Materials Science · Cross-listed from physics.soc-ph
The Sustainability Paradox of Biodegradable Packaging: A Life Cycle Perspective on Chitosan-Based Food Packaging
Maria Cosic, Milena Cosic, Amy Kim et al. (+9 more)
Chitosan-based nanomaterials are being increasingly explored as sustainable alternatives to petroleum-derived food packaging, yet their environmental performance across the full life cycle remains insufficiently understood. This review critically evaluates these systems from a life cycle perspective and examines how material origin, processing pathways, functional performance, and end-of-life behavior collectively influence sustainability outcomes. Beginning with chitin extraction from crustacean waste, key processing steps, including demineralization, deproteinization, and nanoparticle synthesis, are assessed in terms of chemical intensity, energy demand, and associated emissions. Manufacturing routes, including solvent-based and green synthesis approaches, are compared with those of conventional plastics to identify relative environmental burdens. The use phase is analyzed with respect to antimicrobial functionality, shelf life extension, and potential reductions in food waste. End-of-life pathways, including biodegradation and composting, are evaluated alongside uncertainties related to degradation behavior and nanoparticle fate. By synthesizing these stagewise interactions, this review highlights critical trade-offs that are often overlooked in sustainability narratives and examines whether chitosan-based nanomaterials provide net environmental benefits in food packaging applications.
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All authors Maria Cosic, Milena Cosic, Amy Kim, Siddhartha Pahari, Grace Wu, Rayna Zhou, Trisha Daftari, Shaily Gupta, Reyna Michelle Suneel, Srinivasan Latha, Rashi Sharma, Utkarsh Chadha
Soft Condensed Matter
Inverse Laplace Transform for Dynamic Light Scattering: Impact of Regularization
Pierre Pajuelo, Stéphane G. Roux, Adrien Meynard et al. (+3 more)
Dynamic Light Scattering (DLS) analyzes particle dynamics from the autocorrelation functions of scattered light intensity, yet extracting accurate relaxation time distributions from noisy data is challenging. We develop an inverse problem approach to recover this distribution by inverting the Laplace transform with physics-based regularization, called the CONTIN method. We improve it to use it on noisy data, across a wide range of time scales, with a selection of the regularization strength through a data-driven L-curve criterion. Our approach enhances robustness under high noise and reveals multi-scale dynamics in complex systems. Validation is performed on simulated data, compared to the Cramér-Rao bound and to parametric methods, and on experimental data from Carbopol microgels. It demonstrates superior accuracy over parametric methods, especially for broad time distributions. The
algorithm's logarithmic discretization and variance-reduced correlation estimation enhance performance, offering a powerful tool for non-parametric DLS analysis and deeper insights into soft matter dynamics.
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All authors Pierre Pajuelo, Stéphane G. Roux, Adrien Meynard, François Liénard, Éric Freyssingeas, Pierre Borgnat
Statistical Mechanics
Numerical simulations of the spread from the mean of the SLE and Multiple SLE dynamics
Phillip Kim and Vlad Margarint
The Schramm-Loewner
Evolution (SLE) describes a family of fractal curves that arise in the study of the scaling limits of many planar Statistical Physics models. These curves are modeled using the Loewner Differential Equation for the conformal maps $g_t(z)$ with a Brownian motion driver. Using Euler's Method, in the current work we performed numerical experiments to study at a fixed time the quantities $|g_t(z) - \overline{g_t(z)}|$ and $Re(g_t(z)) - Re(\overline{g_t(z)})$, where $Re$ denotes the real part and $\overline{g_t(z)}$ refers to the sample average. These random variables measure the 'spread' of the dynamics from the average behavior at fixed time. One of the scopes of this work is to give numerical predictions for future theoretical investigations on these quantities. When investigating these quantities in the SLE case our experiments predict that the distribution is bimodal when the dynamics started close to the origin, and it can become bell-shaped if the dynamics is started further from the origin. In the second part, we performed experiments for a Multiple SLE model whose driver is Dyson Brownian Motion. Due to singularity in the dynamics of the drivers and the many data points needed, this part is challenging from a computational perspective. In the multiple SLE case, our experiments predict that the distribution is bell-shaped in all cases. In addition, we check the changes in the distributions as we vary the parameter $\kappa$ in the SLE case and $\beta$ in the Multiple SLE case.
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Strongly Correlated Electrons
Coherent manipulation of Kondo Majoranas in two-channel Kondo setups
Yashar Komijani and C. J. Bolech
We study coherent manipulation of Majorana zero modes emerging in overscreened two-channel Kondo systems. Using compactified
lattice models, we show that these interacting Kondo Majoranas support non-local
qubits and admit teleportation, fusion, and braiding operations. In particular, we identify a distinction between non-topological and genuinely topological Y-junction geometries, the latter realizing a non-Abelian
geometric holonomy. Our results establish a proof-of-principle route toward coherent control of non-Abelian anyons beyond conventional free-
fermion platforms.
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Materials Science
Robust AI-Driven Discovery of Electronic Metal Phosphide Semiconductors
Benhao Zhu, Muhammad Faizan, Zewei Li et al. (+4 more)
Metal phosphides have diverse bonding motifs and coordination environments, making them promising for optoelectronic and thermoelectric applications, but their chemical space remains underexplored. Here we report an AI-driven high-throughput discovery workflow that combines generative materials design, machine-learning interatomic potentials, and targeted density functional theory (DFT) calculations. ICSD-derived Wyckoff-site substitution and MatterGen-based conditional structure generation are used to expand the candidate space beyond existing phosphide databases. A domain-finetuned DPA3 machine-learning potential then enables efficient prescreening of thermodynamic and dynamical stability before DFT validation. This workflow identifies 3,574 previously unreported stable phosphide structures, including 196
semiconductors with HSE06 band gaps of 0-3.0 eV. By screening these new
semiconductors together with experimentally known phosphide
semiconductors, we identify 30 promising optoelectronic candidates and 26 promising thermoelectric candidates, including seven newly discovered optoelectronic materials and eight newly discovered thermoelectric materials. These results provide a candidate pool for experimental synthesis and show that combining generative AI with machine-learning interatomic potentials can accelerate the discovery of functional
semiconductor materials.
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All authors Benhao Zhu, Muhammad Faizan, Zewei Li, Wenshuo Li, Feifei Ren, Jiahao Xie, Lijun Zhang
Disordered Systems and Neural Networks · Cross-listed from q-bio.NC
Multifractal Signatures of Ageing and Dementia Development: A Multifractal Space-Filling Curve Analysis
Marta Lotka, Jacek Grela, Zbigniew Drogosz et al. (+2 more)
Multifractality is an effective formalism for quantifying the nonlinear, scale-free properties of complex data. In this study, we propose a novel and efficient methodology, termed Multifractal Space-filling Curve Analysis (MFSCA), for quantifying the correlation structure of multidimensional data. Within this framework, the original multidimensional data - while preserving both local and long-range organisational properties - are projected onto a one-dimensional representation using a fractal space-filling curve. The resulting one-dimensional signal is then analysed using multifractal
algorithms. We demonstrate the utility of the method using both artificially generated multifractal structures and real data. In particular, we apply MFSCA to analyse
magnetic resonance imaging (MRI) data from Alzheimer patients at different stages of dementia. Based on the results, we estimate the multifractal profiles of the
brain for healthy subjects of different ages as well as for dementia patients. The analysis reveals that the spatial organization of
brain structures, as measured by the degree of multifractality, progressively weakens with age and the development of dementia. A transition from multifractality to monofractality is observed both in control groups, when comparing the Young Control and Elderly Control groups, and among dementia subjects of similar age but at different stages of the disease, namely early dementia and mild cognitive impairment. Thus, from the perspective of multiscaling properties, the heterogeneous characteristics of spatial
brain organization deteriorate under worsening conditions, leading to a homogeneous and weakly correlated structure. These findings not only effectively capture key aspects of
brain organisation, but also demonstrate that the multifractality of MRI data can serve as a marker of structural
brain changes.
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All authors Marta Lotka, Jacek Grela, Zbigniew Drogosz, Jeremi K. Ochab, Paweł Oświęcimka
Materials Science
Finite-temperature Fe K-edge X-ray absorption simulations reveal local structural dynamics of an iron(II) photosensitizer in solution and the crystalline phase
Patrick Müller, Lorena Fritsch, Matthias Bauer et al. (+1 more)
Interpreting metal K-edge spectra of flexible photosensitizers requires a structural model that separates electronic signatures from thermal motion, solvent disorder, and
crystal-packing effects. We combine Fe K-edge X-ray absorption measurements with second-generation Car–Parrinello ab initio molecular dynamics and all-electron Gaussian and augmented-plane-wave simulations for an iron(II) N-heterocyclic carbene photosensitizer in acetonitrile solution and in the crystalline phase. Ensemble-averaged spectra reproduce the main near-edge features in both environments and preserve the experimentally observed similarity of the first Fe coordination shell upon dissolution. Comparison with radial distributions extracted from extended fine-structure measurements validates the Fe–N and Fe–C coordination shells sampled by the trajectories, while element-resolved pair distributions explain why higher-shell experimental contrast is rapidly lost. The same dynamical ensembles reveal a broad out-of-plane distribution of the terpyridine nitrogen atom and a nearly octahedral distribution of the Fe-centered coordination planes. The results show that finite-temperature X-ray absorption simulations can provide a compact structural-dynamics picture of molecular transition metal photosensitizers by linking local spectra, solvent-phase ligand motion, and medium-range structural disorder within one trajectory-based description.
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All authors Patrick Müller, Lorena Fritsch, Matthias Bauer, Thomas D. Kühne
Materials Science
Graphlet Histogram Representation Database of Inorganic Crystals
Aaditya Panigrahi, Yanjun Liu, Omri Lesser et al. (+2 more)
Machine learning models for materials property prediction increasingly rely on representations learned end-to-end from large density-functional-theory databases, limiting their applicability when only scarce experimental data are available. Domain-knowledge-driven representations precomputed from
crystal structures alone offer a data-efficient, interpretable alternative, but existing approaches capture at most composition or bonding connectivity and discard local structural geometry. Here, we present Graphlet-MP, a database of graphlet histogram representations for 149,082 inorganic
crystals from the Materials Project (MP). Seventy-nine distributions describe each material over three hierarchical graphlet orders: atomic sites, bonded pairs, and bond-angle triplets, extracted via screened Voronoi tessellation from the crystallographic information file. We provide a complete technical specification of the representation, an Earth Mover's Distance metric for comparing materials in this space, and the full precomputed database. An accompanying open-source codebase enables users to generate graphlet histograms for arbitrary
crystal structures, including experimentally determined ones, and to extend the database to new materials or target properties.
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All authors Aaditya Panigrahi, Yanjun Liu, Omri Lesser, Krishnanand Mallayya, Eun-Ah Kim
Materials Science
Pressure-Driven Structural Phase Competition and Functional Response in Layered LiInP2S6
Xiaochi Xie, Pegah Mohammadi and Sobhit Singh
Understanding how hydrostatic pressure modifies interlayer interactions and competing ionic configurations is essential for controlling the emergent functional properties of layered
quantum materials. Here, using first-principles density-functional theory calculations, we investigate the pressure-dependent structural, mechanical, electronic, and optical properties of three competing LiInP2S6 polymorphs: the monoclinic C2/c phase and the trigonal P-31c phase in both in-layer and in-gap configurations. Our results reveal a pressure-induced structural phase transition from the monoclinic ground-state C2/c phase to a trigonal P-31c in-layer phase at ~0.38 GPa, driven by enhanced interlayer coupling and anisotropic
lattice compression. In contrast, the trigonal P-31c in-gap phase remains energetically unfavorable due to its stronger interlayer ionic interactions and reduced compressibility. All phases remain mechanically stable under compression (0-26 GPa) and exhibit enhanced mechanical
rigidity, elastic wave velocities, and Debye temperatures with increasing pressure. Remarkably, the electronic and optical properties within each phase remain highly robust under pressure, with only moderate changes in the band gap and optical absorption edge (UV-Visible range) under pressure; however, substantial modifications emerge across the pressure-induced structural phase transition. These findings establish LiInP2S6 as a pressure sensitive ionic-vdW material in which subtle changes in interlayer interactions govern structural stability and functional properties.
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Superconductivity
Microscopic Investigation of the Superconducting State in CuCo$_{2}$S$_{4}$: Evidence for an Intermediate-Coupling Fully Gapped Superconductor
K. Panda, A. Bhattacharyya, Liang-Wen Ji et al. (+4 more)
The thiospinel compound CuCo$_2$S$_4$ provides an attractive platform for exploring superconductivity in transition-metal chalcogenide spinels. Here, we report the first microscopic investigation of the superconducting state in CuCo$_2$S$_4$ using
muon spin rotation and relaxation ($\mu$SR) measurements, complemented by magnetization and heat-capacity experiments. The temperature dependence of the superconducting depolarization rate obtained from transverse-field $\mu$SR measurements indicates a fully gapped superconducting order parameter. The extracted gap ratio $2\Delta(0)/(k_{\mathrm{B}}T_\mathrm{SC}) = 3.95(2)$ exceeds the BCS weak-coupling value of 3.53, placing CuCo$_2$S$_4$ in the intermediate electron-
phonon coupling regime. Zero-field $\mu$SR measurements were performed to probe possible time-reversal
symmetry breaking (TRSB) in the superconducting state. Within the experimental resolution, no additional spontaneous internal
magnetic fields are observed below $T_c$. However, due to the presence of a
ferromagnetic impurity phase and the associated fast-relaxing signal component, the sensitivity of the present measurements to weak spontaneous fields is reduced. Consequently, while no evidence for TRSB is detected, its existence cannot be definitively ruled out. Overall, our combined thermodynamic and $\mu$SR results demonstrate that CuCo$_2$S$_4$ exhibits a fully gapped superconducting state with intermediate coupling strength, consistent with conventional $s$-wave superconductivity in this cobalt-based thiospinel system.
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All authors K. Panda, A. Bhattacharyya, Liang-Wen Ji, Jing Li, R. Stewart, D. T. Adroja, Guang-Han Cao
Soft Condensed Matter
Beyond the Markovian limit: Exact solutions for active motion in a power-law viscoelastic bath
Mintu Karmakar, Jure Dobnikar and Ignacio Pagonabarraga
Active particles from bacteria to synthetic microswimmers often navigate viscoelastic media with complex relaxation dynamics. The classical active Brownian model that assumes instantaneous friction is clearly not applicable to describe such motility, while the non-Markovian processes combined with viscoelasticity are relatively unexplored. Here, we develop an analytical theory for an active particle in a power-law viscoelastic medium by solving coupled non-Markovian generalized Langevin equations for translational and rotational degrees of freedom. The viscoelastic
memory results in novel phenomena such as fractional short-time transport, enhanced long-time persistence, and de-correlation of the instantaneous force and the swimmer orientation. We demonstrate that the
memory kernel controls the anomalous scaling exponents, while the activity determines the crossover between sub-diffusive, ballistic and diffusive regimes. Our work provides a framework for theoretical description of biological and synthetic micro swimmers in complex biological and polymeric environments.
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Materials Science
Synthesis and Characterization of Atomically-Sharp Superconductor-Dielectric Interface
Nathan Sitaraman, Zhaslan Baraissov, Alexis Grassl et al. (+4 more)
Modification of
superconductor-dielectric interfaces is known to strongly impact coherence times of superconducting
quantum devices. This relationship is thought to arise from differences in the concentration of "two-level system" defects in the disordered dielectrics and
superconductor-dielectric interfaces; these defects couple to electromagnetic modes in the device and cause dissipation. Zirconium
oxide barrier layers on niobium have emerged as a promising pathway to low-loss interfaces in recent years, evidently due to the crystalline nature of these layers in comparison to the amorphous niobium native
oxide. We explain the unique ability of zirconium
oxide to form a crystalline layer, to maintain a sharp interface with metallic niobium, and to prevent niobium
oxide re-growth in terms of the chemical properties of ZrO$_2$ and the Nb-Zr-O ternary system. We demonstrate a new method to grow air-stable zirconium
oxide layers on niobium with a higher level of crystallinity and a sharper
oxide-metal interface than previously shown, and provide the first comprehensive microscopic analysis of ZrO$_2$ capping layer properties. These developments pave the way toward vital performance advances in superconducting
quantum devices.
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All authors Nathan Sitaraman, Zhaslan Baraissov, Alexis Grassl, Hongbin Yang, Daniel Tong, David Muller, Matthias Liepe
Materials Science
Predicting Defect States: A Quick Screening Protocol for Substitutional Point Defect Engineering
Hyosik Kang and Lukas Muechler
Point defects in crystalline materials play a central role in determining electronic, optical, and
magnetic properties. However, systematic exploration of defect configurations remains computationally expensive because large supercell calculations are required to approximate isolated defects under periodic boundary conditions. We present a unit-
cell-based tight-binding protocol that enables rapid pre-screening of substitutional defects. The protocol extracts Wannier tight-binding
Hamiltonians from small, fully relaxed unit
cells of the host and defect-like systems, replicates the host
Hamiltonian to construct a supercell model, and introduces the defect by modifying only the on-site energies at the substitution site while leaving the hopping parameters unchanged. We validate the protocol across three diverse systems: isostructural substitutional defects in transition-metal dichalcogenides (M$_\mathrm{Mo}$ MoS$_2$, M = Ce, Zr, Nb, Tc, and Ru),
symmetry-breaking carbon substitutions in hexagonal boron nitride (C$_\mathrm{B}$C$_\mathrm{N}$ h-BN), and nitrogen-vacancy (NV$^-$) centers in diamond. These case studies span two-dimensional and three-dimensional hosts, simple substitutions, and substitution-vacancy complexes. In all cases, the protocol successfully captures the number of in-gap states, their degeneracies, and their shallow or deep character relative to host band edges, despite some quantitative deviations in absolute energy positions. We further identify limitations for vacancies of highly electronegative atoms and for charge-state or
spin-polarization effects, both of which involve self-consistent charge redistribution not captured by the protocol.
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Quantum Gases · Cross-listed from nucl-th
Determining universal spectra from probability distributions
Charles Kacir, Joseph Moscoso, Amy Nicholson et al. (+2 more)
The probability distribution of a two-particle correlation function computed over background auxiliary field configurations, used to generate the interactions, has been shown to inform about the spectra of universal $n$-body clusters [1]. Here, we utilize two approaches, a numerical
lattice computation and an analytic expansion in the limit of large numbers of identical species, in an attempt to refine the initial predictions. Exploratory calculations in these directions are presented, and future investigations laid out.
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All authors Charles Kacir, Joseph Moscoso, Amy Nicholson, Thomas R. Richardson, Cade Rodgers
Strongly Correlated Electrons
Correlation-driven phonon renormalisation and the equation of state of $\gamma$-cerium
Yao Wei, Siyu Chen, Evgeny Plekhanov et al. (+3 more)
We investigate the thermodynamic properties of elemental cerium by assessing the crucial role of
phonon free energy within the framework of dynamical mean-field theory (DMFT). While conventional density functional theory (DFT) often fails to capture the intricate energy landscape of $f$-electron materials, our approach integrates many-body electronic correlations with
lattice dynamics to achieve a more rigorous description of the equation of state. We calculate the total energy as a function of the
lattice constant at both the DFT and DFT+DMFT levels, subsequently incorporating the vibrational free energy derived from the
phonon density of states. Our findings reveal that electronic renormalisation of the force constants significantly alters the
phonon spectra, particularly in the strongly correlated $\gamma$-phase. By applying these
phonon corrections to the energy profiles, we observe a substantial refinement in the predicted
equilibrium volumes. Using principal-component-based machine learning, we interpolate
phonon dispersions continuously from a finite set of first-principles calculations and compare them to experiment, finding significantly closer agreement compared to conventional DFT and DFT+U calculations that neglect dynamical many-body correlations. This study underlines the necessity of accounting for both electronic and vibrational
entropy when evaluating the phase stability and structural transitions of lanthanide systems under varying pressures and temperatures.
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All authors Yao Wei, Siyu Chen, Evgeny Plekhanov, Ivan Stich, Cedric Weber, Jan M. Tomczak
Strongly Correlated Electrons
Atomic-scale visualization of the toroidal order in a trimeric Dy(III) single-molecule toroic
Michael J. Jenkins, Madalynn G. Marshall, Jie Xing et al. (+8 more)
Single-molecule toroics (SMTs) offer a unique platform for next-generation
quantum devices utilizing head-to-tail
spin alignments in the compounds. Presence of toroidal moments in SMTs has been essentially based on magnetometry and ab initio calculations. Here, we report observation and probe of the toroidal moment in [Dy3(OH)(teaH2)3(paa)3]Cl(OMe) [teaH3: triethanolamine; paaH: N-(2-pyridyl)-acetoacetamide] from mapping of Dy3+
magnetic susceptibility tensors by polarized
neutron diffraction (PND).
Neutron diffraction under variable
magnetic fields demonstrates field-induced magnetization along the c-axis with toroidal moments anti-parallelly stacked, providing definite proof of the toroidal moment. Magnetometry studies confirm the toroidal ground state. For the first time, the combined use of PND, variable-field
neutron diffraction, ab initio calculations, and magnetometry is introduced as a robust and quantitative methodology to probe molecular-scale toroidal magnetism. This integrated approach overcomes limitations of earlier indirect methods, establishes a
benchmark framework for investigating SMTs, and provides valuable insights for the design of molecular
quantum materials.
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All authors Michael J. Jenkins, Madalynn G. Marshall, Jie Xing, Muthu Satheeshkumar, Brandon D. Watson-Sanders, Xiaoping Wang, Christina M. Hoffmann, Gopalan Rajaraman, Huibo Cao, Rongying Jin, Zi-Ling Xue
Disordered Systems and Neural Networks
Jittery Quantum Boomerang Effect
Pedro Dornelas and Gerson J. Ferreira
We study the dynamics of a
spin-polarized wave packet in a disordered Rashba two-dimensional electron gas and identify a jittery
quantum boomerang effect in which longitudinal and transverse motion return to the origin through fundamentally distinct mechanisms. Starting from an initial state with finite momentum along $x$ and
spin polarized along $z$, we calculate the time
evolution by combining a Chebyshev expansion of the time-
evolution operator with a disorder ensemble average. In the weak-scattering regime, equations of motion derived from the
quantum kinetic equation reproduce the numerical trends and show that
impurity scattering acts as a viscous damping mechanism that suppresses the transient Zitterbewegung and drives the transverse displacement back to $y=0$ at long times. In contrast, the longitudinal dynamics show a Drude-like saturation at weak disorder. These results are consistent with the vanishing intrinsic
spin Hall conductivity in the disordered Rashba model and with experimental observations of a transient intrinsic
spin Hall effect in the time-domain. As disorder increases, the longitudinal dynamics evolve to a partial return toward the origin, which signals a transition from weak antilocalization to Anderson localization in 2D.
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Materials Science
Dominant in-plane anomalous Hall effect in a monoclinic room-temperature ferromagnet
Guoxin Zheng, Arjyama Bordoloi, Mingjun Fan et al. (+6 more)
Ferromagnetic metals are characterized by enhanced dissipationless transverse transport responses via the anomalous
Hall effect, offering a route towards
magnetic sensing and spintronic readout functionalities. In most ferromagnets, the anomalous
Hall current is constrained to lie in the plane perpendicular to the magnetization (or applied
magnetic field). Recently, it has been recognized that selected
symmetries can also permit a
Hall response in a traditionally forbidden configuration, where the
Hall current lies in the same plane as the magnetization, realizing an in-plane anomalous
Hall effect. Reported realizations of this effect, however, are typically much weaker than the conventional
Hall response in the same material. Here, through engineering specific crystallographic mirror
symmetry-breaking, we realize a strongly enhanced in-plane anomalous
Hall response in monoclinic Cr3Te4 with room-temperature ferromagnetism. Remarkably, the in-plane anomalous
Hall signal exceeds the out-of-plane response by a factor of five, with which we demonstrate a unique in-plane field and current sensing functionality. Combined with density functional theory calculations, our results establish low-crystalline-
symmetry ferromagnets with near-
Fermi-level Weyl points as a practical platform for
symmetry-engineered
Hall responses, and point to a route towards room-temperature, geometry-flexible sensing devices.
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All authors Guoxin Zheng, Arjyama Bordoloi, Mingjun Fan, Shunsuke Kitou, Hiraku Saito, Taro Nakajima, Sobhit Singh, Takashi Kurumaji, Linda Ye
Statistical Mechanics · Cross-listed from quant-ph
Scaling law of asymptotic freedom in collective charging of quantum batteries
Gentaro Watanabe, Chunlin Chen and B. Prasanna Venkatesh
We establish a universal scaling law for collective charging of
quantum batteries, independent of microscopic details. We prove that the ergotropy-to-energy ratio approaches unity at least as fast as $\sim N^{-1}$ with the number of batteries $N$, implying generic asymptotic freedom. We further show how the universal $1/N$ scaling can be overcome: when the battery state becomes asymptotically pure, the convergence can be substantially faster, including $\sim N^{-b}$ with $b>1$ and even exponential scaling in $N^2$. Rigorous finite-$N$ upper and lower bounds on the ergotropy-to-energy ratio are further derived, providing nonasymptotic guarantees for the universal $1/N$ scaling.
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