, and allows for an unbiased exploration of coexisting phases at weak and moderate interaction strengths. Unrestricted mean-field calculations of the effective Hamiltonian on large finite lattices are combined with analyses in the thermodynamic limit, revealing a rich phase diagram with extensive regions of coexisting orders. We find that $d$-wave superconductivity coexists with Néel order on the electron-doped side. On the hole-doped side, superconductivity is found to coexist with spiral or stripe magnetic orders. Within the stripe-ordered region, the superconducting order parameter is spatially modulated, with a period that follows the charge modulation of the stripes. Below Van Hove filling, pairing provides the primary energy gain, while the stripe order yields only a small, and hence fragile, additional energy lowering.

We report results of large-scale quantum Monte Carlo (QMC) simulations of graphene. Using cutting-edge algorithmic improvements, we are able to consider spatial volumes, corresponding to 20808 electrons, that allow us to access energy scales of direct relevance to experiments. Using constrained random phase approximation (cRPA) estimates of short-ranged interactions combined with a Coulomb tail, we are able to successfully confront numerical and experimental estimates of the Fermi velocity renormalization, thus fixing the parameters of a microscopic Hamiltonian to exactly reproduce the experimental data. Subsequent comparison of the QMC results with perturbation theory not only show the non-Fermi-liquid character of graphene, but also prove the importance of lattice-scale physics and higher-order perturbative corrections beyond the RPA for the quantitative description of the experimental data for the Fermi velocity renormalization in suspended graphene. The same Monte Carlo technique is subsequently employed to investigate electrical transport by computing the renormalization of the optical conductivity in freestanding graphene. The large lattice sizes and low temperatures accessible in our QMC simulations enable a clear resolution of the Dirac plateau in the spectral function, even in the presence of interactions. We demonstrate that, despite the strong renormalization of the single-particle dispersion, the optical conductivity remains essentially constant, in agreement with current experimental observations. This finding points to the possibility of extending the theorem on the absence of optical conductivity renormalization for the Hubbard model on the hexagonal lattice to the case of long-range interactions.

We introduce a generalized hydrodynamic framework for predicting thermal energy currents in semiconductor nanostructures with characteristic dimensions down to tens of nanometers across a broad temperature range. The evolution of slowly evolving moments of the phonon distribution function is predicted deterministically using a hydrodynamic equation. To accurately capture nonequilibrium effects under extreme confinement, the heat-flux profile is further refined by stochastically modeling higher-order statistical features of the distribution that become significant far from local equilibrium. The formalism, which uses inputs strictly calculated from first principles, is validated against the apparent thermal conductivity of silicon ultrathin films across a wide range of temperatures.

We demonstrate that the charge value of transport mechanisms heavily impacts the validity of thermodynamic uncertainty relations (TURs). Specifically, we show within the framework of full counting statistics, that the recently established quantum TUR can be violated by the presence of transport processes that carry more than one charge, like Andreev reflection processes in normal metal-superconductor junctions. We propose a modified quantum TUR, which incorporates the charge value and demonstrate that this charge-dependent quantum TUR can only be violated if the highest charge transport process exceeds this charge value. In particular, we establish that the breaking of the quantum TUR solely originates from the charge value of the highest charge transport process. Namely, our analytical considerations do not invoke the existence of superconductivity, and these considerations generally hold for noninteracting electronic transport which can be described by the scattering formalism.

Zero-field muon-spin-rotation ($\mu \mathrm{SR}$) experiments and the observation of a spontaneous Nernst effect indicate broken time-reversal symmetry (BTRS) at $T_{\mathrm{c}}^{{\mathbb{Z}}_2}$, slightly above the superconducting transition temperature $T_{\mathrm{c}}^{}$ in ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$, with $x\sim 0.8$. This was interpreted as evidence of a distinct state of matter where BTRS above $T_{\mathrm{c}}^{}$ arises from noncondensed Cooper pairs, which maintain long-range interband correlations. However, an inhomogeneous superconducting state competing with magnetism as the origin of the observed behavior cannot be completely ruled out to date. Here, we present detailed ${}^{75}\mathrm{As}$ nuclear magnetic resonance (NMR) measurements of ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$, with $x=0.766$, yielding $T_{\mathrm{c}}^{{\mathbb{Z}}_2}>T_{\mathrm{c}}^{}$. The NMR data demonstrate that the superconductivity is homogeneous throughout the entire sample volume despite the occurrence of nanoclustering of Ba/K atoms. We observe a suppression of the spin-lattice relaxation rate $1/T_{1}T$ below the characteristic temperature $T^{*}$, well above $T_{\mathrm{c}}^{}$. The specific heat yields somewhat lower values of $T^{*}$, consistent with the microscopic theory of electron quadrupling. The observed monotonous suppression of $T^{*}$ and $T_{\mathrm{c}}^{{\mathbb{Z}}_2}$ in applied magnetic field, the overall temperature dependence of $1/T_{1}T$, the NMR spectral characteristics, and the $\mu \mathrm{SR}$ Knight shift rule out spin magnetism and the proximity to a magnetic instability. We conclude that our data indicate the appearance of pairing fluctuations below $T^{*}$ and that the breaking of time-reversal symmetry represents an unconventional state which is unrelated to spin magnetism.

Single-crystal tellurium provides an elemental platform for examining how electron-phonon coupling and anharmonic phonon scattering appear in Raman spectra and lattice thermal transport. Here, we combine temperature-dependent Raman spectroscopy with first-principles calculations to investigate the phonon response and heat transport of trigonal Te. The calculated electronic structures are strongly modified by spin-orbit coupling, whereas the phonon spectra obtained with and without spin-orbit coupling show only minor differences. Raman measurements reveal a pronounced Fano asymmetry of the $A_1$ mode at low temperature, indicating interference between a discrete phonon and an electronic continuum. The inverse asymmetry parameter $1/q$ decreases toward the Lorentzian limit upon warming, showing that the Fano interference becomes less pronounced at elevated temperatures. Mode-resolved frozen-phonon calculations show that the in-plane $E^1$ and $A_1$ displacements produce appreciable phonon-induced electronic band responses near the $H$ point, where the low-energy bands have strong $p_x/p_y$ character, whereas the $E^2$ displacement produces a weaker response. The temperature-dependent Raman shifts and linewidths are described within a Balkanski-type anharmonic framework, with the linewidths indicating appreciable higher-order anharmonic contributions for selected optical modes. First-principles thermal-transport calculations further show that four-phonon scattering shortens the lifetimes of low-frequency heat-carrying phonons and reduces the lattice thermal conductivity. These results clarify the distinct roles of the mode-selective Fano response and anharmonic phonon scattering in the Raman spectra and lattice thermal transport of single-crystal Te.