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Tuesday 8 September

Materials Science

Davydov Splitting Without a Davydov Pair and Highly Mobile Singlet Excitons in Perylene Red Microcrystals

We investigate the excitonic properties of perylene red microcrystals, whose unit cell contain eight molecules, using both experimental and theoretical methods. Only two of the nominal eight Davydov transitions are experimentally observed, with an apparent splitting of 610 cm$^{-1}$ that is reasonably reproduced by a Frenkel-Holstein exciton model parametrized from density functional theory. A symmetry-based analysis of the eigenstates reveals that the two observed transitions do not belong to the same Davydov pair, so the splitting does not directly report on the Coulomb coupling within the unit cell. The complex mixture of local excitations produces a J-like band for the dominant transitions, favouring exciton transfer through increased spectral overlap. Consistent with this picture, time-correlated single-photon counting of the emission and ultrafast transient absorption spectroscopy show that the dynamics is dominated by highly mobile singlet Frenkel excitons. The exciton mobility extracted from experiment agrees very well with that obtained from Kinetic Monte Carlo simulations, supporting a picture of incoherent hopping transport.
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All authors Chris Rehhagen, Tolibjon Abdurakhmonov, Magnus Frank, Oliver Kühn, Stefan Lochbrunner

High Energy Physics (Phenomenology)

Top-quark flavor-changing neutral currents and charged-lepton flavor violation in a generational three-Higgs-doublet model

In this work, we systematically investigate top-quark flavor-changing neutral-current decays, charged-lepton flavor-violating processes, and coherent $\mu-e$ conversion in the CP-conserving generational three-Higgs-doublet model (G3HDM), including the relevant tree-level and one-loop contributions. The 4839-point sample passes conservative sufficient vacuum-stability conditions, necessary perturbative-unitarity preselection cuts, Higgs-mass and signal-strength constraints, electroweak precision data, the UTfit 2025 neutral-meson-mixing ranges, $\overline{B}\to X_s\gamma$, and $B_s\to\mu^+\mu^-$. In the scanned region, $\mathrm{Br}(t\to c h_1)\leq 1.04\times10^{-5}$ and $\mathrm{Br}(h_1\to\mu\tau)\leq 4.83\times10^{-4}$, while the maximal conversion rates are $\mathrm{CR}(\mu\mathrm{Al}\to e\mathrm{Al})\approx 4.95\times10^{-15}$ and $\mathrm{CR}(\mu\mathrm{Au}\to e\mathrm{Au})\approx 1.05\times10^{-14}$. The dominant collider signals are mainly driven by tree-level flavor-changing neutral-Higgs couplings, which determine the relative strengths of these processes and induce the pronounced enhancement of $t\to c h_1$ and $h_1\to\mu\tau$. Meanwhile, the upper range of the predicted $\mu-e$ conversion rate in aluminum lies within the sensitivity reach of next-generation $\mu-e$ conversion experiments.
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All authors Hao-Ran Ma, Ti-Bin Hou, Yu-Ju Peng, Jin-Lei Yang, Tai-Fu Feng

Superconductivity

Geometry-induced current decrowding in superconducting 3D constrictions

Current crowding is a ubiquitous limitation in nanoscale devices, where confinement and sharp features distort current flow, generating localized current-density hotspots and premature failure. In superconductors, this effect suppresses the order parameter and promotes vortex nucleation, reducing the critical current below the intrinsic depairing limit. Here, we demonstrate that 3D shaping of superconducting nanoarchitectures overcomes geometric current crowding through spatial current redistribution, establishing a geometry-induced decrowding effect. Using finite-element modeling based on the time-dependent Ginzburg–Landau equation, we reveal that 3D constrictions exhibit a pronounced and tunable response to moderate in-plane magnetic fields—a functionality absent in planar geometries. This field-controlled geometry enables critical-current modulation and unveils new vortex-dynamics regimes, including a non-reciprocal critical current. Furthermore, curvature and finite thickness fundamentally alter vortex nucleation by enabling apex-mediated entry of single vortex lines, followed by their 3D splitting into two vortex filaments—a mechanism that does not occur in 2D manifolds. Our findings demonstrate 3D geometric engineering as a design paradigm for superconducting nanoarchitectures, offering control over current distribution and vortex dynamics in devices approaching the depairing limit.
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Quantum Physics

Instantaneous-Frame Theory of Strongly Driven Parametric Gates

Parametric two-qubit gates using tunable couplers are a promising approach to scalable superconducting quantum processors. However, conventional theories formulated in the idle eigenbasis break down in the large-amplitude regime required for fast gate operations. Here, we show that parametric gates are more naturally and accurately described as coherent rotations between instantaneous eigenstates rather than idle eigenstates. This approach may appear counterintuitive, since instantaneous eigenbases are usually associated with adiabatic dynamics. However, they also form a natural moving frame for driven dynamics, where the non-Abelian Berry connection governs transitions and provides a compact description of the exchange and effective $ZZ$ couplings. We show that this theory achieves quantitative agreement with exact numerical simulations across representative tunable-coupler architectures, including double-transmon, capacitively shunted double-transmon, and single-transmon couplers. The description remains accurate even in regimes where idle-frame descriptions fail, while substantially reducing the reliance on computationally demanding exact simulations. These results establish an instantaneous-frame theory of parametric gates beyond conventional idle-frame models and provide a new perspective on driven quantum dynamics.
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All authors Kentaro Kubo, Shinichi Inoue, Jushin Tei, Yinghao Ho, Yasunobu Nakamura, Hayato Goto

Mesoscale and Nanoscale Physics

The First Magic Angle Beyond the Chiral Limit in Twisted Bilayer Graphene

We develop a squared-Hamiltonian description of twisted bilayer graphene beyond the chiral limit to explain why the first magic angle remains robust under lattice relaxation, while higher-order magic angles are strongly destabilized. Starting from the non-chiral Bistritzer–MacDonald model with finite same-sublattice tunneling, we show that lattice relaxation reshapes the effective confinement landscape rather than acting as a simple perturbation of the chiral theory. A central result is that the realistic relaxation-renormalized tunneling ratio lies close to a special confinement point where the oscillatory part of the symmetric confinement potential nearly cancels. This places realistic twisted bilayer graphene near a nearly uniform confinement regime. At the same time, finite same-sublattice tunneling activates an additional inter-sublattice current-like channel that competes with the chiral orbital channel. The first magic angle survives because these confinement and current-like contributions remain balanced, whereas higher-order magic angles lose this balance through stronger remote-band hybridization and enhanced real-space localization around AA regions. Our results provide a single-particle mechanism for the breakdown of the chiral magic-angle hierarchy and clarify why the experimentally relevant first magic angle remains the most stable remnant of the chiral flat-band structure.
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All authors Leonardo A. Navarro-Labastida, Pierre A. Pantaleon, Francisco Guinea, Gerardo G. Naumis

Optics

Thresholdless corner vortex solitons in fractal Sierpiński topological insulators

Quantized vortices are ubiquitous in physics, spanning superconductivity, astrophysics, superfluid condensed matter systems, and nonlinear optics. Yet embedding vorticity into topologically protected nonlinear states has remained a major challenge, with all previously observed corner solitons in higher-order topological insulators (HOTIs) exhibiting only trivial phase distributions. Here, we report on the first realization of stable topological corner vortex solitons in a photonic fractal HOTI. Using an array of laser-written waveguides in the shape of Sierpiński gasket with a controllable distortion, we design linear topological vortex modes, from which nonlinear corner vortex solitons bifurcate. Moreover, we demonstrate that these solitons exhibit exceptional robustness across a broad power range and, unlike vortex solitons in topologically trivial lattices, form without a power threshold. Our results introduce the angular momentum degree of freedom into the physics of topological corner modes, opening prospects for topologically protected vortex-based photonics.
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All authors Yiqi Zhang, Alexander V. Kireev, Victor O. Kompanets, Sergey Y. Alyatkin, Nikita S. Kostyuchenko, Sergei A. Zhuravitskii, Nikolay N. Skryabin, Khalil Sabour, Alexander A. Kalinkin, Yongdong Li, Sergei P. Kulik, Pavlos G. Lagoudakis, Sergey V. Chekalin, Yaroslav V. Kartashov, Victor N. Zadkov

Atomic Physics

Ultralow Frequency Magnetic Comb Using Pulse Modulated Spin Maser

Frequency combs are widely used in fundamental physics and practical applications at various areas. Here we report the realization of an ultralow frequency magnetic comb using a pulse modulated spin maser in the Rb-Xe hybrid gaseous spin system. The magnetic frequency comb has a spectral distribution depending mainly on the pulse duration, the strength and phase shift of the feedback field. A simple theoretical model is given in terms of the spin echo chain and agrees well with the experimental results. The magnetic frequency comb works in an ultralow frequency range and reaches a frequency resolution down to tens of nanohertz.
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All authors Weiyu Wang, Qingchang Li, Erwei Li, Lan Wu, Qianjin Ma, Shougang Zhang, Guobin Liu

Materials Science

EBSDmagus: Managing Multi-Stage Dynamical Electron Backscatter Diffraction Simulations

Dynamical electron backscatter diffraction (EBSD) simulations increasingly support systematic studies of how material and experimental parameters affect diffraction patterns. Investigations that combine composition and crystal structure, including changes in atomic order, with accelerating voltage, temperature, and many orientations or detector conditions can require large sets of related calculations. Several packages provide dynamical EBSD simulation; we use the open-source, scriptable EMsoft programs because they are well suited to large studies on high-performance computing systems. An EMsoft investigation nevertheless spans separate configuration, structure, scattering, and orientation files and several dependent calculation stages. As variations multiply, manual preparation becomes difficult to check, interruptions obscure which results remain usable, and the origin of individual patterns becomes laborious to reconstruct. With EBSDmagus, researchers define fixed and varying parameters once. The software prepares the required EMsoft calculations, reuses compatible intermediate results, and checks that the generated files represent the requested investigation before execution. After execution, it checks the expected outputs and, following an interruption, resumes only unresolved calculations while preserving completed work. A portable run record links each output to its inputs and calculation history, providing the provenance needed when the results are deposited as findable, accessible, interoperable, and reusable (FAIR) data. We assess this approach using a completed representative calculation, a larger study prepared and checked before submission, two observed cluster interruptions, and a controlled job cancellation. During recovery, EBSDmagus retained completed calculations and resubmitted only incomplete work.
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Chemical Physics

Fixed-Dimensional Latent Flow for Generating Variable-Size 3D Molecules

Molecular size is intrinsically linked to composition, structure, and properties, but most 3D molecular generators treat size as a predefined input rather than a quantity to be generated. We introduce Equivariant-Free Transformer-Autoencoded Latent Flow Matching, a two-stage framework that samples a fixed-dimensional latent vector using flow matching and employs an autoregressive Transformer to determine molecular size, atom types, coordinates, and chemical attributes. Canonical atom ordering and rigid-pose alignment enable Transformers without equivariant layers, while decoded attributes guide bond reconstruction. On PCQM4Mv2, unconditional generation yields 87.9% unique and novel molecules passing sanitization and PoseBusters checks, exceeding baselines with lower end-to-end training and sampling time and higher end-to-end throughput. Across ten target HOMO-LUMO gaps, internal ranking retains 30% of the screened candidates and increases the density functional theory-verified hit rate within 0.1 eV of the target from 25.0% to 52.4%, while largely preserving novelty and diversity. These results demonstrate fixed-dimensional latent generation with autoregressive decoding as a practical approach to molecular design without prespecifying size.
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All authors Weichi Yao, Cameron Gruich, Bryan R. Goldsmith, Yixin Wang

Materials Science

Band Structure Modulation of ZrO2 Nanoparticles for Control of CO Adsorption Properties: A Combined Density Functional Theory - Density Functional Tight Binding Study

We present a combined density functional theory (DFT) and density functional tight binding (DFTB) study of zirconia (ZrO2) nanoparticles of experimentally relevant sizes of several nanometers and their interactions with the CO molecule. A hybrid DFTB - Force Field (DFTB-FF) framework is developed, whereby band structure calculations rely on an existing Slater-Koster framework, while the accuracy of structural optimization and adsorption properties is controlled by the introduction of classical long-range interatomic potentials into DFTB instead of the traditional repulsive potentials. Additionally, coordination-dependent Zr-C potentials are introduced to account for the distinct local chemical environments of bulk-like facet sites and under-coordinated tip and edge sites, thereby improving the description of CO adsorption. This hybrid DFTB-FF approach substantially improves the robustness of geometry optimization and provides a practical strategy for extending the applicability of DFTB to complex oxide nanostructures. The calculations reveal that termination stoichiometry can be used to engineer intrinsic, p-type, or n-type electronic structures and thereby tune the adsorption activity of zirconia nanoparticles. While stoichiometric nanoparticles do not activate the C-O bond, low-coordinated sites in Zr-rich (n-type) nanoparticles exhibit chemisorption accompanied by charge donation into a CO antibonding LUMO-derived orbital, resulting in C-O bond activation. These results demonstrate that stoichiometry-controlled electronic structure and under-coordinated surface sites introduced by nanostructuring play a key role in governing the adsorption strength and reactivity of zirconia nanoparticles.
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All authors Kexin Chen, William Dawson, Aulia Sukma Hutama, Takahito Nakajima, Keisuke Kameda, Manabu Ihara, Sergei Manzhos

Superconductivity

$1/3$-Flux Bound States in Multicomponent Superconductor Exhibiting Non-Abelian Statistics of $\mathbb {Z}_3$ Parafermions

Multicomponent superconductors (MSCs) are predicted to host magnetic flux quanta carrying arbitrary fractions of the superconducting flux quantum. Recent advances have raised the expectation that $1/3$ flux quanta may emerge in MSCs with $C_3$ rotational symmetry. Electrons bound to such fractional flux are long regarded to form anyons, yet their explicit braiding statistics remain unexplored. We propose that these $1/3$-flux bound states ($1/3$-FBSs) exhibit the intriguing non-Abelian statistics of $\mathbb{Z}_3$ parafermions. Under no-double-occupancy constraint, two successive braiding operations of $1/3$-FBSs are equivalent to a single $\mathbb Z_3$ parafermion braiding operation. The parafermion parity encoding the braiding outcome can be read out via the fermionic occupation number of the $1/3$-FBSs. Combined with a crossed-Andreev-reflection-induced Hadamard gate, one can realize the complete set of $\mathbb{Z}_3$ parafermion braiding operations.
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All authors Xing-Yu Wu, Y. H. Wang, X. C. Xie, Yijia Wu

High Energy Astrophysical Phenomena

Radio Emission with Dust: A Bow-Shock Interpretation for the TDE Candidate AT 2019avd

Forward shocks produced by interactions between a single ejected blob and the circumnuclear medium (CNM) around a supermassive black hole are widely invoked to explain radio emission from tidal disruption events (TDEs). However, recent observational evidence such as rapidly rising radio emission, double-peaked broadband spectral energy distributions, and declining shock energies at late times poses challenges to this picture. Alternative scenarios have therefore been proposed, including bow shocks arising from outflows colliding with dense clouds, the production of new ejecta at late times, and variations in microphysical parameters. To investigate the roles of these factors, we analyze the long-lived radio flare of the TDE candidate AT 2019avd, for which dust has been identified through mid-infrared observations, using a radio dataset spanning more than five years. We find that a bow-shock scenario can account for the majority of the observed radio emission, while a forward shock, or an additional bow shock involving different clouds, may begin to dominate at late times. However, a forward-shock scenario remains viable when allowing for evolving microphysical parameters and more complex CNM structures. We discuss the remaining tensions in both the outflow-CNM and outflow-cloud interaction scenarios.
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All authors Shuyuan Wei, Yanan Wang, Guobin Mou, Linhui Wu, Santiago del Palacio, Ranieri D. Baldi, Xiaolong Yang, Yang Huang

Quantum Physics · Cross-listed from math.CO

Spectral characterization of the uniform theta graph $\Theta(t,2)$ and classification of 6-periodic Grover walks

We characterize the uniform theta graph $\Theta(t,2)$ by the spectrum of its normalized adjacency matrix, or equivalently, by the spectrum of its normalized Laplacian matrix. We also investigate the periodicity of Grover walks on nonregular graphs, which is closely related to the eigenvalues of the normalized adjacency matrix and those of the time evolution matrix of the Grover walk. We show that the Dutch windmill graph $D_n^{(t)}$ is $2n$-periodic and that the uniform theta graph $\Theta(t,n)$ is $(2n+2)$-periodic. Furthermore, we completely determine the connected $6$-periodic graphs and prove that they are precisely $D_3^{(t)}$ with $t \geq 2$ and $\Theta(t,2)$ with $t \geq 1$.
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Mathematical Physics · Cross-listed from math.PR

Subcritical limits of a 1D KPZ growth surface with finite moments

We consider the one-dimensional random growth surface introduced by Adhikari and Chatterjee [AC24], where the height function is defined recursively via the heights at two neighboring sites and an independent noise term at each space-time point. In the subcritical regime, assuming uniformly bounded eighth moments of the noise variables and appropriate regularity of the driving function, we establish convergence in law of the suitably rescaled height function to the solution of the additive stochastic heat equation. Our approach employs a direct recursive renormalization scheme. A central component is a graph representation of the renormalization terms, which enables the derivation of the high-moment bounds essential to the renormalization procedure.
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All authors Dang-Zheng Liu, Fangzhou Luo, Tian Wu, Yuxuan Zong

General Relativity and Quantum Cosmology

Particle Dynamics and Thermodynamics of a Charged-Like Hairy Black Hole in Extended Gravitational Decoupling

This work focuses on the analysis of particle dynamics and the thermodynamic properties associated with a particular branch of hairy black hole solutions. These solutions are obtained by using extended geometric deformation gravitational decoupling on the seed solution of the Schwarzschild solution. This resulting geometry satisfies the dominant energy condition and the condition $Q^{2}=2\chi\ell M$. We analyze the motion of massive particles and photons in this geometry and the thermodynamics of this solution. In particular, for massless particles, we study the effective potential and determine the radii of the photon sphere and the shadow of the black hole. For massive particles, we calculate the specific energy and angular momentum of circular orbits, the ISCO, the radiative efficiency of an accretion disk, and representative particle trajectories. Finally, we investigate the thermodynamic properties of this hairy black hole geometry, including the Hawking temperature, the Bekenstein-Hawking entropy, and the heat capacity in the fixed-$(Q,\ell)$ ensemble.
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All authors M. Zeeshan Gul, Ahmad Al-Badawi, J. Andrade, Sadia Zahid

General Relativity and Quantum Cosmology

Kerr-referenced analytical parametrization of rotating black holes in dynamical Chern-Simons gravity

Rotating black holes in dynamical Chern-Simons gravity are known nonperturbatively mainly through numerical solutions, which limits their direct use in strong-field applications. We construct a Kerr-referenced analytical parametrization of the near-general-relativistic numerical branch using a compact radial coordinate, low-order angular multipoles, and continued fractions. The resulting two-parameter family accurately reproduces the numerical metric throughout the validated weak-coupling and moderate-spin domain. Independent off-grid solutions confirm that the parametrization retains its accuracy away from the calibration set. We further assess the model through equatorial light rings, critical impact parameters, and full two-dimensional null-geodesic ray tracing without assuming separability. The resulting shadow observables closely match those obtained from the numerical geometries. This parametrization therefore provides a practical analytical metric for geodesic and strong-field calculations without relying on a slow-rotation expansion.
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All authors Sardor Murodov, Bahodir Ahmedov, Bekzod Rahmatov, Javlon Rayimbaev, Bobomurat Ahmedov

High Energy Physics (Phenomenology)

Probing the $B^{0}$ meson in a hot medium with finite chemical potential

We employ QCD two-point sum rules to determine the $B^{0}$ meson mass and decay constant as functions of temperature $T$ and chemical potential $\mu$, using the perturbative spectral density and nonperturbative contributions up to mass dimension five. Numerical analysis indicates that, in the limits $T \to 0$ and $\mu \to 0$, the mass and decay constant of the $B^{0}$ meson are in good agreement with the available experimental data. Furthermore, at fixed chemical potentials, the $B^{0}$ meson mass and decay constant initially increase with temperature, reach their maxima, and then decrease to zero, with the increase being more pronounced for the decay constant than for the mass. The temperatures at which both quantities vanish decrease as the chemical potential increases. At fixed temperatures, both quantities initially increase slightly with chemical potential before decreasing and eventually vanishing, with the mass exhibiting a much weaker decrease in vacuum. The vanishing points of both quantities shift toward lower chemical potentials as the temperature increases, following a trend similar to that observed in the QCD phase diagram. The results obtained in this study are expected to contribute to the interpretation of data from ongoing heavy-ion collision experiments and to the generation of data for future planned experiments focusing on $B$ mesons, quark-gluon plasma signatures, and dense matter environments.
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Materials Science

Flexoelectric Polarization in Wrinkled Janus Transition-Metal Dichalcogenide Monolayers

Strain-gradient engineering via out-of-plane wrinkling offers a powerful route to tune electronic and electromechanical properties in two-dimensional (2D) materials. Here, we systematically investigate the electronic and flexoelectric response of wrinkled Janus MoSSe/MoSeS monolayers using density functional theory (DFT) calculations coupled with continuum elastica modeling. Exploring varying wrinkle sizes and compressive strain levels ($5\%-20\%$), we show that the global out-of-plane polarization follows a linear behavior when parameterized by the projected aspect ratio of the nanowrinkles. On this basis, we develop a physically grounded geometric model incorporating an effective 2D flexoelectric coefficient, which accurately predicts DFT polarizations without requiring higher-order nonlinear parameters. Atom- and orbital-resolved charge density analyses reveal the microscopic origin of this behavior: while the central Mo $4d$-orbital manifold acts as a robust, linear flexoelectric core, local curvature drives continuous, chemically asymmetric charge transfer between the S $3p$ and Se $4p$ sublayer manifolds. Our findings establish clear geometric design rules for harnessing Janus-based flexoelectricity for flexible nanoelectronics and optoelectronics.
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All authors Stefan Velja, Surender Kumar, Domenico Corona, Caterina Cocchi

General Relativity and Quantum Cosmology

Cosmological extra dimensions can mimic dark energy

We present a simple model of a higher dimensional spacetime in which the 4d submanifold is a FLRW metric, while the extra dimensions are compactified to a hypersphere. We calculate the Friedmann equations of this model finding that the extra dimensions interact in a non-trivial way with the 4d submanifold evolution. We prove that what is observed as Dark Energy could be explained only as an effect due to the Gaussian curvature and the expansion rate of the extra dimensions. We calculate the luminosity distance in this model and derive the acceleration parameter q. We find that {\Omega}m and q are in some sense dressed by terms coming from the extra dimensions, thus a Universe filled only with matter (both luminous and dark) and with extra dimensions can explain the observational data. We find that the extra dimensions do not affect the CMB spectrum, but this is not a solution to the Hubble tension. We fit our model to SNe data and find evidence for the presence of 3-4 extra dimensions which are presently contracting.
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Solar and Stellar Astrophysics

Detection of resonant nodes in a pore chromosphere

Active region atmospheres host many oscillatory phenomena. The chromosphere is delimited by steep temperature gradients at the photosphere and transition region, where magnetoacoustic waves are trapped and can form standing oscillations within this resonant cavity. We aim to detect the signature of the resonant nodes of standing waves, which are expected to produce sudden jumps in the oscillatory phase and power dips. Spectroscopic temporal series of H$\alpha$ in a pore were acquired with the Swedish Solar Telescope. The velocity and temperature fluctuations at multiple atmospheric heights were inferred from the analysis of the intensity at many spectral positions along the line wings. Wavelet analysis was employed to characterize the phase differences and the power at different heights. The phase shift between velocity and temperature shows a $\pm90^{\circ}$ value, which is consistent with standing oscillations. Robust evidence of the presence of a nodal layer in the temperature at around the height probed by the intensity at H$\alpha\pm0.30$ Å\ is found, such as the detection of 180$^{\circ}$ jumps in the phase of the temperature oscillations and remarkable power dips at the same atmospheric layer. The exact height of the resonant nodes depends on the spatial location and time. We generally find a mixture of standing and propagating waves. This is consistent with a leaky resonator where waves are partially reflected at the transition region, while some of them are able to propagate into the corona. For the first time, we report the detection and characterization of resonant nodes in the solar chromosphere. This result provides strong observational support for the chromospheric resonant cavity model and paves the way for the development of new seismological techniques to investigate the structure of active region chromospheres.
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Astrophysics of Galaxies

Low-Redshift Interlopers in the SDSS DR16 $z > 5$ Quasar Catalogue

High-redshift quasars ($z>5$) are crucial for studying early black hole growth and reionisation. SDSS Data Release 16 catalogues 655 such sources with multiple redshift estimators. We systematically examined this subsample and found 480 sources (73.2%) where pipeline redshift (Z\_DR16Q) disagrees with the derived estimate (Z\_FIT) significantly. Visual inspection confirms that 125 of these (19.1% of the total) are actually low-redshift active galactic nuclei. In 114 cases, broad Mg\,ii with C\,iii and C\,iv was misidentified as Ly$\alpha$; in 11 cases, H$\alpha$ was misidentified as Ly$\alpha$. This error is immediately revealed by the absence of Gunn–Peterson trough blueward of Ly$\alpha$. Crucially, Z\_FIT recovers correct redshift in the affected sources where emission lines are prominent, indicating the algorithm is sound; the issue lies in community preference for Z\_DR16Q or Z\_SPEC. We urge caution against uncritical use of pipeline redshifts for $z>5$ DR16Q sources and recommend Z\_FIT as the preferred estimator.
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High Energy Physics (Phenomenology)

Near-Threshold Dynamics of ḩi}c1(3872) and T+cc(3875) States via Effective Range Expansion and Monte Carlo Uncertainty Propagation

We investigate the near-threshold structure of the exotic tetraquark candidates $\chi_{c1}(3872)$ and $T_{cc}^+(3875)$ using the effective-range expansion (ERE) and resonance compositeness relations. From the experimental masses and widths, we extract scattering lengths, effective ranges, and molecular compositeness coefficients within a two-channel framework, with uncertainties propagated through $N=50000$ Monte Carlo samples. We find large negative scattering lengths, $a=-8.49^{+0.93}_{-1.05}$ fm for $\chi_{c1}(3872)$ and $a=-14.57^{+1.70}_{-1.66}$ fm for $T_{cc}^+$, and dominant molecular components $X_2>0.95$. The results are robust against variations of compositeness, experimental correlations, and the loop-function subtraction constant. An explicit ERE+CDD-pole analysis finds no physically reasonable bare-state contribution consistent with the observed poles and a natural background. Our results provide strong evidence for predominantly molecular structures of both states and demonstrate the robustness of the ERE approach to near-threshold exotic hadrons.
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All authors A. A. Atangana Likéné, A. Franck Rothen, J. M. Ema'a Ema'a, P. Pradyun Hebbar, Saralees Nadarajah, Tobias Golling, G. H. Ben-Bolie

Statistical Mechanics · Cross-listed from cs.IT

Physics of Information Geometry - Part I: Principle of Least Action on the Probability Simplex

We develop a least-action framework for describing how a probability distribution can evolve from an equilibrium state to a prescribed nonequilibrium state under constrained incremental changes. Taking a Gibbs distribution as the equilibrium reference, the framework gives a direct physical meaning to the geometry of the probability simplex: distance from equilibrium corresponds to nonequilibrium free energy, while changes between successive distributions carry an informational kinetic cost. The Pythagorean structure of relative entropy then provides the central insight of the work. It shows that intermediate distributions chosen via sequential information projections can reduce the kinetic cost of large transitions and establishes an energy-conservation-like relation between the kinetic expenditure along a path and the free energy accumulated in reaching the target distribution. Motivated by this geometry, we construct a greedy least-action path through successive information projections, obtain a closed-form characterization of each projection through the Lambert W function, and establish a finite-step performance guarantee. We further show that state-dependent costs can be incorporated naturally by reshaping the underlying Gibbs reference, providing a thermodynamic interpretation of path penalties as modifications of the effective energy landscape. Together, these results provide a unified view of distributional evolution through least action, information geometry, and nonequilibrium thermodynamics.
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Superconductivity

Néel-Vector Control of the Josephson Diode Effect in $\mathcal{PT}$-symmetric Antiferromagnets

The interplay of superconductivity and magnetism gives rise to rich phenomena in Josephson junctions. In this Letter, we study Josephson junctions formed by conventional $s$-wave superconductors and a $\PT$-symmetric collinear antiferromagnet modeled on CuMnAs. Using microscopic modeling and symmetry analysis, we show that these junctions exhibit both the Josephson diode effect and $\varphi_{0}$-junction states. Remarkably, both effects are controlled by the Néel vector: rotating it by $90^{\circ}$ switches off both, while reversing it switches the diode polarity. To reveal the microscopic mechanism, we develop a channel-resolved scattering theory that accurately captures the anomalous phases and establishes the exact condition for the diode effect. The interplay of the channel current-phase relations yields a sizable diode efficiency, tunable by both the magnitude and direction of the exchange field. Furthermore, a Green-function reduction identifies a single renormalized $\PT$-degenerate band as the transport carrier and precisely reproduces the full current amplitudes. Our work establishes $\PT$-symmetric antiferromagnets as versatile platforms for field-free, highly tunable Josephson diodes and $\varphi_{0}$ junctions.
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All authors Xian-Tang Xu, Xun-Jiang Luo, Mingliang Tian, Ning Hao

Materials Science

Light-facilitated ferroelectric switching in wurtzite crystals

Wurtzite ferroelectrics combine large remanent polarization with full CMOS compatibility, positioning them as a leading platform for next-generation non-volatile memory. Their practical deployment, however, is hindered by an intrinsically large coercive field, rooted in the high energy barrier separating the polar wurtzite phase from the nonpolar hexagonal phase that mediates polarization switching. Here, using first-principles calculations, we propose an alternative, field-free strategy for lowering this barrier: above-bandgap electronic photoexcitation. Taking LaN as a representative wurtzite ferroelectric, we show that light induced carriers dramatically reduce the energy difference between the hexagonal intermediate phase and the wurtzite ground state, sharply reducing the energy barrier to ferroelectric switching. This effect originates from a photoinduced partial metallization of the polar phase, which screens the dipole-dipole interactions that stabilize ferroelectric order and thereby favors the competing nonpolar structure. The robustness of this mechanism is further confirmed for the rocksalt polymorph. Our results establish light as a powerful, non-invasive route to controlling ferroelectric switching in wurtzites, opening a path toward faster, lower-voltage, and more energy-efficient non-volatile memory technologies.
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All authors Ricardo Jiménez-Sánchez, Fernando Salazar, Miguel Cruz-Irisson, Riccardo Rurali, Claudio Cazorla

High Energy Physics (Experiment)

Search for the doubly Cabibbo-suppressed decays $D^0\to K^+\pi^-\eta^\prime$ and $D^+\to K^+\pi^0\eta^\prime$

We present the first search for the doubly Cabibbo-suppressed decays $D^0\to K^+\pi^-\eta^\prime$ and $D^+\to K^+\pi^0\eta^\prime$ using an $e^+e^-$ collision data sample corresponding to an integrated luminosity of 20.3 fb$^{-1}$, collected at a center-of-mass energy of 3.773 GeV with the Beijing Spectrometer III (BESIII) detector at the Beijing Electron-Positron Collider II (BEPCII). No significant signals are observed, and the upper limits on their decay branching fractions are set to be $3.0\times 10^{-5}$ and $2.1\times 10^{-5}$ at the 90% confidence level, respectively. By combining these results with the world-average branching fractions of the corresponding Cabibbo-favored decays, upper limits at the 90% confidence level are obtained on the ratios of doubly Cabibbo-suppressed to Cabibbo-favored branching fractions. The limits are determined to be $1.6\times \tan^4\theta_C$ and $3.7\times \tan^4\theta_C$ for $D^0\to K^+\pi^-\eta^\prime$ and $D^+\to K^+\pi^0\eta^\prime$, respectively, where $\theta_C$ denotes the Cabibbo mixing angle.
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All authors BESIII Collaboration: M. Ablikim, M. N. Achasov, P. Adlarson, X. C. Ai, C. S. Akondi, R. Aliberti, A. Amoroso, L. P. An, Q. An, M. S. Anderson, Y. Bai, O. Bakina, H. R. Bao, X. L. Bao, M. Barbagiovanni, V. Batozskaya, K. Begzsuren, N. Berger, M. Berlowski, M. B. Bertani, D. Bettoni, F. Bianchi, E. Bianco, A. Bortone, I. Boyko, R. A. Briere, A. Brueggemann, D. Cabiati, H. Cai, M. H. Cai, X. Cai, A. Calcaterra, G. F. Cao, N. Cao, X. Cao, S. A. Cetin, X. Y. Chai, J. F. Chang, T. T. Chang, G. R. Che, Y. Z. Che, C. H. Chen, Chao Chen, G. Chen, H. S. Chen, H. Y. Chen, J. D. Chen, M. L. Chen, S. J. Chen, S. M. Chen, T. Chen, W. Chen, X. R. Chen, X. T. Chen, X. Y. Chen, Y. B. Chen, Y. Q. Chen, Z. K. Chen, J. Cheng, L. N. Cheng, S. K. Choi, X. Chu, G. Cibinetto, F. Cossio, J. Cottee-Meldrum, H. L. Dai, J. P. Dai, X. C. Dai, A. Dbeyssi, R. E. de Boer, D. Dedovich, Z. Y. Deng, A. Denig, I. Denisenko, M. Destefanis, F. De Mori, E. Di Fiore, X. X. Ding, Y. Ding, J. Dong, L. Y. Dong, M. Y. Dong, Z. J. Dong, K. B. Du, M. C. Du, S. X. Du, Shaoxu Du, X. L. Du, Y. Q. Du, C. D. Duan, Z. H. Duan, P. Egorov, G. F. Fan, K. X. Fan, Y. H. Fan, Z. F. Fan, J. Fang, Jin Fang, S. S. Fang, W. X. Fang, L. Fava, F. Feldbauer, G. Felici, C. Q. Feng, J. H. Feng, Q. X. Feng, Y. T. Feng, M. Fritsch, C. D. Fu, J. L. Fu, Y. W. Fu, Xu Gao, Y. Gao, Y. N. Gao, Yunong Gao, Z. Gao, S. Garbolino, I. Garzia, L. Ge, L. Q. Ge, P. T. Ge, C. Geng, A. Gilman, K. Goetzen, J. Gollub, J. B. Gong, J. D. Gong, L. Gong, W. X. Gong, W. Gradl, M. Greco, M. D. Gu, M. H. Gu, C. Y. Guan, S. Y. Guan, A. Q. Guo, H. Guo, J. N. Guo, L. B. Guo, M. J. Guo, R. P. Guo, X. Guo, Y. P. Guo, Z. Guo, A. Guskov, J. Gutierrez, J. Y. Han, T. T. Han, X. Han, F. Hanisch, J. Y. Hao, K. D. Hao, X. Q. Hao, F. A. Harris, C. Z. He, K. K. He, K. L. He, F. H. Heinsius, C. H. Heinz, Y. K. Heng, C. Herold, N. D. Hoffman, P. C. Hong, G. Y. Hou, X. T. Hou, Z. L. Hou, H. M. Hu, J. F. Hu, Q. P. Hu, S. L. Hu, T. Hu, Y. Hu, Y. X. Hu, Z. M. Hu, G. S. Huang, L. Q. Huang, P. Huang, X. T. Huang, Y. P. Huang, Y. S. Huang, T. Hussain, N. Hüsken, N. in der Wiesche, Q. Ji, Q. P. Ji, W. Ji, X. B. Ji, X. L. Ji, Y. Y. Ji, L. K. Jia, S. Jia, X. Q. Jia, D. Jiang, H. B. Jiang, S. J. Jiang, X. S. Jiang, Y. Jiang, J. B. Jiao, Z. Jiao, B. W. Jin, L. C. L. Jin, S. Jin, Y. Jin, M. Q. Jing, X. M. Jing, T. Johansson, S. Kabana, X. L. Kang, X. S. Kang, B. C. Ke, A. Khoukaz, O. B. Kolcu, S. H. Kong, B. Kopf, L. Kröger, L. Krümmel, Y. Y. Kuang, M. Kuessner, X. Kui, N. Kumar, A. Kupsc, W. Kühn, L. Y. Lan, Q. Lan, J. Y. Lee, T. T. Lei, M. Lellmann, T. Lenz, C. Li, C. H. Li, C. J. Li, C. K. Li, Cong Li, D. M. Li, F. Li, G. Li, H. B. Li, H. J. Li, H. N. Li, H. P. Li, Hui Li, J. N. Li, J. W. Li, K. Li, K. L. Li, L. J. Li, L. K. Li, Lei Li, M. H. Li, M. R. Li, M. T. Li, P. L. Li, P. R. Li, Q. M. Li, R. Li, S. X. Li, S. Y. Li, Shanshan Li, T. Li, T. Y. Li, W. D. Li, W. G. Li, X. Li, X. H. Li, X. K. Li, X. L. Li, X. Y. Li, X. Z. Li, Y. H. Li, Y. B. Li, Y. C. Li, Y. G. Li, Y. L. Li, Y. P. Li, Yi Li, Z. H. Li, Z. J. Li, Z. L. Li, Z. X. Li, Z. Y. Li, Zaiyi Li, C. Liang, H. Liang, Y. F. Liang, Y. T. Liang, Z. Z. Liang, G. R. Liao, L. B. Liao, M. H. Liao, Y. P. Liao, J. Libby, A. Limphirat, C. X. Lin, D. X. Lin, T. Lin, B. J. Liu, C. Liu, C. X. Liu, F. Liu, F. H. Liu, Feng Liu, G. M. Liu, H. Liu, H. B. Liu, H. M. Liu, Huihui Liu, J. B. Liu, K. Liu, K. Y. Liu, Ke Liu, Kun Liu, L. Liu, L. C. Liu, Lu Liu, M. H. Liu, P. L. Liu, Q. Liu, S. B. Liu, T. Liu, W. T. Liu, X. Liu, X. K. Liu, X. L. Liu, X. P. Liu, X. T. Liu, X. Y. Liu, Y. Liu, Y. B. Liu, Yi Liu, Z. A. Liu, Z. D. Liu, Z. Q. Liu, Z. X. Liu, Z. Y. Liu, X. C. Lou, H. J. Lu, J. G. Lu, X. L. Lu, Y. Lu, Y. H. Lu, Y. P. Lu, C. L. Luo, J. Q. Luo, J. R. Luo, J. S. Luo, M. X. Luo, Q. Q. Luo, T. Luo, X. L. Luo, XiaoRong Lv, Z. Y. Lv, X. R. Lyu, Y. F. Lyu, Y. H. Lyu, C. L. Ma, F. C. Ma, H. L. Ma, Heng Ma, J. L. Ma, L. L. Ma, Q. M. Ma, R. Q. Ma, T. Ma, X. T. Ma, X. Y. Ma, F. E. Maas, M. Maggiora, S. Maity, S. Malde, L. M. Mansur, Y. J. Mao, Z. P. Mao, S. Marcello, A. Marshall, F. M. Melendi, Z. X. Meng, G. Mezzadri, H. Miao, T. J. Min, T. Mineeva, S. W. Ming, R. E. Mitchell, X. H. Mo, A. F. Mohammed, B. Moses, N. Yu. Muchnoi, J. Muskalla, Y. Nefedov, F. Nerling, H. Neuwirth, Z. Ning, S. Nisar, Q. L. Niu, W. D. Niu, Y. Niu, C. Normand, S. L. Olsen, Q. Ouyang, I. V. Ovtin, S. Pacetti, X. Pan, Y. Pan, C. Y. Pang, A. Pathak, Y. P. Pei, Y. Y. Pei, M. Pelizaeus, G. L. Peng, H. P. Peng, X. J. Peng, Y. Y. Peng, K. Peters, K. Petridis, J. L. Ping, R. G. Ping, S. Plura, S. Polikarpov, V. Prasad, L. Pöpping, F. Z. Qi, H. R. Qi, L. Y. Qian, S. Qian, W. B. Qian, C. F. Qiao, J. H. Qiao, J. J. Qin, J. L. Qin, L. Q. Qin, L. Y. Qin, P. B. Qin, X. P. Qin, X. S. Qin, Z. H. Qin, J. F. Qiu, J. Rademacker, K. Ravindran, C. F. Redmer, Y. Ren, A. Rivetti, M. Rolo, G. Rong, S. S. Rong, F. Rosini, Ch. Rosner, M. Q. Ruan, W. R. Ruangyoo, N. Salone, A. Sarantsev, Y. Schelhaas, M. Schernau, K. Schoenning, M. Scodeggio, Jian Shan, W. Shan, X. Y. Shan, Z. J. Shang, J. F. Shangguan, L. G. Shao, M. Shao, C. P. Shen, H. F. Shen, W. H. Shen, X. Y. Shen, B. A. Shi, Ch. Y. Shi, H. Shi, J. L. Shi, J. Y. Shi, S. Shi, X. Shi, X. D. Shi, H. L. Song, J. J. Song, M. H. Song, T. Z. Song, W. M. Song, Y. Song, Y. X. Song, Zirong Song, S. Sosio, S. Spataro, S. Stansilaus, F. Stieler, M. Stolte, J. L. Su, S. S Su, G. B. Sun, G. X. Sun, H. Sun, H. K. Sun, J. F. Sun, K. Sun, L. Sun, R. Sun, S. S. Sun, W. Y. Sun, Y. C. Sun, Y. J. Sun, Y. S. Sun, Y. Z. Sun, Z. Q. Sun, Z. T. Sun, H. Tabaharizato, N. T. Tagsinsit, C. J. Tang, G. Y. Tang, H. Y. Tang, J. Tang, J. J. Tang, L. F. Tang, Y. A. Tang, Z. H. Tang, L. Y. Tao, M. Tat, J. X. Teng, J. Y. Tian, W. H. Tian, Y. Tian, K. Yu. Todyshev, I. Uman, E. van der Smagt, S. W. Wan, X. Wan, B. Wang, B. Q. Wang, Bin Wang, Bo Wang, C. Wang, C. L. Wang, Cong Wang, D. Y. Wang, F. K. Wang, H. J. Wang, H. R. Wang, J. Wang, J. H. Wang, J. J. Wang, J. P. Wang, K. Wang, L. L. Wang, M. Wang, Mi Wang, N. Y. Wang, S. Wang, Shun Wang, W. Wang, W. P. Wang, X. F. Wang, X. L. Wang, X. N. Wang, Xin Wang, Y. Wang, Y.-K. Wang, Y. D. Wang, Y. F. Wang, Y. H. Wang, Y. J. Wang, Y. L. Wang, Yanning Wang, Yaqian Wang, Yi Wang, Yuan Wang, Z. Wang, Z. L. Wang, Z. Q. Wang, Z. Y. Wang, Zhi Wang, D. Wei, D. H. Wei, D. J. Wei, H. R. Wei, F. Weidner, H. R. Wen, S. P. Wen, U. Wiedner, G. Wilkinson, J. F. Wu, L. H. Wu, L. J. Wu, S. G. Wu, S. M. Wu, X. W. Wu, Y. C. Wu, Z. Wu, H. L. Xia, L. Xia, B. H. Xiang, D. Xiao, G. Y. Xiao, H. Xiao, Y. L. Xiao, Z. J. Xiao, C. Xie, K. J. Xie, Y. Xie, Y. Xie, Y. G. Xie, Y. H. Xie, Z. P. Xie, T. Y. Xing, G. F. Xu, H. Y. Xu, Q. J. Xu, Q. N. Xu, X. P. Xu, Y. Xu, Y. C. Xu, Z. S. Xu, F. Yan, J. Z. Yan, L. Yan, W. B. Yan, W. C. Yan, W. H. Yan, X. Q. Yan, Y. Y. Yan, H. J. Yang, H. L. Yang, H. X. Yang, J. H. Yang, L. Y. Yang, N. Yang, R. J. Yang, X. Y. Yang, Y. Yang, Y. G. Yang, Y. H. Yang, Y. Q. Yang, Y. Z. Yang, Z. B. Yang, Z. W. Yang, W. 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Astrophysics of Galaxies

AEON-z5: A Candidate AGN-driven Outflow Enriching the Circumgalactic Medium at $z\simeq5.23$

The dispersal of chemically enriched gas from galaxies into their surroundings is a key process in galaxy evolution, yet direct observational evidence at z>5 remains scarce. We present AEON-z5, a galaxy at z~5.23 in the COSMOS field comprising a compact continuum-emitting core surrounded by an extended, line-dominated ionized nebula. JWST/NIRCam imaging shows that H$\alpha$+[N II] and H$\beta$+[O III] emission extends to projected radii of >4 kpc - 2-2.5 times the typical effective radius at the host stellar mass - reaching the outer ISM and the inner CGM. F444W grism data reveal a highly asymmetric H$\alpha$+[N II] profile, which we interpret as a bipolar outflow and decompose into three kinematic components. The dominant component has a flux-weighted velocity offset of ~+489 km/s and FWHM~630 km/s, with a wing reaching $v_{84}$~792 km/s. A tentative detached feature at $\Delta v_{LOS}$~2800 km/s, detected at 1.5$\sigma$ in the 1D spectrum (2.4$\sigma$ in the 2D fit), may trace an outflow clump. These kinematics are most coherently explained by an AGN in the core. For the extended nebula, we derive a velocity curve shifting redward with radius, reaching 200-400 km/s at $r_p$~1.3-2.5 kpc - a trend that may reflect an accelerating outflow, corotating gas, or recycled inflow viewed in projection. Crucially, the measured [N II]/H$\alpha$ ratios imply near-solar N2-based abundances (0.7-1.0 $Z_\odot$), remaining >0.37 $Z_\odot$ after allowing for AGN excitation and calibration systematics. The combination of large extent, enrichment, and extreme kinematics identifies AEON-z5 as a candidate snapshot of feedback-driven metal transport, offering a direct view of how early AGN activity may redistribute chemically processed gas into the CGM within the first ~1.1 billion years.
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All authors Xiaoyang Wei, Zheng Cai, Shiwu Zhang, Fujiang Yu, Yunjing Wu, Shuaiyi Li, Xiaojing Lin, Mingyu Li, Xuelun Mei

Strongly Correlated Electrons

Kane-Lubensky phonons in Maxwell lattice frustrated Mott insulators

We show that zero-energy gapless Weyl-line phonons of the Kane-Lubensky's type can arise in the three-dimensional Maxwell lattice frustrated Mott insulators through the magnetopological mechanics. In a pyrochlore antiferromagnet with the spin-lattice coupling, a magnetic field selects the spin state whose lattice distortion generates a $P4_3 32$ topological lattice. More crucially, the spin-lattice coupling and the spin configuration cause the bending of the neighbouring bonds, and converts the system into the topological Maxwell lattice. Remarkably, the resulting system is found to host the bulk zero-frequency Weyl-line phonons protected topologically, and these gapless phonons are not Goldstone modes. Unlike the conventional ${C_{\rm ph}\sim T^3}$ for the Goldstone phonons, these one-dimensional zero-mode manifolds yield a characteristic low-temperature phonon specific heat ${C_{\rm ph}\sim T^2}$. Our results could find applications in the Cr-based spinel systems, and moreover, we establish a low-energy platform where an extensive number of topological zero-frequency phonons can strongly couple to other degrees of freedom, opening a route to exotic phonon-mediated phenomena.
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Atomic Physics

Nuclear slowing-down factors in alkali-metal vapors

Nuclear slowing-down factors account for the sharing of angular momentum between the electron and the nucleus in the effective Bloch description of alkali-metal spin dynamics in the spin-exchange-relaxation-free (SERF) regime. For collinear optical pumping and magnetic field, we find that longitudinal and transverse spin dynamics are characterized by different polarization-dependent slowing-down factors. Transverse dynamics are governed by the conventional factor $q(p)$, with $p$ the electron spin polarization, whereas longitudinal relaxation is governed by $q(p)+p\,dq(p)/dp$. The two factors coincide at zero polarization but differ substantially at high polarization, where the single-factor description overestimates the underlying pumping and relaxation rates inferred from longitudinal transients by factors approaching two to four, depending on the nuclear spin. We further derive closed-form expressions, valid at arbitrary polarization, for the residual spin-exchange relaxation of the transverse spin components at finite magnetic field along the pumping axis. This contribution is quadratic in the magnetic field and can become comparable to the zero-field linewidth at fields well within the SERF regime. The results are obtained by perturbation theory on the density-matrix dynamics linearized around the stationary state and are verified by numerical solutions of the microscopic density-matrix equation. These findings refine the effective Bloch description of alkali-metal spin dynamics and are directly relevant to atomic magnetometers and alkali-metal–noble-gas comagnetometers, including those used in precision searches for physics beyond the Standard Model.
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Mathematical Physics

Asymptotics of Ground States for Helium-Like and Bosonic Atoms in a Two-Cluster Region

We study the asymptotic behavior of positive ground states of $N$-particle atomic Coulomb Hamiltonians in a two-cluster region. For the $N$-particle ground state $\psi$, its one-particle density $\rho$, and $(N-1)$-particle ground state $\phi$, we show that for $0<\alpha<1$, as $|x_N|\to\infty$, $\psi(x_1,\dots,x_N)/\sqrt{\rho(x_N)}= \phi(x_1,\dots,x_{N-1})+O(|x_N|^{-2+\alpha} \ln |x_N|)$ uniformly in the region $|x_i|\le |x_N|^{\alpha}$. In the region $|x_i|\le R_0\ln |x_N|$, the convergence rate is $O(|x_N|^{-2} (\ln |x_N|)^2)$. We also establish an $O(|x_N|^{-2})$ bound on the squared $L^2$ error. These results hold both below the essential spectrum and at threshold.
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