We calculate the edge current and its fluctuations, i.e., noise, in a two-dimensional topological superconductor using the $T$-matrix and the Green's function techniques. We show that the current is zero for nonchiral edge states and nonzero for chiral edge states, while the edge noise is nonzero whatever the chirality of the edge states. By applying our results to toy models with chiral edge states, we find that the noise is closely related to the Chern number. The edge noise is nonzero only when the Chern number is nonzero, and the bulk noise exhibits a peak each time the Chern number varies, meaning that there are strong current fluctuations when a topological phase transition occurs. Our results suggest that the bulk noise could be seen as a topological susceptibility. In the extended Qi-Wu-Zhang model, where an edge state is present even though the Chern number is zero, the edge noise no longer follows the same behavior as it does in the nonzero Chern number model. Instead, it is related to other topological invariants, such as the Zak phase.

Twisting magnetic van der Waals materials provides a compelling approach to generate emergent spin textures through moiré engineering. In this study, we show that a twisted bilayer formed by two antiferromagnetic MnSe monolayers exhibits sublayer-resolved $i$-wave altermagnetic spin splitting. Combining first-principles calculations with a symmetry-constrained continuum theory, we reveal that this spin splitting originates from the rotational misalignment between two hexagonal warping parabolic band dispersions of the monolayer, coupled with sublayer-dependent moiré potentials arising from interfacial charge redistribution. While both AA- and ${\mathrm{AA}}^R$-twisted configurations display similar patterns of altermagnetic splitting, the underlying symmetry mechanisms are fundamentally distinct. In AA-twisted configuration, the effect is driven by locally inversion symmetric moiré potentials in the two layers, whereas in ${\mathrm{AA}}^R$-twisted configuration, it is dictated by a global in-plane $C_2$ symmetry that enforces opposite moiré potential phase relations between the magnetic sublayers. Additionally, we observe that altermagnetic splitting diminishes with decreasing twist angle, indicating a weakening of the rotational misalignment between the warping dispersions and a reduction in moiré momentum scale. Our results establish twisted MnSe as a representative system for layered antiferromagnets with spin-layer coupling and anisotropic warping band edge, offering a versatile platform for engineering twist-induced altermagnetic phases.

Nonequilibrium manipulation of quantum materials via electromagnetic dressing provides an on-demand route to tailoring electronic band structures through Floquet engineering. Time- and angle-resolved photoemission spectroscopy offers a direct means to probe these light-dressed electronic states. In such photoemission experiments, electromagnetic dressing can also occur for quasifree electrons outside the material, giving rise to so-called Volkov states. In certain cases, most notably in metallic systems, strong surface screening reduces the penetration of the driving electromagnetic field into the solid, resulting in Volkov contributions that dominate over Floquet ones. This underscores the importance of carefully accounting for the material's dielectric response when analyzing the origin of light-induced band replicas using time- and angle-resolved photoemission spectroscopy. In this work, we systematically investigate the influence of materials' dielectric properties on Floquet-Volkov dressing of semiconductors, focusing on bulk layered van der Waals materials GeS, SnS, and $2\mathrm{H}\text{−}{\mathrm{WSe}}_2$. First, by combining a simple model based on Fresnel equations with an electron-scattering description of Volkov amplitudes, we use polarization-dependent Volkov sideband intensities to extract a lower bound for the real part of the materials' dielectric function. The extracted values typically lie between the reported dielectric constants for monolayer and bulk crystals. Furthermore, we demonstrate that increasing the pump fluence of the infrared pump laser enables the generation of high-order Volkov sidebands, which exhibit clear signatures of nonlinear light-matter interactions in their temporal dynamics, polarization dependence, and angular distributions. Finally, we show that for our experimental geometry, the quasitransparent nature of semiconductors in the below-band-gap driving regime allows the optical pump to propagate within the sample and undergo multiple total internal reflections. This process produces a series of temporally delayed Volkov replicas in pump-probe measurements via electromagnetic dressing of photoelectrons by the evanescent field associated with total internal reflections. Together, these systematic studies uncover several previously unexplored aspects of Floquet-Volkov dressing in solids, highlighting the central role of dielectric screening of the driving field.

Two-dimensional (2D) semiconductors possessing strong spin-orbit coupling (SOC) and nontrivial topological features are central to the development of energy-efficient spintronics. While Weyl physics is extensively studied in semimetals, its realization in atomically thin semiconductors remains limited. Using first-principles calculations, we identify layered ${\mathrm{SnP}}_2{\mathrm{Se}}_6$ as a versatile topological semiconductor platform. In its monolayer form, the intrinsic structural chirality gives rise to a coexistence of Ising-type and Weyl-like (the properties of this material cannot be fully characterized as those of Dirac or Weyl materials) SOC. The latter is characterized by radial hedgehoglike spin textures—a signature of unique manifestation of Weyl-type SOC in a 2D chiral lattice within a 2D semiconductor—resulting in a significant intrinsic spin Hall conductivity (SHC) of up to 10.2 (ℏ/e) S/cm. In multilayer ${\mathrm{SnP}}_2{\mathrm{Se}}_6$, symmetry evolution facilitates a distinct regime where Weyl-like and Rashba spin textures coexist. This transition preserves substantial spin-charge conversion efficiency while introducing chirality-switchable components to the SHC tensor, providing a layer-dependent degree of freedom for spin manipulation. The interplay between strong SOC, topological band features, and structural chirality establishes ${\mathrm{SnP}}_2{\mathrm{Se}}_6$ as a platform for thickness-engineered spintronic devices.

We introduce equations of motion for a parity-violating fermionic mean-field theory (PV-FMFT): a numerically efficient fermionic mean-field theory based on parity-violating fermionic Gaussian states (PV-FGS). This work provides explicit equations of motion for studying the real- and imaginary-time evolution of spin-1/2 Hamiltonians with arbitrary geometries and interactions. We extend previous formulations of parity-preserving fermionic mean-field theory (PP-FMFT) by including fermionic displacement operators in the variational Ansatz. Unlike PP-FMFT, PV-FMFT can be applied to general spin-1/2 Hamiltonians, describe quenches from arbitrary initial spin-1/2 product states, and compute local and nonlocal observables in a straightforward manner at the same modest computational cost as PP-FMFT—scaling as $O\left(N^3\right)$ in the worst case for a system of $N$ spins or fermionic modes. We demonstrate that PV-FMFT can exactly capture the imaginary- and real-time dynamics of noninteracting spin-1/2 Hamiltonians. We then study the postquench dynamics of the one- and two-dimensional Ising model in the presence of longitudinal and transversal fields with PV-FMFT and compute the single site magnetization and correlation functions, and compare them against results from other state-of-the-art numerical approaches. In two-dimensional spin systems, we show that the employed spin-to-fermion mapping can break rotational symmetry within the PV-FMFT description, and we discuss the resulting consequences for the calculated correlation functions. Our work introduces PV-FMFT as a benchmark for other numerical techniques and quantum simulators, and it outlines both its capabilities and its limitations.

Increasing the valley splitting in Si-based heterostructures is critical for improving the performance of semiconductor qubits. This paper establishes that the two low-energy conduction band valleys are not independent parabolic bands. Instead, they originate from the X point of the Brillouin zone, where they are interconnected by a degeneracy protected by the nonsymmorphic symmetry of the diamond lattice. This semi-Dirac-node degeneracy gives rise to the ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_{2^{\prime }}$ bands, which constitute the valley degrees of freedom. By explicitly computing the two-component Bloch functions $X_1^{\pm }$, using the wave vector group at the X point, we determine the transformation properties of the object $(X_1^{+},X_1^{-})$. We demonstrate that these properties are fundamentally different from those of a spinor. Consequently, we introduce the term “valleyor” to emphasize this fundamental distinction. The transformation properties of valleyors induce corresponding transformations of the Pauli matrices ${\tau }_1,{\tau }_2$ and ${\tau }_3$ in the valley space. Determining these transformations allows us to classify possible external perturbations that couple to each valley Pauli matrix, thereby identifying candidates for valley-magnetic fields, $\vec{\mathcal{B}}$. These fields are defined by a Zeeman-like coupling $\vec{\mathcal{B}}·\vec{\tau }$ to the valley degree of freedom. In this way, we identify scenarios where an applied magnetic field $\vec{B}$ can leverage other background fields, such as strain, to generate a valley-magnetic field $\vec{\mathcal{B}}$. This analysis suggests that beyond the well-known mechanism of potential scattering from Ge impurities, there exist additional channels—mediated by combinations of magnetic and strain-induced vector potentials—to control the valley degree of freedom.

Propagation of short pulses through ${\mathrm{Cu}}_2\mathrm{O}$ crystal containing Rydberg excitons is studied with the use of the density matrix formalism and the finite difference time domain method. Effects of the so-called Rydberg blockade are studied extensively, exploring not only a reduction of absorption (bleaching) but also power-dependent changes of dispersive properties of the medium. The role of an exciton lifetime and coherent population oscillations in the dynamics of the system are investigated. A pump-probe setup with two pulses is also studied, showing a good agreement with recent experimental results. Calculations are performed for Rydberg states, both for low $n<10$ and high $n\sim 20$ principal quantum numbers.

Lack of inversion symmetry in metals breaks SU(2) symmetry, which results in spin-splitting of the electronic states at the Fermi level due to various types of spin-orbit coupling (SOC) such as Dresselhaus, Rashba, or Ising (also called valley-Zeeman). This splitting is known to enable both incoherent spin-flip excitations and coherent chiral-spin modes. Another effect of breaking of SU(2) is the introduction of a direct spin-photon interaction. We use this concept to formulate a theory of inelastic scattering of photons from the charge carriers of such a system [electronic Raman scattering (eRS)]. As a result of broken SU(2), we show that the eRS probe, unlike the conventional theory of Raman scattering, couples to spin excitations even without tuning the laser to an internal resonance. We show that the spin-dependent excitations induced by photon scattering are sensitive to the polarization geometries as well as to the spin structure of the Hilbert space of the low-energy states. As a concrete realization, we examine doped/gated graphene on substrates with strong SOC with various compositions of Rashba and valley-Zeeman SOC and compare their spectra with those for a model 2D electron gas (2DEG). The spectra are shown to have a resonant feature in select polarization geometries near the SOC-splitting energy and, importantly, are shown to be different in the two systems. The signal in graphene systems is shown to be stronger than that in a 2DEG by orders of magnitude due to the large Dirac velocity. We also outline how the line shapes from the spectra can be used to infer various components of SOC in the system.

Open-shell nanographenes offer a controlled setting to study correlated magnetism emerging from $\pi$-electron systems. Here, we study non-bipartite Fermi-Hubbard ladders describing oligo(indenoindene) molecules. These feature a gapped, weakly dispersing manifold of quasizero modes in their single-particle spectra, and we show that their low-energy properties can be effectively mapped onto an interacting set of spin-1/2 degrees of freedom. Using density matrix renormalization group simulations of the full Fermi-Hubbard model, we obtain their excitation spectra, entanglement profiles, and spin-spin correlations. We then construct optimized delocalized fermionic modes that act as emergent spins and demonstrate that their interactions are well described by a frustrated $J_1–J_2$ Heisenberg chain. This effective description clarifies how spin degrees of freedom arise and interact in non-bipartite nanographene ladders, providing a compact and accurate representation of their correlated behavior.

Asymptotic behavior of generic Tomonaga-Luttinger liquid in the vicinity of phase-separated regions is known to produce an instability where well-known relativistic conformal field theory (CFT) techniques fail. In this paper, we introduce an analytic paradigm that provides a continuum description of this important issue. We show that there is an emergent Carrollian symmetry when phase separation is reached, and techniques of Carroll CFT, as opposed to its relativistic relative, are central to the understanding of the physics. We work with the analogous spinless fermionic system in this region and capture the transition across this phase separation. Our numerical results corroborate the density-density correlations intrinsically computed using Carroll CFT. We further test the framework in a number of lattice systems, namely the spinless and spinfull fermionic models with distinct microscopic content, and find the same scaling at the transition. We discuss the scope of the framework and broader perspective.

We use high-dimensional bosonization to derive an effective field theory that describes the Pomeranchuck transition in isotropic two-dimensional Fermi liquids. We find that the transition is triggered by the softening of an eigenmode that leads to spontaneous Fermi surface distortion. The resultant theory in terms of this critical mode has dynamical critical exponent $z=2$ and the upper critical dimension is $d_c=4-z=2$. As a result the system is at the upper critical dimension in 2D, resulting in a Gaussian fixed point with a marginally irrelevant quartic perturbation.

We investigate localization transitions in a one-dimensional non-Hermitian quasiperiodic chain with period-2 mosaic modulation and nonreciprocal hopping. We show that tuning the degree of non-Hermiticity can induce a rich reentrant localization behavior: As the hopping asymmetry increases, reentrant localization successively emerges, disappears, resurges, and ultimately vanishes. By mapping the non-Hermitian Hamiltonian to a Hermitian reference chain via an imaginary gauge transformation and performing a transfer-matrix analysis, we introduce an effective Lyapunov exponent ${\gamma }_{\mathrm{eff}}={\gamma }_{\mathrm{qp}}-\alpha$, where ${\gamma }_{\mathrm{qp}}$ denotes the Lyapunov exponent of the Hermitian chain and $\alpha$ characterizes the hopping asymmetry. The interplay between ${\gamma }_{\mathrm{qp}}$ and $\alpha$ reveals that reentrant localization and its resurgence arise from the competition between quasiperiodic disorder and non-Hermiticity. Numerical analyses of participation ratios, single-particle spectra, and spectral winding numbers corroborate these findings.

We investigate the anatomy and complexity of quantum states in Krylov space, in the ergodic and many-body localized (MBL) phases of a disordered, interacting spin chain. The Krylov basis generated by the Hamiltonian from an initial state provides a representation in which the spread of the time-evolving state constitutes a basis-optimized measure of complexity. We show that the long-time Krylov spread complexity sharply distinguishes the two phases. In the ergodic regime, the infinite-time complexity scales linearly with the Fock-space dimension, indicating that the state spreads over a finite fraction of the Krylov chain. By contrast, it grows sublinearly in the MBL regime, implying that the long-time state occupies only a vanishing fraction of the chain. Further, the profile of the infinite-time state along the Krylov chain exhibits a stretched-exponential decay in the MBL regime. This behavior reflects a broad distribution of decay length scales, associated with different eigenstates contributing to the long-time state. Consistently, a large-deviation analysis of the statistics of eigenstate spread complexities shows that while the ergodic regime receives contributions from almost all eigenstates, the complexity in the MBL regime is dominated by a vanishing fraction of eigenstates, which have anomalously large complexity relative to the typical ones.

$\mathrm{CaFe}{\mathrm{O}}_3$ and $\mathrm{CaF}{\mathrm{e}}_{0.95}\mathrm{C}{\mathrm{o}}_{0.05}{\mathrm{O}}_3$ single crystals were grown by combing floating-zone method with high oxygen pressure treatment. Both crystals show charge disproportionation of Fe ($\mathrm{F}{\mathrm{e}}^{4+}\to \mathrm{F}{\mathrm{e}}^{3+}+\mathrm{F}{\mathrm{e}}^{5+}$) accompanied by a metal-to-insulator transition as well as a crystal structural transformation from Pbnm to $P{2}_1/n$. However, the slight introduction of Co significantly suppresses the critical temperature of charge disproportionation from 290 K in $\mathrm{CaFe}{\mathrm{O}}_3$ to 260 K in $\mathrm{CaF}{\mathrm{e}}_{0.95}\mathrm{C}{\mathrm{o}}_{0.05}{\mathrm{O}}_3$. Different from the single antiferromagnetic phase transition as observed in the polycrystalline $\mathrm{CaFe}{\mathrm{O}}_3$, two sequential antiferromagnetic transitions are found to occur in these two single crystals with the Néel temperatures around 119 and 112 K for $\mathrm{CaFe}{\mathrm{O}}_3$ crystal and 109 and 98 K for $\mathrm{CaF}{\mathrm{e}}_{0.95}\mathrm{C}{\mathrm{o}}_{0.05}{\mathrm{O}}_3$ crystal, respectively. Neutron diffraction illustrates the formation of a spiral antiferromagnetic structure. Moreover, as the temperature further decreases to 65 K, a third magnetic transition accompanied by a second electrical transition is observed in the slightly Co-doped $\mathrm{CaF}{\mathrm{e}}_{0.95}\mathrm{C}{\mathrm{o}}_{0.05}{\mathrm{O}}_3$ crystal. First-principles calculations suggest that the introduction of a moderate amount of Co and the peculiar spiral spin texture play an important role in the presence of such new magnetic and electrical transitions.

The copper metaborate $\mathrm{Cu}{\mathrm{B}}_2{\mathrm{O}}_4$ (CBO) has been studied within the parametric space of magnetic field ($B$) and temperature ($T$), showing a series of distinct phases, including commensurate and incommensurate magnetic orders, as well as magnetic soliton phases. However, no single study has simultaneously demonstrated all these phases. Moreover, the existence of additional phases below 2 K or above 2 T for $B\perp c$ has been scarcely explored. This work presents a comprehensive construction of the complex $B-T$ phase diagram for $T=0.1–25$ K and $B=0–9$ T ($B\perp c$), utilizing thermodynamic probes, magnetic measurements, and neutron scattering on a single batch of CBO crystals. Our findings elucidate additional magnetic phases and regimes. First-principles calculations help to gain a deeper understanding of this complex phase diagram. The emergence of various phases within such a narrow $B$ and $T$ regime is attributed to two primary mechanisms: the delicate competition between ferromagnetic and antiferromagnetic interactions, and the weak exchange and superexchange interactions within and between the Cu(A) and Cu(B) sublattices. This competition also leads to a magnetically frustrated ground state at absolute zero where the total magnetic entropy release from high to low temperatures is only 0.76 $R\ln 2$, lower than $R\ln 2$ expected for copper spin-½. Using CBO as a model system, the present work elucidates the intricate mechanisms underlying the formation of complex magnetic phases.

Hematite $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ is a $g$-wave altermagnetic material which has an easy-axis phase and easy-plane weak ferromagnetic phase below and above the Morin transition temperature, respectively. The presence of these phases renders it a good candidate to study the characteristic spin splitting in altermagnets under the impacts of the relativistic effect and finite temperature. In this regard, we calculate the band structure of $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ based on density functional theory (DFT) which also considers the Hubbard $U$ correction and spin-orbit coupling (SOC) effects. Additionally, charge self-consistent DFT plus dynamical mean-field theory (DMFT) calculations are performed at finite temperatures. We find that the altermagnetic spin splitting in $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ is preserved by taking either SOC or the temperature effect into account. Furthermore, we present a numerical simulation of the x-ray magnetic circular dichroism (XMCD) of the $L_{2,3}$ edge of Fe using a combination of DFT and multiplet ligand-field theory. In terms of the different Néel vectors present in $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$, we calculate the x-ray absorption spectroscopy of the $L_{2,3}$ edge of Fe in the form of a conductivity tensor and analyze the XMCD response from the perspective of symmetry. A characteristic XMCD line shape is expected when the Néel vector is along the [010] direction (magnetic point group $2^{\prime }/m^{\prime }$) and the light propagation vector is perpendicular to the Néel vector, which can be further distinguished from the XMCD response originating from weak ferromagnetism with the light propagation vector parallel to the Néel vector.

Polarization-resolved second-harmonic generation (SHG) is reported in highly doped ${\mathrm{Bi}}_1{\mathrm{Te}}_1$, dual topological insulator, demonstrating the role of Rashba spin-split states (RSS) in driving a magnetization-induced nonlinear optical response. Steady-state measurements reveal an enhancement of the SHG signal under circularly polarized probe excitation, evidencing a magnetization-induced susceptibility ${\chi }^{\left(2\right)}\left(M\right)$. Time-resolved SHG precisely disentangles the contributions of the electric-field-induced SHG and the magnetization-induced SHG. A linearly polarized probe measures the relaxation of the screened electric field in the space-charge region. In contrast, with a circularly polarized probe, the signal strongly depends on the pump polarization and even reverses sign, indicating the influence of magnetization. Comparing experimental results with theoretical calculations of second-order conductivity and the induced magnetization of Rashba states under circularly polarized excitation using Floquet formalism yields an estimated Rashba spin-orbit coupling parameter of the order of a few eV Å, evidencing the presence of strong spin-orbit coupling in the system. Moreover, the theoretical framework successfully captures the observed dynamics and fluence dependence, in close agreement with experiments. These results establish ${\mathrm{Bi}}_1{\mathrm{Te}}_1$ as a prototypical platform for exploring magnetization-driven nonlinear phenomena and demonstrate the utility of SHG as a sensitive probe of strong spin-orbit-coupled systems.