Two-dimensional coherent spectroscopy (2DCS) is an established method for characterizing molecules and has been proposed in the terahertz regime as a tool for probing exotic excitations of quantum magnets; however, the precise nature of the coupling between pump field and spin degrees of freedom has remained unclear. Here, we develop a general response theory of 2DCS and show how magnetoelectric as well as polarization couplings contribute to 2DCS in addition to the typically assumed magnetization. We propose experimental protocols to probe the presence of polarization couplings, for instance, from exchange-striction or spin-current mechanism, when the electric field couples to quadratic terms in spin operators. We provide example calculations for the paradigmatic twisted Kitaev chain material ${\mathrm{CoNb}}_2{\mathrm{O}}_6$ and highlight the crucial role of contributions from cross-coupling between polarization and magnetic nonlinear susceptibilities. Our work paves the way for systematic studies of light-matter couplings in quantum magnets and for establishing 2DCS as a versatile tool for probing fractional excitations of exotic magnetic quantum phases.

The spin Hall magnetoresistance (SMR) is widely used to study the interplay between charge and spin currents in bilayers of a magnetic insulator and a normal metal. However, not much is known about the spatial variation of the SMR across the surface of the same sample. In this work, we investigate the statistical distribution of the SMR in hundreds of nominally identical Hall bar structures patterned into prototypical yttrium iron garnet (YIG)/Pt heterostructures. We find a Gaussian-distributed SMR with a narrow standard deviation of approximately $10\%$ of the mean value in each YIG/Pt bilayer studied. However, the variation of the mean SMR between different YIG/Pt samples can be as large as $\sim 35\%$, despite nominally identical fabrication conditions. Additionally, we compare the SMR statistics in YIG/Pt heterostructures prepared in-situ (without breaking the vacuum) and ex-situ (with the YIG layer exposed to air before Pt deposition), as well as in CoFeB/Pt. We find the SMR standard deviation to be systematically smaller for in-situ interfaces. We conclude that on a microscopic level, local variations of the interface quality captured by the spin mixing conductance are the most likely origin for the observed SMR amplitude variations.

We study a broad class of qudit stabilizer codes, termed ${\mathbb{Z}}_N$ bivariate-bicycle (BB) codes, arising either as two-dimensional realizations of modulated gauge theories or as ${\mathbb{Z}}_N$ generalizations of binary BB codes. Our central finding, derived from the polynomial representation, is that the essential topological properties of these ${\mathbb{Z}}_N$ codes can be determined by the properties of their ${\mathbb{Z}}_p$ counterparts, where $p$ are the prime factors of $N$, even when $N$ contains prime powers ($N={\prod }_i{p}_i^{k_i}$). This result yields a significant simplification by leveraging the well-studied framework of codes with prime qudit dimensions. In particular, this insight directly enables the generalization of the algebraic-geometric methods (e.g., the Bernstein-Khovanskii-Kushnirenko theorem) to determine anyon fusion rules in the general qudit situation. Moreover, we elucidate the symmetry-enriched topological (SET) order underlying the quasifractonic behavior in qudit BB codes (including the Delfino-Chamon-You model), resolving the associated anyon mobility puzzle. We also develop an efficient computational algebraic method, based on Gröbner bases over the ring of integers, to determine both the topological order and its SET properties.

Machine learning interatomic potentials (MLIPs) have been widely employed in simulations of materials. In many systems, particularly ionic systems, long-range interactions often dominate and significantly influence their dynamical behavior. However, long-range effects such as Coulomb and van der Waals interactions are not considered in most MLIPs. To address this issue, we propose a method that incorporates long-range effects into local MLIPs using a reciprocal-space neural network. The structure information in real space is firstly transformed into reciprocal space and then encoded into a reciprocal-space potential (RSP) or a reciprocal-space descriptor (RSD) that takes complete atomic interactions into account. The RSP and RSD keep full invariance of Euclidean symmetry and unit-cell choice. Benefiting from the reciprocal-space information, MLIPs are enabled to describe not only Coulomb but also other types of long-range interactions. We demonstrate the advantages of our method using a model NaCl system with Coulomb and van der Waals interactions, ${\mathrm{Ga}}_x{\mathrm{N}}_y$ system, and ${\mathrm{HfO}}_2$ system with defects. At the same time, our approach helps to improve the prediction accuracy of some global properties such as the band gap where atomic interactions beyond local atomic environments play a very important role.

Frustrated magnets host strong quantum fluctuations that can suppress conventional magnetic order and give rise to exotic quantum phases such as spin liquids. In some cases, however, quantum fluctuations lift classical degeneracies and stabilize ordered states via an order-by-quantum-disorder mechanism. The spin-7/2 honeycomb antiferromagnet GdZnPO has recently been proposed as a spiral spin-liquid candidate arising from cooperative fluctuations among a subextensively degenerate manifold of spiral states. Here, we investigate the local magnetization and spin dynamics in GdZnPO using nuclear magnetic resonance. In an intermediate field regime between $\sim 3\mathrm{T}$ and the saturation field $(\sim 12\mathrm{T})$, we observe a spatially uniform magnetization and persistent low-energy spin dynamics down to 0.033 K, with no detectable symmetry breaking, providing spectroscopic evidence for a spin-liquid state. At lower fields below $\sim 3\mathrm{T}$, a weak stripe order emerges below $\sim 0.25$ K; however, strong fluctuations persist, as indicated by a nearly temperature-independent and unusually large spin-lattice relaxation rate in the low-temperature limit. Our results demonstrate that spin-7/2 quantization weakly lifts the spiral degeneracy, stabilizing subtle magnetic order while preserving robust dynamics and spin-liquid phenomenology. These findings establish GdZnPO as a promising platform for exploring spin liquids in high-spin frustrated magnets down to the lowest accessible temperatures.

At very small twist angles, bilayer moiré systems exhibit characteristic stacking domain patterns, where the moiré length scale is determined solely by the twist angle. In contrast, the additional stacking and twisting degrees of freedom in twisted trilayer systems give rise to richer and more intricate domain tessellations. In twisted trilayer hexagonal boron nitride (TTBN), the interplay between polar and nonpolar domains and their domain walls is shown to result in unconventional responses to external electric fields, including electric-field tunability of the moiré-of-moiré or super moiré pattern—features absent in bilayer counterparts. We demonstrate that at the vertices of super moiré domains, TTBN can support arrays of quantum dots hosting localized quantum harmonic oscillator (QHO) states with diverse spatial symmetries. Furthermore, we show that the shape of the array and the spacing between the localized QHO states can be dynamically reconfigured by electric fields, enabling facile switching between fully isolated and strongly coupled regimes. The local potentials for the quantum dot state are predicted to be sufficiently deep to support a series of QHO states with nonzero angular momentum. This tunability enables control over the transport of quantum dot states and their interdot coupling, facillitating long-range quantum state transfer. Combined with the feasibility of large-scale fabrication of homogeneous twisted trilayer materials, these properties position TTBN as a promising platform for a wide range of quantum technologies.

Combining superconducting and magnetic materials is a promising path to generate exotic interface subgap states. In this regard, altermagnetism is particularly interesting because it lifts spin degeneracy while providing tailored anisotropy of spin splittings. Here, we investigate the realization and control of subgap states by using the symmetry contrast between altermagnetic fields and unconventional pairings. When the symmetries of altermagnetism and unconventional superconductivity align, we demonstrate the emergence of bulk zero-energy flat bands as the Bogoliubov Fermi surface, giving rise to a zero-bias conductance peak. The symmetry and strength of $d$-wave altermagnets strongly affect the surface Andreev states from $d$-wave and chiral $d$- and $p$-wave superconductors. As a result, distinct types of subgap states, including curved and flat bands, are realized that can be detected by tunneling spectroscopy. Our results offer a solid route for designing and manipulating subgap states in superconducting systems, which can be useful for functionalizing superconducting devices.

We decompose the intrinsic second-order nonlinear Hall effect (NLHE) of a generic multiband system into its quantum geometric contributions within a fully quantum-mechanical, projector-based formalism. By expanding the nonlinear conductivity in powers of the quasiparticle lifetime $\tau$, we recover the established Berry curvature dipole at order $\tau$ and clarify discrepancies in previous literature concerning the (interband) quantum metric dipole (or Berry curvature polarizability) contribution at order ${\tau }^0$. Crucially, our method reveals an additional contribution at order ${\tau }^0$, determined by the intraband quantum metric dipole (intraQMD), arising from additional virtual interband transitions captured within the fully quantum-mechanical treatment. The intraQMD contribution is generically nonzero in systems with broken time-reversal symmetry and can be distinguished from other geometric contributions by symmetry. Analytical results for low-energy models of topological band crossings, which are hot spots of quantum geometry, demonstrate how band topology influences each contribution. In particular, the intraQMD contribution is especially large in gapped Dirac cones in antiferromagnets. Through a comprehensive symmetry classification of all magnetic space groups, we identify several candidate materials that are expected to exhibit large intrinsic NLHE, including the topological antiferromagnets ${\mathrm{Yb}}_3{\mathrm{Pt}}_4$, CuMnAs, and ${\mathrm{CoNb}}_3{\mathrm{S}}_6$, as well as the nodal-plane material ${\mathrm{MnNb}}_3{\mathrm{S}}_6$.

We study the interplay between superconductivity and altermagnetism in disordered systems using recently derived quantum kinetic transport equations. Starting from this framework, we derive the Ginzburg-Landau free energy and identify, in addition to the conventional pair-breaking term, a coupling between the spin and the spatial variation of the superconducting order parameter. Two distinct effects emerge from this coupling. The first is a nonlinear magnetoelectric effect, in which a supercurrent (i.e., a phase gradient) induces a spin texture; this contribution is quadratic in the phase gradient. The second effect arises when the magnitude, rather than the phase, of the superconducting order parameter varies in space, likewise leading to a finite magnetization. We show that these two contributions compete in the case of an Abrikosov vortex, where both the amplitude and phase of the order parameter vary spatially. The effect associated with amplitude variations also gives rise to a proximity-induced magnetization (PIM) in hybrid structures composed of a superconductor (S) and an altermagnet (AM). Using quasiclassical theory, we analyze the PIM in diffusive S/AM bilayers and S/AM/S Josephson junctions, and determine the induced magnetization profiles. In Josephson junctions, where both the PIM and the nonlinear magnetoelectric effect coexist, we further predict the occurrence of $0-\pi$ transitions.

Magnetic hopfions—three-dimensional topological solitons with knotted spin textures—have recently garnered attention in topological magnetism due to their unique knot topology characterized by the Hopf number $H$. In contrast with two-dimensional skyrmions, which are typically limited to small topological invariants, hopfions can, in principle, be stabilized with arbitrary $H$. However, the nonequilibrium dynamics, especially interconversion between different $H$ values, remain poorly understood. Here, we theoretically investigate the nonequilibrium dynamics of hopfions with various $H$ values by numerically solving the Landau-Lifshitz-Gilbert equation with spin-orbit torque (SOT). For $H=1$, we show that SOT induces both translational and precessional motion, with dynamics sensitive to the initial orientation. For $H=2$, we find that intermediate SOT strengths can forcibly split the hopfion into two $H=1$ hopfions. This behavior is explained by an effective tension picture, derived from the dynamics observed in the $H=1$ case. By comparing the splitting dynamics across different $H$ values, we identify a hierarchical structure governing SOT-driven behavior and use it to predict the dynamics of hopfions with general $H$. Furthermore, we show that, by appropriately scheduling the time dependence of the SOT, it is possible to repeatedly induce both splitting and recombination of hopfions. These results demonstrate the controllability of hopfion topology via SOT and suggest a pathway toward multilevel spintronic devices based on topology switching.

We use muon spin rotation/relaxation/resonance $\left(\mu \mathrm{SR}\right)$ to investigate the magnetic properties of niobium pentoxide $\left({\mathrm{Nb}}_2{\mathrm{O}}_5\right)$ and tantalum pentoxide $\left({\mathrm{Ta}}_2{\mathrm{O}}_5\right)$ thin films. In both oxides, we observe a magnetic response at the lowest available temperature of 2.8 K. This response appears to be structurally dependent: thermally oxidized ${\mathrm{Ta}}_2{\mathrm{O}}_5$ with low crystallinity demonstrates suppressed magnetism, while fully amorphous ${\mathrm{Ta}}_2{\mathrm{O}}_5$ demonstrates local static magnetism. In contrast, amorphous ${\mathrm{Nb}}_2{\mathrm{O}}_5$ is dominated by magnetic fluctuations and is strongly magnetically disordered compared to ${\mathrm{Ta}}_2{\mathrm{O}}_5$. Our results suggest that these fundamental differences in the magnetism of Ta and Nb oxides could explain the performance limitations in superconducting qubits and resonators.

When chiral superfluid ${}^3\mathrm{He}\text{−}\mathrm{A}$ is confined in between parallel plates with an in-plane magnetic field, almost uniform textural domain, where the orbital angular momentum axis $\widehat{\mathit{l}}$ and spin anisotropy axis $\widehat{\mathit{d}}$ are aligned to be perpendicular to the plate surface, is formed. Due to the degeneracy of the reversed directions of anisotropic axes, adjacent domains with a different combination of the two anisotropic axes may coexist. In this case, a thin domain wall appears in between two domains. The existence of multiple kinds of domain walls, which are topologically sustainable objects in a surrounding slowly varying texture, are found through the NMR and MRI measurements. A structure of the domain walls are determined through the spectroscopic and relaxometric study with spatial resolution. In addition to the identification of the domain walls, we have succeeded in capturing a real-space and real-time movie of the motion and pair annihilation of the stacked domain walls.

In a significant conceptual revision to the tenfold classification scheme for topological insulators and superconductors, it was recently demonstrated that most three-dimensional (3D) classes are simultaneously “localizable” in two distinct, but intricately connected ways. (1) There is no obstruction to Wannier localization of all bulk eigenstates, and (2) Almost all surface states can be Anderson localized by arbitrarily weak symmetry-preserving quenched disorder. Here we consider the localizable class CI in 3D, and numerically investigate the stability of surface states. We demonstrate that surface states of a bulk class-CI topological lattice model are fragile in that they can be Anderson localized by the combination of weak quenched randomness and hybridization with an additional trivial 2D band (a trivializing proximity effect, TPE). With the TPE, stronger disorder is more destructive to the surface states of the bulk lattice model. By contrast, without additional bands, the surface states remain extended, exhibiting robust spectrum-wide quantum criticality. We also investigate the fragility of surface states in effective 2D class-CI continuum Dirac theories, including the chiral limit of the Bistritzer-MacDonald model for twisted bilayer graphene. Although the continuum models exhibit signs of Anderson localization near gap edges for weak disorder, stronger disorder instead appears to heal the surface, restoring criticality whilst filling in spectral energy gaps. Our results provide further evidence that effective continuum field theories fail to capture key aspects of surface-state physics in localizable topological phases.

Misfit layered compounds are natural van der Waals heterostructures in which electronically active transition-metal dichalcogenide layers are decoupled by incommensurate blocking layers, enabling bulk realization of quasi-two-dimensional quantum states. Here we investigate the superconducting, transport, and structural properties of the misfit compound ${\left(\mathrm{SnS}\right)}_{1.15}\left(\mathrm{Ta}{\mathrm{S}}_2\right)$ under pressures up to $\sim 150$ GPa. The low-pressure superconducting phase is gradually suppressed and disappears near 14.7 GPa. Remarkably, a distinct superconducting phase reemerges above $\sim 80$ GPa and persists to the highest pressures achieved. This reentrant superconductivity follows a pressure-induced sign reversal of the Hall coefficient near 60 GPa and a nonmonotonic evolution of the normal-state resistance, indicating an intimate connection with pressure-driven electronic reconstruction. Our work demonstrates a pressure-driven electronic reconstruction leading to reentrant superconductivity in a misfit layered compound, establishing pressure as an effective route to engineer superconductivity and electronic states in natural van der Waals heterostructures.

Motivated by experiments on rhombohedral tetralayer graphene showing signs of superconductivity emerging from a valley-polarized normal state, we here analyze theoretically how tunneling spectroscopy can be used to probe the superconducting order parameter of the system. To describe different pairing scenarios on equal footing, we develop a microscopic tunneling approach that can capture arbitrary, including finite-momentum, superconducting order parameters and low-symmetry normal-state Hamiltonians. Our analysis shows that the broken time-reversal symmetry in a single valley leads to unique features in the weak-tunneling regime that are different for commensurate and incommensurate Cooper pair momenta. We further uncover an unconventional spatial dependence of the Andreev conductance, allowing to distinguish between three topologically distinct classes of single-$\mathbf{q}$ pairing states in the system, and compute the signatures of a competing translational-symmetry breaking three-$\mathbf{q}$ superconductor.

We study Landau-level mixing in a time-reversal symmetric Hamiltonian composed of two sets of Landau levels with opposite magnetic field, relevant to moiré minibands in twisted homobilayer transition-metal dichalcogenides in the adiabatic limit, where electrons in opposite valleys have flat Chern bands with opposite Chern numbers. Strong spin-orbit coupling polarizes spins in opposite directions in opposite valleys, separating Coulomb interactions into like-spin ($V^{\uparrow \uparrow }$) and opposite-spin ($V^{\uparrow \downarrow }$). Using degenerate perturbation theory, we compute Landau-level mixing corrections to $V^{\uparrow \uparrow }$ and $V^{\uparrow \downarrow }$ for different filling fractions. In the lowest Landau level, screening exhibits an even-odd effect: $V^{\uparrow \uparrow }$ is reduced more strongly than $V^{\uparrow \downarrow }$ in an even-$m$ angular momentum Haldane pseudopotential and less strongly in odd-$m$ angular momentum ones. In the first Landau level, the short-range part ($m=0,1$) of $V^{\uparrow \downarrow }$ is reduced comparably to $V^{\uparrow \uparrow }$, while the strongest spin anisotropy appears in the $m=2$ pseudopotential. These novel short-range spin correlations have important implications for candidate correlated phases of fractional quantum spin Hall insulators. A distinctive feature of this time-reversal symmetric Hamiltonian, absent in conventional quantum Hall systems, is that spin-flip excitations form localized quasiparticles. We compute their excitation spectrum and predict a nonmonotonic dependence of the ordering temperature of Chern ferromagnetism in ${\mathrm{MoTe}}_2$ on the Landau-level mixing parameter.

The intermediate state of type-I superconductors presents a classic paradigm of modulated pattern formation, arising from the competition between short-range attractive and long-range repulsive vortex-vortex interactions. However, direct visualization and, more importantly, active control over the topology and dynamics of these flux structures have remained significant challenges, limiting our ability to manipulate them for fundamental studies and potential applications. Here, using low-temperature magnetic force microscopy, we achieve direct imaging and controllable manipulation of the flux structures in a high-purity tantalum single crystal. We systematically track the evolution of flux morphology—from tubes to stripes—during flux penetration and expulsion, revealing a pronounced topological hysteresis originating from the geometric barrier. Furthermore, we demonstrate precise local control by using the magnetic tip to drag and merge individual flux tubes and to reconfigure entire stripe domains. Under global alternating current (ac) excitation, we discover a reversible stripe-grid-stripe transition, a dynamic reorganization driven by current-induced flux penetration and pinning effects. The corresponding phase diagram shows that the threshold current decreases with magnetic field but increases with ac frequency. Our work establishes a pathway for active flux manipulation in type-I superconductors, revealing rich dynamics and paving the way for flux-based superconducting devices.

Probing magnetic order in insulators with small magnetization is particularly challenging in the ultrathin limit, where magnetometry and diffraction lose sensitivity. Here we show that topological spin order with vanishingly small magnetization can be electrically detected through an interfacial topological Hall effect in heavy-metal/magnetic-insulator heterostructures. Using Pt coupled to the canted antiferromagnetic insulator hexagonal $\mathrm{LuFe}{\mathrm{O}}_3$ (h-$\mathrm{LuFe}{\mathrm{O}}_3$) we observe an unusually large and robust Hall response arising from the transfer of real-space spin topology across the interface via magnetic proximity effect (MPE). This interfacial signal enables detection of magnetic order in h-$\mathrm{LuFe}{\mathrm{O}}_3$ with an extremely small net magnetization ($0.025{\mu }_{\mathrm{B}}/\mathrm{Fe}$), corresponding to Hall-conductivity-magnetization ratio $\approx 2{\mathrm{V}}^{-1}$, which is one to two orders of magnitude large than other MPE-induced Hall effects and anomalous Hall effects of conducting magnets. This high sensitivity allows detection of magnetic order in h-$\mathrm{LuFe}{\mathrm{O}}_3$ down to a thickness of only 1.5 unit cells. Our results establish an interfacial Hall-based approach for electrically probing topological spin structures in ultrathin insulators and the related interfaces, enabling access to magnetic order in regimes where conventional probes fail.

Superconducting proximity effects in topological insulators enable novel phenomena such as the Josephson diode effect (JDE), yet disentangling surface and bulk contributions remains challenging. Here we report a tunable JDE mediated by topological surface states in ${\mathrm{BiSbTeSe}}_2$ (BSTS) Josephson junctions. Finite-momentum Cooper pairing of spin-helical surface states gives rise to a pronounced JDE under an in-plane magnetic field, with diode efficiency reaching up to 40%. Temperature-dependent measurements reveal that at higher temperatures, the supercurrent is dominated by surface states, resulting in a rapid increase in diode efficiency upon cooling, while the onset of bulk contributions at lower temperatures leads to efficiency saturation. Gate tuning approaching the Dirac point region enhances surface transport, increasing the diode efficiency from 40% to 50%. These results provide strong evidence for a dominant role of topological surface states in the nonreciprocal superconductivity and establish BSTS junctions as a promising platform for tunable, high-performance superconducting diodes.

Electrons in a two-dimensional electron gas under a strong perpendicular magnetic field undergo cyclotron motion, whose quantization yields discrete Landau levels and quantized Larmor radii. The magnetic length—the Larmor radius of the smallest Landau orbit—is only tens of nanometers for fields of a few tesla. We propose a method to probe the spatial structure of Landau orbits via scattering with extreme ultraviolet radiation, whose wavelength naturally matches this scale. Starting from a microscopic Hamiltonian, we derive the far-field spectrum emitted by the optically induced current density. The Landau orbit contributions in the spectrum are suppressed due to scattering at the sample geometry. However, normalizing to the zero-magnetic-field reference defines a Landau-level scattering spectrum that isolates Landau orbit information from the full spectrum. A projection onto Laguerre polynomials allows for reconstructing the probability density distributions of individual Landau-level wave functions, featuring radial maxima at the quantized Larmor radii. This way, we propose an experimental method to determine details of the Landau orbits.