Papers

We develop a backpropagation algorithm for evaluating parameter gradients in quantum circuits using Pauli propagation simulation. The method has computational complexity comparable to that of standard sparse Pauli simulation techniques, while producing gradients whose accuracy is of the same order as the corresponding observable expectation values. By exploiting the reversibility of quantum circuits, the algorithm reduces the memory cost by a factor of $\mathcal{O}(n_\text{param})$ compared with conventional reverse-mode automatic differentiation, where $n_\text{param}$ denotes the number of parameters in the circuit. Compared with finite difference methods, the algorithm is $\mathcal{O}(n_\text{param})$ more efficient in function evaluations. These features enable efficient and accurate classical optimization of quantum circuits for applications such as state preparation and time-evolution compression, while also allowing operator-complexity measures such as the operator stabilizer Rényi entropy to be monitored and regularized during optimization. We demonstrate the method by optimizing low-energy state-preparation circuits for transverse-field Ising models in one, two, and three dimensions and for the three-dimensional Heisenberg model, and by compressing two-dimensional time-evolution circuits.

Coulomb branches of 3-dimensional $N=4$ gauge theories, as defined by Braverman–Finkelberg–Nakajima, form a large and interesting class of symplectic singularities. In this paper, we give a geometric description of these varieties in terms of blowups of toric compactifications, uniformly across the rational, $K$-theoretic, and elliptic settings, which in physics are often associated with 3-, 4-, and 5-dimensional versions of these theories. We use the resulting local description to classify the symplectic leaves of Coulomb branches and describe their transverse slices in terms of the classification of zero-dimensional leaves of certain smaller Coulomb branches. We also discuss related topics, including removing an unnecessary hypothesis from a theorem of Gannon and the author on functoriality for Coulomb branches, and clarifying the relationship with transverse Hilbert schemes by refining the approach of Bielawski and Foscolo.

This study explores the nonlinear interactions between gravitational waves in an expanding universe. Random ensembles of gravitational waves are evolved numerically in cosmological models whose spatial volumes increase by more than an order of magnitude during these simulations. These evolutions show the spectra of the gravitational waves evolving as energy cascades away from the wavelengths at the peak of the initial spectra toward both longer and shorter wavelengths. Evolutions performed at different gravitational wave amplitudes are shown to produce cascades that scale with amplitude in the way predicted by the four-wave scattering models of gravitational-wave turbulence.

Dissipative preparation of quantum order offers a route to superconductivity that does not rely on enhancing attractive interactions. Here we propose a driven-dissipative protocol to prepare superconductivity as a stationary state of a two-dimensional moiré heterostructure. The key ingredient is a bilayer moiré platform in which the layer degree of freedom acts as a pseudospin, allowing the pseudospin structure required for pairing to be implemented through optically induced spatial operations. This preparation scheme requires local dissipation, which we show to arises naturally from weakly dispersive bosonic modes in the heterostructure. In contrast, in the opposite regime of collective dissipation, the same platform exhibits an early-time superradiant burst. Our results establish driven-dissipative moiré heterostructures as a promising platform for preparing superconductivity, while also revealing a connection between steady-state pairing and transient superradiance.

We study the spectra of the accreting X-ray millisecond pulsar SRGA J144459.2-604207 during the 2024 outburst using the Neutron star Interior Composition Explorer (NICER) and Swift observations. The spectra during the outburst decay, reflares, and quiescent state are explored using the absorbed Comptonized model. We find that the spectra during the quiescent state can also be explained using the absorbed power-law and absorbed blackbody model. The spectral evolution of the source is explored as the outburst decays into quiescence. We study the long-term quiescent X-ray activity of the source spanning roughly 45 years and find that the long-term quiescent luminosity may be explained using the deep crustal heating model. We also find that the coronal activity of the companion star alone cannot power the quiescent X-ray luminosity of the source. We place the source on the radio-X-ray luminosity plane and compare its position with other sources. We estimate the propeller luminosity of the source and find that it is smaller than the estimated luminosity during reflares and the quiescent state during the 2024 outburst. Several reflares are detected during the outburst decay, one of which is near-simultaneous with the detection of an ultrafast outflow and radio emission. We explore plausible mechanisms that may power outflow, radio emission and associated jet formation in this accreting binary.

Optical thermodynamics is a newly developed framework that applies principles from statistical mechanics to describe the intricate behavior of weakly nonlinear, multimode photonic systems. Utilizing this theory, the collective dynamics of complex optical arrangements can be systematically uncovered and understood. The purpose of this work is to examine fundamental aspects of optical thermodynamics, including optical pressure, and provide a unified framework that can be applied to effectively any optical setting within the domain of validity of optical thermodynamics. We find that in addition to the conservation laws, the remaining extensive and intensive parameters of the system are naturally provided by the parameters of the propagation constants. At this point, several thermodynamic approaches exist for analyzing optical forces in multimode settings. Here, we develop a new theoretical methodology that unifies these perspectives in a variety of different configurations, irrespective of whether they are discrete or continuous. We apply our theory in four different settings. By studying Su-Schrieffer-Heeger lattices, we elucidate the thermodynamics of polyatomic chains and show that intercell and intracell bonds can display different optical forces. In addition, we provide a thermodynamic formalism to predict and understand the optical pressure at equilibrium arising in arrangements characterized by a continuous index variation and apply our results to graded-index fibers.

Using a robust photon loss model for photonic quantum experiments, we derive $n-$th order parameter-shift rules for computing gradients of Fock boson sampling transition probabilities where $n$ photons are sent into a lossy interferometer. We also show that in general it is not possible to generalize this gradient recipe to the case of Gaussian boson sampling. Only in the specific case where the transmission matrix of the interferometer can be factorized as a diagonal loss matrix premultiplied by a pure unitary it is possible to obtain parameter-shift rules with a finite order given by twice the total number of photons detected. We demonstrate the efficacy of the proposed method by comparing its performance against finite differences on real hardware.

We construct a new family of Calderbank-Shor-Steane (CSS) codes using the generator and parity-check matrices of Low-Density Generator Matrix (LDGM) codes, with row operations applied to both matrices in order to achieve the desired quantum rate. Decoding is performed in an iterative manner, by applying message passing over the associated graph, and discrete Density Evolution (DDE) is used to optimize performance in the depolarizing channel. The proposed construction offers high flexibility and easiness in the design, producing quantum codes that possess excellent error correction capabilities. By properly designing the structure of the code, we are able to control and bound the weight of the stabilizer generators to a small value, which results in codes particularly well suited for fault-tolerant quantum computation. At the same time, these codes achieve very good performance in terms of error correction capability.

The corner transfer matrix renormalization group (CTMRG) is one of the standard contraction methods for infinite projected entangled-pair states (iPEPS), but its computational cost is dominated by repeated truncated singular value decompositions (SVDs). We introduce subspace-iteration CTMRG (SI-CTMRG), a QR-based projector construction that replaces each large-matrix SVD with an SVD of a much smaller matrix. The resulting algorithm shifts the dominant cost from decompositions to tensor contractions, making it highly suited to GPU acceleration and yielding speedups of up to two orders of magnitude over standard CTMRG. We demonstrate the efficiency and accuracy of the method for the triangular-lattice Heisenberg antiferromagnet, reaching state-of-the-art iPEPS results on a single H100 GPU in approximately 10 hours of computation.

The dynamics of premixed flames propagating in two-dimensional and cylindrical channels are investigated using direct numerical simulations of the fully compressible reactive Navier-Stokes equations coupled with conductive heat transfer within the channel walls. The simulations employ a high-order numerical method, detailed chemical kinetics and transport models for a stoichiometric hydrogen-air combustion. The influence of heat losses during the early stages of flame propagation is examined for channels of different aspect ratios, with particular focus on tulip flame formation and its subsequent transition to distorted tulip structures. Heat losses are modelled considering convective heat transfer from the hot combustion products to the inner wall surface, thermal conduction heat transfer through the wall, and convective and radiative heat losses from the outer wall surface to the surroundings. The obtained results are compared with corresponding simulations under adiabatic wall boundary conditions. The simulations reproduce the principal features of flame dynamics observed experimentally, highlighting the combined influence of wall heat losses and geometric confinement on flame dynamics in confined channels.

We study quantum spin correlations in top-quark pair production at future lepton colliders in the presence of a neutral gauge boson from anomaly-free general $U(1)$ extensions of the Standard Model. The process $\ell^+\ell^-\to t\bar t$, with $\ell=e,\mu$, is analyzed through the spin-density matrix including $\gamma$, $Z$ and $Z^\prime$ exchange and their interference. We focus on quantum-information observables such as the sufficient entanglement marker $\mathcal{D}_{\min}$, concurrence, purity and the maximal Clauser-Horne-Shimony-Holt (CHSH) parameter, and compare their behavior with conventional rate information. Within the $U(1)_X$ framework, we consider several representative charge assignments to investigate how different chiral structures influence these observables, with particular emphasis on the $Z^\prime$ resonance region and polarized $e^-e^+$ collisions, where the two allowed initial-state helicity configurations can be selectively enhanced. We show that electron-beam polarization provides a direct handle on the left- and right-handed lepton charges of various $U(1)_X$ scenarios. These results demonstrate that quantum spin observables provide information complementary to cross sections and angular distributions in searches for chiral neutral gauge interactions.

The entanglement-assisted capacity of a quantum channel admits an additive single-letter characterization, implying that joint encodings across channel uses cannot increase the ultimate communication rate. Here, we show that this additive picture does not extend to communication reliability. Specifically, we prove that the Petz-Rényi channel information can be strictly superadditive for every $\alpha\in[1/2,1)$, yielding a genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent, even though the entanglement-assisted capacity remains additive. We establish this phenomenon analytically already for measurement channels, which are entanglement-breaking and have additive unassisted capacity. Remarkably, this strict superadditivity is witnessed by a separable, classically correlated two-copy channel-input marginal, demonstrating that no entanglement between the transmitted systems is required. Our results show that, although correlations across channel uses cannot increase the ultimate rate of entanglement-assisted communication, they can enhance its reliability.

We investigate the thermodynamic topology of charged AdS black holes in a non-commutative spacetime sourced by Lorentzian-smeared matter distributions. Since exact analytical solutions for the critical thermodynamic quantities are not available, we employ a perturbative expansion in the non-commutative parameter and validate the resulting expressions through numerical analysis. Using the generalized off-shell free-energy framework, we explore the topological structure of the thermodynamic phase space and evaluate the corresponding winding number that characterizes the phase transitions. Our results reveal that non-commutative effects introduce qualitative modifications to the thermodynamic behavior compared with the standard Reissner-Nordström AdS black hole. Furthermore, we demonstrate that the bulk and boundary descriptions possess an identical global thermodynamic topology, providing strong evidence for the correspondence between their topological structures. We also investigate the lower bound on the remnant mass implied by the second law of black-hole thermodynamics and observe that non-commutative corrections modify key thermodynamic quantities, with particular emphasis on the entropy and the final black-hole mass.

Star clusters are perfectly suited for studying the formation, evolution, and dynamics of stars and their host galaxies. For this, we need to estimate total masses, radii and shapes of star clusters. These parameters strongly influence their fates in the host galaxies. However, there is no agreement about these characteristics in the literature. The published values widely disagree. We estimated total masses and cluster radii using a very conservative approach: calculating member masses and summing them. Furthermore, we investigated the three-dimensional characteristics of open clusters and their limitations. We used a commonly used grid of evolutionary tracks for different photometric sources, including 2MASS, Gaia, and Pan-STARRS. The interpolation method was tested using synthetic clusters built with ASteCA. To estimate three-dimensional shapes, we used the distribution of the well-known Galactic coordinates and inverse parallaxes. We estimated the total masses and radii of about 7000 open clusters and observed traces of tidal tails. Individual values can be used in future $Gaia$ data releases to obtain even more precise values. The current DR3 allows the study of only three-dimensional shapes up to 500 pc. Beyond this distance, we find a significant needle-like shape due to uncertainties in observed parallaxes.

We report on the ground and numerous excited electronic states. In the ground state the C4n rings are closed-shell systems possessing polyynic structures and can be classified as double anti-aromatic molecules. In their energetically lowest lying triplet state the rings exhibit aromatic cumulenic structures. The overall change in the electronic structures is rather dramatic upon the found moderate geometric changes from polyynic to cumulenic structure. Among others, Hund's rule is violated in both C8 and C12 in their cumulenic structures. We mention that until now, graphene is the only carbon allotrope reported to violate Hund's rule. The reasons for the violation are analyzed. Much effort has been invested to understand the relaxation pathways of the low-lying states leading the C8 from polyynic to cumulenic geometry and vice versa. On its minimum energy path, the first singlet excited state changes from open-shell character in the polyynic structure to a closed-shell state in the cumulenic structure. The cumulenic state lowest in energy is an open-shell singlet which relaxes to the closed-shell polyynic ground state.

We prove a Weyl law for the number of quasinormal modes (QNM) of a Schwarzschild black hole contained in a sector below the real axis. This requires introducing a new pseudodifferential operator calculus tailored to the study of semiclassical spectral problems near threshold energies. Elliptic theory in this calculus can be combined with the method of complex scaling to give uniform resolvent estimates near zero energy for operators that behave at infinity like a semiclassical Schrödinger operator with a repulsive inverse-square potential. Applied to the Regge-Wheeler potential, our methods imply the absence of high angular momentum QNM from a disc whose radius grows linearly with the angular momentum. Together with the asymptotic description of Schwarzschild QNM recently obtained by Hitrik and Zworski, this shows that the number of QNM contained in a small sector below the real axis and with modulus bounded by $\lambda$ grows as $C\lambda^3$. We also study the effect of cutting off the Schwarzschild resolvent away from the event horizon and show that such a cutoff does not lead to any pole cancellations.

Atmospheric compositions of giant exoplanets can retain information about their formation environments, as volatile species condense at different temperatures in protoplanetary discs. We investigated whether the atmospheric compositions of hot Jupiters can constrain their formation locations. We performed planet formation simulations using the ChemComp code, including pebble drift, pebble and gas accretion, planet migration, stellar abundances, and two additional chemical processes: thermal decomposition of refractory organics and CO/CO2 trapping in water ice. We applied this framework to nine observed hot Jupiter systems and compared the resulting atmospheric metallicities (C/H and O/H) with observational constraints. We found that the observed atmospheric abundances of the nine hot Jupiter systems can be reproduced by planets forming at different locations relative to the H2O and CO2 snowlines. Our results suggest that at least six of the nine systems are consistent with formation beyond the H2O snowline. Combined with the observed orbital separations, eccentricities, and spin-orbit obliquities, these inferred formation locations indicate that many systems likely experienced dynamical scattering followed by tidal evolution. Atmospheric abundances, in combination with detailed orbital parameters, can provide a powerful diagnostic of the formation and migration histories of hot Jupiter systems, opening up avenues to understand the origin of giant planets in general.

Establishing the global-in-time smoothness of solutions to the 3D Navier-Stokes equations (NSEs) with large initial data remains a long-standing open problem. In this paper, we prove that the 3D NSEs driven by a suitably chosen transport noise – a physically motivated stochastic perturbation – admit global-in-time smooth solutions with high probability. This regularizing effect holds uniformly for initial data in arbitrarily large balls of subcritical function spaces with positive smoothness.

A method is presented for the translation of acoustic field data from a source to a target region. Field data are represented as spherical harmonic expansions on spheres surrounding the source and target regions respectively and expansions are translated using a “point and shoot” method using the Kirchhoff–Helmholtz integral to carry out an axial translation from one sphere to the other. The principal motivation for the method is its use in a time-domain Fast Multipole Method, and test cases reflective of this application are presented. The method converges to six digits for appropriate values of parameters and for the values of $N$ considered here computational effort scales approximately as $N^{2}$ where $N$ is the order of spherical harmonic expansion for the field data. The method is causal and thus avoids artifacts generated in methods which are not based on intrinsically causal formulations.

Quantum many-body scars provide an exception to generic thermalising dynamics, but their relation to integrability remains unclear. Here we apply the periodic-orbit framework to the integrable XYZ/XXZ spin-1/2 chain, developing an energy-resolved approach that enables us to locate and track periodic orbits across the spectrum. We identify families of scarred orbits and resolve their supporting eigenstates in terms of Bethe-ansatz quantum numbers, revealing a common string core dressed by zero-momentum magnon pairs. This allows us to reconstruct both the scarred eigenstates and the associated towers directly within the Bethe ansatz, and explain the equidistant tower spacing analytically from the decoupling of zero-momentum magnon pairs in the Bethe equations, providing a microscopic realisation of scar phenomenology in an integrable setting. We further show that this structure persists upon breaking integrability with a transverse field, with the periodic orbits continuing to govern the dynamics. Our results establish a direct connection between the periodic-orbit picture of quantum scarring and the algebraic structure of integrable models, showing that key features of scar dynamics can be understood analytically within the Bethe ansatz framework.

We present a comprehensive chemo-dynamical analysis of the old, metal-poor open cluster Berkeley 32 based on Gaia DR3 astrometry and Gaia-ESO Survey DR5.1 spectroscopy. Cluster membership is determined using a Gaussian Mixture Model applied to proper-motion components and trigonometric parallaxes. Isochrone fitting yields an age of 4.9 +/- 0.5 Gyr, a heliocentric distance of 3325 pc, and an extinction of A_V = 0.38 +/- 0.12 mag. Spectroscopic member stars exhibit a mean metallicity of [Fe/H] = -0.39 +/- 0.02 dex, near-solar alpha-element abundances, and a weighted mean radial velocity of V_rad = 106.26 +/- 0.03 km s^-1. The [Y/Mg] chemical clock yields an age of 4.73 +/- 2.39 Gyr, consistent with the isochrone estimate. Orbital integration indicates a moderately eccentric orbit (e = 0.268 +/- 0.004) with a guiding radius of R_g = 8.82 kpc. The inferred chemical birth radius, R_b = 9.82 kpc, together with DeltaR ~ -1 kpc, suggests moderate inward radial migration, while the offsets among R_b, R_g, and R_GC are consistent with both churning and blurring processes. A photometric analysis identifies a binary fraction of f_b = 0.449 +/- 0.017 for systems with mass ratios q >= 0.5, implying a substantial unresolved binary population. Radial cumulative distribution functions further reveal significant mass segregation, with evolved stars more centrally concentrated than main-sequence stars. These results indicate that Berkeley 32 is a dynamically evolved old-disk cluster whose present-day structure and orbit preserve signatures of both internal dynamical evolution and radial migration within the Galactic disk.

The Burst Alert Telescope (BAT) onboard the Neil Gehrels Swift Observatory is capable of localizing gamma-ray transients with arcminute precision, enabling rapid multi-wavelength follow-up. However, onboard triggering is disabled during spacecraft slews, preventing the autonomous detection and localization of transients occurring during these intervals. We present here BAT-GLIMPSE, a fully autonomous, open-source pipeline for the low-latency localization of transient gamma-ray sources in Swift-BAT data. Making use of the BatAnalysis package, the pipeline combines coded-mask imaging and mosaic techniques, automatically selecting the appropriate analysis according to the spacecraft attitude and enabling localization searches during both pointing observations and spacecraft slews. We validate the performance of BAT-GLIMPSE on a sample of 66 GRBs reported in GUANO circulars. The pipeline successfully recovers arcminute positions for 43 events, consistent with published localizations, with typical offsets of $\lesssim5$ arcminutes. Approximately $88\%$ of the GRBs occurring during spacecraft slews are recovered through imaging or mosaic analyses. During the fourth LIGO-Virgo-KAGRA observing run, the role of BAT-GLIMPSE was crucial in the search for gamma-ray counterparts of gravitational waves, particularly in response to pre-merger alerts which triggered the slew of the Swift spacecraft with extremely low latency. Operating synergistically with NITRATES, BAT-GLIMPSE fills the critical gap left by slew intervals and, together, the two pipelines are estimated to double the onboard arcminute-localization rate of Swift-BAT, unlocking the full potential of the Swift mission for time-domain and multi-messenger astrophysics.

We study the spacelike-collinear limit of gauge-theory scattering amplitudes using the Method of Regions. The corresponding splitting amplitude violates strict collinear factorisation through its dependence on the non-collinear partons. While the associated colour dependence has long been known, starting at two loops the splitting amplitude also acquires dependence on their kinematics. We show that this kinematic dependence originates from a unique hidden region present in the asymptotic expansion of the five-point amplitude in the spacelike-collinear limit, but absent in the timelike limit. More generally, we propose that hidden regions provide the mechanism by which crossing-related asymptotic limits cease to be analytically connected. We develop a general algorithm for the systematic identification of hidden regions. Applying it to the five-point amplitude in super Yang-Mills theory, we compute the hidden-region contributions to the complete set of basis integrals and recover the exact kinematically dependent factorisation-violating splitting amplitude. In momentum space, the hidden region is characterised by soft and Glauber loop momenta. This explains why the Wilson-line calculation captures the complete kinematic dependence, thereby accounting for the observed universality across gauge theories.

The concept of finite entanglement scaling forms one of the pillars on which the tensor network ecosystem is built. In this paper, we resolve the open problem of determining the actual perturbations induced by matrix product state approximations of critical systems, and we demonstrate that these can be quite different than the ones predicted by conformal field theory. To that aim, we develop a sparse linear solver to calculate the forward and backward derivatives of 2-dimensional tensor networks with respect to their defining parameters in an implicit way. This algorithm is of independent interest as it provides a primitive for the variational optimization of projected entangled pair states that circumvents the instabilities plaguing traditional automatic differentiation methods.

Hysterons provide a minimal description of memory in driven matter: bistable elements with distinct switching thresholds whose interactions generate hysteresis, avalanches, and return point memory or its violation. Experimental realizations have so far been dominated by solid state mechanical systems, where bistability is usually encoded structurally through buckling, snap through, or geometric incompatibility. Here we realize hysteron physics through a hydrodynamic route. A single elastic fiber anchored in a microfluidic channel becomes bistable through nonlinear elastohydrodynamic feedback: viscous loading deforms the fiber, deformation reshapes hydraulic resistance, and flow redistribution modifies the loading. This feedback produces a fluidic hysteron whose onset is organized by a cusp catastrophe in geometric control parameters. A parallel bypass channel acts as a geometric load line that reshapes, and can even eliminate, bistability while simultaneously mediating long ranged hydraulic interactions between fibers. In arrays, varying a single geometric parameter drives a transition from a non interacting Preisach regime with return point memory to an interacting regime with avalanche like switching and return point memory violation. These results establish a passive hydrodynamic route to hysteron networks, in which memory emerges from flow structure feedback and global hydraulic constraints rather than solid state multistability or external control.

In recent years, quantum lock-in detection has emerged as a promising technique to accurately detect weak signals submerged in background noise. However, the signal-to-noise ratio of existing protocols is severely limited by spectral leakage resulting from control operations implemented in pulse form. Here, we propose a general protocol for realizing quantum lock-in detection by employing successive quantum adiabatic evolution. In our protocol, the signal modulation is achieved by adiabatically controlling the time evolution of the quantum probe, which enables the implementation of sinusoidal modulation functions. The realization of sinusoidal-wave modulation substantially mitigates the problem of spectral leakage and facilitates the extraction of the complete characteristics of the target signals. We present a practical implementation scheme of adiabatic quantum lock-in detection based on nitrogen-vacancy centers in diamond, and demonstrate that the proposed protocol possesses strong resilience against experimental imperfections. Our results establish adiabatic quantum lock-in detection as a robust and experimentally accessible approach to detection of weak alternating signals in noisy environments, thus promoting the advance of real-world quantum sensing technologies.

In CrSBr, as in many crystalline materials, the type and density of defects are expected to strongly influence material behavior. Identifying the underlying atomic defect configurations and controlling their populations during growth are therefore important steps toward understanding and ultimately tailoring its rich magneto-electrical properties. However, systematic control of defects in CrSBr during chemical vapor transport (CVT) growth has not yet been established. Here, we correlate CVT growth conditions with defect concentrations measured using conductive atomic force microscopy (CAFM). We focus on a characteristic defect with a strong electronic fingerprint, labeled D*, and decrease its concentration by up to an order of magnitude through optimized growth conditions. We show that defect densities can be tuned by adjusting precursor stoichiometry, where sulfur- and bromine-rich conditions suppress defect formation, and by lowering the absolute growth temperatures while maintaining the same temperature gradient. Thermodynamic modeling and density functional theory calculations suggest that D* is most consistent with a sulfur-related vacancy complex rather than an isolated point defect. These results provide practical strategies for growing high-quality CrSBr with controlled defect densities.

We introduce a thermodynamic theory of voting and show that it provides a good description of distribution of party votes in EU elections. The theory traces parallels between system energies of coupled nonlinear oscillators and party vote fractions. Such a classical system evolution is characterized by the conservation of total energy and probability norm that leads to the Rayleigh-Jeans (RJ) thermalization and condensation at low energy states. A similar thermalization also describes the wealth inequality in society. This feature belongs to the phenomena of constraint driven condensation known in statistical mechanics. We show that the RJ theory well depicts the Lorenz and Pareto curves obtained from the EU vote results. The theory also recovers the dispersion of votes between candidates of first round presidential elections in France.

Magnetic materials support both time-reversal-even (T-even) and time-reversal-odd (T-odd) spin Hall currents, yet their underlying microscopic origins remain elusive. Here, we elucidate the spin Hall effect (SHE) in collinear ferromagnets by treating spin-orbit coupling (SOC) as a perturbation that breaks spin-group symmetry, thereby revealing how magnetic order activates distinct spin Hall response. To first order in SOC, we identify two dominant T-even SHE mechanisms: a magnetization-independent conventional contribution and a magnetization-dependent channel associated with anomalous Hall charge transport. At the same order, the leading T-odd magnetic spin Hall effect (MSHE) originates from the exchange interaction between the conventional spin current and the local magnetization. At second order in SOC, we further uncover a distinct T-odd planar spin Hall mechanism. Our spin-symmetry analysis is corroborated by first-principles calculations, which reveal a pronounced anisotropic magnetic spin Hall effect whose magnitude can be comparable to the T-even spin Hall conductivity (SHC) when the magnetic moment is tilted away from the principal crystallographic axes. These findings clarify the microscopic origins of the SHC in collinear ferromagnets and pave the way for ferromagnet-based spin current sources with versatile properties in spintronic applications.

Cr2AlC MAX phase is synthesized by high-pressure solid-state annealing and investigated as a candidate platform for optically responsive magnetism. Structural characterization confirms the formation of the Cr2AlC phase, while magnetic and optical-magnetic properties are examined by superconducting quantum interference device (SQUID) magnetometry, electron spin resonance (ESR), and first principles calculations. SQUID magnetometry identifies Cr 2AlC as a weak, field-linear metallic paramagnet dominated by Pauli-like susceptibility of itinerant Cr-derived states. Its non-monotonic temperature dependence is described by an additional contribution from antiferromagnetically coupled Cr-Cr dimers, whereas the low-temperature Curie-like upturn originates from only a trace population of localized Cr centers. Under red-light illumination, SQUID magnetometry does not reveal an intrinsic macroscopic optomagnetic response. In contrast, ESR at 4 K shows a reversible light-induced reduction of a local magnetic signal, but the optically modified spin population corresponds only to several tens of ppm of the Cr sublattice. Ab initio Bethe-Salpeter equation (ai-BSE) calculations combined with the maximally localized Wannier function analysis suggest that optical excitation can redistribute spin polarization between neighboring Cr sites with the opposite local moments. The combined experiment-theory approach therefore establishes the hierarchy of magnetic contributions in Cr 2AlC and identifies the microscopic origin of its local optical sensitivity. This provides a reference for designing MAX phases and related MXenes in which defects, surface terminations or reduced dimensionality may enhance optically active magnetic states.