Valerie Finke

David Baracchi

Martin Giurfa

Ricarda Scheiner

Aurore Avarguès- Weber

Dr Aurore Avarguès-Weber
email: aurore.avargues-weber@univ-tlse3.fr

Evidence of cognitive specialization in an

Keywords:

Introduction

Cognitive skills are often attributed to a species based on the ability of few representative members to pass or fail key cognitive tests. Yet, depicting cognition at a species level ignores the inter-individual variability that may reveal fundamental properties in terms of behavioural plasticity and, in case of eusocial animals, specialization within a species. Indeed, interindividual differences may involve variation in different domains such as motivation to complete the task, choice strategy, personality, or any combination of these factors. However, variable performances between individuals may also be due to intrinsic differences in cognitive abilities, which may be retraced to multiple dimensions, from differences in gene expression to variability in neural population responses and hormonal levels [START_REF] Akhund-Zade | The effect of environmental enrichment on behavioral variability depends on genotype, behavior, and type of enrichment[END_REF][START_REF] Honegger | Idiosyncratic neural coding and neuromodulation of olfactory individuality in Drosophila[END_REF]. Shifting the focus from the species to the individual level in the study of cognitive abilities provides a remarkable opportunity to reveal key underpinning mechanisms. Moreover, it also offers novel perspectives to understand the link between fitness and cognition [START_REF] Raine | The correlation of learning speed and natural foraging success in bumble-bees[END_REF]Thornton & Lukas, 2012;Thornton et al., 2014;[START_REF] Evans | Fast learning in free-foraging bumble bees is negatively correlated with lifetime resource collection[END_REF][START_REF] Boogert | Measuring and understanding individual differences in cognition[END_REF].

Social insects have attracted wide attention due to their remarkable cognitive feats [START_REF] Giurfa | Behavioral and neural analysis of associative learning in the honeybee: a taste from the magic well[END_REF][START_REF] Srinivasan | Honey bees as a model for vision, perception, and cognition[END_REF][START_REF] Avarguès-Weber | Conceptual learning by miniature brains[END_REF][START_REF] Chittka | Bee cognition[END_REF]. The coexistence of individuals with variable cognitive abilities within a social group may be favoured due to the important energetic cost of investing in important learning faculties or problem-solving abilities [START_REF] Mery | A fitness cost of learning ability in Drosophila melanogaster[END_REF][START_REF] Burns | Impulsive bees forage better: the advantage of quick, sometimes inaccurate foraging decisions[END_REF][START_REF] Burns | Hastiness, brain size and predation regime affect the performance of wild guppies in a spatial memory task[END_REF][START_REF] Kawecki | Evolutionary ecology of learning: insights from fruit flies[END_REF][START_REF] Burns | Costs of memory: lessons from 'mini' brains[END_REF][START_REF] Jaumann | Energetic cost of learning and memory can cause cognitive impairment in honeybees[END_REF][START_REF] Kotrschal | Artificial selection on relative brain size in the guppy reveals costs and benefits of evolving a larger brain[END_REF]. Such a coexistence has been documented for example in bumblebees colonies where some individuals consistently make fast but inaccurate foraging decisions while others decide more slowly yet with higher accuracy [START_REF] Chittka | Psychophysics: Bees trade off foraging speed for accuracy[END_REF]. Colony success indeed benefits from the coexistence of costly but highly skilled foragers and cheaper but less accurate animals, as this heterogeneity may improve exploitation of different food sources and information distribution within the colony [START_REF] Burns | Impulsive bees forage better: the advantage of quick, sometimes inaccurate foraging decisions[END_REF][START_REF] Burns | Diversity of speed-accuracy strategies benefits social insects[END_REF][START_REF] Muller | Animal personalities: The advantage of diversity[END_REF][START_REF] Chittka | Speed-accuracy tradeoffs in animal decision making[END_REF]. For example, the distinction between scout foragers searching novel resources and recruited bees relying on social information to exploit massively a unique resource as long as it remains profitable might be based on different cognitive abilities [START_REF] Cook | Individual differences in learning and biogenic amine levels influence the behavioural division between foraging honey bee scouts and recruits[END_REF].

Bees are particularly appealing to study inter-individual cognitive variability as forager bees demonstrate diverse learning abilities ranging from elemental associative tasks to higher-order forms of learning such as categorization, numerical tasks or concept formation, among others [START_REF] Srinivasan | Honey bees as a model for vision, perception, and cognition[END_REF][START_REF] Avarguès-Weber | Visual cognition in social insects[END_REF][START_REF] Avarguès-Weber | Conceptual learning by miniature brains[END_REF][START_REF] Giurfa | Cognition with few neurons: higher-order learning in insects[END_REF][START_REF] Giurfa | An Insect's sense of number[END_REF]. Inter-individual variability has been generally neglected in standard tests of bee learning, which rely on mean group performance. Exceptions to this trend are the identification of individual variability in sucrose responsiveness as an important factor determining individual learning performance in honeybees [START_REF] Scheiner | Tactile learning and the individual evaluation of the reward in honey bees (Apis mellifera L.)[END_REF][START_REF] Scheiner | The effects of genotype, foraging role, and sucrose responsiveness on the tactile learning performance of honey bees (Apis mellifera L.)[END_REF][START_REF] Scheiner | Sensory responsiveness and the effects of equal subjective rewards on tactile learning and memory of honeybees[END_REF] and the analysis of variability in olfactory learning performances on an individual basis [START_REF] Pamir | Average group behavior does not represent individual behavior in classical conditioning of the honeybee[END_REF][START_REF] Pamir | Rapid learning dynamics in individual honeybees during classical conditioning[END_REF]. Both show the importance of focusing on individual performances as group-level analysis may lead to the misinterpretation of response dynamics.

Pioneer studies in bumblebees investigated whether such individual differences are consistent between cognitive tasks. Thus, relative learning performances were compared between visual and olfactory tasks with contradictory results between studies concerning the existence of a correlation [START_REF] Muller | Consistent interindividual differences in discrimination performance by bumblebees in colour, shape and odour learning tasks (Hymenoptera: Apidae: Bombus terrestris)[END_REF][START_REF] Smith | A comparison of visual and olfactory learning performance in the bumblebee Bombus terrestris[END_REF]. Bumblebees ability to solve an elemental discrimination task (A+ vs. B-) seems correlated with the faculty to then learn reversed reward contingencies in a non-elemental reversal learning phase (A-vs. B+) [START_REF] Raine | No trade-off between learning speed and associative flexibility in bumblebees: A reversal learning test with multiple colonies[END_REF]. On the contrary, recent studies on honeybees showed no correlation between individual relative performance in odours discrimination tasks when the reinforcement was appetitive or aversive [START_REF] Junca | Genotypic trade-off between appetitive and aversive capacities in honeybees[END_REF] nor between landmark and olfactory learning (Tait et al., 2019), which suggests some level of cognitive specialization.

Here we analysed individual learning performances in a simple visual discrimination task in which bees had to discriminate a rewarded from a non-rewarded visual target. We determined if learning proficiency was stable over time (three consecutive days) despite interindividual differences. After confirming that inter-individual differences were consistent and unaffected by the kind of visual stimulus used in this task, we determined if performance across visual tasks of different complexity, or across sensory modalities (visual versus olfactory), was correlated. To this end we trained bees along two consecutive tasks: i) the same elemental visual discrimination task described above and either ii) a conceptual visual discrimination based on learning the constant spatial relationships between variable patterns ('choosing the picture presenting an object above/below the other, or to the left/right of the other, independently of the physical properties the objects present') [START_REF] Avarguès-Weber | Simultaneous mastering of two abstract concepts with a miniature brain[END_REF] or ii') a simple olfactory discrimination (a discrimination between a rewarded and a non-rewarded odorant). While the simple visual and olfactory tasks represent basic forms of learning in which two stimuli have unambiguous outcomes (A+ vs. B-), the conceptual task requires transfer to novel unknown stimuli preserving the appropriate spatial relationship, and represents, therefore, a higher-order learning form [START_REF] Giurfa | Cognitive neuroethology: dissecting non-elemental learning in a honeybee brain[END_REF][START_REF] Avarguès-Weber | Conceptual learning by miniature brains[END_REF].

Methods

(a) General procedure

Free-flying honeybees (Apis mellifera) originating from a single hive and trained to forage for nectar on a sucrose gravity feeder were used in all our experiments. Only bees with intact wings were used as wing damage could account for reduced foraging performance [START_REF] Higginson | Accumulating wing damage affects foraging decisions in honeybees (Apis mellifera L.)[END_REF][START_REF] Higginson | Experimental wing damage affects foraging effort and foraging distance in honeybees Apis mellifera[END_REF]. The age of the bees was not controlled. Bees were recruited from the feeder to the setup by offering them a drop of sucrose solution with a concentration (1.8 M) higher than that of the feeder (variable depending on foraging motivation). While drinking, the bees were gently placed at the entrance of our Y-maze setup.

Once satiated, the bees flew back to their hive. Returning bees on their subsequent foraging trip were colour-marked and individually trained in a stepwise fashion to enter the Y-maze to collect a sucrose reward (1.8 M) placed on the back walls of the maze. This pre-training usually took 1-2 hours per bee.

The maze consisted of a pre-chamber, a decision chamber and the two arms in which the stimuli to be learned were presented. The pre-chamber was equipped with a sliding door, which allowed controlling the traffic of foraging bees. Only one marked bee was allowed at a time in the Y-maze. After entering the pre-chamber, the bee could fly into the decision chamber through a hole (Ø 5 cm), leading to the two arms (40 cm x 20 cm x 20 cm, L x H x W) of the apparatus. The stimuli were presented on a UV-reflecting white background covering the back walls (20 x 20 cm) of the arms. Back walls were placed at 15 cm from the centre of the decision chamber. During the learning tasks, bees always had to discriminate a stimulus rewarded with sucrose solution 1.8 M from an alternative stimulus punished with quinine solution 60 mM [START_REF] Avarguès-Weber | Aversive reinforcement improves visual discrimination learning in free-flying honeybees[END_REF]. A choice was scored when the bee approached one stimulus (< 5cm). If the bee chose the rewarded stimulus, it was allowed to drink the sucrose solution ad libitum until it returned to the hive to deliver the sucrose. An incorrect choice led to the tasting of quinine, which was followed by the possibility to move to the alternative arm to collect sucrose on the rewarded stimulus. Only the first choice within a foraging bout was recorded. The sides of the rewarded and punished stimuli were exchanged in a pseudo-random sequence (i.e. a stimulus presented no more than twice consecutively on the same side) throughout all experiments to prevent positional learning. After the last training trial, non-reinforced tests were performed using fresh stimuli.

The tests were repeated twice to swap stimulus side and they were spaced by three refreshing reinforced trials in order to maintain appetitive motivation. Each test lasted 45 seconds during which the contacts the bees made with the surface of the stimuli were recorded. This period is typically used in these tests as it allows uncovering the learning induced by the training; longer periods may result in a switch of choice strategy due to the extinction conditions. The percentage of correct choices for a given test was calculated for each bee using the number of contacts with the stimuli in both repetitions of the test.

(b) Experiment 1: Performance along three consecutive visual elemental discriminations

In this experiment, 18 bees were individually trained during three days to learn three consecutive visual discriminations, one per day. A minimum of two, but more often, at least three data points are typically used to show consistency in individual traits [START_REF] Stamps | The development of animal personality: relevance, concepts and perspectives[END_REF]. We chose to replicate the visual task during three consecutive days to determine the stability of individual performances. We did not extend the measurement period beyond three days to avoid losing bees due to natural death or recruitment to alternative food places. On each day, the bee experienced a 15-trial conditioning (i.e. 15 consecutive visits to the maze) during which it had to learn to discriminate two visual achromatic patterns, one (CS+) being consistently associated with reward (sucrose) and the other (CS-) with punishment (quinine). Training was followed by a test in which the stimuli used for training were presented without reinforcement (Fig. 1A). Training and testing took 1-2 hours per bee.

In the next two days, this procedure was repeated using a new set of visual stimuli every day.

All the bees that completed the experimental schedule returned voluntarily to the setup every day. They were not maintained captive overnight in the laboratory. The bees returned reliably to the experimental set-up throughout the three-day period and only one bee trained on day 1 did not come back on the next day to complete the training sequence.

The stimuli used were 7 x 7 cm black patterns printed on UV-reflecting white paper.

Six different patterns were used, which varied between bees and were presented as counterbalanced pair combinations on each experimental day (Fig. S1). These patterns were originally used in the study of [START_REF] Avarguès-Weber | Simultaneous mastering of two abstract concepts with a miniature brain[END_REF] and could be well resolved by the visual system of honeybees.

(c) Experiment 2: Performance in elemental visual and olfactory discriminations

In this experiment, a novel set of bees (N = 18) were trained consecutively within a day to solve a visual discrimination task and an olfactory discrimination task. The sequence of visual and olfactory tasks was randomized between bees. The tasks were spaced by approximately 30 minutes during which the bees could collect sucrose solution outside of the Y-maze. This delay allowed preparing the setup and stimuli for the next learning task. Both training phases consisted of 15 trials in which the bees had to discriminate between a rewarded stimulus (CS+) associated with sucrose solution and a second stimulus (CS-) associated with a quinine solution (Fig. 2A). Both training phases were followed by a test in which the respective stimuli were presented without reinforcement. The stimuli used in the visual task were the same as those described for experiment 1. The pair combinations were counterbalanced across bees. For the olfactory task, 10 µl of pure odorant (2-Octanol and Limonene, Sigma Aldrich)

were applied to 7x7 cm squares of filter paper. For each trial fresh stimuli were used to assure that the odours could be well perceived throughout training. Between trials, when the bees were absent from the set-up, the Y-maze was ventilated, and the arms of the maze were cleaned with 30% ethanol to remove potential odour residues. In both tasks, reinforcement contingencies were counterbalanced between bees. The whole procedure took between 2-3.5 hours per bee. keeping their alignment (Fig. 3A). By doing this, we ruled out that bees could use either the absolute spatial locations or the centre of gravity as cue to solve the task [START_REF] Avarguès-Weber | Simultaneous mastering of two abstract concepts with a miniature brain[END_REF]. The spatial relation between the discs was consequently the only reliable predictor of the reward. The conditioning phase was followed by non-reinforced tests, in which novel achromatic (black) geometric shapes were used to recreate the trained spatial relations. This allowed examining whether bees learned the spatial concept, irrespective of the stimuli properties. We took special care to choose shapes differing as much as possible from the patterns used in the elemental learning task (Fig. 3A). Different shapes were used to this end (Fig. S1). Stimuli were printed on UV-reflecting copy paper and had a size of 7 x 7 cm.

Training and testing took 3-4.5 hours per bee.

(e) Statistical analysis

Individual bee responses (correct or incorrect) during the acquisition phases were examined using generalized linear mixed models (GLMM) with a binomial error structure -logit-link function -, glmer function of R package lme4 [START_REF] Bates | Fitting linear mixed-effects models using lme4[END_REF]. In the models, the bee's choices (0 or 1) were entered as the dependent variable, while the trial number, the task (Day number (Experiment 1), Visual/Olfactory (Experiment 2) or Elemental/Non-Elemental (Experiment 3)), the stimuli used and the order of the tasks were entered as fixed factors.

Subjects identity (ID) was entered as a random factor to account for the repeated-measure design. Several models were run by testing interactions between factors and by dropping each factor subsequently to select the significant model with the highest explanatory power (i.e. the lowest AIC value) (see supplementary tables).

Performances during the non-reinforced tests were analysed with a generalized linear mixed model (GLMM) with a binomial error structure and logit-link function, where the proportion of correct choices for each bee was entered as dependent variable, while the task and the task order were entered as fixed factors whenever appropriate. The intercept term informed us on whether the mean proportion of correct choices is different from chance level.

Correlations were computed using both the Pearson and Spearman correlation coefficients.

All statistical analyses were performed with R 3.4.2 (R Development Core Team, 2016).

Results

Experiment 1: Performance along three consecutive visual elemental discriminations

Honeybees learned successfully the three elemental visual discriminations between the achromatic visual patterns (Fig. 1A) as they significantly improved their performance during the acquisition phases (GLMM, n = 17, Trial: χ 2 = 5.6, p = 0.02; Fig. 1B, table S1). There was neither a significant effect of the pair of stimuli used (Stimuli: χ 2 = 15.4, p = 0.97; table S1) nor of the training sequence as performance did not improve along the three consecutive visual discriminations (Day: χ 2 = 0.01, p = 0.92; table S1). Accordingly, the performances in the non-reinforced tests (Fig. 1C) were significantly higher than chance in all three days with no significant influence of task repetition (n = 17; Day 1: 76.1 ± 3.3 % (mean ± SEM), Z = 6.17, p < 0.001 (GLM); Day 2: 76.0 ± 2.3 %, Z = 8.79, p < 0.001; Day 3: 72.0 ± 2.7 %, Z = 7.85, p < 0.001; Day: Z = 1.34, p = 0.17 (GLMM)).

Individual learning performances (proportion of correct choices) were highly variable between individuals (Fig. 1D), yet they were consistent along the three days (Day 1 vs. Day 2:

Spearman correlation, r S = 0.62, p = 0.009, Pearson correlation, r p = 0.63, p = 0.007; Day 2 vs. Day 3: r S = 0.66, p = 0.004; r p = 0.78, p < 0.001; Day 1 vs. Day 3: r S = 0.70, p = 0.002; r p = 0.66, p = 0.004; Fig. 1D). This result indicates that despite population variability in learning proficiency, individual proficiency remained stable across days and visual discrimination tasks.

Experiment 2: Performance in elemental visual and olfactory discriminations

Although no significant improvement of performance could be detected over trials (GLMM, Trial: χ 2 = 1.7, p = 0.19; Fig. 2B, table S2), the bees (n = 18) learned both tasks as shown by their performance in the non-reinforced tests where they preferred the correct stimulus, be it visual (74.9 ± 3.2 % of correct choices, GLM, Z = 6.54, p < 0.001; Fig. 2C) or olfactory (84.0 ± 2.6 %, Z = 8.24, p < 0.001; Fig. 2C). Bees were generally more accurate in the olfactory task than in the visual task (GLMM, Task: Z = 3.58, p < 0.001; Fig. 2C).

As in the previous experiment, a high variability in learning proficiency was observed between the individuals trained (Fig. 2D). However, this time no correlation between individual performances was found (Olf. vs. Vis: r S = 0.31, p = 0.21; r p = 0.26, p = 0.29; Fig. 2D), thus showing that individual proficiency is not stable between tasks involving different sensory modalities.

Experiment 3: Performance in visual discriminations of different cognitive complexity

Honeybees improved their performance during the acquisition phase in both the elemental and the conceptual task (GLMM, n = 18, Trial: χ 2 = 9.0, p = 0.003; Fig. 3B, table S3). The task sequence did not affect the bees' performance (Order, χ 2 = 0.05, p = 0.82; Fig. S2, table S3).

Overall the bees' accuracy was higher in the elemental task than in the non-elemental task (Task, χ 2 = 15.0, p < 0.001, Fig. 3B, table S3), a result that is consistent with the different levels of complexity of these tasks. Performance in the non-reinforced tests was significantly higher than chance in both tasks (elemental task: 75.4 ± 2.7 % of correct choices, GLM, Z = 7.18, p < 0.001; non-elemental task: 68.0 ± 2.9 % of correct choices, Z = 4.73, p < 0.001; Fig. 3B). Yet, it was also affected by the complexity of the task (GLMM, Task: Z = 2.36, p = 0.02, Fig. 3B) as test performance was better after the elemental conditioning than after the nonelemental conditioning.

Individual learning proficiency was variable but it correlated between individuals across the two learning tasks (r S = 0.64, p = 0.004; r p = 0.69, p = 0.002; Fig. 3C), with some individuals being consistently more error-prone than others in both tests.

Discussion

Our results highlight the importance of individual variability in cognitive tasks and its relationship with the nature of the task considered. By testing the same bees on consecutive days and tasks that were either similar (Experiment 1) or different (Experiments 2 and 3), we observed an important inter-individual variability in learning performance as in all cases the proportion of correct choices varied at the population level, with some bees being efficient learners and others, on the contrary, poor learners. Importantly, this proficiency was maintained across time when individuals were tested on three similar consecutive visual discriminations (Experiment 1). Thus, the variable response observed within a given task seems to be a consequence of individual stable factors rather than being noise resulting from transitory variability in foraging activity, appetitive motivation or stochasticity in choice persistence. In addition, we showed that proficiency is maintained across elemental and higher-order learning tasks within the same sensory modality (Experiment 3), even if performances were again highly variable at the population level. This result is important as it shows that bees trained within the visual modality will conserve their success irrespective of the cognitive complexity of the task, a problem that has never been addressed so far in invertebrates. Finally, we showed that consistency in performance was not maintained when bees were trained using tasks involving different modalities (vision and olfaction; Experiment 2), thus arguing in favour of within-modality cognitive specialization.

Several factors can account for the interindividual variability observed. Yet, some of them can be ruled out in our study. In our experiments, only nectar foragers captured at a sucrose feeder and consequently motivated for foraging were used, which discards differences due to division of labour and appetitive motivation. In addition, the temporal sequence did neither influence the performance of the bees. We expected that familiarization with the setup and enhanced attention might be promoted by prior training experience, resulting in faster acquisition in subsequent tasks. However, such improvement was not observed in our conditions.

Consistent inter-individual difference in performances maintained within a visual task or across elemental and higher-order visual tasks could have a genetic basis. Drosophila flies from a population selected over several generations on the basis of their good learning abilities in an aversive olfactory task exhibited an equally good performance in a different aversive olfactory task (reinforced by an electric shock rather than a bitter substance), thus highlighting the importance of genetic selection for learning abilities [START_REF] Mery | Experimental evolution of learning ability in fruit flies[END_REF][START_REF] Mery | Experimental evolution of olfactory memory in Drosophila melanogaster[END_REF]. Numerous studies suggest that genetic factors influence cognitive performance in invertebrates [START_REF] Raine | Intercolony variation in learning performance of a wild British bumblebee population Hymenoptera: apidae: Bombus terrestris audax[END_REF][START_REF] Ings | A population comparison of the strength and persistence of innate colour preference and learning speed in the bumblebee Bombus terrestris[END_REF][START_REF] Orr | Differences in LTMforming capability between geographically different strains of Alberta Lymnaea stagnalis are maintained whether they are trained in the lab or in the wild[END_REF][START_REF] Raine | No trade-off between learning speed and associative flexibility in bumblebees: A reversal learning test with multiple colonies[END_REF][START_REF] Scheiner | Comparing the appetitive learning performance of six European honeybee subspecies in a common apiary[END_REF]. The bees of our study originated from a single hive, a fact that reduces but does not abolish the genetic diversity among the bees tested, as different patrilines typically coexist within a hive due to multiple mating of the queen during the nuptial flight. The learning performance of individual worker bees in elemental olfactory tasks can indeed be predicted partially by their patriline [START_REF] Brandes | Estimation of heritability of learning behavior in honeybees (Apis mellifera capensis)[END_REF][START_REF] Bhagavan | Effect of genotype but not of age or caste on olfactory learning performance in the honey bee, Apis mellifera[END_REF][START_REF] Scheiner | Effects of patriline on gustatory responsiveness and olfactory learning in honey bees[END_REF][START_REF] Junca | Genotypic trade-off between appetitive and aversive capacities in honeybees[END_REF]. Genetic variability has a strong impact on responsiveness to appetitive and aversive stimuli such as sucrose or thermic shocks, respectively [START_REF] Scheiner | Effects of patriline on gustatory responsiveness and olfactory learning in honey bees[END_REF][START_REF] Junca | Genotypic trade-off between appetitive and aversive capacities in honeybees[END_REF]. This variable responsiveness translates into variation of performance observed in associative learning protocols in which such stimuli are used as unconditioned stimuli [START_REF] Scheiner | Sensory responsiveness and the effects of equal subjective rewards on tactile learning and memory of honeybees[END_REF][START_REF] Roussel | Reappraising social insect behavior through aversive responsiveness and learning[END_REF][START_REF] Scheiner | Effects of patriline on gustatory responsiveness and olfactory learning in honey bees[END_REF]. Thus, the variable success of foragers co-opted for our experiments could be due to their belonging to different genetic patrilines within the colony.

Variability in learning performance in our study could also be influenced by prior visual experience gathered on a larger time scale than the duration of our experiment during foraging activities. Both age and sensory experience influence brain structural development of forager bees, which, in turn, can modulate learning performance, although mostly in the form of a cognitive decline with aging (Withers et al., 1993;[START_REF] Durst | Development and experience lead to increased volume of subcompartments of the honeybee mushroom body[END_REF][START_REF] Farris | Experience-and age-related outgrowth of intrinsic neurons in the mushroom bodies of the adult worker honeybee[END_REF][START_REF] Münch | In the Laboratory and during Free-Flight: Old Honey Bees Reveal Learning and Extinction Deficits that Mirror Mammalian Functional Decline[END_REF][START_REF] Groh | Age-related plasticity in the synaptic ultrastructure of neurons in the mushroom body calyx of the adult honeybee Apis mellifera[END_REF][START_REF] Scholl | Light exposure leads to reorganization of microglomeruli in the mushroom bodies and influences juvenile hormone levels in the honeybee[END_REF][START_REF] Cabirol | Relationship between brain plasticity, learning and foraging performance in honey bees[END_REF]. The mushroom bodies, the main higher-order structures of the insect brain, show experience-dependent variations in their volume or density of synaptic buttons, following light exposure, age, foraging experience or learning events [START_REF] Hourcade | Longterm memory leads to synaptic reorganization in mushroom bodies: a memory trace in the insect brain?[END_REF][START_REF] Scholl | Light exposure leads to reorganization of microglomeruli in the mushroom bodies and influences juvenile hormone levels in the honeybee[END_REF][START_REF] Cabirol | Experience during early adulthood shapes the learning capacities and the number of synaptic boutons in the mushroom bodies of honey bees (Apis mellifera)[END_REF][START_REF] Cabirol | Relationship between brain plasticity, learning and foraging performance in honey bees[END_REF]. Individual variability in mushroom body development may have an impact on cognitive faculties [START_REF] Li | A possible structural correlate of learning performance on a colour discrimination task in the brain of the bumblebee[END_REF]. Therefore, it is likely that the stability of learning proficiency observed across days or tasks of different complexity relies, at least partially, on neurobiological variability resulting from different life experiences.

Learning differences could also emerge from variations in the processing of the stimuli to be learned such as odours or visual cues. This possibility is supported by our study as we found that some bees were relatively better at learning olfactory cues than visual cues, and vice versa, thus reflecting potential variation in perceptual salience between modalities.

Honeybees are known to differ in their responsiveness to odours [START_REF] Scheiner | Sucrose responsiveness and behavioral plasticity in honey bees (Apis mellifera)[END_REF], which could be retraced to inter-individual differences in the activity of olfactory neural circuits. In fruit flies, for instance, stable inter-individual variability can be found in an odour-preference assay, which translates into consistent inter-individual differences in Ca 2+ activity levels in a key structure of the olfactory circuit, the projection neurons of the antennal lobes [START_REF] Honegger | Idiosyncratic neural coding and neuromodulation of olfactory individuality in Drosophila[END_REF]. Similar arguments could apply to visual processing and its underlying visual circuits.

The question of whether learning abilities correlate across problems of different complexity or sensory domains is particularly relevant for the analysis of brain modularity and for understanding the contributions of different neural circuits to different forms of learning. In honeybees, different brain structures have been associated with different levels of complexity in olfactory learning. The mushroom bodies are required for non-elemental olfactory discrimination tasks such as the negative patterning problem (A+, B+, AB-) [START_REF] Devaud | Neural substrate for higher-order learning in an insect: Mushroom bodies are necessary for configural discriminations[END_REF] or for reversal learning (A+ B-A-B+) [START_REF] Boitard | GABAergic feedback signaling into the calyces of the mushroom bodies enables olfactory reversal learning in honey bees[END_REF] but are dispensable for elemental olfactory discrimination (A+, B+, CD-) [START_REF] Devaud | Neural substrate for higher-order learning in an insect: Mushroom bodies are necessary for configural discriminations[END_REF]. From this perspective, proficiency in non-elemental olfactory learning may not necessarily be correlated with proficiency in elemental olfactory learning, given that these learning forms are mediated by different brain structures. Applying this reasoning to our results in the visual domain suggests that both learning forms, which are highly correlated, may require similar visual circuits/structures. To date, the brain neuropils involved in different forms of visual learning remain unknown due to the difficulty of reproducing successful visual learning in the which would allow coupling with invasive recordings of neural activity [START_REF] Avarguès-Weber | Advances and limitations of visual conditioning protocols in harnessed bees[END_REF]). Yet, given the massive visual afferences to the mushroom bodies [START_REF] Ehmer | Segregation of visual input to the mushroom bodies in the honeybee (Apis mellifera)[END_REF][START_REF] Paulk | Higher order visual input to the mushroom bodies in the bee, Bombus impatiens[END_REF]) and to the central complex [START_REF] Pfeiffer | Organization and functional roles of the central complex in the insect brain[END_REF], participation of these structures is expected. For instance, in Drosophila, both mushroom bodies and the central complex are involved in visual learning depending on the specific task and setup used [START_REF] Liu | Context generalization in Drosophila visual learning requires the mushroom bodies[END_REF][START_REF] Pan | Differential roles of the fan-shaped body and the ellipsoid body in Drosophila visual pattern memory[END_REF][START_REF] Ofstad | Visual place learning in Drosophila melanogaster[END_REF]Vogt et al., 2014;Vogt et al., 2016).

The fact that we observed a positive correlation of performances across an elemental task and a conceptual task in the visual modality suggests that a similar brain circuitry underlies both discriminations. An alternative explanation may be that differences in general visual processing abilities and attentional processes could have a major influence on the performance in both tasks. In humans and rodents, general intelligence has been linked to selective attention and working memory abilities (reviewed in [START_REF] Matzel | Selective attention, working memory, and animal intelligence[END_REF] which encompassed both the storage of information and the processing and integration of information [START_REF] Baddeley | Working memory: looking back and looking forward[END_REF][START_REF] Jarrold | Individual differences in working memory[END_REF]). Increasing evidence indicates that insects are capable of selective attention mediated by several higher-order brain areas (van Swinderen, 2011;[START_REF] Nityananda | Attention-like processes in insects[END_REF].

Shifting our focus from group to individual performances in cognitive studies could thus contribute to the elucidation of the underlying mechanisms (Thornton & Lukas, 2012;[START_REF] Pamir | Rapid learning dynamics in individual honeybees during classical conditioning[END_REF][START_REF] Boogert | Measuring and understanding individual differences in cognition[END_REF][START_REF] Klein | Causes and consequences of individual forager variability in social bees[END_REF]. It also raises fascinating ecological questions such as the possible existence of cognitive specialization between workers from different patrilines in a hive. In honeybees, the best learners might deal with more demanding, complex tasks and concentrate their foraging effort towards the exploration of novel food sources. Others might simply copy the former and use social information when facing difficult tasks as occurs in bumblebees [START_REF] Baracchi | Foraging bumblebees use social cues more when the task is difficult[END_REF]. This would explain the known differentiation between scouts and recruits [START_REF] Biesmeijer | Exploration and exploitation of food sources by social insect colonies: a revision of the scout-recruit concept[END_REF][START_REF] Beekman | What makes a honeybee scout?[END_REF]. The diversity of foraging strategies within a colony has been shown to increase its fitness [START_REF] Burns | Impulsive bees forage better: the advantage of quick, sometimes inaccurate foraging decisions[END_REF][START_REF] Burns | Diversity of speed-accuracy strategies benefits social insects[END_REF][START_REF] Jeanson | Interindividual variability in social insectsproximate causes and ultimate consequences[END_REF][START_REF] Klein | Causes and consequences of individual forager variability in social bees[END_REF] but, interestingly, the best learners are not necessarily the best foragers as demonstrated in bumblebees, where bad learners foraged for a longer time frame and collected more resources, potentially due to the energetic cost associated with cognition [START_REF] Evans | Fast learning in free-foraging bumble bees is negatively correlated with lifetime resource collection[END_REF]. Thus, the complex interplay between interindividual cognitive skill diversity, task allocation and colony fitness remains to be fully elucidated.

( d )

 d Experiment 3: Performance in visual discriminations of different cognitive complexityAn additional group of bees (N = 18) were trained within the same day along two successive visual tasks, one elemental discrimination similar to the ones described in Experiment 1 and one non-elemental conceptual discrimination based on spatial relational learning[START_REF] Avarguès-Weber | Simultaneous mastering of two abstract concepts with a miniature brain[END_REF]. The sequence of the elemental and non-elemental tasks was randomized between bees. Both tasks were spaced by approximately 30 minutes during which the bees had access to a sucrose solution outside of the training apparatus. The tasks differed in the number of conditioning trials (15 for the elemental task and 30 for the non-elemental task) due to their different complexity. Training lengths were decided to ensure significant learning of the majority of bees. The procedure and stimuli of the elemental task were identical to those used in experiment 1 or in the visual task of experiment 2. In the non-elemental task, the bees were trained to discriminate between two composite images, each consisting of two coloured discs (Ø 7 cm), but arranged in two different spatial configurations -above/below (discs aligned vertically, i.e. one above/below the other) and left/right (discs aligned horizontally, i.e. one to the left/right of the other). The discs differed only in their chromatic properties. They were cut from uncoated HKS papers (K+E Stuttgart, Stuttgart-Feuerbach, Germany; 1N, 3N, 29N, 32N, 48N, 71N; Fig.S1). The reinforcement contingency (above/below+ or left/right+) was counterbalanced between bees. The colour of the discs and their positions within the back walls of the Y-maze were pseudo-randomized over trials, yet

Figures legends Figure 1 :

 legends1 Figures legends

Figure 2 :

 2 Figure 2: Experiment 2: Comparison of performance in a visual and an olfactory elemental discrimination. A. Schematic presentation of the tasks submitted to the bees. B Acquisition curves expressed as proportion of correct choices ± SEM of forager bees (n = 18) confronted with a visual task (V; red curve) and an olfactory task (O; blue curve) consisting of 15 trials. No significant improvement of performance could be detected over trials (GLMM, Trial: χ 2 = 1.7, p = 0.19). C. Choice accuracy expressed as proportion of correct choices ± SEM of forager bees confronted with the V and the O tasks in the non-reinforced learning test. Bees' performance was above chance level for both tasks (GLM, Vision: Z = 6.54, p < 0.001; Olfaction: Z = 8.24, p < 0.001). D. Correlation between performances (proportion of correct choices) of individual bees in the non-reinforced learning tests. Each dot shows data for one individual bee. The blue line represents the regression line while the blue shadow indicates the 95% confidence interval. No correlation between individual performances was observed (Spearman correlation, r S = 0.31, p = 0.21).

Figure 3 :

 3 Figure 3: Experiment 3: Comparison of performance in an Elemental and a Non-Elemental visual task. A. Schematic presentation of the visual learning tasks submitted to the bees. B Acquisition curves expressed as proportion of correct choices ± SEM of forager bees (n = 18) confronted with an Elemental task (E; red curve) consisting of 15 trials and a Non-Elemental task (NE; blue curve) consisting of 30 trials. There was a significant improvement in performance over trial repetition (GLMM, n = 18, Trial: χ 2 = 9.0, p = 0.003). C. Choice accuracy expressed as proportion of correct choices ± SEM of forager bees confronted with the E and the NE tasks in the non-reinforced learning test. Bees' performance was significantly above chance level in both tasks (GLM, n = 18, Elemental: Z = 7.18, p < 0.001; Non-Elemental: Z = 4.73, p < 0.001). D. Correlation between performances (proportion of

Acknowledgements

We are grateful to Dr. Jürgen Schramme for providing access to the research apiary of Mainz University.

Funding

Our work was supported by the Fyssen foundation, the French National Research Center (CNRS), the Institut Universitaire de France and the University Paul Sabatier of Toulouse.

MG was funded by an ERC Advanced Grant "Cognibrains". VF was supported by a doctoral scholarship of the German Academic Scholarship Foundation. DB is currently supported by the Rita Levi Montalcini Fellowship (ID PGR15YTFW9) provided by MIUR.

Authors' contributions

AAW conceived the project. VF and AAW designed the experiment. VF performed the experiment. VF, AAW and DB analysed the data. VF, DB, MG, RS and AAW participated in the interpretation of the results and in the redaction of the manuscript.

Data accessibility

The datasets supporting this article are accessible on Dryad:

https://datadryad.org/stash/share/xpGwh3DQUlu7QVY0afLB6w9Zg4fF63hqJzEIXI_SZrk

Competing interests

We declare no competing interests.

Ethical statement

Our protocols comply with standard welfare practice in our field. The bees were not manipulated and free to visit our apparatus. The experiment involved bees from an apiary dedicated to research.


Jacques Miltat
email: jacques.miltat@universite-paris-saclay.fr

André Thiaville

Lattice-Compliant Simulations of Anti-Ferromagnetic Textures and their Response to Spin-Orbit Torques

Spin textures in perfectly ordered antiferromagnetic ultra-thin films are evaluated on the basis of atomistic simulations and compared to the results of recent extensions of the (l, m) decomposition model. A comparison between similar textures supported by either a bcc or a fcc lattice shows that the sole inclusion of symmetric Heisenberg exchange interactions suffices to provide, when geometry commands, a natural canting between neighboring s ↑ and s ↓ spins within a texture. Moreover, a pure 'at lattice points' dipole model leads to natural symmetries in clear disagreement with expectations from the (l, m) model. Inversely, properties such as wall mobility under spin orbit torques depending mostly on wall geometrical characteristics such as chirality and chirality axis orientation prove, as anticipated, primarily lattice immune.

I. INTRODUCTION

C. Kittel and F. Keffer's analysis of antiferromagnetic resonance [1,2], revealed i) the existence of a resonance frequency in zero field, ii) the coexistence of several resonance modes, all with large frequencies w.r.t. the ferromagnetic case. These concepts, established in a two spin-lattices approach, have received a full experimental confirmation in the case of e.g. MnF 2 [3,4], NiO [5,6] or CrCl 3 [7]. Large frequencies, in the Ter-aHz range, due to precession around the sum of the exchange and anisotropy fields, the latter potentially incorporating dipole-dipole interactions, entail large domain wall velocities at low damping. Unusual wall velocities, up to 20 km/sec, have indeed been experimentally observed in rare-earth orthoferrites, not exactly antiferromagnets though, but weak ferromagnets as early as 1978 (see [START_REF] Bar'yaktar | Dynamics of Topological Solitons, Experiments and Theory[END_REF]). Recently, a possible electrical control of the order parameter in antiferromagnets with potential applications to spintronics has attracted a sustained interest both in metallic [9][10][11][12][13][14][15][16] and insulating antiferromagnetic compounds [17][18][19][20][21].

In ferromagnets (F), micromagnetic simulations have more than often proved instrumental in deciphering responses of small ferromagnetic elements to fields or to spin transfer torques (STT), both in the frequency and time domains. In antiferromagnets (AF), the sole fact that spin-orbit torques (SOT) may reverse at the atomic scale [9,22] ought to promote atomistic simulations of the spin dynamics. Such codes exist, e.g. [23][24][25][26]. However, although numerous applications of atomistic codes may be traced back in the case of F's, fewer examples deal with AF's [26][27][28][29] beyond the 1D spin chain limit [30][31][32]. The spin orientation in AF's is characterized by the superposition of spatially fast variables (spin inversion on adjacent sites) and slow variables (e.g. in the presence of an AF texture). Efforts to separate the fast from the slow variables start with the (l, m) and Haldane's decomposi-tions [33][34][35]. Haldane's decomposition retains a general staggered field character for the transverse spin component. In the words of Tveten et al. [36], Haldane's canting field represents the dynamic magnetization induced by temporal variations of the order parameter. The (l, m) decomposition, l = 1 2 (ŝ ↑ -ŝ↓ ), m = 1 2 (ŝ ↑ + ŝ↓ ) allows to envisage the new variables l and m as continuous variables. Models using the (l, m) decomposition implicitly assume, however, l, the so-called Néel vector, and m, the magnetization, to be located at the same node within a regular mesh in spite of the initial need to associate two specific AF coupled spins with distinct locations within the primary lattice. As a consequence, Papanicolaou [30] is lead, in his mesoscale treatment of 1D textures, to introduce asymmetrical exchange interactions in order to allow for some canting within a texture (a wall in this case) between neighbouring s ↑ and s ↓ spins. An extension to the 3D case [36] appears, however, questionable since tantamount to the introduction of isotropic asymmetric exchange interactions where none such exist apart from surface and/or interface Dzyaloshinskii-Moriya (DM) interactions.

In this work, we analyze characteristic spin textures in AF thin films within an atomistic model focusing on two crystal structures, the bcc and the fcc cubic Bravais lattices, and provide values for their energy, concentrating in a first step on the exchange and anisotropy energies. In a second step, the introduction of dipole-dipole interactions allows to quantify their role in the stability of the AF order and the generation of (minute) stray fields above a free surface. We examine next the conditions in which the numerical computation load may be alleviated via what we term "lattice inflation" or "swelling". Lastly, we analyse the role of spin-orbit torques on specific textures in the presence of DM interactions (see [22,37]).

II. ATOMISTIC MODEL

The atomistic model Hamiltonian includes Heisenberg exchange interactions, anisotropy, the Zeeman and dipole-dipole interactions energies as well as Dzyaloshinskii-Moriya interactions:

H = H Exch + H K + H Ha + H Dip + H DMI H Exch = - 1 2 i =j J ij (ŝ i • ŝj ) H K = k 1 i f (s 2 i,
x , s 2 i,y , s 2 i,z ) + k 2 ....

H Ha = -µ s i (B a • ŝi ) H Dip = - 1 2 µ s i (B i Dip • ŝi ) H DMI = 1 2 i =j D ij • (ŝ i × ŝj ) (1)
where, ŝi is a unit vector collinear with the local moment, index i. B a and B Dip are the applied and dipole field, respectively. The anisotropy constants k are expressed in terms of an energy whereas J ij (F > 0; AF < 0) is defined as an energy per bond, hence the factor 1 2 in front of the exchange energy. We choose here the meV as the energy unit and express all fields B in Tesla (T), or fraction thereof, whereas µ s is expressed in Bohr magneton translated into an energy per T (1 µ B ≈ 5.8 10 -2 meV.T -1 ). Lastly, D ij is the Dzyaloshinskii vector, D ij = d DMI (û ij × ẑ) in the case of interface driven DM interactions with the highest symmetry, ûij is the unit vector along the line joining sites i and j in the surface plane and ẑ the normal to the surface. d DMI is also defined as an energy per bond.

The equation of motion is an exact analog to the Landau-Lifshitz-Gilbert (LLG) equation of magnetization motion within micromagnetics, namely,

dŝ dt = -γ(ŝ × B Eff ) + αγ ŝ × dŝ dt + τ SOT (2)
where, γ is the gyromagnetic ratio, α the damping parameter, τ SOT the spin-orbit torque and B Eff , the effective field that, classically, reads:

B Eff = - 1 µ S δH δŝ (3)
In these equations, both the exchange field and the dipole field mirror the lattice symmetries.

We consider here two simple cubic lattices, the bcc and the fcc lattices, with as background antiferromagnetic materials Cr on the one hand and NiO on the other. Chromium has been widely studied at the time of the discovery of the Giant Magnetoresistance [39,40] and, later, in the context of exchange coupling. In their study of 3d transition metal monolayers on various (001) surfaces, including Ag, Au and Pd, Blügel et al. [41,42] envisage two possibilities for a Cr overlayer: either a F p(1×1) or an AF c(2×2) spin ordering and conclude that both are stable with, however, a slightly lower energy for the AF ordering. On the other hand, as soon as the thickness of a free standing layer exceeds 3 monolayers, the staggered stacking of F (001) planes becomes more stable. A largely amplified moment at the interface has been obtained from ab initio calculations; it also appears that Cr layers thin enough are devoid of incommensurate spin density waves (see [43] for a timely review). High quality Cr layers have also been grown on W [44]. For bulk Cr, the spin orientation is in-plane with preferred directions along [100] and [010] in the (001) plane. We adopt this general view, without, however, imposing an exalted moment at the interface. In the bcc lattice, each ↑ spin has 8 ↓ nearest neighbours (n.n.) and vice versa, as recalled in Fig. 1. The exchange field limited to n.n. for a site with its full environment thus reads: bcc :

B i Exch = + 1 µ S j=1,N (=8) J ij ŝj (4)
According to Eqn.4, the exchange effective field aligns spin i with minus the average spin orientation of its 8 n.n.(J ij < 0). This sole mechanism allows for the lattice induced canting of n.n. spins within a texture as shown below (Section III). We also adopt a uniform value for the Cr moment, a symbolic value µ S = 1µ B , somewhat exceeding the value stemming from first principle calculations. We retain a value of -15 meV for the exchange energy per link, in agreement with [45].

Various forms of anisotropy have been used, with for the bcc lattice, for instance, the cubic anisotropy (K cc f (α 2 , β 2 , γ 2 )) and a general orthorhombic anisotropy reading K u cos 2 θ + K iP sin 4 θ cos 2 (2φ), where α, β, γ are the director cosines of the spin orientation in the cubic axes, θ and φ the polar and azimuthal angle of the spin orientation owing to the specific orientation of the computation box. The exact form of the anisotropy will be made explicit with each computation result below.

NiO is another well-studied AF material, sharing the structure of MnO, the first synthesized antiferromagnetic oxide [46]. In the search for high quality supported NiO or CoO films, Ag(001) and MgO have played a leading role due to their minute lattice parameter mismatch [19,20,[47][48][49][50]. NiO growth has, however, also been achieved with SrTiO 3 as a substrate [18] as well as Pt(111) [17]. Several first principles studies of magnetic oxides appeared during the last decade [51][52][53]. Only one of them [51] evoques a possible reduction of the Ni moment for the first grown layer over Ag(001). All agree on the AF2 spin ordering within NiO or CoO, meaning staggered F (111) planes, and on the existence of exchange interactions with nearest neighbours (n.n.), at distance a/ √ 2, and next nearest neighbours (n.n.n.), at distance a, corresponding, respectively to 90 • and 180

• (Ni 2+ - O 2--Ni 2+ ) bonds.
Values obtained for the exchange energies J 1 (n.n.) and J 2 (n.n.n.), however, somewhat fluctuate. We consider here perfect (111) NiO epilayers and adopt the exchange parameters provided long ago by neutron scattering techniques [54], namely J 1 = +1.39 and +1.35 meV for (ferromagnetic) n.n. interactions within the (111) plane and (ferromagnetic) n.n. interactions outof-plane, respectively, and J 2 = -19.05 meV for (antiferromagnetic) n.n.n. interactions (see Fig. 2). With 6 n.n. in-plane, 6 n.n. out-of-plane, and 6 n.n.n. for a site with its full environment, the exchange field reads:

fcc : B i Exch = + J iP 1 µ S j=1,6 (ŝ iP j ) + J ooP 1 µ S j=1,6 (ŝ ooP j ) + J 2 µ S j=1,6 (ŝ j ) (5)
where, iP and ooP stand for in-plane and out-of-plane, respectively. Due to the AF2 character of antiferromagnetism in NiO, ooP n.n. exchange interactions are frustrated.

We also adopt a uniform value for the Ni ++ moment, a symbolic value µ S = 2µ B , only marginally exceeding values obtained from first principles calculations. Anisotropy is dealt with on the same footing as in the Cr case, namely a generalized orthorhombic anisotropy reading K u cos 2 θ + K iP sin 6 θ cos 2 (3φ) in order to keep track with the six-fold anisotropy in the (111) plane.

Numerical simulations below have all been performed by means of in-house codes ported to graphical processing units (GPU's). Epilayers contain 16 and 12 atomic layers in the bcc (interlayer spacing: a/2) and fcc (interlayer spacing a √ 3/3 )lattices, respectively, where a is the cubic cell parameter. Double precision has been used throughout.

III. N ÉEL (BEND) AND HEAD-TO-HEAD (SPLAY) WALLS

We consider in this section two types of walls in AF systems where the prevailing anisotropy tends to maintain spins in-plane with respect to the geometry of the slab, namely Néel and Head-to-Head (resp. Tail-to-Tail) walls that, in the terminology used in nematic liquid crystals, may be termed Bend and Splay walls [START_REF] Kléman | Points, Lines and Walls[END_REF]. At this stage, only the exchange and anisotropy terms in the Hamiltonian are being considered. In the absence of dipole fields, such walls are anticipated to entail equivalent energies.

A. bcc Spin lattice

Specifically, in the axes of Fig. 3, the anisotropy energy now reads:

E k = k u s 2 z + k iP (s 2 x -s 2 y ) 2 (6)
with k u and k iP > 0. We adopt also here symbolic values for the anisotropy constants, namely k u = 0.1 meV, k iP = 0.05 meV. Under such conditions, spins remain in the (001) plane and rotate gradually from the [100] to the [010] easy axis as the result of the competition between exchange interactions and anisotropy (see Fig. 3) [56].

The global AF ordering between successive (001) planes is, however, preserved. Similar remarks apply to the Splay wall spin texture displayed in Fig. 4. The Bend and Splay 90 • wall profiles displayed in Fig. 5, prove essentially similar, as anticipated. The roles of the s x and s y spin components are simply swapped. The lattice geometry commands an offset between the spin locations along equivalent rows located in adjacent (001) planes, as captured in Fig. 3,a. Plotting minus s x and s y of the ↓ plane along the spin components of the ↑ plane, or vice versa, clearly reveals the continuous character of the spin rotation across the two sub-lattices, or, equivalently, the continuous spin-canting between AF coupled neighbouring spins within a texture (see the insets in Fig. 5).

Before addressing the wall energy issue, it is worth identifying what may be attributed to the texture (curvature) within the exchange energy. Looking back at Eqn.1, any uniform orientation of the order parameter yields for the bcc lattice an energy 4 J ij < 0 per general site. Therefore the wall contribution to the exchange energy is

-1 2 J ij i ( j=1,N (ŝ i • ŝj ) + N ),
where N is the number of n.n., N = 8 for a spin with its full environment within a bcc lattice.

With this in mind, the wall exchange and anisotropy energies are found to be almost strictly equal, namely 872.393 meV and 871.078 meV, respectively, for a L × t wall surface, where L is the wall length and t the slab thickness, namely (130 a √ 2/2) × (16 a/2) for this computation. Note that the volume over which the wall energy is computed represents only a fraction of the computation volume in order to minimize edge effects. With a = 0.2884 nm (Cr), the wall energy amounts to ≈ 28.504 meV/nm 2 . The Splay wall profile s x in Fig. 5,b has been fitted with a function characteristic of a Bloch wall in a ferromagnet. The excellent fit provides a precise value for the width parameter of that model Bloch wall, namely ∆ * = 1.713 nm. Wall width combined with energy provide effective values for the continuum limit of the exchange and anisotropy constants, respectively A * = 12.207 meV/nm and K iP = 4.160 meV/nm 3 . The discussion of such estimates is deferred to the end of the section.

B. fcc Spin lattice

We solely consider below walls separating S(or Spin)domains, i.e. domains belonging to a single T(or Twin)domain [START_REF] Roth | [END_REF]58]. A T-domain is characterized by a unique {111} AF2 ordering. Four T-domains may coexist, corresponding to (111), 111 , 111 or 111 F-ordered planes, respectively. In the axes of Fig. 6, the anisotropy energy may be expressed as:

E k = k u s 2 z + k iP s 2 x (s 2 x -3s 2 y ) 2 (7)
Adopting here also symbolic values for the anisotropy constants, namely k u = 0.1 meV, k iP = 0.05 meV [59], spins remain confined to the (111) plane and rotate gradually from the 211 to the 121 easy axis, resulting in a 60 • Néel wall (see Fig. 6). Due to a specific distribution of the easy axes (6-fold symmetry), the equivalent Splay wall decomposes spontaneously into two 60 • Splay walls with the same handedness as shown by the wall profiles in Fig. 7.

In full similarity with the bcc case, the Néel wall profile may be fitted by a pure Bloch wall profile as displayed in Fig. 7. In contradistinction to the bcc case, however, no canting exists between equivalent spins belonging to two successive (111) planes as shown in the insets of Fig. 7. Had we, however, considered a Néel wall parallel to the 112 plane, then spin canting between AF coupled neighbouring spins would have been recovered.

The contribution of exchange interactions to the wall energy proves more complex than in the bcc case due to the existence of frustrated n.n. interactions. However, when limiting the evaluation of the exchange energy to n.n.n. interactions, the expression derived in the bcc case remains valid with J ij = J 2 and N = 6. The recorded exchange and anisotropy energies amount to 586.198 meV and 609.352 meV, respectively. With a global thickness close to 2.9 nm, a lattice parameter a = 0.41705 nm, and consequently,

d nn = a √ 2/2 = 0.2949 nm, d nnn = d nn √ 2 =
a, the 60 • Néel wall energy amounts to 20.254 meV/nm 2 whereas the wall profile fit yields an effective wall width parameter equal to ∆ * = 1.8814 nm. Wall width combined with energy provide effective values for the continuum limit of the exchange and anisotropy constants, respectively A * = 9.540 meV/nm and K iP = 2.687 meV/nm 3 .

C. Wall width and energy analysis

We have dealt above with 90 • and 60 • Néel walls in a bcc and a fcc lattice, respectively. It is straightforward to show that, due to a prevailing uniaxial anisotropy, spins are confined to their respective easy planes, namely the (001) and the (111) plane. The anisotropy energy may thus be reduced to k iP sin 2 2φ in the bcc case and k iP sin 2 3φ for the fcc lattice. Defining A and K as the exchange and anisotropy constants in the continuum limit, the energy density of these walls may be written as ε = A (dφ/dx) 2 +K sin 2 2φ or ε = A (dφ/dx) 2 +K sin 2 3φ owing to lattice, or, globally as ε = A * (du/dx) 2 + K sin 2 u, where u = 2φ, A * = A/4 in the bcc case, u = 3φ, A * = A/9 in the fcc case. The energy density expression in u entails a Bloch-type 180 • wall with width parameter ∆ * = A * /K and energy 4 √ A * K. The migration from a discrete spin system to a continuum is, in the ferromagnetic case, done classically as follows. Choosing here, as an example, the bcc lattice and referring back to the unit cell geometry depicted in Fig. 1, a Taylor expansion up to the second order of spins s(r) located at the apices of the unit cell may be performed w.r.t. the spin at the cell centre, ŝ0 , with, as a result, an exchange field reading:

B i Exch = J µ s j=1,N (ŝ j ) = J µ s (Nŝ 0 + a 2 ∇ 2 ŝ| 0 ) (8)
The laplacian term at site 0 and ŝ0 are the only surviving terms in the Taylor expansion after summing-up over the 8 n.n.. The ŝ0 field component exerts no torque on ŝ0 itself and, thus, may, in the ferromagnetic case, be discarded. In the continuum limit, the exchange field reads B Exch = (2A/M s )∇ 2 m, be it for the bcc or fcc lattice. Since M s , the saturation magnetization, is, for a bcc lattice, equal to 2µ s /a 3 , one finds A = J/a, a still being the lattice cell parameter.

In order to treat the AF case on a similar footing, let us first consider a F spin chain (1D) embedding a wall. If is the angle between two adjacent spins, the contribution to the exchange energy of that spin pair is -J F cos( ). For constructing the equivalent AF spin chain, just revert one spin over two. The energy per spin pair becomes -J AF cos(π ± ). In the spirit of Haldane [35], now define the order parameter as l = (-1) i s, where i is the spin index along the chain. For any equivalent spin pair along the chain, the exchange energy is either -J AF cos(π ± ), applying to s, or -|J AF |cos( ), applying to l. The argument is easily extended to 3D textures, with, as a particular example, l = (-1) k s in the case of the AF structures considered above, where k is the atomic layer index. The continuum limit values of the exchange and anisotropy constants remain therefore linked to their discrete counterparts as, bcc (M s = 2µ s /a 3 ):

A = |J|/a, K = 2k/a 3 , fcc (M s = 4µ s /a 3 ): A = |2J|/a, K = 4k/a 3 .
From the fit of the wall profile and the wall energy computation, the following parameters were extracted, A = 4A * = 48.827 meV/nm and K = 4.160 meV/nm 3 in the bcc case, A = 9A * = 85.860 meV/nm and K = 2.687 meV/nm 3 in the fcc case, to be compared to bcc: 52.012 meV/nm and 4.169 meV/nm 3 , fcc: 91.356 meV/nm and 2.757 meV/nm 3 , respectively. It may be noticed that, perhaps unsurprisingly, the agreement between respective anisotropy values is better than between exchange parameters. Altogether, however, agreement is considered as satisfactory.

IV. DIPOLE FIELDS

In their group theory analysis of dipolar crystals, Luttinger and Tisza [60,61] first recall that the field at a point r due to a point-like dipole at the origin is linear in the dipole's moment p, classically:

B(r) = µ 0 4π 3(p • r)r r 5 - p r 3 (9)
where p = µ s ŝ. It ensues that the field displays the same symmetry as the generating dipole array. The dipolar energy then is a quadratic function of the dipoles vector components and finding the energy associated to various ordering modes ultimately reduces to an eigenvalue problem. It has been shown [60][61][62] that, for an infinitely extended bcc lattice, the dipole field is strictly zero at lattice points provided the dipoles are oriented along any of the cube axes. For a fcc lattice, with AF2 ordering (i.e. F-ordered {111} planes), the dipolar energy proves minimum for moments uniformly ordered in the {111} plane, irrespective of the spin orientation within the plane (see also [5]).

A. Uniform order parameter

Dealing with ultra-thin epilayers inherently leads to truncations in the dipole fields sums, hence a dipole contribution to the overall energy at variance from the bulk. We first evaluate the dipolar field lattice response when the order parameter is uniform, i.e. in the absence of any spin texture. Note that, in this context, the (l, m) decomposition assumes the system to be devoid of any field, assuming the m distribution to be the sole source of dipolar field. The main results may be summarized as follows: i) bcc lattice : for a spin direction along a 100 direction within the epilayer plane, the field decays rapidly from the outer surfaces, in agreement with the full 3D analysis of Luttinger and Tisza [60,61]. A fairly large amplitude field however still subsists within the outmost atomic layers. The surviving field is dipole aligned (Fig. 8), a property found to be independent of the spin orientation in the (001) plane; ii) fcc lattice : for a spin direction within the epilayer plane, (111) here, the dipole field is also, dipole oriented. In agreement with the 3D analysis [60,61], the dipole field only weakly depends on the distance from outer surfaces (Fig. 9). For all these cases, the dipole field stabilizes the spin orientation.

A different picture emerges when the dipoles are oriented along the epilayer normal. In the bcc lattice, a sizeable field is still to be found in the layers closest to the surfaces (Fig. 10). The field is now, however, oriented opposite to the local moment and the field asymmetry between the bottom and the top surfaces is a consequence of the even number of atomic planes across the thickness, implying a spin orientation reversal between the outmost atomic layers. In the fcc lattice (Fig. 11), the field is now also oriented opposite to the local spin, and its modulus increases gently only in the immediate vicinity of the surfaces. Thusly, for dipoles oriented along the epilayer normal, the dipole field acts for an ultra-thin AF epilayer as a local demagnetizing field.

It is clear that the dipolar energy reduces to a local anisotropy term when dealing with infinitely extended crystals. Is it the case, too, when dealing with ultrathin epilayers ? It seems so since, both for the bcc or fcc lattices, dipolar interactions are consistent with the existence of an easy plane perpendicular to the epilayer normal. An anisotropy of the form k D s 2 z in the computation axes of Figs. 3,6 seems therefore adequate, provided k D be a lattice specific function of the altitude z within the AF layer.

B. Spin Textures

Up to this point, we have only considered a uniform order parameter. Do the dipolar properties above remain true in the presence of a spin texture, as assumed, long ago, by Yamada [63,64] in his study of walls in NiO ? Instead of attempting to answer globally to that question, we notice that the spin textures considered up to now are strictly planar. It follows that the z component of the dipole field is directly linked to the spin distribution In each F-ordered atomic layer parallel to the surface, a Splay wall gives rise to so-called σ charges, a Néel wall to dipolar or π charges as schematically shown in Fig. 14. Charges change sign with each unit increase or decrease of the layer index. In order to deal with fully compensated AF epilayers, we have solely considered an even number of atomic layers, be it for a bcc or a fcc lattice. Then, for any atomic layer in Fig. 14, pair summation, starting with the two layers adjacent to that under consideration, does not lead to a full balance of the fields. Thusly, a monopolar z component field is expected for a Splay wall, dipolar for a Néel wall, the sign of which is defined by the sign of charges closest to the interface and the free surface. It has been checked that the z field component reverses with a global reversal of the spin orientation.

According to the (l, m) decomposition scheme, the exact opposite result would have been reached. Although at first puzzling, this result is easily understood after realizing that, since l ⊥ m, the corresponding m wall is, for a Splay wall, akin a Néel wall and vice versa (see e.g. Fig. 7 in [36]). As such, however, the pure 'at lattice points' point dipole theory is not free of critics, either, as noticed early in [60] when alluding to weak disorder. The very fact that the 'at lattice points' picture proves sensitive to the exact local dipoles positions means a restriction to low temperatures.

Practically, dipole fields evaluation highly benefits from Fast Fourier Transforms (FFT) techniques as soon as the dipole distribution may be dispatched into as necessary a number of ferromagnetic tetragonal sublattices. The bcc case is simple, with only two sublattices. The fcc case proves much more demanding due to the incompatibility between an ABC sequence of (111) planes and the bi-(111) planes AF ordering, meaning a 6 (111) planes sequence, as recalled in Fig. 11. This added complexity may, however, rather easily be mastered. In this work, FFT techniques have been implemented overall and care has been taken to make sure that the FFT dipole fields evaluations were exactly equivalent to direct sum estimates. It remains nevertheless true that dipole fields estimates remain sensitive to the boundary conditions imposed by a finite size 'computation box'. In the computation axes of Figs.3 or 6, the y and z computation box limits are physically linked to the AF stripe width and epilayer thickness, respectively. The x limits are computation specific and care needs to be experienced in order to make sure that the x limits have the minimal incidence on the final results. Altogether, dipole fields exert a rather tiny influence on the converged wall textures, up to the point, actually, where it proves difficult to display their influence at the graph level, except for the s z spin component that remains smaller than 10 -4 under the prevailing anisotropy as displayed in Fig. 15 (the noise level in these computations is of the order of 10 -15 ). Similar results are obtained for Splay walls within a fcc lattice whereas Néel walls display bimodal s z profiles for both lattice types.

On the other hand, dipole fields allow for an estimate of the fringing field below the interface or above the free surface of the AF epilayer that may cast new insights into AF wall observability by means of, for instance, NV microscopy (e.g. [65] and loc.cit.). Fig. 16 displays the x and z components of the fringing field above the free surface corresponding to a) a Splay wall in a bcc lattice and b) a Néel wall in a fcc lattice. In the Splay wall case, the z component of the fringing field outside and wall-induced dipole field inside (Fig. 12) are both primarily unimodal, changing sign, however, upon crossing the free surface, in relation with the wall charge distribution just below the free surface (Fig. 14). Similar arguments hold true for the Néel wall. Such features, characteristic of ferromagnets, therefore find their counterpart in AF's, at least in layered AF's, such as those considered here. As anticipated, the fringing field decays rapidly with distance from the surface: the Splay wall z field component drops to less than 0.15 mT at 15 nm from the surface and its Néel wall counterpart to roughly the same value at only 5 nm.

A last remark ought to be made at the end of this section if wishing to find which part of the dipolar energy ought to be attributed to a wall. The only suggestion, here, is to perform the difference between the dipolar energies linked to two identical volumes, one containing a fully relaxed wall, the second solely containing the two domains separated by an abrupt transition. Following this procedure for instance for the Splay wall in a bcc lattice, one finds ∆E Dip = -92.02 -(-94.94) = 2.9 meV, i.e. a value small w.r.t. the exchange or anisotropy energies within the same volume (Section III A).

V. LATTICE INFLATION AS A ROUTE TOWARDS AF MICROMAGNETICS

Although the exchange energy per link is clearly associated to a precise distance between n.n. or n.n.n., no length explicitly enters the expressions of the exchange or anisotropy effective fields. It follows that lattice inflation, or swelling, allows for the computation of AF textures in larger volumes whilst preserving the native ordering within a given lattice. Defining the swelling ratio as S w , a pure number, the material parameters need to be scaled as a ⇒ S w a,

J ij ⇒ S w J ij , d DMI ⇒ S w d DMI and µ s ⇒ S 3 w µ s , k ⇒ S 3
w k in order to keep their continuum limit equivalents constant. We stress that characteristic lengths such as the wall width are left untouched in the swelling process. The wall energy per unit area is equally conserved. No rescaling of time needs to be envisaged, leaving an open access to the full physical frequency range, in particular those large frequencies imposed by the exchange field. Inversely, the swelling process introduces a cut-off in the spin-wave (or phonon) spectrum. Such properties evoke micromagnetics in ferromagnets, with, however, one difference: in F's, the continuum limit is commonly viewed as an intermediary step between atomistic modeling and projection onto a suitable mesh, the size of which is conditioned by the characteristic lengths of the system. Here, lattice-compliant inflation allows the intermediary step to be skipped. Swelling ratios in the range 2 -5 may easily be used, thereby potentially reducing the number of lattice points by a factor S 3 w for a given physical volume.

Validation of the lattice-compliant inflation scheme in the static limit may be gained from the inspection of Fig. 17 comparing wall profiles for two values of the swelling ratio, in both lattices. Only a slight global translation of the profiles w.r.t. the native lattice profile has been allowed for in order to keep a common wall centre. The agreement between profiles proves excellent.

Unfortunately, the inclusion of dipole-dipole interactions is more problematic due to the alternate nature of the dipole fields sums in AF's and their truncation in ultra-thin layers. Reducing the dipolar energy to a local anisotropy energy certainly helps, at least when almost constant across the thickness as such is the case for the fcc lattice. Reducing dipolar energy to anisotropy implies, however, the neglect of wall specific dipolar interactions. Those have been shown above to be small, but nevertheless real.

VI. ACTION OF SPIN-ORBIT TORQUES

Recently investigated metallic AF compounds include CuMnAs, Mn 2 Au, and IrMn or PtMn [9][10][11][12][13][14][15][16]. The collinear antiferromagnets CuMnAs and Mn 2 Au have in common a tetragonal unit cell and a "staggered" stacking of (001) planes meaning alternate spin orientations in successive (001) planes. Insulating AF compounds include CoO and NiO and the weak ferromagnet Fe 2 O 3 [17][18][19][20][21]. Depending, however, on the substrate orientation, these collinear antiferromagnets may exhibit different preferred orientations of the order parameter. For instance, if grown on Pt(111) or MgO(111), NiO exhibits preferred spin orientations in the (111) plane [17]. If grown on MgO(001), the preferred orientations in strained CoO or NiO epilayers belong anew to the growth plane, and follow 110 directions [18][19][20]49]. Note that, in neither case, is the spin ordering precisely known. If the general picture pertaining to bulk NiO or CoO holds true, i.e. alternate spin orientations in successive {111} planes, then the [110] and [110] directions are, in any given (001) plane, inequivalent in terms of spin stacking as demonstrated experimentally at the free surface of a NiO crystal [66].

Conducting and insulating AF's also differ in the nature of the current induced spin-torque. In the metallic structures CuMnAs, Mn 2 Au, the charge current is meant to flow homogeneously through the AF layer. For the insulating compounds, the current flows within an adjacent heavy metal layer, Pt most often. In one scenario, the spin polarization of carriers flowing across the heavy metal layer splits in the presence of spin-orbit interactions (Spin Hall Effect) and the associated spin-current may be absorbed at the interface [67,68]. A charge current flowing parallel to an interface with naturally broken inversion symmetry may also, in the presence of spinorbit interactions, generate spin currents [69]. Spin-orbit interactions imply a correlation between the conduction electrons momentum and their spin. Such concepts extended to conducting antiferromagnets with adequate structural characteristics lead to a spin-polarization unbalance per sub-lattice, with, as a consequence, a modulation of the carriers spin-polarization at the atomic scale [9,70].

From a purely geometrical point of view, however, torques acting on the localized moments may be simply differentiated as field-like, τ FL ∝ ŝ × p, and damping-like, τ DL ∝ ŝ × (ŝ × p), where ŝ and p are unit vectors along the localized moment and the polarization of the local spin current, respectively [69]. Application to antiferromagnets swiftly reveals the kind of interchange occurring between the field-like and damping-like torques whether p is uniform or reverses with ŝ, see Fig. 18.

The first experiments [9,17] attempted to demonstrate a direct electrical control of the AF order parameter by spin-orbit torques (SOT). Shortly after, however, the observation of domains in CuMnAs proved that wall motion indeed offers an alternative mechanism [11]. Similarly, in NiO, spin-orbit torques resulting from the application of current pulses were shown to lead to important reorganisations of the domain pattern [71]. On the other hand, current pulses in these early experiments prove altogether alarmingly long compared to typical precession times in AF's.

We consider first below the action of SOT's on the spin orientation within small volumes with uniform order parameter, second, the action of SOT's on certain wall types in the presence of DM interactions, still within the lattice-compliant formulation of the present work. More precisely, we now consider epitaxial AF layers with a bcc or fcc lattice in contact with a heavy metal layer. We assume the current to flow solely in the heavy metal layer: in the axes of Figs.3 or 6, a charge current flowing along the x direction produces a spin current polarized along the y direction. We do not envisage here the more complex case of systems like CuMnAs with their staggered spin-orbit fields (see [72]).

Fields corresponding to the SOT's may now be added to the effective field, namely 3/2 for the (111) interface of a fcc lattice. The above definition implies the SOT to be applied into a single atomic layer within the AF, namely that located at the interface. In the following, we stick to that definition, exchange interactions relaying the torque in the upper AF layers. Distributing the torque would have been equally possible. The difference between the two approaches actually remains immaterial.

B DL = B D [ŝ × ŷ]; B FL = B F ŷ; B D = 2
Finally, DM interactions have been experimentally shown to lead to remarkable spin textures in ultra-thin films such as spin spirals, homochiral successive walls or a skyrmion lattice [73][74][75]. The present implementation into an atomistic code adopts the 3 ions mechanism of Levy and Fert [76], owing to which two spin carrying ions belong to the interface plane, one ion, non-magnetic, to the layer just below/above. We also consider at first a fully crystalline stack, so that the non-magnetic ion remains in an epitaxial relation with the spin carrying layer. Practically, for a (001) interface within a bcc lattice, the spin carrying ions are n.n.n.'s and the non magnetic ion a n.n.. For each link in the interface plane, there exist two n.n. non magnetic ions in the heavy metal, justifying the use of the expression D ij = d DMI (û ij × ẑ) in the DM interactions energy. For a (111) interface within a fcc lattice, neighbours in the interface plane are n.n.'s whilst n.n.n.'s control exchange interactions within the AF layer. More importantly, for each link in the interface plane, there exists a single n.n. non magnetic ion [77] so that D ij does not any more belong to the interface plane. Moreover, two successive links around a given spin at the interface share a common n.n. non-magnetic ion. It follows that the out of plane component of the Dzyaloshinskii vector alternates at the atomic scale in full similarity to the Co/Pt(111) case. Although calling for further insight, we prefer at this stage to keep a unique definition of the Dzyaloshinskii vector. Then, for both lattices, the two magnetic ions belong to the interface and so does D ij . It ensues that a gain in DM interactions energy may only be achieved if the spin orientation leaves the interface plane.

All calculations below have been performed for purely indicative parameters, namely |d DMI | = 1.5 meV, J = -50 GA/m 2 , Φ SH = 0.1, χ = 0 or 0.25 and damping parameter α = 0.01.

A. Uniform order parameter

The main result in this section is that, switching-on the current only leads, whether χ = 0 or 0.25, to a minute reorientation of the spins, in full similarity with the ferromagnetic case. This minute reorientation occurs within a few ps for damping parameters in the 10 -2 range. For experimentally accessible currents, a direct control of the AF order parameter [78] proves simply out of reach, except, perhaps, for situations where huge spin currents result from the fast demagnetization of an adjacent fer- romagnetic layer [26].

Obviously, a drastically different scenario develops if, instead of starting with a spin orientation at equilibrium, one starts with a metastable equilibrium state. Such would be the case should the anisotropy landscape be altered due to e.g. rising thermal strains [79]. Sadly so, however, the SOT becomes simply superfluous.

B. Spin textures

Néel type walls within a ferromagnet (F) with perpendicular anisotropy have been shown to move with speeds up to a few 100 m/s under the action of SOT's in the presence of DM interactions [80]. More recently, walls within F stripes with an in-plane magnetization were also shown to move, albeit with a lesser velocity, due to cycloidal distortions of the domain wall structure associated with DM interactions [81]. In their 1D study of simple walls in AF's, Shiino et al. do find that (x, z) plane Néel and Splay walls (resp. z and x axis uniaxial anisotropy) propagate with similar velocities under SOT's in the presence of DM interactions ( [31] and associated Supplemental Material).

The results of the present study are similar. In a nutshell, fully planar ((x, y) plane) spin textures, whether Néel or Splay do not propagate under SOT's, both in the bcc or fcc lattice, even in the presence of DM interactions. These results are in line with a previous remark stating that DM interactions may only have a bearing when the out-of-plane spin component ceases to be zero and dipole field interactions are simply too weak to provide the nec-essary action. On the other hand, wall structures in AF's potentially leading to a high mobility under SOT's may be thought of.

In the bcc lattice, such walls include, for a (001) interface and a (010) easy plane, a 180 • Néel (Splay) wall in a system with leading perpendicular ([100] axis uniaxial) anisotropy, and for a (110) interface, a 90 • Néel or Splay wall. For the whole set of spin textures, (001) remains the ferromagnetic plane. The first two and last two structures are characterized by fully uncompensated (the 1st layer is ferromagnetic) and fully compensated interfaces, respectively. More precisely, each ↑ spin belonging to the (110) bcc interface has four ↓ n.n.n.'s, and vice versa. All these walls are chiral with a chirality axis along ŷ and, thus, couple efficiently to the damping-like SO field. Note that the wall chirality proves common to the various F sub-lattices. Results are summarized in Fig. 19. For these simulations, an initial clockwise (cw) chirality has been imposed to the spin textures (a)-(c), anticlockwise (ccw) for (d). Then, the correct sign is assigned to the DM interactions parameter d DMI so as to lead to a gain in DM energy. Physically, this would mean a particular choice for the heavy metal in contact with the AF layer. The inspection of Fig. 19 indicates that, for a cw chirality, d DMI needs to be > 0 in the case of a ferromagnetic interface, < 0 for a fully compensated AF interface, as a direct consequence of the vector product entering the expression of the DM energy. The sign of d DMI also needs to reverse with chirality. Owing to conventions valid here, a negative current density pushes a cw texture towards x < 0, a ccw texture towards x > 0. The analogy between ferromagnets [80] and antiferromagnets is thus, here also, striking.

Current densities in this work are in the low end of the density spectrum addressed in [31]. Furthermore, a rather conservative value of the damping parameter has been chosen (α = 10 -2 ). Wall velocities in the present lattice-compliant approach of AF's prove therefore equivalent to velocities reached in [31], i.e. velocities in the km/s range for α = 10 -3 . Pending a detailed analysis, the velocity scatter in Fig. 19 is to be attributed mostly to the variation in wall width, itself a function of the anisotropy landscape.

Lastly, we consider, for the fcc lattice, two spin textures, a Néel and a Splay wall, schematically represented in Fig. 20 and equivalent to the textures in Fig. 19 c-d. These still are spin textures belonging to a single (111) AF2 ordered Twin domain but now with a (110) interface. Similarly to the bcc case, the interface is compensated with also each ↑ spin belonging to the interface having four ↓ neighbours, and vice versa. The exact geometry proves, however, different since two of the ↓ spins are n.n.'s, two n.n.n.'s. Although possibly calling for the definition of two DMI parameters, we treat them here indifferently. Both of these walls are found to propagate effortless under damping-like SOT's provided n.n. exchange interactions are turned off [START_REF]The competition between DM and Heisenberg exchange interactions inducing non-collinear and collinear spin structures, respectively, may lead to complex magnetic ground states[END_REF]. As above, variations in velocity, here minute, are ascribable to anisotropy induced wall width modulations. Summarizing, domain wall mobility under SOT's in antiferromagnets is clearly related to wall geometrical characteristics, in full analogy with ferromagnets. It is therefore not surprising that wall mobility under SOT be largely independent of the subtending lattice. Similarly to the F case, successive partial angle walls in AF's (90 • in bcc or, 60 • or 120 • in fcc structures) may not be homochiral, and thus, may only move towards or away from each other under SOT's.

VII. OUTLOOK

In the absence of DM interactions, wall textures in ferromagnets are often the result of competing interactions between anisotropy characteristics and sample specific demagnetizing field distributions, exchange interactions being mostly isotropic. In AF's, anisotropy and latticecompliant exchange interactions set the rules. In particular, magnetostrictive spontaneous deformations and, at a first level, strain-induced anisotropy play a particular role. The latter have been neglected in this work as, except for the most recent work [79], in many previous studies and a suitable continuum mechanics remedy beyond strain-induced anisotropy needs to be found. On the other hand, this work casts new insights into the role of lattice-compliant exchange interactions and 'at lattice points' dipole fields. Lattice-compliant Heisenberg exchange interactions are the only needed exchange interactions in the bulk of AF's with inversion symmetry. Owing to lattice and wall geometry, these may, or not, induce phase lags in the spin canting between sub-lattices. Although complex, 'at lattice points' dipole-dipole interactions respect basic symmetries. Neither of these characteristics is met in the (l, m) decomposition models developed recently. Both for exchange and dipoles interactions, the lattice-compliant and the (l, m) decomposition model differ basically in the location assignment of two neighbouring AF coupled spins and the difference appears irreducible. An alternative, lattice-compliant, route towards AF micromagnetics is also being proposed. A finely tuned comparison with multipole-inspired models [START_REF] Suzuki | [END_REF] probably proves at this stage desirable.

FIG. 1 .

 1 FIG. 1. bcc environment : A ↓ spin has 4 ↑ nearest neighbours in the (001) plane below, 4 in the (001) plane above. The vertical (c) axis in the figure coincides with the [001] direction. Spin orientation owing to the sole legibility. Figure drawn with VESTA [38].

FIG. 2 .

 2 FIG. 2. fcc environment : Top perspective view of Ni ions belonging to 3 successive (111) planes (depth indicated by the atoms size). The c axis is normal to the plane of the figure and coincides with the [111] direction. a) For any given Ni ion within a given (111) plane, there exist 6 nearest neighbours (n.n.) in plane; b) n.n. arrangement around a given Ni ion one (111) plane up, one plane down; c) Ibid for next nearest neighbours (n.n.n.). Figures generated with VESTA[38].

FIG. 3 .

 3 FIG. 3. AF Néel wall in a bcc spin lattice. The [001] direction is normal to the plane of the figure and all spins tend to belong to the easy (001) plane where the easy directions are the cubic axes. The spin texture is displayed around the mid-plane of a 16 monolayer (ML) stack (≈ 2.3 nm thick), one (001) plane with ↑ spins (c), one plane above with ↓ spins (b). a) Spin location projection onto the (001) plane and axes definition. Only the spins belonging to rows with full symbols in a) are represented, i.e. one half of the spins, where red, blue symbols correspond to ↑ and ↓ spins, respectively. The relative shift between b) and c) reflects the symmetry properties of the bcc lattice. Distance between two spins, along x or y: a √ 2, where a is the lattice parameter.

FIG. 4 .

 4 FIG. 4. AF Splay wall in a bcc spin lattice. a) ↓ spins plane, b) ↑ spins plane. Same representation conventions as in Fig.3.

FIG. 5 .

 5 FIG. 5. bcc spin lattice: a) Néel (Bend) and b) Splay wall profiles corresponding to Figs.3 and 4, respectively. The large open symbols in b) correspond to a fit by a classical Bloch wall profile. Insets: superposition of -sx,y, ↓ plane (open symbols) and sx,y, ↑ plane (full symbols).

FIG. 6 .

 6 FIG. 6. AF Néel (Bend) wall in a fcc spin lattice. The [111] direction is normal to the plane of the figure and all spins belong to the easy (111) plane where the easy directions are the 211 directions in the (111) plane. The spin texture is displayed around the mid-plane of a 12 monolayer (ML) stack (≈ 3 nm thick), one (111) plane with ↑ spins (c), one plane above with ↓ spins (b). a) Spin location projected onto the (111) plane and axes definition. In agreement with the geometry of the AF unit cell, only the spins located at positions labeled with full symbols in a) are represented in b) and c), where full red, blue symbols correspond to ↑ and ↓ spins, respectively. Here, the shift between b) and c) takes place along the y directions and amounts to dnn √ 3/3, where dnn is the distance between nearest neighbours. Distances between two spins in the representation: dnn and dnn √ 3, along x and y, respectively.

FIG. 7 .

 7 FIG. 7. fcc spin lattice: a) Néel (Bend) wall profile corresponding to Fig.6 and b) decomposed Splay wall profile. The large open symbols in a) correspond to a fit by a classical Bloch wall profile. Insets: Ibid Fig.5.

FIG. 8 .

 8 FIG. 8. bcc spin lattice: a,b) Uniform spin distribution (sx = sy, sz = 0), left, and dipole field, right; a) Upper, b) Lower (001) plane. c) log of the dipole field modulus (T) across layer thickness.

FIG. 9 .

 9 FIG. 9. fcc spin lattice: a,b) Uniform spin distribution (|sx| = √ 3/2, |sy| = 1/2, sz = 0), left, and dipole field, right; a) Upper, b) Lower (111) plane. c) Dipole field modulus (T) across layer thickness.

FIG. 10 .FIG. 11 .

 1011 FIG. 10. bcc spin lattice: Uniform spin distribution (sx = sy = 0, |sz| = 1), left, and dipole field, right, in a (x, z) plane cross section representation embedding the whole layer thickness t. a) is located one plane beneath b). c) Log of the dipole field modulus (T) across layer thickness.

FIG. 12 .

 12 FIG. 12. Splay wall specific z dipole field component, a) bcc, b) fcc spin lattice, in layers n (top surface), n -1, ..., n/2 + 1 vs distance from wall center (nm). Decreasing symbol size with layer index (cf inset in a)). Left scale: spin components in the top atomic layer; right scale: z dipole field component (mT).

FIG. 13 .

 13 FIG. 13. Néel wall specific z dipole field component, a) bcc, b) fcc spin lattice, in layers n (top surface), n -1, ..., n/2 + 1 vs distance from wall center (nm). Decreasing symbol size with layer index. Left scale: spin components in the top atomic layer; right scale: z field component (mT).

FIG. 14 .

 14 FIG. 14. Schematic magnetic charge distribution across a wall in successive atomic layers and resulting wall induced dipole field (large red arrows). a) Splay wall, b) Néel wall.

15 FIG. 15 .

 1515 FIG. 15. Splay wall profile in a bcc lattice after relaxation including anisotropy, exchange and dipole-dipole interactions. Left scale: sx and sy spin components in the two top atomic layers, index 15 (red symbols) and 16 (blue symbols); right scale: sz spin component in the 8 topmost atomic layers indexed 16, 15, ..., 9, the missing profiles being obtained by symmetry around the epilayer mid-plane.

FIG. 16 .

 16 FIG. 16. Dipole field distribution above the free surface for a) a Splay wall in a bcc lattice, b) a Néel wall in a fcc lattice. Left scale: Bx (mT); right scale: Bz (mT). Bx profiles at distances a) 1, 2 .., 6 nm, b) 0.5, 1, .., 3 nm from the free surface; Bz profiles at surface level and distances a) 1, 2 .., 6 nm, b) 0.5, 1, .., 3 nm from the free surface.

FIG. 17 .

 17 FIG.17. Splay (bcc) and Néel (fcc) wall profiles as a function of lattice swelling ratio Sw. Continuous lines: native lattice, + and × symbols: Sw = 2, full red and blue symbols: Sw = 3.

FIG. 18 .

 18 FIG. 18. Action of the field-like (a,c) and damping-like (b,d) spin-orbit torques on an AF coupled bi-spin (SA, SB) in the case of a,b) a uniform polarization p of the spin current, c,d) a staggered polarization. Black arrows: initial spin orientation; red or blue arrows: spin orientation upon application of the torque.

A

  µs Φ SH ; B F = χB D where, B D and B F are the moduli of the damping-like and field-like Spin Hall fields and χ the ratio between them, J is the current density in the heavy metal layer and A the area per spin in the interface plane, e the electron charge (e < 0) and Φ SH the Spin Hall angle. For the bcc lattice, A is equal to a 2 for a (001) interface, d 2 nn √

FIG. 19 .

 19 FIG. 19. Spin textures and velocity under SOT. Left: four different spin textures within a bcc lattice with ŷ chirality axis. a,b) (001) interface, (010) easy plane and a) [001], b) [100] easy axis; c,d) (110) interface, 100 easy axes within the (001) plane. Right: wall velocity under spin-orbit torques. Simulation parameters: J = -50 GA/m 2 , ΦSH = 0.1, χ = 0 and α = 0.01. dDMI = +1.5 meV(a, b, d), -1.5 meV(c).

FIG. 20 .

 20 FIG. 20. Spin textures and velocity under SOT. Left: two different spin textures within a fcc lattice with ŷ chirality axis, (110) interface and 211 easy axes within the (111) plane. Right: wall velocity under spin-orbit torques. Simulation parameters: J1 = 0, J = -50 GA/m 2 , ΦSH = 0.1, χ = 0 and α = 0.01. dDMI = -1.5 meV(a), +1.5 meV(b).

ACKNOWLEDGMENTS

This work was supported by the French Agence Nationale de la Recherche (project PIAF, ANR-17-CE09-0030). Enriching discussions with A. Mougin, M. Viret, F. Piéchon and P. Simon, as well as a genuinely critical reading of the manuscript by S. Rohart are gratefully acknowledged.