The measurement of the luminosity recorded by the CMS detector installed at LHC interaction point 5, using proton–proton collisions at  R(t) of events produced per unit of time Luminosity, mentclass2pt{minim1 to 15% , using dA central component of the physics program at the LHC consists of measurements that can precisely test the validity of standard model (SM) predictions, e.g., cross sections for the production of electroweak gauge bosons , 5 or toStable luminosity information is also crucial to the efforts of the LHC operators to optimize the performance of the accelerator , 9. In tAn absolute luminosity scale is obtained with good accuracy using the direct method of van der Meer (vdM) scans –13. In tTo address these challenges, CMS employs a multifaceted approach, in which measurements from various individual subsystems are used to produce a final luminosity value with high precision, good linearity, and stability. Several methods and independent detectors are used to provide redundancy and to minimize any bias originating from detector effects.The LHC orbit is divided into a total of 3564 time windows 25involved . Furtherinvolved . The buninvolved .For Run 2 of the LHC, the period from 2015 to 2018 featuring wo years , in whicly small , and henment}mb  , the pilThis paper is structured as follows. In Sect. The CMS detector is a multipurpose apparatus designed to study high-n system , which iEvents of interest for physics are selected using a two-tiered trigger system . The firSeveral subdetectors, although not part of the main CMS data acquisition (DAQ) system, provide additional inputs  to the triggering system. The two beam monitors closest to the IP for each LHC experiment, the Beam Pick-up Timing for eXperiments (BPTX) detectors , are resrequired .The knowledge of the integrated luminosity requires stability over long periods of time, and hence benefits greatly from redundant measurements whose combination can lead to an improved precision. To that end, several upgrades were completed during the first LHC long shutdown (LS1), the transition period between LHC Run 1 (2009–2012) and Run 2. The main luminosity subdetectors (luminometers) in Run 1 were the silicon pixel detector and the HF. The HF back-end electronics, which were upgraded during LS1, consist of two independent readout systems: a primary readout over optical links for physics data taking, and a secondary readout using Ethernet links, explicitly reserved for luminosity data. In addition, two other luminometers were designed, constructed, and commissioned: the Pixel Luminosity Telescope (PLT)  and the The luminometers, along with the accompanying algorithms used to estimate the instantaneous luminosity in Run 2, are briefly described in the following. Figure pythia version 8.223 [Geant4  [The pixel cluster counting (PCC) method, which uses the mean number of pixel clusters in the silicon pixel detector, exploits the very large number of pixels in the inner part of the CMS tracking system. The number of pixels in 2015–2016 was about on 8.223  with theon 8.223 , 26 tunebout 0.5 , indicatOnly the components (modules) of the pixel subdetector that are stable for the entire period of data taking are used for the PCC rate measurements, excluding pixel modules known to be defective or significantly affected by the limited size of the readout buffer . The meaThe primary vertex counting (PVC) method uses the vertices that have been reconstructed using the tracks in the CMS detector. For this method, a good primary vertex is defined to be one with 11 or more tracks. This requirement is sufficient to suppress spurious vertices , and resThe PVC method is simple and robust, but suffers from mild nonlinearity effects when there are many collisions in a single bunch crossing. There are two competing effects. In one effect, primary vertices from two collisions occurring close to one another in space are merged, leading to an undercounting of vertices. In the other effect, the very large numbers of tracks associated with numerous collisions can produce spurious vertices, leading to overcounting. The precision with which these effects are understood falls short of the The HF luminosity measurement uses the separate readout described above, so the measurement can be performed at the full 40The computation of the HF observable is based on the occupancy method (HFOC). In this method, the fraction of channel measurements above an analog-to-digital converter (ADC) threshold is used for each bunch slot in a configurable time window. The ADC threshold is set high enough to avoid most noise and as low as possible otherwise. Both the ADC threshold and the integration time of the histograms between readouts are configurable, but they were fixed during data taking in 2015 and 2016. The number of valid measurements is also stored, so the fraction of events with hits above threshold can be computed.The PLT is a dedicated system for measuring luminosity using silicon pixel sensors, installed in Run 2, at the beginning of 2015. There are a total of 48 sensors arranged into 16 “telescopes”, eight at either end of CMS outside the pixel endcap. Each telescope contains three sensor planes which are arranged in a triplet that faces the IP. The sensors measure The BCM1F measures luminosity and beam-induced background separately. It consists of a total of 24 sensors mounted on the same carriage as the PLT. Single-crystal diamond sensors are used with split-pad metallization. Each sensor has two readout channels to keep the overall occupancy low, given the experimental conditions in Run 2. The BCM1F features a fast readout with 6.25The luminosity measurement based on the DT muon detector  is basedThe Radiation Monitoring System for the Environment and Safety (RAMSES) is a monitoring subsystem of the unified supervisory CERN system , 31. TheWhile not designed as a luminometer, the two chambers with the highest rates (designated PMIL55X14 and PMIL55X15) have been used to produce a luminosity measurement with good linearity and stability over time. However, similarly to the DT luminosity measurement, the overall rates are too low for bunch-by-bunch measurements or extracting an absolute calibration during vdM scans. The RAMSES luminosity is thus calibrated to the normalized PCC luminosity measurement and is used as an additional measurement for assessing the luminometer stability with time.Each bunch crossing gives rise to a certain number of To measure the instantaneous luminosity, we use the fact that Two algorithms have been developed for extracting the instantaneous luminosity. One method is rate-scaling, where the raw rate of observables is scaled with calibration constants to the luminosity. Rearranging Eqs.  and 3),,3), one The second method (zero counting) uses the average fraction of bunch crossings where no observables in a detector are produced. This zero fraction is then used to infer the mean number of observables per bunch crossing. The principal advantage of the zero-counting method is that it is not affected by cases where two or more separate signals overlap in the detector and produce only one reconstructed observable.p, then the probability of no observables seen in a bunch crossing with k interactions is thus simply p does not need to be known beforehand, since it is effectively absorbed in Assuming that the probability of no observables in a single collision is g to Eq. :6\documeAny luminometer requires an externally determined absolute calibration. Approximate  periods , a more Beam-separation scans are therefore performed to obtain calibrated  the ISR , extende the ISR , and havmentclasspt{minima the ISR . The keyx and y represent the horizontal and vertical coordinates in the plane transverse to the beams, and z.The instantaneous luminosity for a single colliding bunch pair in a colliding-beam accelerator is given by:x and y [If one assumes that the bunch profiles can be factorized into terms depending only on x and y , 11, theons, Eq.  reduces t the IP .w in the x direction, or an amount h in the y direction, as x direction, in which w is varied in a systematic way from y direction, the effective beam-overlap height is given byy coordinate. The effective value of We designate the luminosity when the beams are displaced with respect to each other by an amount  in Ref. , when a mentclasspt{minimaIn the LHC, bunches typically cross at a small angle mentclass2pt{minimWhile The total beam intensities are measured with the DC current transformers (DCCT) , and thex-y scan pairs. Figure The vdM scan program at IP 5 consists of a series of x and then in the y direction, although sometimes the pair is performed in the opposite order. In the vdM scans, the two beams are separated by up to 6Typical scan sessions consist of at least three vdM scan pairs, with one scan in each of the transverse coordinates per pair. There are two at the start of the fill and another at the end of fill. In the absence of systematic effects, all scans are expected to produce compatible results. In each pair, the scans are typically performed first in the x direction and once in the y direction, with a total of 10 steps of 60Dedicated length scale calibration (LSC) scans ; when the beams are displaced, rather than being head-on, a beam deflection and change in  at IP 5 , 45, andThe vdM method requires an accurate knowledge of the beam separation. Possible differences in the absolute scale between the nominal beam separation produced by the steering of the LHC magnets, as used in Eqs.  and 12)12), and The LHC beam currents are measured by dedicated devices. The FBCT system is used to measure the current of individual bunches in 25j denotes the BCID) is therefore given by:Both the DCCT and FBCT measurements are sensitive to additional charges outside the actual colliding bunch. These components must be measured and subtracted. The LHC radio frequency (RF) cavities operate at 400s in Eq. . The amoThe spurious charge is measured by the LHC LDM system, which provides a precise longitudinal distribution of the beam charge with a time resolution of 90The ghost charge is also measured using the beam-gas imaging method , 46, 47,Although the LHC beam orbits are generally stable during a fill, even a small variation (either random or systematic in nature) in the beam positions during scans can significantly affect the resulting calibrations. The beam positions are measured primarily using the DOROS BPM system. The LHC arc BPMs, when possible, are used to confirm the stability of the orbits during the scan.x and y in three 15To measure the orbit drift, we use the beam position measurements in x and y directions [x direction and 2.8 in the y direction. The measurements from the DOROS BPMs are integrated over all bunches. Therefore, the observed beam-beam deflection may be overestimated because of the inclusion of noncolliding, nondeflected bunches. In this analysis, a reduction factor of 0.6 is thus applied on top of the geometric factor in both years, which is the approximate fraction of the total number of bunches in the vdM fills that collide at IP 5. The orbit drift, as described above, is also subtracted from the single-beam DOROS measurements before forming the actual beam separation. Finally, an additional length scale correction is made to DOROS data for each beam and in both of the two transverse directions. The calibration using vertices, both for DOROS and nominal LHC positions, determines only the average length scale for the two beams. The calibrations of each beam are also not necessarily the same for the two sets of data. Therefore, a final, relative calibration of the DOROS data is made to align each beam in both transverse directions to the scale of the LHC beams. Figure y scans in 2015 and 2016 as well as the residuals per beam in a single scan, which shows symmetric behavior. The resulting impact on At each scan step, the actual beam separation can be also affected by systematic or random deviations of the beam positions from their nominal settings, which, in turn, impact the observed rate at each scan point. The magnitude of this potential bias is evaluated from consecutive single-beam orbit measurements at IP 5, provided by the DOROS BPMs and with a duration of a few seconds each. They are further corrected for the beam-beam effects  and the detector rate (incoherent) via the change of the beam shapes.The closed orbits of the bunches in the scans are shifted coherently by the angular kick induced by their electromagnetic repulsion, resulting in an increase in the absolute beam separation. The size of this additional beam-beam deflection depends on the transverse beam size, bunch intensities, collision optics, and separation between the orbits of colliding bunches. It is calculated based on the Bassetti–Erskine formalism for the electric field of elliptically distributed bunches, as discussed in Ref. . The orbmentclass2pt{minimThe incoherent effect corresponds to the change of the proton bunch density distribution functions potheses , 45. ConThe total beam-beam correction  overlap integral of the single-bunch distributions. The luminous region is modeled by a three-dimensional ellipsoid whose parameters  are extracted from an unbinned maximum likelihood fit of a three-dimensional Gaussian function to the spatial distribution of the primary vertices [In this method, which was inspired by Ref. , the lumvertices . The verj, are the sum of three individual Gaussian distributions y during the x-separation beam scan, which is indicative of nonfactorization. The goodness of fit is better than The bunch profiles This procedure is applied to all  scans in fills 4266 and 4954, and the results are summarized in Fig. The calibration scans described in the previous sections are performed with a small number of well-separated proton bunches with low bunch intensity. In contrast, during nominal conditions, the collision rate is generally maximized to produce large data sets for physics measurements and searches. This section describes the corrections that are applied to uncalibrated luminometer rates to ensure that the final luminosity values are accurate. These corrections, summarized in Table The measurements in most detectors have out-of-time pileup contributions that do not arise from the in-time n is proportional to The T1 afterglow generally only impacts the following bunch crossing because electronic signals tend to decline exponentially and hence two bunches later  model can be constructed that gives the response for a specific bunch crossing, accounting for contributions from all other 3563 bunch crossing slots. This model is referred to as the single-bunch response (SBR). The SBR for HFOC luminosity is taken directly from data in a reference fill with After the corrections from the SBR are applied, empty bunch slots, where there are no collisions, should have a rate of zero. For PCC, the SBR is determined by optimizing The empty bunch slots are also used to estimate the residual afterglow after the full set of corrections is applied. The corrected rate in the first empty bunch slot after a colliding bunch slot is used to estimate the residual T1 response. Likewise, the 2nd to 30th empty bunch slots are used to estimate the residual T2 effect. This procedure is performed for the entire 2015 and 2016 data sets for PCC and HFOC luminosity measurements. A window covering all residuals over the course of each data set is used as the systematic uncertainty in the final corrections. The resulting uncertainty for PCC in the two corrections is These types of per-bunch luminosity corrections are applied for PCC, HFOC, and BCM1F, whereas PLT is almost completely background free and no such correction is needed. Since the DT and RAMSES measurements integrate over all bunch crossings, out-of-time pileup corrections can only be applied on average to the integrated rates. For DT these amount to 0–1%, while no corrections are applied to RAMSES.A second type of T1 afterglow affects the HFOC luminosity. This is the case where the afterglow from a preceding bunch and the signal from the current bunch are both under the threshold to be counted as a hit, but their sum exceeds the threshold. This effect is referred to as the “bunch train effect”, because it affects only active bunches preceded by other active bunches . The method previously described for estimating T1 afterglow does not include this contribution. This effect is measured in a dedicated study comparing the double ratio of the leading bunch in a train relative to the second bunch for HFOC divided by the same ratio for PCC. A single correction model with magnitude 1–5%, linearly increasing with instantaneous luminosity, is determined utilizing most valid data from 2016.Radiation damage can affect the detector response by reducing efficiency, increasing noise, or both. Noise is typically a small effect for most luminometers, but reduced response in detectors due to radiation damage can have significant (percent-level) effects, and so corrections are required. Corrections are measured against a stable benchmark relative to the performance at or near the vdM scans, and are applied to A further efficiency correction is necessary for the PCC measurement. The pixel detector has a static internal memory buffer for data storage before the trigger decision is taken. When the buffer is filled, the oldest data overflows and are lost. This effect is proportional to the total instantaneous luminosity, and it can be estimated by studying the frequency of missing pixel clusters in otherwise well-reconstructed tracks . In 2016In the absence of out-of-time pileup, the PCC luminosity is expected to be linear, according to simulations, so no corrections are applied. Moreover, the ratios of PCC to both DT and RAMSES luminosity measurements are highly compatible as a function of the instantaneous luminosity without any corrections. The HFOC response in 2015 and 2016, on the other hand, exhibits significant nonlinearity compared to the other luminometers. The main source of nonlinearity is the uncalibrated ADC-to-charge conversion applied at the time of data taking. Data from fill 5416, which exhibit a wide range of instantaneous luminosity, are used to model the correction for HFOC with a fourth-order polynomial. This smooth function extrapolates to the As described in Sect. After the rate corrections are applied . In a perfectly stable system, the fractional rates among different subcomponents would exhibit no variation with time. Figure However, this method cannot detect global shifts in After the exclusion of invalidated data, which amount to We make use of two methods for assessing the detector linearity. The primary method compares the ratio of the instantaneous luminosity from two luminometers per fill as a function of the instantaneous luminosity, which is estimated from the numerator. A first-order polynomial fit is performed and the slope is extracted. The slopes per fill are then studied as a function of time. No significant deviation over time is observed between DT/PCC or RAMSES/PCC and HFOC/PCC, DT/PCC, or RAMSES/PCC in 2015 and 2016, respectively.To estimate the uncertainty, the fitted slopes are weighted according to the per-fill integrated luminosity. The mean values deviate slightly from 0, and the largest deviation is the systematic uncertainty in the linearity of PCC luminosity. Figure The alternative method makes use of the entire data set throughout the year, and extracts a single relative slope with a first-order polynomial fit. To remove effects from variations in the absolute luminosity scale over time, the per-fill ratios are shifted such that their extrapolation at zero luminosity is unity. The results are consistent with the primary method described above.“normalization” uncertainty in the absolute luminosity scale, “integration” uncertainty associated with The magnitudes of the corrections applied to the absolute normalization from the vdM calibration are listed in Table For each data set, final rate corrections and final calibrations are applied to data in small time windows of The dominant sources of integration uncertainty arise from the linearity and stability of the primary relative to secondary luminosity measurements over the course of each year  scans has been presented for data from proton–proton collisions recorded by the CMS experiment in 2015 and 2016 when all subdetectors were fully operational. The main sources of systematic uncertainty are related to residual differences between the measured beam positions and the ones provided by the operational settings of the LHC magnets, the factorizability of the transverse spatial distributions of proton bunches, and the modeling of effects on the proton distributions due to electromagnetic interactions among protons in the colliding bunches. When applying the vdM calibration to the entire data-taking period, the relative stability and linearity of luminosity subdetectors (luminometers) are included in the uncertainty in the integrated luminosity measurement as well.The resulting relative precision in the calibration from the vdM scans is 1.3 (1.0)% in 2015 (2016) at The final precision is among the best achieved at bunched-beam hadron colliders. Advanced techniques are used to estimate and correct for the bias associated with the beam position monitoring at the scale of In the coming years, a similarly precise calibration of the real-time luminosity delivered to the LHC will become increasingly important for standard operations. Under those conditions, the impact of out-of-time pileup effects is expected to be larger, but in principle they can be mitigated using techniques described in this paper.


Metarhizium brunneum. Here, we investigate whether a contamination of the environment can also influence the behaviour of foragers, both at the individual and collective level. In a Y-maze, Myrmica rubra ants had the choice to forage on two prey patches, one of which containing sporulating items. Unexpectedly, the nearby presence of sporulating bodies did not deter foragers nor prevent them from retrieving palatable prey. Ant colonies exploited both prey patches equally, without further mortality resulting from foraging on the contaminated area. Thus, a contamination of the environment did not prompt an active avoidance by foragers of which the activity depended primarily on the food characteristics. Generalist entomopathogenic fungi such as M. brunneum in the area around the nest appear more to be of a nuisance to ant foragers than a major selective force driving them to adopt avoidance strategies. We discuss the cost–benefit balance derived from the fine-tuning of strategies of pathogen avoidance in ants.Eusocial insects are exposed to a wide range of pathogens while foraging outside their nest. We know that opportunistic scavenging ants are able to assess the sanitary state of food and to discriminate a prey which died from infection by the entomopathogenic fungus  Food exploitation relies on the cooperation between the foragers, which explore nest surroundings and retrieve the discovered resources, and the internal workers, which store, process and distribute food inside the nest. In this cooperative exploitation of environmental resources, foragers play a key role because they regulate the dynamics of nestmate recruitment and orient the choice made by the whole colony among several alternative resources. The level of exploitation of a newly discovered food source will then depend on the decision of successful foragers to actively recruit nestmates by direct contacts and/or by the laying a pheromone trail on their way back to the nest3. The release of a recruitment signal by foragers, which will trigger a collective exploitation of food, depends on numerous factors such as the carbohydrate concentration of a food droplet7, the quantity of food10, its nature (e.g. prey or sugar solution)13, its distance from the nest15 but also the colony's current nutritional status18. A fine-tuning of recruitment signals by foragers allows an efficient adjustment of food exploitation to the current needs of the colony and usually leads the colony to choose the most rewarding sources (but see20).Insect societies prove to be highly efficient to discover and exploit available food resources in their environment21 and may eventually develop a fidelity to stable and rewarding food sites. Such a site fidelity has been observed in many different ants such as ponerine ants (Dinoponera quadriceps22), harvester ants  and myrmicine ants (Myrmica rubra26). Route learning coupled to fidelity to a foraging site minimizes the travel time, reduces the likelihood of foragers to be lost, speeds up the dynamics of food exploitation and eventually favours the choice of this site by the whole colony. Spatial fidelity of foragers increases with food abundance27, and depends on food nature with a greater tendency to thoroughly explore an area where they previously found carbohydrates instead of insect prey24.Besides the social information mediated by pheromone trails, ant individuals improve their navigational skills over successive foraging trips28 and the ambient temperature can influence the running speed and the searching behaviour of foragers29. Besides, biotic factors such as the presence of predators or competitors are known to act upon ants’ foraging, both at the individual and collective level. For example, predators (Formica sanguinea) near a food source influence the choice of Lasius pallitarsis and Myrmica incompleta foragers, which abandon a rich source when the risks incurred outweigh the expected energy benefits30. Likewise, competitors can alter foraging in different ways by either making the ants avoiding the location31, by increasing recruitment to food32 or by eliciting no response33.A large number of studies have thus demonstrated that the foraging behaviour of ants closely depends on the characteristics of available resources as well as on the current needs of the colony. Far less is known about the impact of the surrounding environment as a driving force that can shape ants’ foraging responses. As regards to the abiotic environment, the soil substrate can modify the collective choice of a foraging routeSolenopsis invicta34 and Pheidole dentata35). Likewise, microparasites such as fungi, virus or bacteria could shape the strategies of food exploitation in ants. These microparasites are assumed to exert a strong selective pressure on the insect societies, namely because they are composed of genetically-related individuals that live at high densities in a confined nest36. To counter sanitary risks, insect societies rely on social immunity which encompasses adaptive cooperative defences against pathogens having evolved at the physiological, behavioural and organisational levels37. Social immunity is namely achieved through grooming38, use of antimicrobial substances on nest materials39, discarding of waste42, removal of dead nestmates44 or self-exclusion of moribund workers46. A first line of defence against pathogens consists in the prevention of their entry inside the colony, notably by avoiding to retrieve contaminated food items. This sanitary control is particularly important for scavenging ant species since dying prey or insect cadavers commonly host a wide range of entomopathogenic microorganisms.In addition to competitors and predators, parasites near a food source or on foraging routes represent another major risk, not only for the foragers themselves but for the colony as a whole. For example, the presence of parasitoid flies near the ant nest leads to a decrease in its foraging activity  and the potential selection of a food source depending on the nearby presence of sporulating cadavers. We assume that a time-and spatially-limited search by individual foragers will occur over conidia-contaminated areas since it would reduce the sanitary risks incurred.In this study, we investigate whether the propensity of left = 18; Nright = 27; binomial test: P = 0.23), suggesting there was no orientation bias due to external factors in the surrounding environment. Furthermore, exploring ants did not seem to be repelled nor attracted from a distance by sporulating items. Indeed, foragers that first explored the uninfected foraging area (UFA) were as numerous as those that first headed towards the contaminated foraging area (CFA) . During this first trip, we observed a very small number of grooming behaviours , which occurred equally on the UFA and on the CFA (7 grooming events on each area) and which lasted as long on the UFA  as on the CFA  .When making their first trip to the Y maze, individuals did not preferentially explore the left or the right platform (NUFA = 17 trips) as to the CFA (NCFA = 11 trips) . We also found that the propensity for an individual to return several times to the Y-maze was not altered by its exposure to a contaminated environment. Indeed, there was no significant relationship between the proportion of visits made by a forager to the CFA and its total number of foraging trips . Overall, foragers stayed on each foraging platform for a similar amount of time regardless of the sanitary state of the environment . This staying time however decreased over the successive foraging trips, most probably reflecting a reduced exploration of a better-known area and an improved foraging efficiency .During the allocated time (i.e. 90 min), individual foragers made a highly variable number of trips towards the Y-maze, which ranged from 1 to 26 trips . In total, a prey retrieval was as likely to occur when the ants visited the UFA  or the CFA . As expected, the sporulating items on the CFA were never collected by the foragers.Concerning the exploitation of the prey patches, the majority of the visits made to a foraging area  resulted in the retrieval of a cold-killed fly. Interestingly, the decision of individual foragers to retrieve a prey was not influenced by the vicinity of sporulating items . This means that, globally, the foragers were as likely to visit each of the two platforms regardless of the presence of sporulating items.Ant individuals may differ in their propensity to detect pathogen and to avoid a contaminated area. Therefore, we examined whether some individuals showed a preference to forage in one of the two areas. A preference index was calculated for each forager by averaging its scores when visiting the UFA (score of + 1) and the CFA (score of − 1). We found that the distribution of the individual preference indices followed a normal distribution . By taking into account all the trips (two trips or more) made by a forager, we estimated the spatial fidelity of each individual by counting the number of times it changed or sticked to the same area over two successive trips. We found that the fidelity indices of individual ants (N = 39 ants) were not normally distributed  and that their median value was significantly higher than 0 . Furthermore, the ants did not learn to avoid risky areas over time since the proportion of foragers first visiting the CFA never differed from 0.5 across their successive trips to the Y-maze  or the CFA . Thus, the propensity of foragers to return to the same foraging area was not influenced by its associated sanitary risk . Thus, colonies mobilized a similar number of workers towards the two patches of prey regardless of the presence of sporulating items in their vicinity.We compared the 5-min flows of ants that reached each foraging area. The flows of ants towards each platform increased rapidly during 1 h and then decreased until the end of the experiment. The flows of ant foragers significantly changed over time .Only conidia-free prey were retrieved from the UFA and CFA. The dynamics of prey retrieval did not significantly differ depending on the sanitary risks of the foraging environment , sporulating nestmates (in Formica rufa51) or items covered with fungal conidia . The level of pathogen avoidance however differs according to the feeding ecology of the ant species50. Species that store food in their nest are more likely to avoid retrieving items covered with conidia than species that quickly consume the collected food. The stage of development of the pathogens can also influence the foraging behaviour of ant workers. While Myrmica rubra foragers may retrieve an insect that was artificially covered with a huge amount of conidia after its death, they consistently avoid to take a sporulating prey that bears conidia due to succumbing from infection48. Far less is known about the ability of workers to detect sanitary risks that are not directly associated to the food itself but instead, to the environment where foraging is taking place. We found that the vicinity of infectious sporulating items does not alter the willingness of Myrmica rubra ants to retrieve conidia-free prey. Likewise, we found no influence of sanitary risks on the collective choice of a foraging area nor on the staying time of ant workers on it. These results go against the expected avoidance and limitation of time spent by foragers on a potentially health-hazardous environment.Once brought by foragers inside the nest, pathogen-bearing food items become a sanitary threat for the whole colony and may eventually lead to a disease outbreak. Therefore, foragers usually avoid to retrieve sporulating insects . In the present study, the incidental contamination of foragers’ body by about 30,000 conidia per individual did not trigger more self-grooming responses and led to low rates of mortality with no dead nestmates showing any signs of fungal infection. This suggests that self-or allo-grooming of foragers may have taken place inside the nest, thereby contributing to reduce the amount of conidia on their body to sublethal doses.To counter the risks of getting infected when exploiting food in an environment contaminated by pathogens, ant foragers are expected to display hygienic behaviours. However, in several ant species, grooming events have been reported only when ant’s body was contaminated by very high doses of entomopathogenic fungi  or when conidia were applied on soil instead of a more artificial substrate63, ant colonies suffered from far lower rates of infection and mortality. Interestingly, a 1-year field study64 on various species of leaf cutting ants also found less than 0.1% of live individuals being infected, despite the high levels of Metarhizium fungus in the soil inhabited by the colonies. Finally, there is no record of natural epizootic events in ant colonies and natural infections are rather isolated and very rare (reviewed in 57). Taken all together, these studies suggest that the complexity of a more realistic environment makes the actual exposure of ants to pathogens and the incurred sanitary risks much lower than commonly reported in laboratory studies. Furthermore, the social life of ants also contributes to their resistance against microparasites. Pathogenicity is typically a sigmoidal function with threshold doses leading towards a maximal mortality65. Cooperative behaviour between workers inside the nest may contribute to keep the amount of parasites below the threshold of a significant pathogen-induced mortality. This can be achieved by a careful inspection and grooming of workers that returned to the nest67. Besides the active removal of parasites by mutual grooming, frequent contacts of foragers with nestmates could lead to a passive transmission of conidia and hence to dilution effects reducing the per-ant capita exposure to sublethal doses of pathogens65. One may even consider the possibility of delayed benefits such as certain forms of acquired immunity. This would confer an increased specific individual resistance and a higher survival rate after a first exposure to pathogens . Nevertheless, immune priming in ants is still under debate73 and awaiting to be investigated in Myrmica rubra species.The lack of fungus-induced mortality as well as the lack of avoidance of a contaminated environment on behalf of Metarhizium actually act as a strong selective force on ant societies is another matter, for which the answer needs to be mitigated. As regards to foraging, several ant species have evolved efficient and non-specific behavioural strategies that limit nest contamination such as forgoing the retrieval of sporulating dead insects51 or fastly discarding wastes42. On the other hand, we found that the contamination of the environment where foraging takes place does not influence the behaviour of ants. The vicinity of the fungal pathogens near exploited food sources does not deter ant workers nor alter their spatial fidelity to these foraged areas. This unveils that more naturalistic conditions for ants to get infected, such as walking close to conidia-releasing items, expose the ants only to sublethal doses of pathogens. The resulting sanitary risks can be too low for ants to display a systematic avoidance of contaminated zones, what would be costly in terms of restriction of areas still available for foraging. The survival, the susceptibility and the behavioural response to generalist pathogens is known to differ among ant species, namely according to their feeding ecology60. Here, we suggest that the circumstances under which ants encounter microparasites also contributes to the high variability of their avoidance strategy. In the case of foraging, the presence of generalist fungal conidia near food sources seemed to be more of a daily nuisance to the ants, rather than a major threat for the survival of ant societies.It is commonly accepted that parasites and pathogens represent a potential threat shaping the life history traits of the host species. Whether generalist entomopathogenic fungi such as Myrmica rubra is a polygynous and monomorphic ant species living in temperate zones of Europe. This species can be found in a wide range of environments, with a preference for semi-humid habitats such as stumps of dead trees, loose soil under stones or meadows75.Metarhizium brunneum is a generalist entomopathogenic fungus commonly present in the soil, including in the nesting sites of M. rubra ants59. Once in contact with the insect host, M. brunneum conidia attach themselves to its body surface and develop hyphae that pierce its cuticle by releasing fungal enzymes. The fungus mycelium then grows inside the insect body, what ultimately causes its death. Several days after the host death, the fungus begins to sporulate over the corpse and produces new mature conidia that will be released in the environment76.M. rubra were collected in June 2019 in Falisolle  in Belgium. They were placed in plastic boxes (40 × 25 × 7 cm) whose edges were coated with Fluon to prevent ants from escaping. We kept colonies in the laboratory at a constant temperature of 21 ± 2 °C, at 50 ± 5% relative humidity and at a photoperiod of 12L–12D. The nests were made of glass test tubes with a water reservoir and covered with a red plastic filter. We supplied colonies ad libitum with water and sugar (0.3 M sucrose solution) and with mealworms (Tenebrio molitor) twice a week as a source of proteins.Colonies of Drosophila melanogaster fruit flies . Fruit flies were reared on a home-made food substrate  at a room temperature of 21 °C and a 50% relative humidity. We dropped raffia strings on the mixture in order to facilitate the pupation of larvae. We renewed the food substrate once per month.As a food source, we used 77). The conidia-covered flies were then isolated in a plastic box with water and sugar, with death occurring approximately 4–5 days later. Their corpses were collected on the day of the insect death, washed with ethanol and then rinsed with distilled water according to the Lacey method78 to prevent the growth of other opportunistic microorganisms. We then placed the fly corpses on a moist filter paper in Petri dishes hermetically sealed and kept them in a thermostatic cabinet at 25 °C for a period of 2–7 days until the fungus started to sporulate.As a source of contamination, we used sporulating fruit flies. They were obtained by vortexing living flies in an Eppendorf with a sporulating fly cadaver previously (four times during 5 s at 1500 RPM according to protocol in Ref.2) where the prey patches were deposited. The ant nest was connected to the Y-maze by an access ramp (15 × 1.5 cm). A removable part (1.5 cm × 1.5 cm) was cut on the common trunk at 2 cm from the access ramp. When required for following up ant individuals, this removable part allowed to give access to a single worker at a time . During the entire experiment, the Y-maze was recorded from above using a webcam Logitech Pro HD C920.The experimental set-up consisted in a Y-maze leading to two separate platforms. The Y-maze was composed of a common trunk (10 × 1.5 cm) and two branches (8 × 1.5 cm), forming an angle of 60° and each leading to a foraging area (9 cm2) representing a triangle or a square, were placed on the side of each branch.In order to facilitate ants’ navigation and memorization of food location, we added visual cues to the experimental setup: (1) a white LED light source was centred between the two branches of the Y-maze and (2) two black pictograms of identical area , the food source consisted in 15 cold-killed and conidia-free flies that were evenly distributed over the area. On the other platform referred as the contaminated foraging area (CFA), the food source also consisted in 15 cold-killed and conidia-free flies but they were deposited close to 10 7 conidia. Indeed, the number of conidia covering the body of a sporulating fly was estimated to around 106 spores following the procedure described in Ref.49. In order to prevent any unwanted contamination of the UFA by conidia dispersed through air flows, all the experiments were carried out inside a closed box under still air conditions. Besides, the renewal of papers over the Y-maze between successive trips as well as the very few number of ants that explored first the CFA and then the UFA (6% out of 287 trips) made a contamination of UFA due to spores incidentally transported by walking foragers very unlikely.The ten sporulating items on the CFA represented approximatively a level of contamination of 10Experiments were carried out on ten colonies composed of 200 workers , one queen and 50 larvae (second and third stages). In order to motivate foragers to search for prey, colonies were starved for 3 days before the experiment by depriving them from mealworms and sugar sources.The day before the experiment, 25 individual foragers that were wandering outside the colony were gently removed. After being chilled at − 10 °C for 30 s, they were marked with a unique colour code on the abdomen (two dots of enamel paint marker Edding™ 8750). Once marked, the ants were isolated for 3 h to avoid paint removal by allo-grooming nestmates, and then were reintroduced inside their colony.For the individual experiments, a single marked forager was allowed to enter the Y-maze at a time and was followed over its consecutive trips to the Y-maze. The experiment started as soon as a marked ant entered the Y-maze and reached the bifurcation leading to the two foraging platforms. The experimenter then raised the removable bridge to prevent any further passage of other workers. The experimenter identified the ant in real time thanks to its colour code. Once the ant had visited a foraging platform and had returned to the common trunk of the bridge, the experimenter allowed this ant to exit the Y-maze by placing again the removable section of bridge .At most 5 min were needed for the ant to reach one of the two platforms. Then, it was allowed, for at most 20 min, to inspect the prey source and to exit the Y-maze. Preliminary experiments showed that this amount of time was sufficient for 93% of the individuals (N = 13) to freely explore the foraging platform. We allowed each marked ant to return on the Y-maze as many times as it wanted. The consecutive trips made by the same ant were followed for a maximal duration of 90 min after its first entry on the maze.During the experiment, the observer noted which platforms were explored by the marked ant over its successive foraging trips on the Y-maze. We defined a trip as the exploration made by the ant from its entry on the Y-maze until its exit. We defined a visit as the exploration made by the ant over one of the two platforms. A worker could thus do several visits during the same trip when it explored several times either one or the two platforms before exiting the Y-maze. The number of visits to the conidia-contaminated area allowed to assess whether sanitary risks alter the exploration as well as the spatial fidelity of workers to a foraging site. Likewise, the number of trips to the Y-maze quantified the persistence of ant individuals in being engaged in a foraging activity. We also examined whether the presence of sporulating items on the CFA reduces the level of food exploitation -here the number of retrieved flies. A fly was considered as being retrieved when being transported until the end of the Y-maze.In order to exclude any effect of trail pheromone on ants’ orientation and choice of a foraging platform, the common section as well as the branches of the Y-maze were covered with a white paper which was renewed between successive observations of individual foragers. Furthermore, in order to avoid any left–right bias on ants’ choice of a food source, the location of the contaminated foraging platform was switched for half of the ten replicates.Each experiment lasted at most 3 h. Every time a worker retrieved a prey, we re-adjusted the number of palatable flies on the platform in order to prevent any bias due to food depletion.In total, we followed up 45 marked ants over the ten tested colonies  or close to ten sporulating items (CFA). Once the colony was connected to the Y-maze, workers were free to explore it, to forage on the platforms and to collect dead flies for a total duration of 3 h. The number of available conidia-free prey was not re-adjusted during the experiment.We recorded the Y-maze throughout the experiment with a webcam (Logitech Pro HD C920) to observe ants’ behaviour. On video recordings, we quantified the dynamics of prey retrieval as well as the flows of ants, i.e. the number of foragers entering each foraging area over successive 5-min time intervals.78) and placed them in a temperature-controlled cabinet in order to check whether M. brunneum fungus had caused the ants’ death.After the end of the experiment, we measured the daily mortality of workers in each colony during 14 days. We washed the ant corpses . Then, we put these workers in an Eppendorf containing 1 mL of Triton X 0.05% solution and vortexed it five times for 10 s. We removed the workers and centrifugated the solution containing the conidia during 10 min at 1600 rpm. After removing the supernatant, we added 500 µL of Tween 20 solution and we scattered the pellet by vortexing it. Finally, by counting the number of conidia with a Thomas’s cell under a microscope, we estimated that the foragers’ body was covered by around 30,000 conidia per capita.79 and all tests were two-tailed (excepted when specified) with a significance level of alpha = 0.05. Data were expressed as medians, 1st and 3rd quartiles , except for the flows of ants towards a foraging area which were averaged (mean + SD). All the figures were achieved with the package “ggplot2”80.All data were analysed with R software (version 4.0.2)To check whether external factors in the laboratory had biased the orientation of ants, we tested whether the proportion of workers that first discovered either the left or the right platform differed from random by using a binomial test. Likewise, we examined whether ants were attracted or repelled from a distance by sporulating items on CFA by analysing with a binomial test whether the proportion of foragers that first explored the UFA or the CFA differed from 0.5. We also compared with a Mann–Whitney test the total duration of grooming events performed by foragers while first visiting the UFA or the CFA.MASS”81). To study whether individuals headed toward the same foraging area during their second trip, we performed a GLMM  by including the sanitary state (UFA vs CFA) as a fixed factor and the individual ID nested in colonial ID as random factors. This GLMM was carried out with the R-packages “lme4”82 and “lmerTest”83. We checked that data met the models’ assumption and showed no under or over-dispersion based on model deviance/degrees of freedom values. When significant, we performed pairwise comparisons with Tukey adjustment of p values, using the R-packages “emmeans”84 and “multcomp”85.We checked whether the level of exposure to sporulating items altered the willingness of workers to stay engaged in foraging. We thus used a Spearman test that correlated the proportion of visits made by each worker toward the CFA to its total number of trips on the Y-maze. We also investigated whether the time spent by foragers on a foraging platform depended on its contamination and changed across the successive trips to the Y-maze. To this aim, we carried out a general linear mixed model (GLMM) analysis with the sanitary state of the platform (UFA vs CFA) and the rank of the trip (1st to 8th trip done by the ant) as fixed factors and the individual ID nested in colonial ID as random factors (quasi-Poisson distribution) , we calculated a preference index that reflected its propensity to forage on one of the two platforms. This preference index was calculated as follows: a score of 1 was allotted to each visit to the UFA, a score of − 1 to each visit to the CFA and these scores were averaged on all the visits made by the forager. A preference index close to − 1 or to 1 means that the individual visited mainly the CFA or the UFA respectively. A preference index of 0 means that a worker equally visited both foraging areas. As the distribution of preference indices met the normality conditions (Shapiro-Wilk’s test), we performed a Student's t-test to compare it with a normal distribution centred on a theoretical value of 0 (corresponding to a no-preference).We also investigated whether individual foragers developed a spatial fidelity to one foraging area regardless of its sanitary state. For each new trip made by a forager on the Y-maze, we observed whether the first-visited platform was the same as the one it had left on the previous trip. If the ant successively visited the same platform, we assigned a score of 1 and if the ant switched to the other one, we assigned a score of − 1. We calculated the fidelity index of an individual by averaging the scores over all the visits made by the individual. A fidelity index of − 1 means that the ant systematically shifts from one foraging area to the other while an index of 1 accounts for an ant being perfectly faithful to a single platform. As the fidelity indices were not normally distributed, we used an one-sample Wilcoxon signed rank test (one-tailed) to check whether the fidelity indices of individual foragers were higher than the theoretical value of 0 expected if the ant was as likely to return to the same platform as to switch to the other one. Since exposure to the CFA may alter the spatial fidelity of foragers, a Spearman correlation test correlated the fidelity indices of workers and the number of visits they made on the CFA. We also used two binomial tests to check whether (1) the proportion of workers returning to the previously visiting foraging area as well as (2) the proportion of workers that first visited the CFA differed from 0.5 on each of the first eight trips to the Y-maze.We measured the flow of ants arriving at either the CFA or the UFA every 5 min for 3 h of experiment. These flows were analysed with a GLMM  by including the sanitary state of the area as well as the time as fixed factors and the colony ID as a random factor . We also compared the retrieval time of the first prey as well as the last prey between the UFA and CFA by using a Wilcoxon signed rank test.Changes over time in the number of flies still available on each platform were compared for UFA and CFA conditions by using a log-rank test (with the R package “Supplementary Information.


The authors would like to make a correction to their published paper .On page 13, in the online version of this article, the “fluorescence image in Figure 4D″ was mistakenly misplaced with a fluorescence image of Figure 2C during submission/revision. The authors would like to change Figure 4 with a new corrected version (with replaced The change does not affect the scientific results.The rest of the manuscript does not need to be changed. The authors would like to apologize for any inconvenience caused. The original manuscript has been updated.


Proteins fulfill complex and diverse biological functions through the controlled atomic motions of their structures . The protein composition is given by its amino-acid sequence, which was assumed to encode the function. However, the discovery of functional sequence variants proved that the functional encoding does not come down to the sequence, otherwise a change in the sequence would mean a change of function. Likewise, the discovery that function is fulfilled by a set of structures and not by a unique structure showed that the functional encoding does not come down to the structure either. That leaves us with the possibility that a set of atomic motions, achievable by different sequences and different structures, encodes a specific function. Thanks to the exponential growth in annual depositions in the Protein Data Bank of protein tridimensional structures at atomic resolutions, network models using the Cartesian coordinates of atoms of a protein structure as input have been used over 20 years to investigate protein features. Combining networks with experimental measures or with Molecular Dynamics (MD) simulations and using typical or ad-hoc network measures is well suited to decipher the link between protein dynamics and function. One perspective is to consider static structures alone as alternatives to address the question and find network measures relevant to dynamics that can be subsequently used for mining and classification of dynamic sequence changes functionally robust, adaptable or faulty. This way the set of dynamics that fulfill a function over a diversity of sequences and structures will be determined. Proteins fulfill complex and diverse biological functions through the controlled atomic motions of their structures . The proThe next alternative is that sets of atomic motions, achievable by different sequences and different structures, encode a specific function . This isNetwork science is appropriate to study system dynamics from protein structures because it offers multiple avenues to study the complex spatiotemporal relationships between interacting entities . IntegraThe advantages of network based models in probing protein dynamics come from the inference of amino acid and atomic links from the structure. Now, one on-going question is to clarify why network measures pinpoint functional residues  or distinguish disease mutations from the rest of the residues in order to better understand what properties amino acids have in a structure that make them functionally tolerant to mutations or not. The comparison of network measures and network models over proteins spanning large dynamics scales from enzyme to pore-forming toxins and over their sequence variants will help validating network measures as hallmarks of functional dynamics and of functional dynamic perturbations related to diseases.One alternative perspective to network integrative approaches is to find network measures that are relevant to functional dynamics simply from protein structures. This implies a network measure probing collective slow motions and therefore shared across proteins and independent of amino acid features as observed from global modes . In addiWe consider the neighborhoods of each amino acid of a protein as potentially relevant to the problem. This is because on average over its neighbors every amino acid makes moderate and similar number of atomic interactions, a property shared by many different proteins . In addiTo analyze neighborhoods, a protein structure is modeled by an amino acid network (AAN), where the nodes are amino acids and the links are atomic interactions between amino acids, inferred from atomic proximity . The spaOur case study is the third PDZ domain of the synaptic protein PDS-95 (PDB 1BE9) chosen because the functional impact of most of its single amino-acid mutations is known from experiments as well as some double mutations . Thus, tStarting from the Protein Data Bank (PDB) data, protein structures are modeled using the Amino Acid Network (AAN), an established model in Computational Biology . The AANi and j are connected if there exists at least one couple of atoms, one belonging to i and one belonging to j, at a distance lower or equal to a given threshold (Cutoff distance c),Nodes of the AAN: Each node in the AAN corresponds to one amino acid of the protein’s structure named according to the protein sequence:c = 5 Å, that is a threshold for chemical interactions, the link weights measure the number of atomic interactions between two amino acids.Link weights of the AAN: Each link is weighted according to the number of atomic couples that respect the cutoff condition:ki, defined as the number of amino acid neighbors of a node i, measures the amino-acid packing around the amino acid i, referred to as the amino-acid neighborhood. The node weight wi is defined as the sum over all the weights of the links that connect the node i to its neighbors (N(i) the set of neighbors of node i). The node weight measures the atomic packing around the amino acid i, referred to as the atomic neighborhood.Packing around amino acids: In the AAN, the node degree Cutoff distance: Different cutoff distances are used in this study such that the packing around each amino acid is described at different length-scale via the neighborhoods at variable cutoffs. The cutoffs are integers and range from 3 to 11 Å such that the packing within chemical reach (≤5 Å) and above chemical reach (>5 Å) are monitored. The rational is to distinguish amino acids by their ‘chemical’ neighborhoods and above chemical reach neighborhoods to probe space occupancy involved in multiple spatiotemporal scales.Plateau versus linear degree dependencies: The degree dependency with the cutoff is plotted. Some amino acids show a linear dependency of the degree with the cutoff and are referred to as linear amino acids. Some amino acids exhibit a plateau over some cutoffs determined with the derivative (here just Δk because the cutoffs are consecutive integers) equals to zero or equals to one if and only if the derivative at cutoffs before the plateau is equal to four or more. We refer to these amino acids as plateau amino acids.Amino acid side chain length classification: Amino acids are classified by side-chain lengths as follows. Side chain length <3 Å are small amino acids , side chain length between 3 Å ≤ length <5 Å are medium amino acids  and side chain length ≥5 Å are big amino acids .1D-barcode: The 1D-barcode represents the degree dependency with the cutoff (plateau- or linear-) of the amino acids of the 1BE9 sequence (1D).2D-barcode: The 2D-barcode represents the degree dependency with the cutoff of the amino acids of the 1BE9 sequence (1D) at variable cutoffs (2D).The AAN of the PDZ domain of the synaptic protein PDS-95 is generated using the PDB 1BE9 and the weights and degrees of every amino acid nodes are computed at different cutoff distances (Methods). We recall that the weight of a node describes its atomic packing, referred to as its atomic neighborhood while the degree of a node describes its amino-acid packing referred to as its amino-acid neighborhood. The atomic neighborhood takes into account the features of the amino acids. The weight and degree cutoff dependencies are plotted to investigate how atoms and amino acid neighbors occupy the space around each amino acid at different scales in the protein structure. This is a proxy of the dynamics as the more space occupied the less space left available for atomic motions.The weight cutoff dependencies are quadratic indicating that the weight (the number of atomic interactions) increases with the square of the cutoff distance, i.e., with a surface of contact between atoms rather than a volume of contacts . The degTen nodes have a plateau between 4 and 5 Å, twenty between 5 and 6 Å, one between 6 and 7 Å and six over 4 to 6 Å .All amino-acid types observed in 1BE9 except histidine which appears only twice, adopt plateau and linear neighborhoods . MethionTo have a plateau or a linear degree cutoff dependency for one amino acid type implies customizing neighborhoods as illustrated by the higher number of big amino acid neighbors for plateau amino acids than linear amino acids . This meWe can see from this single case 2D barcode classes of neighborhoods in terms of neighbor sizes which anticipate classes of dynamics and of responses upon mutations supporting the possibility to use the data for mining dynamics and its relation to function . Some poThe study shows that amino acid neighborhoods and not only amino acids and amino acid pairs, contain information relevant to protein dynamics, opening a new perspective to explore the link between dynamics and function. The 1D barcode of, for example, an enzyme and a pore-forming toxin can be compared to determine common and distinct features which can in turn be analyzed with the 2D barcode to survey both the neighborhood diversity of the common 1D barcode features and of the distinct 1D barcode features assuming the former identifies positions functionally insensitive (protein aspecific) and the later positions functionally sensitive (protein specific). The analysis of each protein variants can be used to validate the assumption as well as database analysis.This will contribute to diagnosing dynamic functional faults and dynamic functional diversity.


The interface between a solid and vacuum can become electronically distinct from the bulk. This feature, encountered in the case of quantum Hall effect, has a manifestation in insulators with topologically protected metallic surface states. Non-trivial Berry curvature of the Bloch waves or periodically driven perturbation are known to generate it. Here, by studying the angle-dependent magnetoresistance in prismatic bismuth crystals of different shapes, we detect a robust surface contribution to electric conductivity when the magnetic field is aligned parallel to a two-dimensional boundary between the three-dimensional crystal and vacuum. The effect is absent in antimony, which has an identical crystal symmetry, a similar Fermi surface structure and equally ballistic carriers, but an inverted band symmetry and a topological invariant of opposite sign. Our observation confirms that the boundary interrupting the cyclotron orbits remains metallic in bismuth, which is in agreement with what was predicted by Azbel decades ago. However, the absence of the effect in antimony indicates an intimate link between band symmetry and this boundary conductance. The topology of the surface states of a bismuth crystal remains an ongoing debate. Here, the authors observe surface electric conductivity with a magnetic field parallel to the two-dimensional boundary between the three-dimensional bismuth crystal and vacuum, but this effect is absent in antimony crystals indicating a link between band symmetry and boundary conductance. The long Fermi wavelength of its carriers extends over several tens of lattice parameters. Therefore, only extended defects (such as dislocations) can decay the charge current. In crystals lacking such spatially extended disorder, carriers become ballistic4, and their mobility (and as we will see below their magnetoresistance) easily exceeds any other solid hitherto explored5.Bismuth is a semimetal with an extremely low density of highly mobile carriers of both signs6 can explain the complex angle-dependent Landau spectrum of electrons and holes7. The approach successfully accounts for the total evacuation of one or two electron pockets at strong magnetic fields aligned along different axes8. The angle-dependent magnetoresistance and its rich structure9 are also accessible to semiclassical transport theory treating mobility as a tensor9.Recent research has shown that an extended Dirac Hamiltonian combined to the Fermi surface structure derived by a tight-binding model11 is yet to be understood. Such a ‘nematicity’ was also observed on the surface of bismuth crystals12, as well as in other solids13. In addition to ‘valley nemeticity’15, other theoretical possibilities for its origin were proposed16.Open questions remain, however. The origin of the loss of rotational threefold symmetry in presence of strong magnetic fields18, its consequences for the metallic surface states in bismuth19 and the evolution of the latter with thickness and Sb substitutions20. The topology of surface states and their status in the trivial/non-trivial dichotomy has been a subject of ongoing debate27. A recent popular theory identifies bismuth as a higher-order topological solid with topologically protected hinge states23. This hypothesis has been invoked to explain the experimental observation of ballistic transport in micrometer-long bismuth nanowires28.Another open question is the topology of the electron wave functionωcτ ≫ 1). It confirms that the interruption of cyclotron orbits at the boundary of a macroscopic three-dimensional crystal can generate a highly conducting boundary state in which bulk magnetoresistance is absent30.Here, we present a study of magnetoresistance in prismatic crystals of bismuth with ballistic carriers with unexpected consequences for both these issues. By choosing specific crystallographic planes as faces of the prisms, we uncover a specific contribution to electric conductivity when the magnetic field is aligned parallel to a two-dimensional boundary between the three-dimensional crystal and vacuum. The absence of this effect in antimony crystals of identical shapes points to the role played by the band structure topology in tuning the edge-bulk correspondence in macroscopic crystals in the high-field limit . For Bi, ρ0 = 0.18 μΩ cm corresponds to an average mobility of <μe + μh> = 1.2 × 108 cm2 V−1 s−1. For Sb, a residual resistivity of ρ0 = 0.07 μΩ cm corresponds to an average mobility of <μe + μh> = 1.7 × 107 cm2 V−1 s−1. These mobilities exceed those of the samples used in previous studies of magnetoresistance on Bi10 and Sb34.The bismuth (Bi) and the antimony (Sb) crystals used in this study are listed in Table 35.Note that the average mobility deduced from residual resistivity ignores the fact that in presence of anisotropic Fermi pockets of electrons and holes, the electrons and holes in different pockets and along different orientations have different mobilities. It is safe to assume that some carriers are ballistic given the size dependence of the residual resistivity36 were small pieces cut from larger crystals and hosted extended scattering centers such as twin boundaries and dislocations. The presence of such disorder led to a shorter electronic mean free path and a significantly lower magnetoresistance compared to the crystals studied here.Carrier mobility in Bi is probably the highest known in any solid. Bi crystals with a RRR of ≃600 were reported in old scientific literature , binary (C2), and bisectrix (C1)2. As seen in Fig. 1 and three C2 axes, which are equivalent upon 2π/3 rotation. Our experiments consisted in measuring the magnetoresistance of Bi and Sb crystals with the electric current applied along C3 and the magnetic field rotating in the trigonal plane. The orientation of the magnetic field is given by θ, which is defined as the angle between the field and the C1 axis. As reported previously10, despite the constant angle between current and field, orbital magnetoresistance varies with angle reflecting the in-plane anisotropy of the Fermi velocity in the three electron pockets.Bismuth and antimony crystallize in the rhombohedral A7 crystal structure shown in Fig.  binary C, and bisσ at a magnetic field of 12 T and 2 K in a pair of Bi crystals tailored identically as triangular prisms. . The only difference between the two crystals was that in one case each of the three square faces of the prism was a binary (C2) plane, while in the other case, it was a bisectrix (C1) plane. The angle-dependent magnetoresistance is clearly different in the two crystals. In one there are conductivity peaks each time the field is along a bisectrix axis. In the other, there are minima (instead of maxima) at the same field orientations.Our main observation is illustrated in Fig. The same experiment was performed in a pair of Sb crystals tailored in the same way as the bismuth crystals. As one can see in the figure, no difference is visible between the two crystals.Thus, angle-dependent magnetoresistance depends on the choice of specific crystal planes as faces of each prism in bismuth crystals, but not in antimony crystals. We reproduced this observation in two other pairs of Bi crystals and one other pair of Sb crystals.10. As reported previously10, this effect emerges only at sufficiently low temperature and high magnetic field. Let us now consider the amplitude of the required magnetic field.The shape dependence of the orbital magnetoresistance sheds light on the origin of the loss of the threefold symmetry in bismuth crystals of various geometryB = 14 T and B = 0.2 T. The non-trivial evolution of angle-dependent magnetoconductivity can be quantitatively described by invoking the anisotropy of the effective mass and the evolution of scattering time and carrier density among pockets with temperature and magnetic field. At B = 0.2 T and T = 40 K, angle-dependent magnetoconductivity displays a star-like shape. With cooling, the anisotropy is lowered due to a partial compensation of the mass anisotropy by the emerging anisotropy in the scattering time4. At B = 14 T, the anisotropy of mobility is compensated by an anisotropy in the distribution of carriers among the three pockets41.Figure B = 14 T, all electrons are confined to their lowest Landau level, but not at B = 0.2 T. This implies that the observed shape dependence of magnetoconductivity does not require proximity to the quantum limit ℏωc ≈ EF, but the passage to the high-field limit ωcτ > 1.As can be seen in Fig. τD, extracted from quantum oscillations is almost fifty times shorter than the transport scattering rate τtr in Bi42. Such a large difference between τtr and τD has been observed in several other dilute metals44. The semiclassical high-field limit (ωcτtr ≈ 1) is satisfied when the cyclotron radius becomes shorter than the mean free path. The quantum high-field limit (ωcτD ≈ 1) is satisfied when the distance between Landau levels becomes smaller than the broadening caused by temperature and disorder. At T = 40 K, the first criterion is satisfied, but not the second and there is no shape dependence. As the sample is cooled down, the shape dependence and quantum oscillations emerge concomitantly  around the trigonal axis with respect to the two crystallographic planes . On top of these peaks, in all the samples there is another set of peaks, which appear when the magnetic field is parallel to a face of the sample.Figure π/3 and there are six of them. The difference between the two is that in one case the surface peaks and the binary peaks are concomitant and in the other there is a π/6 shift between the two sets of peaks. In prisms with a square cross section, on the other hand, the additional peaks appear with a periodicity of π/2 and there are four of them. They occur each time the field is parallel to one of the four faces. If this face happens to be parallel to the binary axis , than the peak is more pronounced, as shown in Fig. In the two triangular prisms, these peaks appear with a periodicity of This observation clearly establishes that the additional contribution to conductivity emerges when the magnetic field  lies parallel to a two-dimensional boundary of the three-dimensional sample. Moreover, it does not matter at all for this boundary to be a specific crystal plane. The magnitude of the additional contribution remains the same when the boundary in question is the binary plane, the bisectrix plane, or a low-symmetry plane.Having demonstrated that the boundary conductivity emerges whenever the magnetic field is parallel to one of the surfaces of a prismatic crystal, let us now consider its evolution with magnetic field.The relative contribution of the boundary conductance to the total conductivity can be estimated by subtracting the angle-dependent conductivity in two triangular prisms with different crystal planes as faces. This assumes that bulk magnetoconductivity is identical in the two, which is reasonable, but subject to caution given the slight difference in mobility, which implies a difference in the expected magnitude of orbital magnetoresistance.r alternates between 0.2 and −0.2. Vertical red stripes show the excess conductivity in sample 1 and vertical blue stripes show the excess conductivity in sample 2. Remarkably, the width of the stripes or their color does not vary with increasing magnetic field. The relative amplitude of the excess boundary conductivity does not change even when the field increases by two orders of magnitude and the amplitude of conductivity decreases by almost four orders of magnitude.Figure r on angle at B = 0.2 T and B = 10 T. It is striking to observe how the two curves superpose on each other, in spite of a 50-fold change in magnetic field. Thus, the correction to conductivity brought by the B ∥ surface configuration both in amplitude and in angular dependence does not evolve with magnetic length Figure q as a free fitting parameter. The fact that a simple cosine fits the data implies that the excess conductivity detected here is not singular, as observed in other contexts45. Moreover, we find that when the surface becoming parallel to the magnetic field was a bisectrix crystallographic plane q = 20.3 ± 3 and when it was a binary crystallographic plane, it was q = 8.1 ± 1. q quantifies the sharpness of the peak, presumably caused by the anisotropy of the relevant length scales. Now, the Fermi momentum and wavelength of electrons is fourteen times longer along the bisectrix (C1) axis than along the binary axis (C2)6. This is a consequence of the huge (200-fold) anisotropy of the in-plane electron mass. Therefore, the significant difference between the angular width of the excess conductivity brings us to suspect a key role played by the Fermi wavelength of electron pockets in any plausible scenario.The data presented in Fig. 30. This idea provides the departing point of a plausible scenario to explain our observation. In the semiclassical picture, the magnetic field bends the electron trajectory. When the cyclotron radius is shorter than the mean free path of carriers, there is a large magnetoresistance. In semimetals, this magnetoresistance does not saturate even in the high-field limit. Now, at the surface of the sample, the cyclotron orbits are interrupted and what matters is the scattering of the carriers by the edge. If their collision results in a specular reflection then the conductivity at the edge is much higher. As a result, most of the current will flow near the boundaries of the sample where orbital magnetoresistance is absent. This phenomenon was dubbed ’static skin effect’ in analogy with the skin effect in metals. The latter refers to the fact that the density of an alternating current (AC) is largest near the surface of the conductor and decreases exponentially with increasing depth. Note, however, that the conductivity profile in the ’static’ version of the skin effect has a completely different origin.Decades ago, Azbel and Peschanskii put forward the concept of a static skin effect at high magnetic fields in metalsThe large magnetoresistance of bismuth can be understood in the semiclassical picture of cyclotron orbits shrinking with increasing magnetic field Fig. a. When t30. As illustrated in Fig. vs. edge. One may argue that these two tendencies may approximately cancel out generating an additional conductivity which does not vary much with the amplitude of the field. However, a perfect cancellation would be mysterious given the difference in the evolution of the magnetoresistance and the cyclotron radius. In addition, how to explain the indifference of the angular width of the peak to the magnitude of the magnetic field? The observation implies that the current profile has remained the same despite a 50-fold shrink in the size of the cyclotron radius.However, this semiclassical scenario fails to explain two key observations. The first is the fact that, as we saw above, the amplitude of the effect is unchanged when the magnetic field is changed by a factor of 50 , primed (V″) parameters, respectively. The crystal lattice has two sub-lattices, i.e., the unit cell includes two atoms. The third neighbor of the original atom is the closest neighbor on the same sub-lattice and both atoms reside in the same monolayer. The three first and the second neighbor atoms belong to a different sub-lattice and lie in other monolayers above and below the original atom  and antisymmetric (La) with respect to space inversion, where the eigenvalues of parity operator are +1 for Ls and −1 for La. As one can see in panels a and b of Fig. L-point is symmetric and the valence band is antisymmetric, while the inverse is true in the case of antimony.In 2007, BiT, three X, and three L) points in the Brillouin zone  This is exactly what happens in Sb. However, the SOC can alter this energy hierarchy. In Fig. L-point for Bi and Sb as a function of the magnitude of SOC using the Liu-Allen model. The conduction and valence bands of Sb are hardly affected and the energy hierarchy is unchanged by the SOC. On the other hand, the hierarchy is inverted by the SOC for Bi. This hierarchy alternation happens only in Bi, because the band gap  is much smaller and the SOC is larger than in Sb. Actually, the bands of Sb would be inverted if the SOC were unrealistically large (~20 eV).In a conventional Peierls transition, the energy of the symmetric band is lower than that of the antisymmetric bandLs and La bands are distinguished by the parity inversion51.Interestingly, the L-point originates from the Peierls distortion52. The real-space image of the wave functions of conduction and valence bands are depicted in Fig. 850. Peierls distortion can be understood as a dimerization, where the lattice is distorted to generate pairs as seen in Fig. ψs) is given by the periodic array of bonding orbitals, whereas the wave function of the antisymmetric band (ψa) is given by that of anti-bonding orbitals  By the normal reflection, the wave vector of electrons changes as 30 is a semiclassical approach which does not take into account the phase of the electrons’ wave function. The interruption of cyclotron orbits was framed in a specular-diffusive dichotomy. If the reflection is perfectly specular, momentum is conserved and there is an additional contribution to conductivity. The effect will weaken if the reflection becomes partially diffusive. However, reflected electron waves can interfere with incoming waves. The electronic Fabry interferometers employed in two-dimensional electron gases53 are an eloquent demonstration of this fact.The static skin effect pictureIn a quantum treatment of the interruption of the cyclotron orbits by sample boundaries, it is crucial to consider the fate of the electron wave function and its phase following a mirror transformation. As we saw above, quasi-particles residing in the electron pockets have opposite symmetries in Bi and in Sb. The symmetric conduction band of bismuth, in contrast to the antisymmetric band of Sb, allows a constructive interference upon a mirror reflection at the boundary. However, it remains to be seen how this difference survives in presence of quantizing magnetic field.54, which is to be affected by the zero-field anisotropy of the Fermi momentum. The anisotropic cyclotron orbits of Bi surface states have recently become accessible to experiment, thanks to scanning tunneling microscopy studies12.Even in the simple isotropic case, large-index Landau wave functions have a non-trivial angular distribution in real space33. At zero magnetic field, the electronic surface states of a solid are distinct from bulk states by the abrupt interruption of the lattice. There is another distinction, which emerges at high magnetic field. The surface states are periodically disturbed by cyclotron orbits of the bulk. This leads to a spatio-temporal discrete translation symmetry56. An edge atom at a given position and time is not instantaneously equivalent to its neighbor. One of the two may be perturbed by an electron from the bulk in its cyclotron orbit, in contrast to the other. On the other hand, the two atoms remain equivalent if the temporal periodicity is taken into account. In other words, the discrete symmetries of space and time become intertwined: It is tempting to draw an analogy between the present context and Floquet systems in which topological protection is provided by a periodic perturbation23. This was put forward to explain the origin of ballistic transport in Bi nanowires detected by superconducting proximity studies28. Note that the boundary ballistic transport detected in our experiment is two-dimensional and does not appear to arise from one-dimensional channels expected in the case of topologically protected hinge states23.One of the motivations of the present study was the theoretical proposal that bismuth is a higher-order non-trivial topological systemIn conclusion, we found that in bulk crystals of bismuth, there is a robust contribution to conductance when the magnetic field is aligned parallel to a two-dimensional boundary of the sample. The absence of this effect in antimony implies that the difference in symmetry of the conduction band has a significant outcome.A satisfactory explanation of our results is missing and remains a challenge for theory. While the ‘static skin effect’ explains the existence of boundary conductance in a macroscopic crystal, it fails to explain its absence in antimony as well as the robust behavior of the conducting channel in presence of strong magnetic fields. We note that the semiclassical ‘static skin effect’ has not yet been formulated in a quantum-mechanical frame incorporating the known contrast between the parity of the Bloch waves in Bi and in Sb at the L-point. We argued that the latter may affect reflection at the crystal boundaries. To the best of our knowledge, this has not yet been addressed by theory.10 finds a natural explanation. It may also be invoked to explain the loss of symmetry seen by thermodynamic probes11. The anomaly caused by the evacuation of a Landau level is a van-Hove singularity with a cut-off due to disorder and finite size. The latter correlates with the shape of the sample. This would imply that the finite size cut-off of the van-Hove singularity may be different for different field orientations. Thermodynamic measurements on samples with different shapes and different sizes will be instructive to check this. Finally, our observation may indicate that boundaries of a bismuth crystal in presence of magnetic field provide a topological barrier. This would provide a possible solution to the puzzle of distinct chemical potentials between twinned crystals of bismuth7.Our result has implications for several puzzling observations previously reported in bismuth. The loss of threefold symmetry in transport measurementsBi and Sb crystals were commercially obtained through MaTecK GmbH, which oriented and cut them to the desired shape and dimensions. The sample surface was not polished and did not go through any other specific treatment. The experiments were performed in two different locations (Ewha University and ESPCI) and with two different set-ups. A home-made set-up was used in Ewha University and a Quantum Design PPMS was used in ESPCI Paris. In both cases, resistivity was measured with a standard 4-wire configuration and electrical contacts were made with silver paste. In order to ensure the homogeneity of charge flow, current electrodes were applied to a thick layer of silver paste (Dupont 4929N) covering most of the end surfaces of the prism except for a small circular area in the center. Then a small island of silver paste was created in the center and voltage electrodes were placed on it. At Ewha University, magnetoresistance was measured with an AC method with a typical current of 100 μA at frequencies between 13 and 17 Hz. For resistivity measurements as a function of temperature, a DC method was used with a typical current of 1 mA. In Paris, measurements were performed using currents between 1 mA and 5 mA and the resistivity option of the PPMS in AC drive mode (50 Hz square wave excitation).Supplementary informationPeer Review File


In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi–Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity. One of the important areas of quantum information theory refers to an entropic picture of quantum states and operations. It is well known that the entropic uncertainty principle can be applied in quantum key distribution protocols ,2 in ordIn quantum information theory, the relative entropy ant role  and can  testing . The quaThe first approach to defining the entropy of a quantum channel was proposed by Życzkowski , where tLet The set of all such mapping The von Neumann entropy of nterpart .(3)D(ρ∥ formulaH(ρ)=−TrρThe definition of quantum relative entropy can be extended to the case of quantum channels in the following manner .5)D(Φ∥D(Φ∥(5)D=logXThe main focus of this work is to find instances that saturate the inequality in Lemma 1. We will mainly focus on the study of unital qubit channels.In this section we will focus our attention on unital qubit channels, that is Equation  is achieTheorem 1.Let We need to stress here that a non-unital quantum channel Equation  is achieThe remainder of this section contains technical lemmas which combined give the proof of Theorem 1.W is a unitary matrix. Thus, we can skip the unitary operations in our further investigations. We may perform the optimization taking into account only the parameter p which quantifies the amount of entanglement between the input qubits. In order to further simplify notation we will writeX and p. Hence, we formulate the first lemma.A generic two-qubit state can be written asLemma 2.Let Equation . Then D((λ))andγ=(R⊗1l)(In this paper, we discuss two approaches to entropic quantification of quantum channels. We begin our studies with a formulation of a lemma, which describes a relation between the entropy of quantum channels proposed by Gour and Wilde  and entrWe also provide a conjecture backed by numerical experiments that both formulas provide the same results up to an additive constant in the case of large system size.