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At least 1,387 records · Page 77
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Finite and infinite matrix product states for Gutzwiller projected mean-field wave functions

Matrix product states (MPS) and “dressed” ground states of quadratic mean fields (e.g., Gutzwiller projected Slater determinants) are both important classes of variational wave functions. This latter class has played important roles in understanding superconductivity and quantum spin liquids. We present a method to obtain both the finite and infinite MPS (iMPS) representation of the ground state of an arbitrary fermionic quadratic mean-field Hamiltonian (which in the simplest case is a Slater determinant and in the most general case is a Pfaffian). We also show how to represent products of such states (e.g., determinants times Pfaffians). From this representation one can project to single occupancy and evaluate the entanglement spectra after Gutzwiller projection. We then obtain the MPS and iMPS representation of Gutzwiller projected mean-field states that arise from the variational slave-fermion approach to the S = 1 bilinear-biquadratic quantum spin chain. To accomplish this, we develop an approach to orthogonalize degenerate iMPS to find all the states in the degenerate ground-state manifold. We find the energies of the MPS and iMPS states match the variational energies closely, indicating the method is accurate and there is minimal loss due to truncation error. We then present an exploration of the entanglement spectra of projected slave-fermion states, exploring their qualitative features and finding good qualitative agreement with the respective exact ground-state spectra found from density matrix renormalization group.

1-dimensional spin chains↗
DOE OSTI2021

Few-nucleon matrix elements in pionless effective field theory in a finite volume

Here, pionless effective field theory in a finite volume (FVEFT $\notπ$ ) is investigated as a framework for the analysis of multinucleon spectra and matrix elements calculated in lattice QCD (LQCD). By combining FVEFT π with the stochastic variational method, the spectra of nuclei with atomic number A ∈ {2, 3} are matched to existing finite-volume LQCD calculations at heavier-than-physical quark masses corresponding to a pion mass m $\notπ$ = 806 MeV , thereby enabling infinite-volume binding energies to be determined using infinite-volume variational calculations. Based on the variational wave functions that are constructed in this approach, the finite-volume matrix elements of various local operators are computed in FVEFT $\notπ$ and matched to LQCD calculations of the corresponding QCD operators in the same volume, thereby determining the relevant one- and two-body effective field theory counterterms and enabling an extrapolation of the LQCD matrix elements to infinite volume. As examples, the scalar, tensor, and axial matrix elements are considered, in addition to the magnetic moments and the isovector longitudinal momentum fraction.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Detecting dark matter substructures on small scales with fast radio bursts

The matter power spectrum is only weakly constrained on subgalactic scales, while physics beyond the Standard Model can leave unique imprints, especially on sub-parsec scales. We propose measuring the arrival-time difference of fast radio bursts (FRBs) along two adjacent sightlines as a new probe to dark matter substructures on scales down to ∼ 1 AU . We discuss two observational scenarios in which it may be possible to place interesting constraints on such models through the monitoring of repeating FRB sources: (i) By sending radio receivers to space to form a baseline of tens of AU or more and measuring the temporal variation of the arrival-time difference between receivers. (ii) By measuring the temporal variation of the arrival-time difference between two lensed images of one strongly lensed repeater. In both scenarios, obtaining interesting constraints requires correlating the voltage time series to measure the radio signal arrival time to sub-nanosecond precision. We find that two radio dishes separated by 20 AU may be sensitive to the enhancement of small-scale structures at ∼ 10 - 8 M ⊙ masses in the QCD axion dark matter scenario, or from an early epoch of matter domination with a reheating temperature up to 60 MeV. Other dark matter models, such as those composed of ∼ 10 - 13 M ⊙ primordial black holes produced during inflation, would also be probed by this method. We further show that a strong lensing situation of multiple images provides an equivalent ∼ 2000 AU ( σ v / 10 3 km s - 1 ) ( δ t / 10 yr ) baseline, for a typical velocity of dark matter substructures σ v and an observational time span δ . This is much more sensitive, but with the uncertainty that intervening decoherence from the interstellar medium may degrade the timing precision, and that spatial variation in the FRB emission spot may result in confounding signals. We show that the lensing magnifications of Type Ia supernovae constrain a similar quantity to such FRB timing, with present limits being equivalent to ruling out the same parameter space that would be probed by a 0.14 AU baseline.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Superhorizon isocurvature fluctuations relax tensions

Here, we present a new class of models that have potential to alleviate tensions present in the cosmological data today. We postulate about variation in the sound horizon scale on superhorizon scales, i.e., on scales that are larger than that of the present observable low-redshift Universe (≳ 1Gpc) while at the same time smaller than the largest scales probed by the cosmic microwave background (CMB) (≲ 10Gpc). In this scenario, CMB peaks are naturally smoothed as preferred by the Planck data, while at the same time the low-redshift baryon acoustic oscillation calibration is partially decoupled from the CMB. Taking superhorizon variations in baryon fraction as an example and using approximate modeling, we find improvement in the best fit Planck power spectrum model Δχ 2 ~ 6 for 1 extra degree of freedom with the relevant extension parameter 10 3 σ b = 2.10 ± 0.60, implying about 10% variations in baryon fraction across the Universe. At the same time, S 8 drops by about 1 sigma, easing tension with weak lensing surveys. We find that the combination of Planck 2018 data, eBOSS BAO data, and Riess et al. distance ladder Hubble parameter determination produce a satisfactory fit in our model if we allow for a phantom dark energy equation of state.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Heat transfer and transport property contrast effects on the compressible Rayleigh-Taylor instability

In extreme environments, the Rayleigh-Taylor instability (RTI) may occur under large variations in density and temperature and with fluid transport properties strongly dependent on temperature. Direct numerical simulations of the 3D fully compressible RTI are conducted, examining the idealized configuration of a hotter, less dense fluid pushing against a colder, denser fluid. Various temperature ratios and transport property configurations are explored to examine how heat conduction, large variations in transport properties, and sudden changes in transport properties can affect the evolution of the mixing layer. Nonuniform fluid expansion and contraction induced by heat transfer can significantly affect local density differences and overall instability growth, causing profile asymmetries about the initial interface for flow and mixing statistics. The departures from classical self-similar development of the instability along with misalignment between regions of mixing and regions of most intense turbulent activity caused by both heat transfer and transport property contrasts are examined. After sudden changes in fluid transport properties, which may occur as a result of rapid heating (e.g., in inertial confinement fusion), the flow quickly responds and begins to relax towards quasi-self-similar late-time evolution. For many dynamical quantities such as vorticity and dissipation, this late-time evolution resembles that of the configuration that already started with the final transport property magnitudes, suggesting that these quantities depend only on the transport properties and not on past flow history, provided that the density field distributions for the flows remain similar. On the other hand, the mixing evolution after the transport property change is unique, implying that both property magnitudes and previous history are impactful on the mixing. In conclusion, these simulations demonstrate how various temperature-related effects are extremely important to consider in compressible RTI flows with large temperature variations.

42 ENGINEERING↗
DOE OSTI2021

Nonmonotonic Energy Dependence of Net-Proton Number Fluctuations

Nonmonotonic variation with collision energy (√ s NN) of the moments of the net-baryon number distribution in heavy-ion collisions, related to the correlation length and the susceptibilities of the system, is suggested as a signature for the quantum chromodynamics critical point. Here, we report the first evidence of a nonmonotonic variation in the kurtosis times variance of the net-proton number (proxy for net-baryon number) distribution as a function of √ s NN with 3.1 σ significance for head-on (central) gold-on-gold (Au+Au) collisions measured solenoidal tracker at Relativistic Heavy Ion Collider. Data in noncentral Au+Au collisions and models of heavy-ion collisions without a critical point show a monotonic variation as a function of √ s NN.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Dynamic, symmetry-preserving, and hardware-adaptable circuits for quantum computing many-body states and correlators of the Anderson impurity model

We present a hardware-reconfigurable ansatz on N q -qubits for the variational preparation of many-body states of the Anderson impurity model (AIM) with N imp + N bath = N q /2 sites, which conserves total charge and spin z component within each variational search subspace. The many-body ground state of the AIM is determined as the minimum over all minima of O(N$^2_ q$) distinct charge-spin sectors. Hamiltonian expectation values are shown to require ω(N q ) < N meas. $\leqslant$ O(N imp N bath ) symmetry-preserving, parallelizable measurement circuits, each amenable to postselection. To obtain the one-particle impurity Green’s function we show how initial Krylov vectors can be computed via midcircuit measurement and how Lanczos iterations can be computed using the symmetry-preserving ansatz. For a single-impurity Anderson model with a number of bath sites increasing from one to seven, we show using numerical emulation that the ease of variational ground-state preparation is suggestive of linear scaling in circuit depth and subquartic scaling in optimizer complexity. We therefore expect that, combined with time-dependent methods for Green’s function computation, our ansatz provides a useful tool to account for electronic correlations on early fault-tolerant processors. Finally, with a view towards computing real materials properties of interest like magnetic susceptibilities and electron-hole propagators, we provide a straightforward method to compute many-body, time-dependent correlation functions using a combination of time evolution, midcircuit measurement-conditioned operations, and the Hadamard test.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Phase-field model of alloy solidification far from chemical equilibrium at the solid-liquid interface

We further develop a recently introduced phase-field model of far-from-equilibrium alloy solidification under additive manufacturing conditions [K. Ji et al., Phys. Rev. Lett. 130, 026203 (2023)]. This model utilizes enhanced solute diffusivity within the spatially diffuse interface region to quantitatively capture solute trapping with a larger interface width, thereby making simulations on experimentally relevant length and timescales computationally feasible. The main developments presented here include testing the robustness of different variational formulations, extending the model to concentrated alloys by incorporating solid and liquid free energies from thermodynamic databases, as illustrated for hypoeutectic Al-Ag alloys with CALPHAD, extending convergence tests as a function of interface width to 3D, and carrying out simulations in both 2D and 3D to examine existing theories of microstructure development. Our results indicate that the simplest variational formulation that interpolates the bulk free-energy density between its solid and liquid forms is the most robust. Remarkably, for hypoeutectic Al-Ag alloys, this formulation yields a high-velocity nonequilibrium phase diagram that is independent of interface width, thereby demonstrating that the framework of enhanced solute diffusivity can be nontrivially extended to concentrated alloys. Other variational formulations have restricted ranges of materials or processing parameters that can be reliably modeled. We use 2D simulations to construct high-velocity microstructure selection maps for dilute Al-Cu alloys. The results validate the important role of latent heat rejection at the interface and extend the limited predictions of linear stability analysis [A. Karma and A. Sarkissian, Phys. Rev. E 47, 513 (1993)] and sharp-interface 1D simulations to fully nonlinear regimes. Furthermore, 3D simulations, carried out using a computationally tractable axisymmetric cellular/dendritic interface shape, demonstrate a good convergence similar to that observed in 2D as a function of interface width. Full 3D simulations, in turn, reveal that the standard theory of absolute stability is a good predictor of the upper critical velocity beyond which steady-state growth becomes unstable, despite the different morphological manifestations of this instability in 2D and 3D.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Impact of Open Communication Networks on Load Frequency Control with Plug-In Electric Vehicles by Cyber-Physical Dynamic Co-Simulation

With the increasing electrification of the transportation sector to achieve the carbon neutrality objective, despite the challenges of charging electric vehicles (EV), there are also opportunities through smart charging EVs to improve system frequency stability; however, EV control technologies might require nontraditional communication support. This paper investigates the impacts of communication variations of EV on power system load frequency control through a cyber-physical dynamic system (CPDS) co-simulation. Here, the CPDS is built upon our previously developed transmission-and-distribution dynamic co-simulation model with the added communication variation functions (i.e., delay and packet loss). The case studies consider multiple communication variation scenarios when the system experiences an N-1 generation trip contingency. The scenarios include communication delays and packet loss using both homogeneous and heterogeneous assumptions. The outcomes of this work can help improve EV frequency regulation services and provide robust and effective tests for different load frequency control algorithms of the future power systems.

ADVANCED PROPULSION SYSTEMS,ENERGY PLANNING, POLIC↗
DOE OSTI2022

Scaling Superconducting Quantum Computers with Chiplet Architectures

Fixed-frequency transmon quantum computers (QCs) have advanced in coherence times, addressability, and gate fidelities. Unfortunately, these devices are restricted by the number of on-chip qubits, capping processing power and slowing progress toward fault-tolerance. Although emerging transmon devices feature over 100 qubits, building QCs large enough for meaningful demonstrations of quantum advantage requires overcoming many design challenges. For example, today’s transmon qubits suffer from significant variation due to limited precision in fabrication. As a result, barring significant improvements in current fabrication techniques, scaling QCs by building ever larger individual chips with more qubits is hampered by device variation. Severe device variation that degrades QC performance is referred to as a defect. Here, we focus on a specific defect known as a frequency collision. When transmon frequencies collide, their difference falls within a range that limits two-qubit gate fidelity. Frequency collisions occur with greater probability on larger QCs, causing collision-free yields to decline as the number of on-chip qubits increases. As a solution, we propose exploiting the higher yields associated with smaller QCs by integrating quantum chiplets within quantum multi-chip modules (MCMs). Yield, gate performance, and application-based analysis show the feasibility of QC scaling through modularity. Our results demonstrate that chiplet architectures, relative to monolithic designs, benefit from average yield improvements ranging from 9.6 – 92.6 × for ≲5 qubit machines. In addition, our simulations explore the design space of chiplet systems and discover configurations that demonstrate average two-qubit gate infidelity reductions that are at best 0.815 × their monolithic counterpart. Lastly, we observe that carefully-selected modular systems achieve fidelity improvements on a range of benchmark circuits.

quantum architecture↗
DOE OSTI2022

Hybrid Quantum-Classical Neural Networks

Deep learning is one of the most successful and far-reaching strategies used in machine learning today. However, the scale and utility of neural networks is still greatly limited by the current hardware used to train them. These concerns have become increasingly pressing as conventional computers are soon expected to approach the physical limitations that will slow their performance improvements in the near future. For these reasons, scientists have begun to explore alternative computing platforms, like quantum computers, for training neural networks. In recent years, variational quantum circuits have emerged as one of the most successful approaches to quantum deep learning on noisy intermediate scale quantum devices. We propose a hybrid quantum-classical neural network architecture where each neuron is a variational quantum circuit. We empirically analyze the performance of this hybrid neural network on a series of binary classification data sets using a simulated IBM universal quantum computer and a state-of-the-art IBM universal quantum computer. On the simulated hardware, we observe that the hybrid neural network achieves around 10% higher classification accuracy and 20% better minimization of the cost function than an individual variational quantum circuit. On the quantum hardware, we observe that each model only performs well when the qubit and gate count is sufficiently small.

Arthur, Davis↗
DOE OSTI2021

Microbial community composition controls carbon flux across litter types in early phase of litter decomposition

Leaf litter decomposition is a major carbon input to soil, making it a target for increasing soil carbon storage through microbiome engineering. We expand upon previous findings to show with multiple leaf litter types that microbial composition can drive variation in carbon flow from litter decomposition and specific microbial community features are associated with synonymous patterns of carbon flow among litter types. Although plant litter type selects for different decomposer communities, within a litter type, microbial composition drives variation in the quantity of dissolved organic carbon (DOC) measured at the end of the decomposition period. Bacterial richness was negatively correlated with DOC quantity, supporting our hypothesis that across multiple litter types there are common microbial traits linked to carbon flow patterns. Variation in DOC abundance (i.e. high versus low DOC) driven by microbial composition is tentatively due to differences in bacterial metabolism of labile compounds, rather than catabolism of non-labile substrates such as lignin. The temporal asynchrony of metabolic processes across litter types may be a substantial impediment to discovering more microbial features common to synonymous patterns of carbon flow among litters. Overall, our findings support the concept that carbon flow may be programmed by manipulating microbial community composition.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2022

Bacterial community dynamics explain carbon mineralization and assimilation in soils of different land-use history

Soil dwelling microorganisms are key players in the terrestrial carbon cycle, driving both the degradation and stabilization of soil organic matter. Bacterial community structure and function vary with respect to land use; yet the ecological drivers of this variation remain poorly described and difficult to predict. For this work, we conducted a multi-substrate DNA-stable isotope probing experiment across cropland, old-field, and forest habitats to link carbon mineralization dynamics with the dynamics of bacterial growth and carbon assimilation. We tracked the movement of 13 C derived from five distinct carbon sources as it was assimilated into bacterial DNA over time. We show that carbon mineralization, community composition, and carbon assimilation dynamics all differed with respect to land use. We also show that microbial community dynamics affect carbon assimilation dynamics and are associated with soil DNA content. Soil DNA yield is easy to measure and may be useful in predicting microbial community dynamics linked to soil carbon cycling. Soil dwelling microorganisms are key players in the terrestrial carbon cycle, driving both the degradation and stabilization of soil organic matter. Microbial communities vary with respect to land use, but we still have an incomplete understanding of how variation in community structure links to variation in community function. DNA stable isotope probing (DNA-SIP) is a high-resolution method that can identify specific microbial taxa that assimilate carbon in situ. We conducted a large-scale multi-substrate DNA-SIP experiment to explore differences in bacterial activity across land-use regimes. We show that microbial community dynamics vary with land use, that these dynamics are linked to soil carbon cycling, and that they are associated with easily measured soil properties.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2022

Interactively Assessing Disentanglement in GANs

Abstract Generative adversarial networks (GAN) have witnessed tremendous growth in recent years, demonstrating wide applicability in many domains. However, GANs remain notoriously difficult for people to interpret, particularly for modern GANs capable of generating photo‐realistic imagery. In this work we contribute a visual analytics approach for GAN interpretability, where we focus on the analysis and visualization of GAN disentanglement. Disentanglement is concerned with the ability to control content produced by a GAN along a small number of distinct, yet semantic, factors of variation. The goal of our approach is to shed insight on GAN disentanglement, above and beyond coarse summaries, instead permitting a deeper analysis of the data distribution modeled by a GAN. Our visualization allows one to assess a single factor of variation in terms of groupings and trends in the data distribution, where our analysis seeks to relate the learned representation space of GANs with attribute‐based semantic scoring of images produced by GANs. Through use‐cases, we show that our visualization is effective in assessing disentanglement, allowing one to quickly recognize a factor of variation and its overall quality. In addition, we show how our approach can highlight potential dataset biases learned by GANs.

Jeong, Sangwon↗
DOE OSTI2021

Invasion and rapid adaptation of guppies ( Poecilia reticulata ) across the Hawaiian Archipelago

How much does natural selection, as opposed to genetic drift, admixture, and gene flow, contribute to the evolution of invasive species following introduction to a new environment? Here we assess how evolution can shape biological invasions by examining population genomic variation in non-native guppies (Poecilia reticulata) introduced to the Hawaiian Islands approximately a century ago. By examining 18 invasive populations from four Hawaiian islands and four populations from the native range in northern South America, we reconstructed the history of introductions and evaluated population structure as well as the extent of ongoing gene flow across watersheds and among islands. Patterns of differentiation indicate that guppies have developed significant population structure, with little natural or human-mediated gene flow having occurred among populations following introduction. Demographic modeling and admixture graph analyses together suggest that guppies were initially introduced to O‘ahu and Maui and then translocated to Hawai‘i and Kaua‘i. We detected evidence for only one introduction event from the native range, implying that any adaptive evolution in introduced populations likely utilized the genetic variation present in the founding population. Environmental association tests accounting for population structure identified loci exhibiting signatures of adaptive variation related to predators and landscape characteristics but not nutrient regimes. When paired with high estimates of effective population sizes and detectable population structure, the presence of environment-associated loci supports the role of natural selection in shaping contemporary evolution of Hawaiian guppy populations. Our findings indicate that local adaptation may engender invasion success, particularly in species with life histories that facilitate rapid evolution. Finally, evidence of low gene flow between populations suggests that removal could be an effective approach to control invasive guppies across the Hawaiian archipelago.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2021

Global patterns of forest autotrophic carbon fluxes

Carbon (C) fixation, allocation, and metabolism by trees set the basis for energy and material flows in forest ecosystems and define their interactions with Earth’s changing climate. However, we lack a cohesive synthesis on how forest carbon fluxes vary globally with respect to climate and one another. Here, we draw upon 1,319 records from the Global Forest Carbon Database (ForC), representing all major forest types and the nine most significant autotrophic carbon fluxes, to comprehensively explore how C cycling in mature, undisturbed forests varies with latitude and climate on a global scale. We show that, across all flux variables analyzed, C cycling decreases continuously with absolute latitude – a finding that confirms multiple previous studies but contradicts the idea that net primary productivity of temperate forests rivals that of tropical forests. C flux variables generally displayed similar trends across latitude and multiple climate variables, with no differences in allocation detected at this global scale. Temperature variables in general, and mean annual temperature and temperature seasonality in particular, were the best univariate predictors of C flux, explaining 19 - 71% of variation in the C fluxes analyzed. The effects of temperature were modified by moisture availability, with C flux reduced under hot and dry conditions and sometimes under very high precipitation. C fluxes increased with growing season length, but this was never the best univariate predictor. Within the growing season, the influence of climate on C cycling was small but significant for a number of flux variables. These findings clarify how forest C flux varies with latitude and climate on a global scale. In a period of accelerating climatic change, this improved understanding of the fundamental climatic controls on forest C cycling sets a foundation for understanding patterns of change.

Banbury Morgan, Rebecca↗
DOE OSTI2022

Vegetation clumping modulates global photosynthesis through adjusting canopy light environment

Abstract The spatial dispersion of photoelements within a vegetation canopy, quantified by the clumping index (CI), directly regulates the within‐canopy light environment and photosynthesis rate, but is not commonly implemented in terrestrial biosphere models to estimate the ecosystem carbon cycle. A few global CI products have been developed recently with remote sensing measurements, making it possible to examine the global impacts of CI. This study deployed CI in the radiative transfer scheme of the Community Land Model version 5 (CLM5) and used the revised CLM5 to quantitatively evaluate the extent to which CI can affect canopy absorbed radiation and gross primary production (GPP), and for the first time, considering the uncertainty and seasonal variation of CI with multiple remote sensing products. Compared to the results without considering the CI impact, the revised CLM5 estimated that sunlit canopy absorbed up to 9%–15% and 23%–34% less direct and diffuse radiation, respectively, while shaded canopy absorbed 3%–18% more diffuse radiation across different biome types. The CI impacts on canopy light conditions included changes in canopy light absorption, and sunlit–shaded leaf area fraction related to nitrogen distribution and thus the maximum rate of Rubisco carboxylase activity ( V cmax ), which together decreased photosynthesis in sunlit canopy by 5.9–7.2 PgC year −1 while enhanced photosynthesis by 6.9–8.2 PgC year −1 in shaded canopy. With higher light use efficiency of shaded leaves, shaded canopy increased photosynthesis compensated and exceeded the lost photosynthesis in sunlit canopy, resulting in 1.0 ± 0.12 PgC year −1 net increase in GPP. The uncertainty of GPP due to the different input CI datasets was much larger than that caused by CI seasonal variations, and was up to 50% of the magnitude of GPP interannual variations in the tropical regions. This study highlights the necessity of considering the impacts of CI and its uncertainty in terrestrial biosphere models.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2021

An integrated framework of plant form and function: the belowground perspective

Summary Plant trait variation drives plant function, community composition and ecosystem processes. However, our current understanding of trait variation disproportionately relies on aboveground observations. Here we integrate root traits into the global framework of plant form and function. We developed and tested an overarching conceptual framework that integrates two recently identified root trait gradients with a well‐established aboveground plant trait framework. We confronted our novel framework with published relationships between above‐ and belowground trait analogues and with multivariate analyses of above‐ and belowground traits of 2510 species. Our traits represent the leaf and root conservation gradients (specific leaf area, leaf and root nitrogen concentration, and root tissue density), the root collaboration gradient (root diameter and specific root length) and the plant size gradient (plant height and rooting depth). We found that an integrated, whole‐plant trait space required as much as four axes. The two main axes represented the fast–slow ‘conservation’ gradient on which leaf and fine‐root traits were well aligned, and the ‘collaboration’ gradient in roots. The two additional axes were separate, orthogonal plant size axes for height and rooting depth. This perspective on the multidimensional nature of plant trait variation better encompasses plant function and influence on the surrounding environment.

59 BASIC BIOLOGICAL SCIENCES↗