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At least 235 records · Page 13

Quantum gravity phenomenology at the dawn of the multi-messenger era—A review

The exploration of the universe has recently entered a new era thanks to the multi-messenger paradigm, characterized by a continuous increase in the quantity and quality of experimental data that is obtained by the detection of the various cosmic messengers (photons, neutrinos, cosmic rays and gravitational waves) from numerous origins. They give us information about their sources in the universe and the properties of the intergalactic medium. Moreover, multi-messenger astronomy opens up the possibility to search for phenomenological signatures of quantum gravity. On the one hand, the most energetic events allow us to test our physical theories at energy regimes which are not directly accessible in accelerators; on the other hand, tiny effects in the propagation of very high energy particles could be amplified by cosmological distances. After decades of merely theoretical investigations, the possibility of obtaining phenomenological indications of Planck-scale effects is a revolutionary step in the quest for a quantum theory of gravity, but it requires cooperation between different communities of physicists (both theoretical and experimental). This review, prepared within the COST Action CA18108 “Quantum gravity phenomenology in the multi-messenger approach”, is aimed at promoting this cooperation by giving a state-of-the art account of the interdisciplinary expertise that is needed in the effective search of quantum gravity footprints in the production, propagation and detection of cosmic messengers.

79 ASTRONOMY AND ASTROPHYSICS↗

A survey of techniques for optimizing transformer inference

Recent years have seen a phenomenal rise in the performance and applications of transformer neural networks. The family of transformer networks, including Bidirectional Encoder Representations from Transformer (BERT), Generative Pretrained Transformer (GPT) and Vision Transformer (ViT), have shown their effectiveness across Natural Language Processing (NLP) and Computer Vision (CV) domains. Transformer-based networks such as ChatGPT have impacted the lives of common men. However, the quest for high predictive performance has led to an exponential increase in transformers' memory and compute footprint. Researchers have proposed techniques to optimize transformer inference at all levels of abstraction. Further, this paper presents a comprehensive survey of techniques for optimizing the inference phase of transformer networks. We survey techniques such as knowledge distillation, pruning, quantization, neural architecture search and lightweight network design at the algorithmic level. We further review hardware-level optimization techniques and the design of novel hardware accelerators for transformers. We summarize the quantitative results on the number of parameters/FLOPs and the accuracy of several models/techniques to showcase the tradeoff exercised by them. We also outline future directions in this rapidly evolving field of research. We believe that this survey will educate both novice and seasoned researchers and also spark a plethora of research efforts in this field.

97 MATHEMATICS AND COMPUTING↗

Scripting a new dialogue between diazotrophs and crops

Diazotrophs are bacteria and archaea that can reduce atmospheric dinitrogen (N 2 ) into ammonium. Plant–diazotroph interactions have been explored for over a century as a nitrogen (N) source for crops to improve agricultural productivity and sustainability. This scientific quest has generated much information about the molecular mechanisms underlying the function, assembly, and regulation of nitrogenase, ammonium assimilation, and plant–diazotroph interactions. This review presents various approaches to manipulating N fixation activity, ammonium release by diazotrophs, and plant–diazotroph interactions. Here, we discuss the research avenues explored in this area, propose potential future routes, emphasizing engineering at the metabolic level via biorthogonal signaling, and conclude by highlighting the importance of biocontrol measures and public acceptance.

59 BASIC BIOLOGICAL SCIENCES↗

Photoinduced Anion Segregation in Mixed Halide Perovskites

Alloyed lead halide perovskites have taken a dominant role in the quest for third-generation solar cells. This is due to optimal light-harvesting properties, which can be tuned across the visible spectrum by mixing halide (X = Cl – , Br – , and I – ) anions and A + cations (A + = FA + , MA + , and Cs + ). Durability issues related to ion movement within the perovskite lattice, however, impede large-scale commercialization. Uniformly mixed halide perovskites [e.g., APb(I 1– x Br x ) 3 ] reversibly segregate into narrow bandgap I-rich and wide bandgap Br-rich domains during continuous visible illumination. Subsequent I-rich domains reduce local open circuit voltages and decrease mixed halide perovskite solar cell power conversion efficiencies. In this review, we assess the known effects of halide segregation on the structural and optical properties of mixed halide materials, discuss ongoing research to suppress the phenomenon, and finally provide a mechanistic overview of its underlying origins.

36 MATERIALS SCIENCE↗

Entangled Photon Spectroscopy

Conspectus The enhanced interest in quantum-related phenomena has provided new opportunities for chemists to push the limits of detection and analysis of chemical processes. As some have called this the second quantum revolution, a time has come to apply the rules learned from previous research in quantum phenomena toward new methods and technologies important to chemists. While there has been great interest recently in quantum information science (QIS), the quest to understand how nonclassical states of light interact with matter has been ongoing for more than two decades. Our entry into this field started around this time with the use of materials to produce nonclassical states of light. Here, the process of multiphoton absorption led to photon-number squeezed states of light, where the photon statistics are sub-Poissonian. In addition to the great interest in generating squeezed states of light, there was also interest in the formation of entangled states of light. While much of the effort is still in foundational physics, there are numerous new avenues as to how quantum entanglement can be applied to spectroscopy, imaging, and sensing. These opportunities could have a large impact on the chemical community for a broad spectrum of applications. Here, we discuss the use of entangled (or quantum) light for spectroscopy as well as applications in microscopy and interferometry. The potential benefits of the use of quantum light are discussed in detail. From the first experiments in porphyrin dendrimer systems by Dr. Dong-Ik Lee in our group to the measurements of the entangled two photon absorption cross sections of biological systems such as flavoproteins, the usefulness of entangled light for spectroscopy has been illustrated. These early measurements led the way to more advanced measurements of the unique characteristics of both entangled light and the entangled photon absorption cross-section, which provides new control knobs for manipulating excited states in molecules. The first reports of fluorescence-induced entangled processes were in organic chromophores where the entangled photon cross-section was measured. These results would later have widespread impact in applications such as entangled two-photon microscopy. From our design, construction and implementation of a quantum entangled photon excited microscope, important imaging capabilities were achieved at an unprecedented low excitation intensity of 10 7 photons/s, which is 6 orders of magnitude lower than the excitation level for the classical two-photon image. New reports have also illustrated an advantage of nonclassical light in Raman imaging as well. From a standpoint of more precise measurements, the use of entangled photons in quantum interferometry may offer new opportunities for chemistry research. Experiments that combine molecular spectroscopy and quantum interferometry, by utilizing the correlations of entangled photons in a Hong–Ou–Mandel (HOM) interferometer, have been carried out. The initial experiment showed that the HOM signal is sensitive to the presence of a resonant organic sample placed in one arm of the interferometer. In addition, parameters such as the dephasing time have been obtained with the opportunity for even more advanced phenomenology in the future.

36 MATERIALS SCIENCE↗

Access to Ru(IV)–Ru(V) and Ru(V)–Ru(VI) Redox in Layered Li 7 RuO 6 via Intercalation Reactions

The push to increase the capacity of oxides to store charge as Li-ion battery cathodes has challenged our knowledge of the boundaries of redox chemistry and its relationship to the atomic structure. Layered phases with Ru are now classical models in the quest for refined theories. Current understanding dictates that the highest formal state that can be reached in them is Ru(V), after which activity centered at the oxide anions is triggered. Yet, this picture is challenged by this investigation of the intercalation chemistry of Li 7 RuO 6 . Coupling structural and spectroscopic analysis with computational simulations, we show that Li 7 RuO 6 is capable of highly reversible lithiation to Li 8 RuO 6 through the Ru(V)/Ru(IV) redox couple. Further, Li 7 RuO 6 can also undergo anodic Li deintercalation, showing clear evidence of the reversible formation of Ru(VI) with octahedral coordination, an unusual state that is not accessible in other layered oxides. The results highlight the versatility of the Ru-O bond to undergo distinct redox transitions depending on the specific layered arrangements. They enrich our understanding of redox chemistry in solids while underscoring the need for descriptions of charge compensation that reflect the nuance of covalent interactions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Mitigating Calendar Aging in Si-NMC Batteries with Advanced Dual-Salt Glyme Electrolytes

In addressing the critical challenge of calendar aging in silicon (Si)-based lithium-ion batteries, this study introduces a groundbreaking strategy utilizing glyme-type dual-salt electrolytes (lithium bis(trifluoromethanesulfonyl)imide [LiTFSI] and lithium difluoro(oxalato)borate [LiDFOB]). These electrolytes are demonstrated to significantly mitigate parasitic reactions and capacity loss in Si-NMC (lithium nickel manganese cobalt oxide) full cells, especially when compared with traditional carbonate-based electrolytes. Further, our exhaustive mechanistic analysis reveals that such electrolytes not only preserve the integrity of the Si anode but also improve the cathode/electrolyte interphases (CEI) through the formation of a conformal coating on the high-voltage cathode surface. This dual-salt approach, enhanced by the addition of a phosphate additive, effectively decelerates calendar aging, marking a substantial advance in the quest for durable and reliable Si-based energy storage technologies. The findings underscore the vital role of electrolyte composition in extending the calendar life of Si batteries, offering an alternative avenue toward maximizing the performance and longevity of next-generation Li-Si batteries.

36 MATERIALS SCIENCE↗

Unconventional Thermoelectric Materials for Energy Harvesting and Sensing Applications

Heat is an abundant but often wasted source of energy. Thus, harvesting just a portion of this tremendous amount of energy holds significant promise for a more sustainable society. While traditional solid-state inorganic semiconductors have dominated the research stage on thermal-to-electrical energy conversion, carbon-based semiconductors have recently attracted a great deal of attention as potential thermoelectric materials for low-temperature energy harvesting, primarily driven by the high abundance of their atomic elements, ease of processing/manufacturing, and intrinsically low thermal conductivity. This quest for new materials has resulted in the discovery of several new kinds of thermoelectric materials and concepts capable of converting a heat flux into an electrical current by means of various types of particles transporting the electric charge: (i) electrons, (ii) ions, and (iii) redox molecules. This has contributed to expanding the applications envisaged for thermoelectric materials far beyond simple conversion of heat into electricity. This is the motivation behind this review. This work is divided in three sections. In the first section, we present the basic principle of the thermoelectric effects when the particles transporting the electric charge are electrons, ions, and redox molecules and describe the conceptual differences between the three thermodiffusion phenomena. In the second section, we review the efforts made on developing devices exploiting these three effects and give a thorough understanding of what limits their performance. In the third section, we review the state-of-the-art thermoelectric materials investigated so far and provide a comprehensive understanding of what limits charge and energy transport in each of these classes of materials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Facile Solvent-Free Synthesis of Manganese Nickel-Layered Double Hydroxide for Sustainable Water-Splitting Applications

The quest for efficient and sustainable water-splitting electrocatalysts has led to the development of a novel bifunctional material, manganese nickel-layered double hydroxide (MnNi-LDH), which demonstrates promising performance for both the oxygen evolution reaction (OER) and hydrogen evolution reaction (HER). Although manganese-based materials are less explored than other transition metals, they offer significant potential owing to their widespread availability, affordability, and customizable electronic characteristics. MnNi-LDH exhibits a nanosheet morphology and a layered structure, which collectively provide numerous accessible active sites and facilitate efficient charge transfer and mass transport. Characterization using X-ray diffraction (XRD), X-ray photoelectron spectroscopy (XPS), Fourier-transform infrared spectroscopy (FTIR), scanning electron microscopy, and transmission electron microscopy reveals the structural and compositional properties of MnNi-LDH. The oxidation states of Mn and Ni, as determined by XPS, play a crucial role in improving the catalytic activity. Notably, MnNi-LDH demonstrates low overpotentials of 187 mV for OER and 225 mV for HER at 10 mA/cm 2 current density comparable to conventional catalysts. Long-term stability tests show minimal degradation in cell performance over 50 h, with a current density drop of only 0.6153% per hour for the OER and 0.37% per hour for the HER. Further, these findings highlight the potential of MnNi-LDH as a promising and environmentally friendly bifunctional electrocatalyst for water splitting, contributing to the advancement of renewable energy sources.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Robust Tensor Hypercontraction of the Particle–Particle Ladder Term in Equation-of-Motion Coupled Cluster Theory

One method of representing a high-rank tensor as a (hyper-)product of lower-rank tensors is the tensor hypercontraction (THC) method of Hohenstein et al. This strategy has been found to be useful for reducing the polynomial scaling of coupled-cluster methods by representation of a four-dimensional tensor of electron-repulsion integrals in terms of five two-dimensional matrices. Pierce et al. have already shown that the application of a robust form of THC to the particle–particle ladder (PPL) term reduces the cost of this term in couple-cluster singles and doubles (CCSD) from O(N 6 ) to O(N 5 ) with negligible errors in energy with respect to the density-fitted variant. In this work, we have implemented the least-squares variant of THC (LS-THC) which does not require a nonlinear tensor factorization, including the robust form (R-LS-THC), for the calculation of the excitation and electron attachment energies using equation-of-motion coupled cluster methods EOMEE-CCSD and EOMEA-CCSD, respectively. We have benchmarked the effect of the R-LS-THC-PPL approximation on excitation energies using the comprehensive QUEST database and the accuracy of electron attachment energies using the NAB22 database. Here, we find that errors on the order of 1 meV are achievable with a reduction in total calculation time of approximately 5x.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Machine Learning a Simple Interpretable Short-Range Potential for Silica

A wide array of models, spanning from computationally expensive ab initio methods to a spectrum of force-field approaches, have been developed and employed to probe silica polymorphs and understand growth processes and atomic-level dynamical transitions in silica. However, the quest for a model capable of making accurate predictions with high computational efficiency for various silica polymorphs is still ongoing. Recent developments in short-range machine-learned models, such as GAP and NNPScan, have shown promise in providing reasonable descriptions of silica, but their computational cost remains high compared to force fields such as BKS which are based on simple interpretable functional forms. Here, in this study, we build on the recent success of our reinforcement learning (RL) workflow to derive a new set of optimal parameters for a promising short-range BKS-based model proposed by Soules. We use RL to navigate the eight-dimensional parameter space of the Soules potential using an experimental training data set that includes both local and global structural features from approximately 21 experimentally realized silica polymorphs, including high density phases and porous zeolites. We compare the performance of our machine-learned ML-Soules model with other high quality models including our recent machine-learned parametrization of BKS (ML-BKS), a machine-learned potential (GAP), as well as predictions of ab initio calculations with the highly fidelity SCAN functional. The ML-Soules accurately captures the relative energetic ordering of various polymorphs as well as their structural features at a significantly reduced computational expense. The ML-Soules model also reasonably captures the structure, density, and elastic constants of quartz, as well as metastable silica polymorphs. We further discuss the limitations of the Soules functional form and propose potential enhancements, including the incorporation of additional three-body terms and/or the utilization of different short-ranged functional forms to achieve greater accuracy for both global and local features in the modeling of silica while retaining low computational cost.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Flash Formation of I-Rich Clusters during Multistage Halide Segregation Studied in MAPbI 1.5 Br 1.5

Wide band gap mixed halide perovskites such as MAPb(I 1–x Br x ) 3 sparked great research interest because of their outstanding optoelectronic properties, ease of fabrication, and band gap tunability. Their application thus far is however limited by light-induced halide segregation in which microscopic clusters with a high iodide content are formed and act as recombination centers. The key mechanism(s) underlying this halide segregation process are still debated. Here, we present a study on the photoluminescence evolution in MAPb(I 1.5 Br 1.5 ) perovskites with varying microstructures under constant illumination at room temperature and at elevated temperature. Our findings reveal a more complicated picture of the segregation mechanism occurring in three stages instead of two as commonly reported. The process starts with a flash formation of I-rich domains. Following is a rapid blue shift before the gradual and typically observed red shift occurs. The evolution of the three stages is fully reversible in the dark and is also present at elevated temperatures (50 °C). We explain the existence of multiple stages during light-induced halide segregation by natural compositional fluctuations of the halides and the formation of halide clusters with a dynamically changing distribution in I–Br content. The variation in the I–Br ratio depends on the grain size and film heterogeneity. Overall, these findings add further details in the quest of unraveling the underlying segregation mechanism(s), which need to be identified to stabilize halides in wide band gap perovskites.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Computational Optimization of Room Temperature Usable Capacity for Hydrogen Storage in MFU-4-Type Metal–Organic Frameworks via Pairwise Metal Substitutions

The efficient storage of hydrogen is a critical challenge in the quest for sustainable energy solutions. Current adsorbent-based methods achieve satisfactory storage densities predominantly under cryogenic temperatures and/or high pressures, which imposes problems with cost-efficient and safe implementation of this technology. Materials that can bind hydrogen gas reversibly at ambient temperatures and more moderate pressures could play a pivotal role in enabling hydrogen-powered technologies. In this study, we use reliable computational modeling to investigate two synthetically feasible paths for tuning the enthalpy of H2 binding in MFU-4-type metal–organic frameworks (MOFs), aiming to maximize usable capacity. This study examines MIM4 IICl3(bta)6 (bta– = benzotriazolate) Kuratowski-type clusters as a model for strong binding sites in MFU-4l frameworks. We systematically evaluate the impact of separately tuning the central MII metal ion (which plays a structural role) and the peripheral MI metal ion (which binds the substrate) on the energetics of H2 binding. Our computational study reveals that H2 binding at an MI site mostly follows the trend AgI < CuI < NiI < CoI < AuI while a larger central MII site generally weakens the H2 binding at a MI site. Importantly, we have identified three new combinations of MI and MII to achieve high fractional usable capacities of the total H2 adsorbed under a pressure swing from 5 to 100 bar at room temperature. Additionally, we examine the nature of the binding interaction between the peripheral metal atom and the hydrogen molecule. While charge transfer predominantly induces this interaction, for several atom combinations, a change in the polarization (associated with variations in the ionic radius of the MI binding atom) is another important factor for adjusting the strength of the interaction. We suggest that the proposed compositions of Kuratowski-type clusters are highly desirable synthetic targets for future laboratory study.

Tkachenko, Nikolay V↗

Tip-Enhanced Multipolar Raman Scattering

We record nanoscale chemical images of thiobenzonitrile (TBN)-functionalized plasmonic gold nanocubes via tip-enhanced Raman spectroscopy (TERS). The spatially averaged optical response is dominated by conventional (dipolar) TERS scattering from TBN but also contains weaker spectral signatures in the 1225-1500 cm-1 region. The weak optical signatures dominate some of the recorded single pixel spectra. We can uniquely assign these Raman forbidden transitions to multipolar Raman scattering, which implicates spatially varying enhanced electric field gradients at plasmonic tip-sample nanojunctions. Specifically, we can assign observations of tip-enhanced electric dipole-magnetic dipole as well as electric dipole-electric quadrupole driven transitions. These signals need to be understood and accounted for in the interpretation of TERS spectral images, particularly in on-going quests aimed at chemical reaction mapping via TERS.

Wang, Chih-Feng↗

Toward the Practical Use of Cobalt-Free Lithium-Ion Batteries by an Advanced Ether-Based Electrolyte

The criticality of cobalt (Co) has been motivating the quest for Co-free positive electrode materials for building lithium (Li)-ion batteries. However, the Co-free positive electrode materials usually suffer from relatively fast capacity decay when coupled with conventional LiPF 6 -organocarbonates electrolytes. To address this issue, a 1,2-dimethoxyethane (DME) based localized high concentration electrolyte (LHCE) was developed and evaluated in a Co-free Li-ion cell chemistry (Graphite||LiNi 0.96 Mg 0.02 Ti 0.02 O 2 ). Extraordinary capacity retentions were achieved with the LHCE in coin cells (95.3%), single layer pouch cells (79.4%) and high capacity loading double layer pouch cells (70.9 %) after being operated within the voltage range of 2.5-4.4 V for 500 charge/discharge cycles. The capacity retentions of counterpart cells using LiPF6 based conventional electrolyte only reached 61.1%, 57.2% and 59.8%, respectively. Mechanistic studies reveal that the superior electrode/electrolyte interphases formed by the LHCE and the intrinsic chemical stability of the LHCE account for the excellent electrochemical performance in the Co-free Li-ion cells.

25 ENERGY STORAGE↗

Programmable Photoluminescence via Intrinsic and DNA-Fluorophore Association in a Mixed Cluster Heterometallic MOF

In this report a rapid and facile design strategy to create a highly complex optical tag with programmable, multimodal photoluminescent properties is described. This was achieved via intrinsic and DNA-fluorophore hidden signatures. As a first covert feature of the tag, an intricate novel heterometallic near-infrared (NIR)-emitting mesoporous metal–organic framework (MOF) was designed and synthesized. The material is constructed from two chemically distinct, homometallic hexanuclear clusters based on Nd and Yb. Uniquely, the Nd-based cluster is observed here for the first time in a MOF and consists of two staggered Nd μ3-oxo trimers. To generate controlled, multimodal, and tailorable emission with difficult to counterfeit features, the NIR-emissive MOF was post-synthetically modified via a fluorescent DNA oligo labeling design strategy. The surface attachment of several distinct fluorophores, including the simultaneous attachment of up to three distinct fluorescently labeled oligos was achieved, with excitation and emission properties across the visible spectrum (480-800 nm). The DNA inclusion as a secondary covert element in the tag was demonstrated via the detection of SYBR Gold dye association. Importantly, the approach implemented here serves as a rapid and tailorable way to encrypt distinct information in a facile and modular fashion and provides an innovative technology in the quest toward complex optical tags.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Intrinsically Conductive {pi}‑d Conjugated Layers with Co–N4 Active Sites for Efficient Nitrate Electrocatalysis and Zinc-Nitrate Batteries

Electrochemical synthesis of ammonia from nitrate has been extensively investigated as a potential alternative to the energy-intensive Haber-Bosch process. This approach not only operates under ambient conditions but also simultaneously removes nitrate contaminants while producing ammonia as a value-added product. However, the ongoing quest lies in designing an efficient electrocatalyst that achieves a high ammonia yield rate, high selectivity, and long-term stability. Herein, we report the outstanding performance of a Co–N4 coordinated π-d layered Co3(HITP)2 (HITP = 2,3,6,7,10,11-hexaiminotriphenylene) in nitrate electrocatalysis. The unique combination of abundant Co–N4 active sites and superior electrical conductivity enables significant electrocatalytic activity, delivering a maximum ammonia yield rate of 56.8 mg cm–2 h–1 at −0.8 V vs RHE and a Faradaic efficiency of ∼91% at −0.4 V vs RHE. Mechanistic analysis reveals that alkaline conditions accelerate water dissociation to generate adsorbed hydrogen intermediates (H*), which are utilized by Co–N4 sites to drive the stepwise hydrogenation of nitrate to ammonia while suppressing competing hydrogen evolution reaction (HER) pathways. Furthermore, integration of this catalyst into a zinc-nitrate battery resulted in a maximum power density of 5.3 mW cm–2 and an open-circuit potential of ∼1.45 V. These results highlight the potential of π-d conjugated Co–N4 materials as an efficient catalyst for both environmental remediation and energy conversion.

Namvar, shahrirar↗

Enhancing the Optically Detected Magnetic Resonance Signal of Organic Molecular Qubits

In quantum information science and sensing, electron spins are often purified into a specific polarization through an optical-spin interface, a process known as optically detected magnetic resonance (ODMR). Diamond-NV centers and transition metals are both excellent platforms for these so-called color centers, while metal-free molecular analogues are also gaining popularity for their extended polarization lifetimes, milder environmental impacts, and reduced costs. In our earlier attempt at designing such organic high-spin π-diradicals, we proposed to spin-polarize by shelving triplet M S = ±1 populations as singlets. This was recently verified by experiments albeit with low ODMR contrasts of <1% at temperatures above 5 K. In this work, we propose to improve the ODMR signal by moving singlet populations back into the triplet M S = 0 sublevel, designing a true carbon-based molecular analogue to the NV center. Our proposal is based upon transition-orbital and group-theoretical analyses of beyond-nearest-neighbor spin–orbit couplings, which are further confirmed by ab initio calculations of a realistic trityl-based radical dimer. Microkinetic analyses point toward high ODMR contrasts of around 30% under experimentally feasible conditions, a stark improvement from previous works. Finally, in our quest toward ground-state optically addressable molecular spin qubits, we exemplify how our symmetry-based design avoids Zeeman-induced singlet–triplet mixings, setting the scene for realizing electron spin qubit gates.

Group theory↗