My Dung Jusselme
email: jusselme@gmail.com

Charlotte Pruvost

Eric Motard

Stéphanie Giusti-Miller

Sophie Frechault

Vanessa Alphonse

Clarisse Balland-Bolou-Bi

Isabelle Dajoz

Dr Philippe Mora
email: mora@u-pec.fr

Stéphanie Giusti- Miller

Clarisse Balland-Bolou- Bi

Dr My Dung

Increasing the ability of a green roof to provide ecosystem services by adding organic matter and earthworms

Keywords: Green roofs, Earthworms, Compost, Microbial activity, Ecosystem services

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Under urbanization pressure and climate change, ecological services provided by urban ecosystems are endangered [START_REF] Mcmichael | Climate change and human health: Present and future risks[END_REF]. To counteract this tendency, green roofs are increasingly set up on top of buildings. This technology has become increasingly important over the last 20 years with thousands of green roofs being installed worldwide, especially in countries with temperate and subtropical climates [START_REF] Brenneisen | Space for urban wildlife: Designing green roofs as habitats in Switzerland[END_REF].Green roofs are created to provide a range of ecosystem services, from rainwater retention to thermal insulation. They also provide habitats for biodiversity among which invertebrates such as insects, spiders and soil arthropods, whose diversity is in relation with the composition of the plant community and growth substrate (MacIvor and [START_REF] Macivor | Invertebrates on Green Roofs[END_REF]. Indeed, studies sampling invertebrate communities on a variety of green roof types have found many hundreds of species [START_REF] Brenneisen | Space for urban wildlife: Designing green roofs as habitats in Switzerland[END_REF][START_REF] Schindler | Factors Influencing Arthropod Diversity on Green Roofs[END_REF][START_REF] Ksiazek | An assessment of pollen limitation on Chicago green roofs[END_REF]. Most of them colonize green roofs during their installation (as they are already present in the growing substrate) or by their own means. Many of these invertebrates are involved in ecosystem functions such as plant pollination and reproduction, contributions to food webs, and the maintenance and stabilization of soil through the degradation of organic matter [START_REF] Vijayaraghavan | Green roofs: A critical review on the role of components, benefits, limitations and trends[END_REF][START_REF] Wang | Are green roofs a source or sink of runoff pollutants?[END_REF][START_REF] Zhang | Green roof storage capacity can be more important than evapotranspiration for retention performance[END_REF].

Green roofs are comprised of vegetation placed in a growth medium, called substrate [START_REF] Oberndorfer | Green Roofs as Urban Ecosystems: Ecological Structures, Functions, and Services[END_REF] on top of building roofs. In addition to the harsh microclimate on rooftops, plant communities in green roof system are often subject to low nutrient availability of the substrate [START_REF] Rowe | Assessment of heat-expanded slate and fertility requirements in green roof substrates[END_REF]. As the nature of the substrate directly influences plant growth and the performance of the green roofs [START_REF] Young | Importance of different components of green roof substrate on plant growth and physiological performance[END_REF], choosing an appropriate substrate is, therefore, critical for the sustainability of a green roof. However, the substrates used for green roofs have depauperate biological activities because they are sterilized prior to installation or are derived mostly from dry mineral materials [START_REF] John | Colonization of green roof plants by mycorrhizal and root endophytic fungi[END_REF]. More and more research shows the critical relationships between plants and their microbial associates, called "holobionts" are essential for plants to deal with various environmental stressors [START_REF] Vandenkoornhuyse | The importance of the microbiome of the plant holobiont[END_REF]. Moreover, both detritivores and microorganisms play an important role in soil fertility because they are responsible for the recycling of nutrients from organic matter [START_REF] Madsen | Identifying microorganisms responsible for ecologically significant biogeochemical processes[END_REF] and increased microbial activity facilitates the production of readily available forms of nutrients [START_REF] Trasar-Cepeda | Thermodynamic parameters of enzymes in grassland soils from Galicia, NW Spain[END_REF]. Improving biological activities in the substrate might thus improve growing conditions and therefore reinforce plant health [START_REF] Molineux | Manipulating soil microbial communities in extensive green roof substrates[END_REF]Molineux et al., 2015a[START_REF] Molineux | Are microbial communities in green roof substrates comparable to those in post-industrial sites?-a preliminary study[END_REF].

Earthworms act as soil engineers (Lavelle, 2002a;[START_REF] Scheu | Effects of the presence and community composition of earthworms on microbial community functioning[END_REF][START_REF] Brown | Regulation of soil organic matter dynamics and microbial activityin the drilosphere and the role of interactionswith other edaphic functional domains[END_REF] and they are known to enhance carbon and nitrogen mineralization through direct or indirect effect on the composition, distribution and activity of soil microorganisms [START_REF] Scheu | Effects of the presence and community composition of earthworms on microbial community functioning[END_REF]Aira and Domínguez, 2011). Furthermore, earthworms can survive to harsh environmental conditions such as highly polluted soils [START_REF] Jusselme | Increased lead availability and enzyme activities in root-adhering soil of Lantana camara during phytoextraction in the presence of earthworms[END_REF] or extreme rainfall events [START_REF] Andriuzzi | Anecic earthworms (Lumbricus terrestris) alleviate negative effects of extreme rainfall events on soil and plants in field mesocosms[END_REF], making them good candidates for the improvement of biological activities of green roofs substrates. Actually, earthworms are not commonly found on green roofs because of the shallow substrate depth in combination with frequent drought conditions. The use of earthworms in this context is thus dependent on the characteristics of the substrate, including its composition and depth.

In recent years, utilization of compost issued from green waste material produced in cities has emerged as a way of improving soil quality while protecting the environment. [START_REF] Weber | Agricultural and ecological aspects of a sandy soil as affected by the application of municipal solid waste composts[END_REF] have shown that the application of compost improved soil fertility and increased cation exchange capacity in a sandy soil. This also improves water retention and soil structure by increasing the stability of soil aggregates [START_REF] Celik | Effects of compost, mycorrhiza, manure and fertilizer on some physical properties of a Chromoxerert soil[END_REF]. A recent study focused on using compost to increase the long-term stability of soil organic matter [START_REF] Bass | Soil properties, greenhouse gas emissions and crop yield under compost, biochar and co-composted biochar in two tropical agronomic systems[END_REF].

Thus adding compost to green roof systems may enhance the substrate biological activities.

Ondoño et al. (2014) compared the mix of compost and brick to compost and soil media and

showed the first-mentioned had far less microbial activity and fewer nutrients than the lastmentioned.

The aim of this study was to test if the addition of earthworms and compost may improve the biological and chemical properties of a standard green roof substrate. We conducted a mesocosm experiment using a plant community comprising five species grown on a commercial green roof substrate, with or without the addition of vermicompost and earthworms. The mesocosms were set up on a rooftop and at ground level. We recorded the weather conditions, the physiological characteristics of the plants, the activity of the microbial communities in the substrate and the pollinator abundance and visitation rates to the plant community. We tested three hypotheses: (i) the roof microclimate affects the green roof ecosystem, (ii) adding earthworms and compost stimulates microbial activities and improves plant development in the green roof system, and (iii) adding earthworms and compost supports ecosystem services such as pollination.

Materials and methods

Experimental design and sample collection

The experiment was conducted on the University of Paris Est Créteil (UPEC) campus, Val de Marne (48°47'17.9''N 2°26'36.0''E). To test the effect of roof microclimate on plant development, mesocosms were set at two heights: (i) on the roof, 20 m high ("R") and (ii) at ground level ("G") (Fig. 1).

The mesocosms (120 x 80 x 33 cm) were filled randomly with either: (i) a low complexity substrate (LC), comprising a 25 cm layer of a commercial green roof growing medium (Substrate Type E, Optigreen, Germany); (ii) a high complexity substrate (HC), with a 5 cm layer of vermicompost and earthworms on top of 20 cm of the same commercial medium.

Each combination of height and substrate was replicated five times. According to the manufacturer's data sheet, the composition of the substrate is variable according to the production site. The main constituents are expanded shale, expanded clay, stone, crushed bricks, perlite and compost from green waste. The main characteristics of the substrate are a pH comprised between 6 and 8, salt, organic substances content < 65 g/l, permeability to water ≥ 0.6 mm/min and total porosity comprised between 60 and 70 %. The vermicompost was obtained by adding earthworms Eisenia foetida and Eisenia Andrei to locally sourced green waste from the Val de Marne district. In addition to the vermicompost, each HC mesocosms was inoculated with 53 (about 170 g) adult earthworms, Lumbricus terrestris. This species digs temporary deep burrows and returns to the surface to feed, playing an important role in the incorporation of the vermicompost into the substrate. Worms were provided by the Lombri Carraz Company (France). The initial pH at a 1:5 soil-to-water ratio was 8.5 for the LC mesocosms and 6.0 for the HC mesocosms. (Carthusian pink or German pink, Caryoplyllaceae). The plants were selected for their suitability for green roofs in the Ile-de-France region as well as for their insect-pollination system (except K. glauca which is wind-pollinated). There were five plants of each species in each mesocosm. The plants within each plant mixture were randomly distributed in each mesocosm.

Weather conditions

The light level, the ultraviolet index, the air temperature, the relative humidity and the wind speed were measured with a Pro WMR86 weather station (Oregon Scientific®, France) at each height. The data were recorded during June, July and August 2015 to characterize microclimates on the roof and at ground level with a minimum of 27 measurements for each parameter, height and time of day (morning 9 am and afternoon 4 pm).

Plant parameters

During the experiment, the plant parameters (vegetative cover, above-ground biomass, H. maximum, C. jacea, and L. corniculatus leaf chlorophyll content, H. maximum nectar volume and nectar sugar concentration) were measured from June to August 2015.

All images were taken from 1 meter above the surface of the substrate. The vegetative cover was determined by processing photographs with Image J (the number of "green" pixels divided by the total number of pixels). The chlorophyll concentration (µmol.m 2 ) was measured for three species of plants, H. maximum, C. jacea, and L. corniculatus using an SPAD 502 Plus chlorophyll meter (Minolta, Japan). In each mesocosm with 5 plants of each species, three measurements for each plant were taken on three randomly chosen leaves and in addition, at least one node from the growing tip. At the end of the experiment, the aboveground biomass per plant per mesocosm was collected by cutting at the level of the root collar, oven-dried at 70°C for 48h and weighed.

The nectar volume and the nectar sugar content were measured on H.maximum flowers following the protocol described in [START_REF] Flacher | Competition with wind-pollinated plant species alters floral traits of insect-pollinated plant species[END_REF]. Measurements were carried out over three successive weeks (one measurement per week) from 7 th to 30 th August 2015. The nectar volume was measured early in the morning on three flowers per plant and five plants per mesocosm. To prevent visits by insects, flowers were covered with mesh bags the evening before the measurements. For each plant, three open flowers were collected at random. For each flower, the nectar was collected using microcapillary tubes (0.5 μl, end to end Minicaps, Hirschmann Laborgeräte, Germany) and the nectar volume was calculated by measuring the length of liquid in the microcapillary tube with a digital caliper. The nectar sugar concentration (estimated as a sucrose equivalent) was determined using a hand-held refractometer (Eclipse E-line 80, Bellingham Stanley Ltd., UK) which was calibrated using a 30° brix sucrose solution. If necessary, the nectar was diluted in Milli-Q water before the measurement. The mean amount of sugar produced per flower, in each mesocosm, was estimated by multiplying the mean sugar concentration by the mean volume of nectar per flower.

Physical and chemical properties

The physical and chemical properties of the substrate were measured at the INRA Soil Analysis Laboratory (Arras, France) in accordance with ISO/IEC 17025.

The substrate samples were passed through a 2-mm sieve prior to analysis. The substrate pH was measured using a 1:5 soil-to-water ratio. The cation exchange capacity (CEC) was determined by ammonium acetate saturation. The total organic carbon content was determined by rapid titration. NO3 -and NH4 + were extracted by shaking in 0.5M KCl and NO3 -and NH4 + measured by colorimetry. The available phosphorus was determined by colorimetry using the phosphomolybdic blue reaction.

Microbial activity in substrates 2.5.1. Enzyme activity

The total microbial activity of the substrates was determined by fluorescein diacetate (FDA) hydrolysis (Schnurer and Rosswall, 1982). The mineralization of organic phosphorus was evaluated by the measurement of alkaline phosphatase (ALP) and acid phosphatase (ACP) activities [START_REF] Nelson | Part 2[END_REF]. The assays were conducted in quintuplicate.

For FDA assay, the wells in a 96 well microplate were loaded with 100 µL of substrate sample solution, 50 µL of citrate phosphate buffer and 50 µL of FDA solution (3g L -1 ). The microplates were incubated for 2 h at 37°C, shaken at 900 rpm. At the end of incubation the microplates were centrifuged at 1500 rpm for 10 min (Sigma 2-5, Fisher Bioblock, France). 100 µL of the supernatant was transferred to a new microplate with 100 µL of citrate phosphate buffer at 4°C. The absorbance at 490 nm was measured using a BioTek EL800 Universal plate reader (Bio-Tek Instruments, Winooski, VT). The FDA hydrolyzed was expressed as μg fluorescein g -1 substrate h -1 .

For alkaline and acid phosphatase activity assays, the wells in a 96 well microplate were loaded with 50 µL substrate sample solution, 25 µL borate buffer (pH 5 for ADP and pH 9 for ALP) and 0.5 µL of p-nitrophhenyl phosphate solution (10g L -1 ). The microplates were incubated for 2 h at 37°C, shaken at 150 rpm. 125 µL 2% Na2CO3 solution was added to stop the reaction. The microplates were centrifuged at 1500 rpm for 10 min (Sigma 2-5, Fisher Bioblock, France). 50 µL of the supernatant was transferred to a new microplate with 25 µL 2% Na2CO3 solution. The absorbance at 405 nm was measured using a BioTek EL800 Universal plate reader (Bio-Tek Instruments, Winooski, VT). The ADP and ALP enzyme activities were expressed as μg p-nitrophenol g -1 substrate h -1 .

Microbial respiration

The microbial respiration (MR) was measured as proposed by [START_REF] Garcia | Microbial activity in soils under mediterranean environmental conditions[END_REF]. We placed 2 g of substrate in hermetically sealed flasks and incubated for 4 days at 28°C to reactivate the microorganisms. The flasks were then moistened to 70 % of water holding capacity (WHC) with 9 g L -1 NaCl solution. Emitted CO2 was measured every 6 h by chromatography (490 Micro GC, Agilent, France). The cumulative CO2emitted was calculated and the respiration rate was expressed as mg CO2-C kg -1 dry substrate h -1 .

Microbial metabolic activity and diversity

96-well EcoPlates (AES Laboratory France) were used to assess the microbial metabolic activity of the substrates. Each Ecoplates TM had three replicates of 31 different environmentally significant carbon sources and three control wells [START_REF] Garland | Patterns of Potential Rhizosphere C Source Utilization Communities By[END_REF]. Each well of an EcoPlate (AES Laboratory France) was filled with 150 µL substrate sample solution and the plates were incubated in darkness on a shaker (150 rpm) at 20°C for 3 days. The color development was measured as absorbance at 595 nm using a Microplate spectrophotometer (Xenius, Safas SA, Monaco) every 24 h for one week. Data were normalized using the average wellcolor development (AWCD) as described by [START_REF] Garland | Classification and characterisation of heterotrophic microbial communities on the basis of pattern of community-level sole-carbon-source utilization[END_REF]:

AWCD=Σ (C-R)/n.
where C is the color developed within each well (optical density measurement), R is the color of the control well, and n is the number of substrates in each replicate (n=31). All negative values were considered to be zero in the subsequent data analyses. Diversity indexes (Shannon index and Richness) were calculated as suggested by John C. Zak (1994).

Floral visitation rates

Over 3 consecutive weeks during the flowering period in July and August 2015, we recorded the total number of pollinators that visited flowers. On days when weather conditions were favorable for pollination (i.e. wind speed < 15 km/h, without rain), pollinator visits were recorded for a period of 5 minutes per mesocosm, 3 days per week.

Statistical analyses

All data was analyzed using Statgraphics (Centurion XVI, Sigma Plus, France). Before testing for significant differences between the treatments, the normality of the data was tested using the Kolmogorov Smirnov test (P < 0.05).

We used one-way ANOVA (alpha = 0.05) to analyze the effect of the height on the meteorological parameters and the effect of the substrate on pollinators.

We used two-way ANOVA (alpha = 0.05) to analyze the effects of the substrate, the height and their interaction on physical, chemical, biological, and plant parameters.

We used Pearson's product-moment coefficient (r) to test whether the variations in microbial activity were correlated with the weather conditions and with the chemical properties of the substrate. We tested for linear dependences between local weather conditions (temperature, light intensity, air humidity and wind speed), substrate properties (pH, soil C/N ratio, organic matter, N and P concentrations, water content), microbial activities (enzyme activities, respiration, metabolic activities) and plant parameters (height, diameter, vegetative cover, chlorophyll, nectar volume and sugar concentration).

Results

Weather conditions

The mean temperatures were not significantly different between heights but differed significantly between the morning and afternoon with a slight increase in the afternoon for both heights (ground, F = 5.0, P < 0.05 and roof, F = 6.1, P < 0.05) (Table 1). The light intensity on the roof was significantly higher than at ground level in the afternoon only (Table 1). The UV index on the roof was always higher than at ground level. The wind speed in the morning and afternoon was not significantly different between heights. However, the variation of wind speed on the roof was significant higher than at ground level (F = 10.6, P < 0.01, Table 1). The relative humidity was not significantly different between heights but was significant higher in the morning than in the afternoon (F = 14.9, P < 0.001, Table 1).

3.2. Physical and chemical characteristics of the substrates at the end of the experiment Whatever the location, the HC substrate had significantly higher availability of mineral nutrients (C, N and P), higher field capacity (F = 60.5, P < 0.001), higher water content (F= 121.3, P < 0.001), and lower pH (F = 66.4, P < 0.001) for both heights (Table 2). The available N-NO3 -was significantly higher on the roof (F = 14.1, P < 0.05) (Table2).

The HC substrate C content was 58% (G, ground) and 68% (R, roof) higher than for the LC substrate, while the HC substrate total N content was 56% (G) and 68% (R) higher. The HC substrate available N-NH4 + was 69% (G) and 89% (R) higher and N-NO3 -was 94% (G) and 96% (R) higher. The HC substrate field capacity was 23% (G) and 55% (R) higher than for the LC substrate (Table 2).

Plant parameters

All plants looked healthy throughout the experiment (i.e. no signs of pathogen or herbivore attacks).The HC substrates had 55% (R) and 67% (G) greater aboveground plant biomass, 16% (R) and 7% (G) greater vegetative cover, and 53% (R) and 41% (G) more H. maximum buds than the LC substrates (Table 3).

For the HC substrate, the chlorophyll contents of H. maximum, C. jacea and L. corniculatus were significantly higher (30%) than for the LC substrates at both heights (Table 3). The mean nectar volume per flower for H. maximum was 73% (R) and 68% (G) higher and the nectar sugar concentration was 65% (R) and 66% (G) higher (Table 3).

The HC substrates had significantly higher values for all plant parameters: above-ground biomass (F = 21.6, P < 0.001), vegetative cover (F = 63.7, P < 0.001), H. maximum buds number (F = 65.8, P < 0.001), chlorophyll content of H. maximum, C. jacea and L. corniculatus (F = 100.5, P < 0.001; F = 29.2, P < 0.001; F = 84.4, P < 0.001, respectively), H. maximum mean nectar volume (F = 32.8, P < 0.001) and nectar sugar concentration (F = 12.6, P < 0.05). Height had a significant effect, for both substrates, on vegetative cover and H. maximum mean nectar concentration, with lower values at the ground level, and H. maximum chlorophyll content, with higher values at ground level (Table 3).

Microbial activity and diversity

The flowering phase is a critical period in which plants produce many compounds that will shape the composition and activities within the microbial community. We therefore tested the differences between HC and LC substrates in both the vegetative and flowering phases separately.

The microbial respiration in the vegetative phase was 76 (R) and 92 (G) mg CO2 kg -1 DW h -1 for the HC substrate, which was significantly higher than 39 (R) and 49 (G) mg CO2 kg -1 DW h -1 for the LC substrate (Table 4). The results in the flowering phase were similar.

The HC substrates had a significantly higher metabolic activity, expressed as average well color development (AWCD), independently of the height (F = 16.0, P < 0.001, Table 4). The Shannon diversity index (F = 7.6, P < 0.01) and richness (F = 8.8, P < 0.01) of the microbial community were significantly higher in the HC substrates, but there was no significant interaction between height and substrate.

Generally the HC substrates had significantly higher enzyme activities than the LC substrates in both time periods (Table 4). In the vegetative phase, the FDA measured in HC substrates were 2 (R) and 6 (G) Ug -1 DW which were significantly higher than in the LC substrates (~ 1 U g -1 DW).In the flowering phase, the acid and alkaline phosphatase activities in HC were significantly higher than those in LC (Table 4). The substrate had a significant effect (F = 24.2, P < 0.001 for FDA, F = 8.2, P < 0.01 for ACP, and F = 4.9, P < 0.05 for ALP).

Floral visitation by pollinators

Solitary bees, honey bees and bumblebees were the most common insect pollinators observed foraging on the mesocosms. The mean number of visits and pollinator abundances were significantly (F = 32.4; P < 0.01 and F = 15.7; P < 0.05) higher for HC substrates than for LC substrates (Fig. 2A,B).

3.6. Relationship between physical and chemical parameters, microbial activity and plant parameters ACP and ALP activity were positively correlated with available P. FDA activity was positively correlated with available N, P, and C content and total N. Microbial respiration and AWCD were positively correlated with all measured chemical properties. The plant parameters (biomass, cover, volume of nectar and nectar sugar concentration) were positively correlated with the total and available nutrient content (C, N and P) (Table 4).

Discussion

Effect of microclimate on the vegetation in the green roof system

Green roofs are known to be exposed to a variety of harsh abiotic factors such as intense solar radiation, temperature, humidity and wind speed (Fulthorpe et al., 2018); however there is limited knowledge on the effect of these microclimate parameters on plant growth on roofs. In our study, we observed that the vegetative cover and above-ground plant biomass on a roof were lower than those at ground level for either substrate. This can be explained by the higher light levels on the roof top than at ground level. Light is an absolute requirement for plants growth and development, and different plants have different light requirements. An increase in light will increase the rate of photosynthesis; but excessive light intensity may scorch the leaves and reduce growth [START_REF] Sargent | Effect of Light Intensity on the Development of the Photosynthetic Mechanism[END_REF][START_REF] Thut | International Journal of Biological Papers Effect of Light Intensity on the Growth of Dioscoreophyllum[END_REF]. Moreover, humidity seemed more important on the roof than on the ground, even though not significantly so, but this may have affected soil moisture and plants.

We also observed that H. maximum nectar sugar concentration was higher on the roof than at ground level, but there was no significant difference in mean nectar volume per flower. A higher nectar sugar concentration in H. maximum can be related to the abiotic factors. In fact, abiotic parameters such as light, water, temperature, and CO2 concentration can have important impacts on plant physiology that can in turn affect the nectar quantity and quality (Gardener and Gillman, 2001a; [START_REF] Burkle | The effects of nutrient addition on floral characters and pollination in two subalpine plants, Ipomopsis aggregata and Linum lewisii[END_REF]. In soil, the availability of nutrients can indirectly affect flower production and nectar [START_REF] Smith | Mycorrhizal Symbiosis, Mycorrhizal Symbiosis[END_REF].Foliar dust deposition may also be more important on the roof than on the ground, which could supply supplementary nutrients for plants. Overall, the higher levels of plant nutrients (N, N-NO3 - and P, P-PO4 2-) on the roof may result in the higher concentration of sugar in the nectar of H. maximum flowers growing on the roof.

Effect of adding vermicompost and earthworms on the green roof ecosystem

It is known that plants in green roofs encounter harsh meteorological factors and low nutrient availability in the substrate [START_REF] Oberndorfer | Green Roofs as Urban Ecosystems: Ecological Structures, Functions, and Services[END_REF][START_REF] Rowe | Green roofs as a means of pollution abatement[END_REF]. The idea of this study was to use the synergic interaction between earthworms and vermicompost in order to improve the physico-chemical and biological parameters of the substrate. Results show that the application of vermicompost and earthworms increased substrate field capacity by about 50% and humidity by about 80%. This is an important factor as it is directly involved in the cooling capacity of green roofs as well as in the development of the plant and animal communities in the substrate [START_REF] Getter | Carbon sequestration potential of extensive green roofs[END_REF][START_REF] Macivor | Performance evaluation of native plants suited to extensive green roof conditions in a maritime climate[END_REF].Our results also show that the addition of vermicompost and earthworms improved the quantities of nutriments and organic matter of the substrate with an increase by about 68 and 58% in organic matter, 68 and 58% in organic C, as well as 68 and 56% in total nitrogen observed in the substrate at rooftop and ground levels, respectively. Moreover, as earthworms are known to affect the distribution of microorganisms and microbial biomass [START_REF] Brown | How do earthworms affect microfloral and faunal community diversity?[END_REF], we hypothesized that the effects of earthworms on the plant growth resulted in part from their impact on soil microorganisms. Indeed, our results show that inoculating the substrate with earthworms and vermicompost resulted in higher values for microbial respiration and enzyme activity in the substrate. Although we recorded increased microbial activity in the substrate in the presence of earthworms and vermicompost, the mechanisms involved are unclear. However, Lavelle, (2002b) assumed that microorganisms are largely dormant in the soil awaiting suitable conditions that earthworms can provide as they burrow and cast through soil, secreting substances such as mucus resulting in microbial reactivation and stimulation.

As well as more available N and P, supporting the hypothesis that higher microbial activity would lead to more available nutrients for plants. ACP and ALP activity were positively correlated with available P. Similar results have been reported for earthworm casts, but it is not known whether the increase in activity is due to earthworm-derived enzymes or to increased microbial activity [START_REF] Satchell | Short Communication Stimulation of Microbial Phosphatase[END_REF]. Greater ALP activity in the presence of earthworms would increase the hydrolysis rate of organic P compounds, increasing the available P released [START_REF] Aira | Earthworms strongly modify microbial biomass and activity triggering enzymatic activities during vermicomposting independently of the application rates of pig slurry[END_REF]. N and P are major nutrients for plant growth, and expanding availability of these nutrients should increase the above-ground plant biomass, vegetative cover, chlorophyll content, nectar volume and nectar sugar concentration [START_REF] Flacher | Competition with wind-pollinated plant species alters floral traits of insect-pollinated plant species[END_REF]. We suggest that earthworms transformed this organic matter into plant-available nutrients [START_REF] Villenave | Effects of earthworms on soil organic matter and nutrient dynamics following earthworm inoculation in field experimental situations[END_REF]. This was confirmed by the positive correlations between the plant parameters and the available nutrients (Table 5). Thus, using a combination of vermicompost and anecic earthworms favors plant-growth in a green roof ecosystem.

Nectar, the main source of energy for pollinators, plays a central role in the behavior and composition of pollinator communities [START_REF] Baude | Litter inputs and plant interactions affect nectar sugar content[END_REF]. Many studies have investigated the role of below-ground communities on above-ground biota including pollinators [START_REF] Barber | How do belowground organisms influence plantpollinator interactions?[END_REF]. Green roofs are known to provide habitats for pollinators, particularly, bees and butterflies [START_REF] Ksiazek | An assessment of pollen limitation on Chicago green roofs[END_REF]. Soil nutrient availability to plants is known to influence the quantity and quality of nectar and can lead to changes in pollinator visitation [START_REF] Burkle | The effects of nutrient addition on floral characters and pollination in two subalpine plants, Ipomopsis aggregata and Linum lewisii[END_REF]. However, less is known about the effect of compost and earthworms on nectar production and pollinator behavior in a green roof ecosystem. In our study, the H. maximum nectar sugar concentration was higher with added vermicompost and earthworms. This led to an increase in the abundance of flower visitors (solitary bees, honey bees and bumblebees) as well as frequency of pollinator's visits. [START_REF] Kudo | Floral and inflorescence effects on variation in pollen removal and seed production among six legume species[END_REF] suggested that nectar availability and floral display are the most important traits influencing attractiveness of plants to pollinators. Plants also interact with a diverse set of substrate biota and these interactions have the potential to affect nutrient uptake by plants [START_REF] Smith | Mycorrhizal Symbiosis, Mycorrhizal Symbiosis[END_REF]. Consequently this can indirectly affect flower production [START_REF] Burkle | The effects of nutrient addition on floral characters and pollination in two subalpine plants, Ipomopsis aggregata and Linum lewisii[END_REF], pollen [START_REF] Lau | Effects of soil phosphorus on pollen production, pollen size, pollen phosphorus content, and the ability to sire seeds in Cucurbita pepo (Cucurbitaceae)[END_REF] and nectar (Gardener and Gillman, 2001b;[START_REF] Becklin | Soil fungal effects on floral signals, rewards, and aboveground interactions in an alpine pollination web[END_REF].

MacIvor and [START_REF] Macivor | Invertebrates on Green Roofs[END_REF] mentioned that despite their important role in soil, earthworms are rarely encountered on green roofs because of their limited ability to disperse and also due to lack of moisture at peak summer periods or freezing temperatures in winter.

Furthermore, the shallow depth of extensive green roofs substrate (< 15 cm) restricts their ability to move to deeper soil during extreme weather conditions. In our case, the substrate depth was 25 cm including 5 cm of vermicompost which created favorable conditions for the development of the earthworms, enabling them to move to deeper areas during dry or cold conditions, thus enhancing their survival probability and the degree to which they provide ecosystem services. In our experimental design, despite no direct estimates of earthworm survival, the presence of turricules at the surface of the substrate two years after the start of the experiment is a good indicator of the survival of the earthworms in the substrate.

Practical implications

Composting is an easy and promising alternative for producing organic fertilizer from food and green waste. For many local authorities, implementing a food waste collection provides a realistic solution to meet regulatory targets for diverting biodegradable waste from landfill and increasing recycling and composting rates. There are several possibilities for food waste collections: the choice will depend on the current waste collection organization, the quantities required, budget, and proximity of treatment facilities, political acceptability and public support. Overall, utilizing food waste in composting is a promising approach which will not only help fulfill fertilizer demand but also provide sustainable waste management.

Although the addition of compost was shown to increase the quality of green roof substrates, there are other consequences leading to some practical recommendations. Firstly, compost is generally not stable over the medium and long-term: it must be regularly reapplied over extended periods in order to maintain organic matter levels [START_REF] Bates | Effects of varying organic matter content on the development of green roof vegetation: A six year experiment[END_REF].

Secondly, home-composting systems may be a source of anthropogenic greenhouse gas emissions [START_REF] Lou | The impact of landfilling and composting on greenhouse gas emissions -A review[END_REF]. Therefore, special attention must be paid to reduce greenhouse gas emissions during composting. Nevertheless, several studies [START_REF] Li | Effect of greenroof on ambient CO2 concentration[END_REF][START_REF] Morau | Life Cycle Analysis of Green Roof Implemented in a Global South Low-income Country[END_REF][START_REF] Kuronuma | CO2 Payoffof extensive green roofs with different vegetation species[END_REF] have shown that when compost is used on green roofs, this may lower CO2 emission in the atmosphere.

The uses of compost are growing as the means of improving its chemical and physical qualities are increased. The combination substrate/compost/earthworms on green roofs can help reduce the risk of flooding in urban areas by increasing rainwater retention [START_REF] Graceson | The water retention capabilities of growing media for green roofs[END_REF], and green roofs seem to be a sink for anthropogenic gas emission [START_REF] Li | Effect of greenroof on ambient CO2 concentration[END_REF][START_REF] Morau | Life Cycle Analysis of Green Roof Implemented in a Global South Low-income Country[END_REF]. The addition of earthworms may allow improving the distribution of the substrate and compost within the substrate and may also improve soil structure, fertility and microbial activity [START_REF] Paz-Ferreiro | The effect of biochar and its interaction with the earthworm Pontoscolex corethrurus on soil microbial community structure in tropical soils[END_REF]. Overall, this may indirectly contribute to preserving biodiversity in cities.

Conclusions

This study showed that adding vermicompost and anecic earthworms to a commercial green roof substrate will (i) increase available N and P, (ii) increase alkaline and acid phosphatase and fluorescein diacetate hydrolysis activities, and (iii) increase microbial respiration and C source metabolic rates. The increased enzyme activity in the amended substrate increased the available N and P which are important nutrients for plant growth. Plant above-ground biomass, nectar volume and nectar sugar concentration were, therefore, increased, leading to higher pollinator visitation. To our knowledge, this is the first experiment to show that amending green roof substrates with earthworms and vermicompost can have a significant effect on green roof ecosystem function, through plant biomass production and plantpollinator interactions. The results confirm our hypothesis that amending commercial green roof substrates with earthworms and compost can improve the biological quality of the substrate.

Captions

  The plant community comprised five perennial species Hylotelephium maximum L. (Holub, Grand Orpine Sedum telephium var maximum L., Crassulaceae); Centaurea jacea L. (Brown knapweed, Asteraceae), Lotus corniculatus L. (Bird's-foot trefoil, Fabaceae), Koeleria glauca (Spreng.) DC. (Blue hair grass, Poaceae), Dianthus carthusianorum L.

Figure 1 :

 1 Figure 1: Experimental design at the University of Paris Est Créteil, Val de Marne (Ile-de-

Figure 2 :

 2 Figure 2: Number of pollinators visiting flowers (A) and number of floral visits by pollinators

Figure 1 :Figure 2 :

 12 Figure 1: Experimental design at the University of Paris Est Créteil, Val de Marne (Ile-de-France, south east of Paris)

Table 1 :

 1 Meteorological parameters of the Rooftop and Ground locations in the morning and afternoon.

	Meteorological	Morning		Afternoon
	parameters	Ground	Rooftop	Ground	Rooftop
	Mean temperature (°C)	30.7 ± 7.2a	29.2 ± 9.2a	36.6 ± 8.3b	34.8 ± 11.3b
	Light intensity (lux)	23.8 ± 8.2a	20.8 ± 4.9a	57.2 ± 10.3b	70.1 ± 13.8c
	UV index	2.0 ± 0.5a	3.5 ± 0.5b	4.5 ± 0.7c	6.2 ± 1.5d
	Mean wind speed (m/s)	3.4 ± 1.8a	5.7 ± 2.1b	2.8 ± 0.7a	6.3 ± 1.2b
	Variation in wind speed	7.1± 2.1a	10.3 ± 1.8b	6.8 ± 1.9a	11.9 ± 3.2b
	Humidity (%)	54.3 ± 10.5b 50.5 ± 13.8b	35.6 ± 8.3a	37.7 ± 10.3a
	In each column, numbers followed by different letters are significantly different (p < 0.05)
	according to Fischer's PLSD-test. Mean values (n = 27) ± standard error.

Table 2 :

 2 Physical and chemical properties of the substrate in "LC", low complexity, and "HC", height complexity, treatments at two locations at the end of the experiment and F-values of a two-way Anova measuring the effects of earthworm and compost on plant parameters. OM, organic matter; N: nitrogen; N-soil, total soil nitrogen; NO -3, soil nitrate content; NH4 + , soil ammonium content; P2O5, soil phosphorus content.

	Location Treatment	pH	OM	Organic C	Total N	N-NH4 +	N-NO3 -	P-P2O5	Field capacity	Soil humidity
				(g/kg)	(g/kg)	(100mg/kg)	( mg/kg)	(mg/kg)	(mg/kg)	(g/kg)	(g/kg)
	Rooftop	LC	8.1 ± 0.06b	79.7 ± 23.3a	46.1 ± 13.5a	30.7 ± 1.0a 41.6 ±2.5a	31.2 ± 16.2a	108.3 ± 45.7a	21.7 ± 4.9a	17.2 ± 1.5a
	Rooftop	HC	7.8 ± 0.02a	250.3 ± 66.6b	144.7 ± 13.8b	96.6 ± 3.4c 137.7 ± 21.2b	716.0 ± 60.3c	862.0 ± 40.9b	48.8 ± 2.3b	93.2 ± 4.0b
	Ground	LC	8.1 ± 0.10b	87.9 ± 28.5a	50.7 ± 16.4a	33.3 ± 1.2a 38.1 ± 4.6a	19.6 ± 6.2a	135.0 ± 37.6a	24.4 ± 2.1a	9.6 ± 7.4a
	Ground	HC	7.9 ± 0.03a	211.3 ± 35.9b	122.4 ± 20.1b	76.6 ± 2.1b 199.0 ± 81.3c	352.3 ± 81.3b	639.3 ± 161.6b 34.5 ± 5.3b	90.4 ± 14.9b
	Location (A)	1.2ns	1.1ns	1.1ns	2.7ns	1.4ns	14.1**	6.9ns	4.6ns	3.1ns
	Treatments (B)	66.4***	98.8***	99.7***	104.9***	28.0***	103.3***	407.1***	60.5***	121.3***
		A x B	1.2ns	2.6ns	2.5ns	4.5ns	1.8ns	12.8**	16.0**	17.6**	1.8ns

In each column, numbers followed by different letters are significantly different (p < 0.05) according to Fischer's PLSD-test. Mean values (n = 5) ± standard error. F values and significance level: ns: not significant; *p < 0.05; **p < 0.01; ***p < 0.001.

Table 3 :

 3 Plant parameters for "LC", low complexity, and "HC", high complexity, treatments at rooftop and on the ground at the end of the experiment and F-values of a two-way Anova measuring the effects of earthworm and compost on plant parameters.

	Location Treatment	Above-ground	Vegetative	Number of	Chlorophyll	Chlorophyll	Chlorophyll	Volume of nectar	[sugar] in
			biomass plant	cover	bourgeons	content (µmol.m 2 )	content (µmol.m 2 )	content (µmol.m 2 )	in flowers	flowers
			(g)	(%)	(Sedum)	(Sedum)	(Centaurea)	(Lotier)	(Sedum)	(Sedum)
	Rooftop	LC	129.5 ± 28.7a	77.7 ± 5.2a	20.7 ± 2.6a	22.7 ± 5.3a	32.5 ± 2.3a	37.5 ± 6.7a	0.16 ± 0.06a	17.5 ± 5.0a
	Rooftop	HC	289.2 ± 39.6b	92.9 ± 9.8b	44.4 ± 3.4b	34.4 ± 6.4 b	43.4 ± 4.9b	45.4 ± 7.9b	0.53 ± 0.07b	49.8 ± 4.8c
	Ground	LC	120.2 ± 26.1a	88.5 ± 8.4a	24.1 ± 2.6a	38.5 ± 10.3b	32.3 ± 5.2a	39.9 ± 5.9a	0.17 ± 0.02a	12.0 ± 2.3a
	Ground	HC	369.2 ± 43.9c	94.8 ± 9.2b	40.8 ± 2.1b	56.9 ± 9.7c	41.8 ± 4.5b	45.1 ± 10.4b	0.59 ± 0.06b	40.4 ± 3.0b
	Location (A)	0.6ns	12.3**	0.1ns	163.4***	2.6ns	2.2ns	2.3ns	8.3*
	Treatments (B)	21.6***	63.7***	65.8***	100.5***	29.2***	84.4***	32.8***	12.6**
		A x B	1.4ns	10.9**	0.01ns	4.9*	12.6***	3.6ns	2.8ns	0.2ns

In each column, numbers followed by different letters are significantly different (p < 0.05) according to Fischer's PLSD-test. Mean values (n=5) ± standard error. F values and significance level: ns: not significant; *p < 0.05; **p < 0.01; ***p < 0.001.

Table 4 :

 4 Microbial activities (respiration, Biolog TM and enzyme) of the "LC", low complexity, and "HC", height complexity, substrates at two locations at different growth stages. In each column, numbers followed by different letters are significantly different (P < 0.05) according to Fischer's PLSD-test. Mean values (n = 5) ± standard error. F values and significance level: ns: not significant; *P < 0.05; **P < 0.01; ***P < 0.001.

	Stage of plant	Localization Treatment Respiration	AWCD	Richness (R) Shannon (H)	FDA	ACP	ALP
	Vegetative	Rooftop	LC	39.3 ± 4.2a	0.28 ± 0.02b	11.8 ± 2.2b	1.21 ± 0.06c	0.69 ± 0.3a	7.6 ± 2.1b	12.1 ± 5.5d
		Rooftop	HC	75.9 ± 4.1d	0.34 ± 0.07c	15.4 ± 2.1c	1.41 ± 0.23d	3.32 ± 0.8c	14.5 ± 4.3c	43.8 ± 11.5f
		Ground	LC	48.6 ± 8.2ab 0.21 ± 0.04ab	10.8 ± 1.8ab	1.02 ± 0.13b	1.10 ± 0.8ab	8.8 ± 6.9b	20.6 ± 17.8e
		Ground	HC	92.0 ± 1.9e	0.35 ± 0.07c	16.2 ± 4.2c	1.48 ± 0.23d	7.00 ± 3.7e	20.7 ± 9.0d	51.5 ± 17.3f
	Floral	Rooftop	LC	37.9 ± 2.1a	0.16 ± 0.05a	8.4 ± 2.7a	0.81 ± 0.15b	3.80 ± 0.6c	2.8 ± 1.9a	2.9 ± 2.6a
		Rooftop	HC	67.2 ± 2.4c	0.23 ± 0.09ab	9.2 ± 2.9ab	0.90 ± 0.27ab	4.94 ± 1.6c	2.8 ± 0.8a	6.1 ± 2.5b
		Ground	LC	44.0 ± 2.2ab 0.13 ± 0.09a	6.2 ± 3.8a	0.64 ± 0.29a	1.11 ± 1.3a	2.4 ± 1.3a	3.7 ± 1.7a
		Ground	HC	63.8 ± 3.9c	0.28 ± 0.10b	11.0 ± 3.8b	1.01 ± 0.39b	4.35 ± 0.2cd	8.5 ± 2.7b	8.8 ± 4.1c
		Localization (A)	3.14ns	0.07ns	0.01ns	0.21ns	0.4ns	1.2ns	0.7ns
		Treatments (B)		66.1***	16.0***	8.8**	7.6**	24.2***	8.2**	4.9*
		(A x B)*Stage of plant	0.03ns	1.76ns	1.4ns	1.6ns	3.5ns	1.3ns	0.02ns

LC: low complexity; HC: high complexity; AWCD: Average well color development; FDA: Fluorescein diacetate hydrolysis; ADP: Acid phosphatase; ALP: Alkaline phosphatase.

Table 5 :

 5 Correlation matrix between parameters from different treatments with and without earthworms and compost at Ground and Rooftop levels. Average well color development; FDA: Fluorescein diacetate hydrolysis; ADP: Acid phosphatase; ALP: Alkaline phosphatase; C: Carbon; N: Nitrogen; NO -3, soil nitrate content; NH4 + , soil ammonium content; P2O5, soil available phosphorus content. F values and significance level: ns: not significant; *P < 0.05; **P < 0.01; ***P < 0.001 according to Fischer's PLSD-test.

		Organic matter C content	Total N	Available N (NH4 + ) Available N (NO3 -) Available P (PO4 2-)
	Acid phosphatase activity	0.73**	0.74**	0.73**	0.67*	0.63*	0.75**
	Alkaline phosphatase activity 0.81**	0.81**	0.78**	0.91***	0.65*	0.78**
	FDA activity	0.67*	0.67*	0.62*	0.88***	NS	0.62*
	Microbial respiration	0.89***	0.89***	0.86***	0.89***	0.72**	0.87***
	AWCD	0.67*	0.67*	0.63*	0.79**	0.57*	0.73**
	Plant biomass	0.76**	0.76**	0.74**	0.75**	0.63*	0.79**
	Plant cover	0.78**	0.78**	0.76**	0.70*	0.63*	0.74**
	Nectar volume	0.89***	0.89***	0.89***	0.77**	0.84***	0.94***
	Amount of sucrose in nectar 0.88***	0.88***	0.88***	0.70*	0.88***	0.93***
	AWCD:

Acknowledgements 20

This research was supported by Université Paris-Est Créteil (UPEC) in France, in the framework of an initiation of federative research projects.


Revaz Ramazashvili

Zeeman spin-orbit coupling in antiferromagnetic conductors

This article is a brief review of Zeeman spin-orbit coupling, arising in a low-carrier commensurate Néel antiferromagnet subject to magnetic field. The field tends to lift the degeneracy of the electron spectrum. However, a hidden symmetry protects double degeneracy of Bloch eigenstates at special momenta in the Brillouin zone. The effective transverse g-factor vanishes at such points, and thus acquires a substantial momentum dependence, which turns a textbook Zeeman term into a spinorbit coupling. After describing the symmetry underpinnings of the Zeeman spin-orbit coupling, I compare it with its intrinsic counterparts such as Rashba coupling, and then show how Zeeman spin-orbit coupling may survive in the presence of intrinsic spin-orbit coupling. Finally, I outline some of the likely experimental manifestations of Zeeman spin-orbit coupling, and compare it with similar phenomena in other settings such as semiconducting quantum wells.

I. INTRODUCTION

Spin-orbit coupling, the central character of this Special Issue, appears in non-relativistic quantum mechanics as only a vestige of relativity, in the form of the Pauli term H P in the Schrödinger Hamiltonian [START_REF] Kittel | Quantum Theory of Solids[END_REF]

H P = 4m 2 0 c 2 σ • p × ∇V (r), (1)
where is the Planck constant, c the speed of light, m 0 is the free electron mass, p is the electron momentum, σ its spin, and V (r) its potential energy as a function of the electron coordinate r. In the low-energy expansion, the H P arrives as a second-order term in the expansion in weak coupling constant α = e 2 / c ≈ 1/137. In this sense, spin-orbit coupling is indeed small.

Yet in solids, spin-orbit coupling is responsible for a variety of fundamental phenomena. In magnets, it may induce magnetocrystalline anisotropy, whereby spontaneous magnetization acquires a preferred set of directions with respect to the crystal axes. [START_REF] Landau | Electrodynamics of Continuous Media[END_REF][START_REF] Cullity | Introduction to Magnetic Materials[END_REF] In antiferromagnets, spin-orbit coupling may give rise to "weak" ferromagnetism. [START_REF] Landau | Electrodynamics of Continuous Media[END_REF][START_REF] Dzialoshinskii | [END_REF][5][6] In transition metal compounds, coupling of spin, orbital and structural degrees of freedom leads to a multitude of unusual phases. [7] In Mott insulators, spin-orbit coupling may produce interesting effects such as realization of an effective Heisenberg-Kitaev model. [8] Last but not the least, in an innocuous band insulator spin-orbit coupling may bring to life topologically non-trivial electron states, that have been a subject of much attention. [9,10] In all of these cases, spin-orbit coupling is intrinsic: it acts in the absence of any external perturbation applied to the crystal. By contrast, the present article is devoted to the Zeeman spin-orbit coupling, that may appear in a low-carrier Néel antiferromagnet, subject to magnetic field. Zeeman spin-orbit coupling is thus a particular example of the Zeeman effect. At the same time, it entangles orbital motion of an electron with its spin, and hence represents a true spin-orbit coupling. Zeeman spin-orbit coupling is proportional to the applied magnetic field, and thus is inherently tunable. This distinguishes it from intrinsic spin-orbit coupling, that acts in the absence of any external field, and that most articles in this Topical Issue focus on.

By virtue of the Zeeman spin-orbit coupling, magnetic field splits a single doublydegenerate electron band into two bands that are non-degenerate almost everywhere in the Brillouin zone. While this is indeed just a form of the Zeeman effect, there is also a similarity here to how, in the absence of inversion symmetry, intrinsic spin-orbit coupling lifts double degeneracy of an electron band in a non-magnetic crystal. This similarity provides a useful perspective, hence the article opens with Section II, that presents a basic overview of degeneracies in a non-magnetic crystal, with and without external magnetic field. Section III does the same for a commensurate collinear Néel antiferromagnet, and elucidates the analogy with non-magnetic case. Most of the Section III is a pedagogical presentation of the degeneracies expected in the absence of intrinsic spin-orbit coupling, where the electron spin is entirely decoupled from its orbital motion. [11,12] The subsection III.c then analyses spectral degeneracies that may appear in transverse magnetic field in the presence of intrinsic spin-orbit coupling. Section IV outlines the implications in the linear order in the field, introduces the Zeeman spin-orbit coupling and recapitulates some of its properties.

Section V addresses some of the likely experimental manifestations of the Zeeman spin-orbit coupling, while Section VI offers a brief comparison with other settings, and an outlook.

The review covers early studies of the Zeeman spin-orbit coupling in antiferromagnets as well as a case that remains largely unexplored, where antiferromagnetism coexists with substantial intrinsic spin-orbit coupling. I attempted to make the presentation pedagogical and coherent, and hope the reader will find such a review useful.

II. SPECTRAL SYMMETRIES IN ZERO FIELD II.a A non-magnetic crystal

Key insight into spectral degeneracies can be gained by simple symmetry arguments.

Consider a non-magnetic crystal, that is one symmetric under time reversal θ. Kramers theorem [START_REF] Kittel | Quantum Theory of Solids[END_REF] tells us, that every single-electron Bloch eigenstate |p at momentum p has a degenerate orthogonal partner θ|p at momentum -p. Attentive reader will instantly recognize that these special momenta p * are nothing but the "time reversal-invariant momenta" (TRIM), that have played crucial role in the analysis of topological insulators. [14] However, the role TRIM may play in the appearance of spectral degeneracies was appreciated long before the advent of topological electron systems. [15][16][17] If, in addition to time reversal, the crystal is symmetric with respect to inversion I, then of the form

H SO = d p • σ, (2)
where d p is a real pseudovector, that is an odd function of momentum p, and σ is the electron spin. Depending on the crystal structure and chemical composition, spin-orbit coupling may take a multitude of forms, whose review would go far beyond the subject of the present article. A concise analysis of symmetry properties of spin-orbit coupling can be found in Refs. [20,21]. An early example of such a coupling was introduced by E. I.

Rashba. [22,[START_REF] Rashba | Proc. Int. Conf. Semiconductor Physics[END_REF] II.b Rashba spin-orbit coupling Rashba spin-orbit coupling [22][START_REF] Rashba | Proc. Int. Conf. Semiconductor Physics[END_REF][START_REF] Yu | [END_REF][25] is a form of the H SO of the Eq. ( 2) in semiconductor quantum wells and heterostructures, where band mismatch at the interface produces electron states that are localized in the transverse direction, but behave as Bloch states in the two dimensions along the interface. For such states, at low momenta, the H SO in the Eq. [START_REF] Landau | Electrodynamics of Continuous Media[END_REF] reduces to

H R = α R (σ • p × n), (3)
where α R is a material-specific constant, p and σ are the electron momentum and spin operators, and n is the normal to the interface.

What may be the order of magnitude of the α R ? For a naïve estimate, let us turn to the textbook Pauli spin-orbit coupling term of the Eq. [START_REF] Kittel | Quantum Theory of Solids[END_REF] problem in its own right, as is extracting α R from experimental data. [START_REF] Yu | [END_REF]25,[START_REF] Winkler | Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems[END_REF]28] In most materials of experimental interest, the ratio α R /c (c being the speed of light) falls in the range α R /c ∼ 10 -4 ÷ 10 -2 . The example of Rashba coupling will prove useful below, as a reference point for the strength of the Zeeman spin-orbit coupling.

III. SPECTRAL SYMMETRIES OF A COLLINEAR N ÉEL ANTIFERROMAGNET III.a Collinear Néel antiferromagnet in zero field

Before turning to the Zeeman spin-orbit coupling, let us define the class of materials, where it is expected to appear -a collinear commensurate Néel antiferromagnet. At each point r in the sample, such an antiferromagnet is characterized by spontaneous local magnetization density M(r), that changes sign upon translation by a period a of the crystal lattice: M(r + a) = -M(r). The adjective "collinear" implies that, everywhere in the sample, M(r) points along or opposite a single direction, one and the same throughout the sample.

In the following, M(r) will be treated as static; both thermal and quantum fluctuations of magnetic order are thus entirely neglected. This will restrict us to the bulk of the antiferromagnetic phase, far from any phase transition, thermal or quantum. In particular, this requires temperatures far below both the Néel temperature and the electron bandwidth. In practice, this also implies the ordered moment, noticeable on the scale of the Bohr magneton.

Some of potentially relevant materials were discussed in the Ref. [12].

First, it is helpful to understand the spectral degeneracies of such a magnet in zero field, and there is a perfect analogy here with what happens in a non-magnetic crystal, whose spectral symmetries were recapitulated in the Subsection II.a. The local magnetization density M(r) couples to the electron spin σ via the Néel exchange coupling

H N = ∆ r • σ,
where the ∆ r is proportional to M(r) and thus inherits its transformation properties: ∆ r changes sign upon lattice translation T a as well as upon time reversal θ. Thus, while neither T a nor θ is a symmetry of the Néel state, their product θT a is indeed a symmetry.

In the presence of inversion symmetry, action of I and of θT a on an exact Bloch eigenstate In a non-centrosymmetric antiferromagnetic crystal, there is no single universal symmetry to protect a degeneracy of Bloch eigenstates at an arbitrary momentum. Moreover, in an antiferromagnet, the Kramers orthogonality relation p|θ|p = -p|θ|p for a non-magnetic crystal is replaced by a less stringent condition

p|θT a |p = -e -2ip•a p|θT a |p (4)
for the scalar product of a given Bloch eigenstate |p at momentum p and its symmetry partner θT a |p at momentum -p. The momenta p such that p = -p + Q (where Q is an antiferromagnetic reciprocal lattice vector) now fall into two classes. [START_REF] Bègue | [END_REF]31] The first class are those p, where e -2ip•a = 1 (such as the relevant points at the non-magnetic Brillouin zone boundary, and the point Γ at the center of the Brillouin zone); they host a doublet of degenerate states. By contrast, for those p, where e -2ip•a = -1, the θT a protects no degeneracy.

Up to the distinction outlined in the preceding paragraph, the analogy between a collinear commensurate Néel antiferromagnet and a paramagnet is complete. In a centrosymmetric material, all bands are doubly degenerate at all momenta in the Brillouin zone, be it an antiferromagnet or a non-magnetic crystal. In the absence of inversion symmetry, the electron bands are non-degenerate, with the exception of special momenta in the Brillouin zone, where the time reversal θ (in a paramagnet) or θT a (in an antiferromagnet) protect a double degeneracy. Near such a momentum, the two split sub-bands can be described by an effective

Hamiltonian with an intrinsic spin-orbit coupling term (2) with d p = 0 at the degeneracy point.

Described above, symmetry-protected degeneracies in a commensurate Néel antiferromagnet were understood over half a century ago. [START_REF] Herring | Magnetism[END_REF] However, symmetry in the presence of magnetic field was not explored until much later.

III.b Collinear centrosymmetric Néel antiferromagnet in magnetic field

A fundamental difference between a non-magnetic crystal and an antiferromagnet resides in how the double degeneracy of Bloch eigenstates at a given momentum is lifted by magnetic field. In a non-magnetic crystal, the Zeeman term H Z = (H • σ) tends to lift the double degeneracy everywhere in the Brillouin zone. In the presence of inversion symmetry, this is illustrated in the leftmost panel of the Fig. 1.

In an antiferromagnet, the situation is more interesting. Here, the field is applied to a system that already has a preferred direction, defined by the staggered exchange field ∆ r .

Thus, the lifting of the double degeneracy by magnetic field H may be sensitive to how H is oriented relative to ∆ r . It turns out, that a longitudinal field H ∆ r lifts the degeneracy everywhere in the Brillouin zone, as illustrated in the leftmost panel of the Fig. 1. This is hardly surprising. Much more interestingly, in a transverse field H ⊥ ∆ r , the system retains enough symmetry to protect double degeneracy of Bloch eigenstates at a special set of momenta in the Brillouin zone.

This symmetry becomes apparent as soon as one pictures a collinear centrosymmetric

Néel state in a transverse field H, as sketched in the Fig. 2. In a lattice model, the two Néel sublattices simply tilt towards H, as do the ∆ r in the upper left corner of the figure and the ∆ r+a in its lower right corner. Magnetic moment along the field is invariant upon translation by a, while the moment along the initial direction of the staggered magnetization changes sign. More generally, if the physics is not restricted to lattice sites, local magnetization is a function of continuous coordinate r and, in a transverse field, ∆ r is no longer collinear.

inherits the Néel translation antisymmetry: ∆ r+a = -∆ r . By contrast, the component ∆ ⊥ r along the applied field represents the field-induced magnetic moment and is translationally symmetric: Remarkably, this can be undone by a single uniform rotation U n (π), a symmetry of the zero-field state. [START_REF] Again | the absence of H SO = d p • σ, since the latter varies under U n (π)[END_REF] As a result, the combination θT a U n (π) is a symmetry of the Néel state in a finite transverse field. Its action on a Bloch eigenstate |p produces a degenerate partner eigenstate at momentum -p. Does this symmetry lead to a degeneracy at a given momentum? It does. Anti-unitarity of θT a U n (π) leads to an analogue of the Kramers orthogonality relation [11,12] p|θT a U n (π)|p = e -2ip•a p|θT a U n (π)|p (5) between the Bloch eigenstate |p at momentum p and its symmetry partner state at -p.

∆ ⊥ r+a = ∆ ⊥ r . π ∆ ∆ θ π ∆ θ π ∆ = ∆ H r+a r n U ( ) n r r U ( ) r+a r n a T U ( ) r U ( ) n
Those p * that are equivalent to their opposite modulo a reciprocal lattice vector Q of the antiferromagnetic state (p * = -p * + Q), and for whom the exponent in the right-hand side of the Eq. ( 5) is different from unity, host a doublet of degenerate states.

The equation p * = -p * + Q implies that such momenta p * lie at the antiferromagnetic Brillouin zone boundary. However, for those p * that also belong to the non-magnetic Brillouin zone boundary, [START_REF]By non-magnetic Brillouin zone boundary I imply the zone boundary of the Hamiltonian with the Néel exchange term H N = ∆ r • σ notionally set to zero[END_REF] the exponent e -2ip•a in the r.h.s. of the Eq. ( 5) equals unity, and thus such a p * does not host a symmetry-protected degeneracy. The precise geometry of the set of p * depends on the conspiracy between the crystal symmetry and the periodicity of the antiferromagnetic order. A number of possible examples were discussed in the Ref. [12].

In a one-dimensional doubly-commensurate antiferromagnet, the degeneracy is guaranteed at p * = ±π/2, as illustrated in the Fig. 3. The arguments above demonstrated how, at a special set of momenta in the Brillouin zone, symmetry protects double degeneracy of the electron spectrum in a centrosymmetric Néel antiferromagnet, subject to a finite transverse magnetic field. However, these arguments tacitly implied both the Néel order and the electron spin to be decoupled from the underlying crystal lattice. In other words, the intrinsic spin-orbit coupling H SO of the Eq. ( 2) was neglected altogether.

-π - π 2 π 2 π p ϵ(p) -π - π 2 π 2 π p ϵ(p
Indeed, as a (formally) small relativistic correction, the H SO may often be ignored. At the same time, the opposite limit of a significant spin-orbit coupling is of considerable interest:

firstly, intrinsic spin-orbit coupling grows rapidly with the atomic number Z. In many antiferromagnets, this alone may rule out the possibility of neglecting H SO . Secondly, spinorbit coupling plays a key role in topological properties of condensed matter and, recently, a body of work has been devoted to topological properties of antiferromagnetic insulatorssee Ref. [START_REF] Mong | [END_REF] and the subsequent work by several groups of authors.

Topologically non-trivial electron properties are intimately related to degeneracies in the electron spectrum. In a Néel antiferromagnet without intrinsic spin-orbit coupling, such degeneracies are symmetry-protected even in a finite transverse magnetic field [11,12]. But could symmetry-protected degeneracies be present in an antiferromagnet with substantial intrinsic spin-orbit coupling? [36] At first sight, this is unlikely. As we saw in Subsections II.a and III.a, intrinsic spin-orbit coupling tends to lift the spectral degeneracy everywhere, except for a special set of points in the Brillouin zone. It seems that magnetic field could only lift any remaining degeneracy.

Yet, this is not necessarily the case, as shown below.

To advance further, we must analyze the symmetries of a Hamiltonian that involves, in addition to a common "non-magnetic" part, three key terms. x + p 2 y ]), the high-symmetry direction is the z-axis. Below, I consider the case of the ∆ r pointing along the z-axis of the d BHZ p . In fact, this choice was implicitly made already in prior work on antiferromagnetic topological insulators, [START_REF] Mong | [END_REF] where the ∆ r was chosen to point along the symmetry axis ẑ of the Bernevig-Hughes-Zhang (BHZ) model [37] Now, what happens in a field H ⊥ , transverse with respect to n? Without H BHZ SO , the product θT a U n (π) was a symmetry in such a transverse field. The H SO is symmetric under both the θ and T a , but not under U n (φ): as we saw in the previous paragraph, to make a symmetry, U n (φ) had to be combined with the orbital rotation R n (-φ) around the same axis. As a result, a generic Hamiltonian, involving the H BHZ SO , the Néel exchange H N = ∆ r •σ and the transverse field H ⊥ • σ, is invariant under the combination θT a U n (π)R n (π). Does this symmetry induce a degeneracy? It does, if [θT a U n (π)R n (π)] 2 = 1. Which, in turn, depends on the mutual orientation of a and n.

If a ⊥ n, then [T a R n (π)] 2 = 1, hence [θT a U n (π)R n (π)] 2 = 1,
and the symmetry θT a U n (π)R n (π) induces no degeneracy. [START_REF]As a side remark, notice that, for a ⊥ n, the state θT a U n (π)R n (π)|p resides at the same momentum p, as the state |p : T b [θT a U n (π)R n (π)] |p = [[END_REF] In two dimensions, a ⊥ n is the only possibility. Thus, in a two-dimensional Néel antiferromagnet with intrinsic spin-orbit coupling, transverse magnetic field lifts the Kramers degeneracy.

In three dimensions, the a and n may be parallel; then T a commutes with R n (π), and

[T a R n (π)] 2 = T 2a . As a result, when acting on a Bloch eigenstate |p , [θT a U n (π)R n (π)] 2 =
T 2a = e 2ip•a . Also, in contrast to the case a ⊥ n, for a n the state θT a U n (π)R n (π)|p resides at the momentum -p. [START_REF][END_REF] In the antiferromagnetic Brillouin zone, the momentum planes p • a = ±π/2 are equivalent, and for such momenta [θT a U n (π)R n (π)] 2 = T 2a = e iπ = -1. Thus, at such momenta, the symmetry θT a U n (π)R n (π) protects double degeneracy in a transverse magnetic field. This is illustrated in the Fig.

∆ r • σ d p • σ (d p + ∆ r ) • σ (d p + ∆ r + H ⊥ ) • σ T 2a T a T 2a T 2a θT a θ θT a U n (φ) U n (φ)R n (-φ) U n (φ)R n (-φ) θT a U n (π)R n (π)
R n (ψ)

IV. ZEEMAN SPIN-ORBIT COUPLING

The arguments of the preceding section have demonstrated a rather peculiar phenomenon: in a commensurate Néel antiferromagnet subject to transverse magnetic field, Bloch eigenstates remain doubly degenerate at a set of special momenta in the Brillouin zone. This degeneracy is protected by symmetry even in a finite field -and, therefore, holds to any order in the field.

A fundamental consequence of this degeneracy appears already in the first order in the field, that is in the form of the effective Zeeman term H ef f Z . By its very nature, antiferromagnet has a special direction, set by the staggered magnetization. Of course, other special directions may exist -defined, for instance, by the crystal structure, but I will first describe the limit where there are none, that is in the absence of any intrinsic spin-orbit coupling.

In the latter case, the only anisotropy of H ef f Z is set by the orientation of the magnetic field H relative to the unit vector n of the staggered magnetization:

H ef f Z = - µ B 2 g (H • σ) + g ⊥ (H ⊥ • σ) .
Here, µ B is the Bohr magneton, the H ≡ n(H • n) and H ⊥ ≡ H -H are the longitudinal and transverse components of the field with respect to the unit vector n of the staggered magnetization, and the g and g ⊥ are the longitudinal and transverse g-factors, respectively. Now, in a transverse field, double degeneracy of Bloch eigenstates at certain special momenta p = p * in the Brillouin zone means that the g ⊥ must vanish at such momenta. Not being identically equal to zero, the g ⊥ must, therefore, substantially depend on momentum p, and the H ef f Z shall be re-written as

H ef f Z = - µ B 2 g (H • σ) + g ⊥ (p)(H ⊥ • σ) . ( 6
)
Momentum dependence of the second term in the r.h.s. above, along with the presence of electron spin σ, turns the textbook Zeeman term into a veritable spin-orbit coupling that is the subject of the present article.

The g is a constant, while general properties of the g ⊥ (p) are as follows. [12] In the absence of intrinsic spin-orbit coupling, the g ⊥ (p) vanishes on a manifold, defined by the equation g ⊥ (p) = 0 in the d-dimensional Brillouin zone. This "degeneracy manifold" is (d-1)-dimensional. In one dimension, it comprises a set of special points. In two dimensions -a set of special lines. [START_REF] Ramazashvili | [END_REF]41] In three dimensions, it forms a set of special surfaces.

Nevertheless, over most of the Brillouin zone, the g ⊥ (p) is close to ±1, and differs from these two values only within a momentum range of the order of ξ a around the degeneracy manifold. Here ξ is of the order of the antiferromagnetic coherence length [12], and is large on the scale of the lattice spacing a. Variation of g ⊥ (p) is thus limited to a momentum range that is small compared with the size of the Brillouin zone. Within a momentum range of about ξ around the manifold g ⊥ (p) = 0, the g ⊥ (p) varies linearly with momentum: g ⊥ (p) ∼ pξ/ , where p is measured along the local normal to the degeneracy manifold. Such a variation of g ⊥ (p) is illustrated in the Fig. 5. This simplest form of the Zeeman spin-orbit coupling ( 6) is linear in momentum, as is the intrinsic Rashba spin-orbit coupling [START_REF] Cullity | Introduction to Magnetic Materials[END_REF]. Thus it is instructive to compare the two. Firstly, the Zeeman spin-orbit coupling field d ZSO (p) = -µ B 2 H ⊥ pξ points along the H ⊥ . By contrast, the spin-orbit field d R (p) = α R p × n of the Rashba coupling is manifestly non-collinear.

Secondly, let us compare the two coefficients of the linear momentum dependence.

For the Rashba coupling, the coefficient is given by the α R . For an elemental material with atomic number Z, the naïve order-of-magnitude estimate of the Section II.b gave In practice, α R /c ∼ 10 -4 ÷ 10 -2 .

α R ∼ e 2 • Z
By contrast, the Zeeman spin-orbit coefficient relative to c is given by α

ZSO /c ∼ µ B H ⊥ ξ c ∼ v F c µ B H ⊥ ∆ ∼ e 2 c µ B H ⊥ ∆ ∼ 1 137 µ B H ⊥ ∆
, where the Fermi velocity v F and the antiferromagnetic coherence length ξ were estimated as per v F ∼ e 2 and ξ ∼ ∆/v F , with ∆ being the antiferromagnetic gap in the electron spectrum. Notice that α ZSO /c carries only a single weak coupling constant α ≈ 1/137 compared with α 3 in the estimate of α R /c.

The ratio of the two couplings is thus α

ZSO /α R ∼ c Ze 2 2 µ B H ⊥ ∆ ∼ 137 Z 2 µ B H ⊥ ∆ .
The present theory applies only in the limit µ B H ⊥ ∆ 1. However, the factor 137 Z 2 is generally large. While this is, indeed, only a hand-waving argument, it indicates that even in a relatively weak field µ B H ⊥ ∆ 1, Zeeman spin-orbit coupling may be comparable to a rather substantial Rashba coupling.

Near higher-symmetry points in the Brillouin zone, the degeneracy manifold may cross itself; in this case the g ⊥ (p) shows a more interesting behavior. For example, in a twodimensional antiferromagnet on a square lattice, the degeneracy lines coincide with the magnetic Brillouin zone boundary p x + p y = ±π and -p x + p y = ±π. Near the corner point (0, π) and (π, 0) of the magnetic Brillouin zone (point X), one finds g ⊥ (p) ∝ p 2

x -p 2 y . At such a point, a degenerate isotropic band extremum (p) = p 2 /2m is split by transverse field into two ellipsoid sub-bands

± (p) = [1 ± H ⊥ /∆] p 2 x /2m + [1 ∓ H ⊥ /∆] p 2 y /2m
, where the energy scale ∆ characterizes g ⊥ (p) near point X. Such a behavior is shown in the Fig. 6. [START_REF] Ramazashvili | [END_REF] The same figure also illustrates linear band splitting near point Σ = (π/2, π/2), where a band minimum ± (p) = p 2

x /2m x + p 2 y /2m y is split by transverse field H ⊥ as per

± (p) = p 2 x /2m x + p 2 y /2m y -pyξ (H ⊥ • σ).
The degenerate band minimum splits into two identical non-degenerate minima, shifted with respect to each other in momentum space:

± (p) = p 2 x /2m x + p y -myξ (H ⊥ • σ) 2 /2m y .
What is the leading term in the momentum expansion of the Zeeman spin-orbit coupling g ⊥ (p)(H ⊥ • σ) around the degeneracy planes p z = ±π/2, in the presence of an intrinsic spin-orbit coupling, described in the Subsection III.c? The symmetry of g ⊥ (p)(H ⊥ • σ) must match that of the Hamiltonian. In particular, g ⊥ (p)(H ⊥ • σ) must be invariant under θT a U n (π)R n (π). This implies g ⊥ (p x , p y , p z ) = g * ⊥ (p x , p y , -p z ), while the degeneracy means g ⊥ (p x , p y , ±π/2) = 0. Since g ⊥ (p)(H ⊥ • σ) is Hermitian, a linear term in p z with an imaginary coefficient is forbidden. The leading term allowed is thus

H ef f Z ∝ p 2 z (H ⊥ • σ).

V. EXPERIMENTAL MANIFESTATIONS

Zeeman spin-orbit coupling may manifest itself in a number of ways, that all stem from field-induced entanglement of the electron spin with its orbital motion. As a result, the Landau level spectrum and its Zeeman splitting acquire an unusual dependence on the field orientation [42][43][44] that, in turn, has a number of experimental consequences.

V.a Magnetic quantum oscillations as a diagnostic tool

One such consequence is that, in a purely transverse field, Landau levels undergo no Zeeman splitting [42][43][44] -if the carrier pocket is centered at a symmetry-protected degeneracy point such as those shown in the Fig. 6. [START_REF] Ramazashvili | [END_REF]41] In magnetic quantum oscillations, Zeeman splitting of the Landau levels produces the so-called 'spin zeros' [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF], that is special field orientations, where the oscillation amplitude vanishes. In an antiferromagnetic insulator, absence of spin zeros may thus constrain the precise position of the band extremum in the Brillouin zone, or even pinpoint it exactly. [START_REF] Ramazashvili | [END_REF]41] This could be particularly useful in antiferromagnets of complex structure and chemical composition, where the location of the band extrema in the Brillouin zone may be less than obvious.

V.b Zeeman electric-dipole resonance

In addition to thermodynamic measurements such as magnetic quantum oscillations, the Landau level spectrum can be studied by resonant spectroscopy. According to textbook, AC electric field excites spin-conserving cyclotron resonance (CR) transitions between adjacent

Landau levels, at the Larmor frequency, whereas AC magnetic field excites spin flip (spin resonance, ESR) transitions at the Zeeman frequency, at a fixed Landau level. Intrinsic spinorbit coupling enriches this simple picture rather dramatically: it allows one to induce ESR transitions by an AC electric rather than magnetic field -and, more generally, to induce transitions at combined frequencies. Predicted by E. I. Rashba over fifty years ago, [20] such transitions are called "electric-dipole spin resonance" or "combined resonance". In case of Zeeman spin-orbit coupling, such transitions may be called Zeeman electric-dipole resonance; their theory was developed in the Ref. [44].

The term 'electric-dipole resonance' implies that the resonance arises from a dipole term eE • r in the Hamiltonian, where E is the external electric field, and r is the displacement.

The resonance matrix elements are thus determined by the characteristic scale of r. For the textbook ESR, this length scale is given by the Compton length λ C = mc ≈ 0.4 pm.

By contrast, for the Zeeman electric-dipole resonance, the length scale in question is the antiferromagnetic coherence length ξ = v F /∆, [44] where ∆ is the antiferromagnetic gap in the electron spectrum.

Thus, matrix elements of Zeeman electric-dipole resonance exceed those of ESR by about c e 2 • F ∆ ≈ 137• F ∆ , or at least by two orders of magnitude. Resonance absorption is proportional to the square of the transition matrix element; thus the absorption due to electric excitation of spin transitions exceeds that of ESR at least by four orders of magnitude.

Last but not the least, Zeeman electric-dipole resonance absorption shows a non-trivial dependence on the orientation of the AC electric field with respect to the crystal axes, and on the orientation of the DC magnetic field with respect to the staggered magnetization.

VI. SUMMARY AND OUTLOOK

As argued above, Zeeman spin-orbit coupling in a Néel antiferromagnet is induced by magnetic field and arises due to a hidden symmetry, that protects double degeneracy of electron eigenstates at special momenta in the Brillouin zone. These special momenta form a degeneracy manifold, whose dimensionality is reduced against that of the Brillouin zone.

Limited to special momenta, the degeneracy means that the transverse g-factor acquires a substantial momentum dependence which is, however, limited to a relatively small part of the Brillouin zone. Therefore, observation of Zeeman spin-orbit coupling requires that carriers be limited to this small part of the Brillouin zone. Which, in turn, suggests lowcarrier antiferromagnetic conductors as a likely system with Zeeman spin-orbit coupling. An extended discussion of relevant materials and experimental constraints may be found in the Refs. [12,44].

VI.a Zeeman spin-orbit coupling in semiconducting quantum wells

To gain a better perspective, it is instructive to look for Zeeman spin-orbit coupling in materials other than antiferromagnets. A welcome example is provided by III-V direct band gap semiconductors of zinc-blende structure, such as GaAs and InAs. Before turning to details, let us recall that, in the simplest case of well-separated conduction and valence bands, it is the intrinsic spin-orbit coupling that may render the g-tensor momentum-dependent. In a centrosymmetric system, spin-orbit coupling has no intraband matrix elements [20,21],

and thus the corrections it induces to the g-tensor are small at least in the measure of ∆ SO /E 0 1, where ∆ SO is the interband matrix element of the intrinsic spin-orbit coupling, and E 0 is the band gap. [START_REF] Roth | [END_REF] The same small ratio limits the relative variation of the g-tensor across the Brillouin zone.

This picture becomes more involved for touching bands, and also if other bands are present nearby, which is the case at the Γ-point of bulk GaAs and InAs. Here, the conduction band is made mostly of s-orbital states, whereas the holes are of p-orbital nature. At the Γ-point, the six hole p-states split into a J = 3/2 quartet and a J = 1/2 doublet, separated from the quartet by the 'spin-orbit gap' ∆ 0 . Upon leaving the Γ-point, the quartet splits into two doubly degenerate bands: the predominantly J z = ±3/2 'heavy hole' (HH) band and its predominantly J z = ±1/2 'light hole' (LH) counterpart. Yet, in bulk GaAs, the electron g-factor remains isotropic, with g ≈ -0.44, whereas in bulk InAs g ≈ -15; in both materials relative variation of g across the Brillouin zone is of the order of 0.04. [START_REF] Ivchenko | Superlattices and Other Heterostructures[END_REF] Effective anisotropy can be enhanced by crossing over from a bulk sample to nearly two- dimensional semiconducting quantum well. Due to a finite size of the well, hole motion in the growth direction becomes quantized, and interplay of the reduced dimensionality with the p-state anisotropy of holes brings about many interesting effects, that have been a subject of active ongoing study. [START_REF] Winkler | Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems[END_REF][START_REF] Ivchenko | Superlattices and Other Heterostructures[END_REF][START_REF] Dyakonov | Spin Physics in Semiconductors[END_REF] Here, we will be interested only in the properties of the hole g-tensor and, for simplicity, will consider only field along the plane of the quantum well. Level separation due to finite size of the well in the growth direction will be assumed to greatly exceed all the other relevant energy scales such as the spin-orbit gap ∆ 0 and the hole cyclotron frequencies.

Zeeman splitting of both the 'heavy holes' HH (J z = ±3/2 at momentum k = 0) and 'light holes' LH (J z = ±1/2 at k = 0) in magnetic field H arises [START_REF] Winkler | Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems[END_REF] from the following two terms:

H = -2µ B κ(J • H) + q(J 3 x H x + J 3 y H y + J 3 z H z ) , (7)
where µ B is the Bohr magneton, vector J = (J x , J y , J z ) is made of the 4 × 4 matrices of the angular momentum J = 3/2, and κ and q are parameters of the Luttinger Hamiltonian. [49] Notice that the first term above is spherically symmetric, whereas the second term arises because the bulk symmetry is cubic rather than spherical. Accordingly, the coefficient κ is of the order of unity, whereas q tends to be small: in GaAs, κ ≈ 1.2 and q ≈ 0.04. [START_REF] Ivchenko | Superlattices and Other Heterostructures[END_REF] For light holes at k = 0, the first term in the Eq. ( 7) produces non-zero splitting proportional to κ; the light hole g-factor is thus of the order of unity. [50] By contrast, for heavy holes at the Γ-point δJ z = 3, and the first term of the Eq. ( 7) does not contribute to the Zeeman effect. Zeeman splitting of heavy holes arises from the second term in the Eq. ( 7) and is small in the measure of q ≈ 0.04. [50] At non-zero k, admixture of J z = ±1/2 components to the heavy holes produces contributions to the g factor, proportional k 2 , k 4 and so on:

g(k) = g 0 + (ka) 2 g 2 (φ) + (ka) 4 g 4 (φ) + ..., (8)
where a is mostly defined by the characteristic width of the well in the growth direction, Secondly, in a quantum well Zeeman spin-orbit coupling appears on the background of a non-zero constant g-factor: one may divide the in-plane components of the g-tensor into a momentum-independent term g 0 = 0 and a momentum-dependent part δg(k), so that g(k) = g 0 + δg(k), with δg(0) = 0. The functional form of δg(k) and its scale relative to g 0 depend on the width of the well and its growth direction. In GaAs quantum wells, some studies have found δg(k) g 0 , [51], while others [START_REF] Winkler | Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems[END_REF]52,55] found the scales of δg(k)

g 0 ∝ q is a constant,
and g 0 to be comparable for most growth directions -and numerically small, with typical measured values of g in the plane being of the order of 10 -2 . [27, 51, 52, 55] Notionally reducing δg(k) at a constant g 0 (for instance, by increasing the width of the quantum well), this picture can be continuously tuned to the textbook case of the constant momentumindependent g-tensor, and Zeeman spin-orbit coupling can thus be "switched off". In a Néel antiferromagnet, symmetry-protected zeros of the transverse g-factor are qualitatively different: here g 0 = 0 while the characteristic scale of δg(k) is unity, and Zeeman spin-orbit coupling can be "switched off" only by destroying the Néel state.

Quantum wells grown in the high-symmetry direction [111] are an exception to the above.

Such wells are symmetric under 2π/3 rotation around the growth axis, whereas the second term in the Eq. ( 7) is not, and thus its Zeeman matrix elements vanish: in such a well, g 0 = 0 is protected by the symmetry of growth direction, just as in an antiferromagnet g 0 = 0 is protected by a hidden anti-unitary symmetry.

[56] However, in contrast to an antiferromagnet, the momentum-dependent part δg(k) in GaAs is still small compared with unity. Much greater bulk g-factor (g ≈ 15) makes InAs quantum wells promising in this regard. [START_REF] Matsuo | [END_REF] Thirdly, III-V semiconductors of zinc-blende structure lack inversion center. Therefore, in a generic quantum well, both HH and LH bands are split by Dresselhaus and Rashba spinorbit couplings. [58,59] This splitting may be significant, thus obscuring the Zeeman effect in general and Zeeman spin-orbit coupling in particular. By contrast, in a centrosymmetric Néel antiferromagnet such effects are absent. Moreover, as we saw in Section III.c, in an antiferromagnet Zeeman spin-orbit coupling may be symmetry-protected even in the presence of intrinsic spin-orbit coupling.

To summarize, both weakly-doped antiferromagnetic insulators and III-V semiconducting quantum wells have their stronger and weaker points for the study of Zeeman spin-orbit coupling. GaAs quantum wells have been studied in great detail. At the same time, typical values of in-plane hole g-factor in this material are small. Semiconductors with greater g-factor, such as InAs, hold promise in this regard. By contrast, in a low-carrier Néel antiferromagnet, Zeeman spin-orbit coupling is protected by symmetry. At the same time, many such materials have not yet been as well-characterized as III-V semiconductors. Clearly, more studies are called for.

VI.b Other possibilities

One may ask whether analogues of the Zeeman spin-orbit coupling may arise in a different physical context. Indeed, one such possibility is known in the cold-atom physics, where synthetic spin-orbit coupling may be generated and tuned in an experiment, involving laserinduced Raman transitions between two internal states of an atom playing the role of its 'pseudospin' (for a review, see [60] and references therein). Synthetic spin-orbit coupling can be presented as an artificial gauge field. Just as Zeeman spin-orbit coupling, synthetic spin-orbit coupling in cold-atom setups is tunable, which holds great promise in spite of numerous experimental challenges [60].

Another closely related and very interesting development involves non-symmorphic magnetic crystals [61,62], that may give rise to degeneracies akin to those described aboveand, therefore, to some of the similar effects. So far explored very little, this direction may bring some positive surprises.

C is complex conjugation. This implies θ 2 = -1. If θ is a symmetry and |ψ is an eigenstate, then |ψ and θ|ψ are degenerate. But are they linearly independent? For O = 1, the equality θ 2 = -1 makes the Eqn. ( 10) read ψ|θ|ψ = -ψ|θ|ψ = 0, demonstrating orthogonality of the |ψ and θ|ψ and thus proving the Kramers theorem: in a time reversal-invariant system, every single-electron level is at least doubly degenerate.

The Kramers theorem above relies on the θ 2 = -1 property of spin-1/2 particles. By contrast, in a Néel antiferromagnet, some of the relevant anti-unitary symmetry operators such as θT a , θT a U n (π) and θT a U n (π)R n (π) do not square to a C-number -but, rather, to an operator: for instance [θT a ] 2 = -T 2a . However, the Bloch states in question are eigenstates of such squared operators, with the eigenvalue different from unity. This guarantees degeneracy, as mentioned immediately below the Eq. ( 10).

[ 13 ]

 13 But are there degenerate states at a given momentum? For a generic momentum p in the Brillouin zone, time reversal symmetry alone does not protect such a degeneracy -with an important exception of special momenta p * , that are equivalent to their opposite up to a reciprocal lattice vector Q, so that -Up * = p * + Q, where U is a point symmetry of the lattice. For such a p * , the state Uθ|p * resides at momentum -Up * = p * + Q. The states Uθ|p * and |p are degenerate and orthogonal. In the nomenclature of the Brillouin zone, the p * and -Up * = p * + Q are one and the same momentum -which, therefore, hosts two degenerate orthogonal eigenstates. This can be most simply illustrated in one dimension, where U is the identity, and p * = 0 (the Γ-point) and p * = ±π are the only such special momenta. At all other momenta, the eigenstates are non-degenerate, as shown in the central panel of the Fig.1.

Fig. 1 .FIG. 1 :

 11 Fig.1. As we saw above, breaking the inversion symmetry while retaining time reversal has a similar yet weaker effect: the double degeneracy at a given momentum is lifted everywhere except for those special momenta, that are equal to their opposite in the Brillouin zone. At the same time, a Bloch eigenstate |p at an arbitrary momentum p still has a degenerate orthogonal partner θ|p at momentum -p.These simple arguments show that, in a non-magnetic crystal, it is the inversion asymmetry that lifts the double degeneracy of a given electron band almost everywhere in the Brillouin zone. The effect can be encapsulated in an intrinsic spin-orbit coupling term H SO

  |p at an arbitrary momentum p generates a quartet of mutually orthogonal degenerate eigenstates: the doublet of |p and IθT a |p at momentum p, and the doublet of I|p and θT a |p at -p, as shown in the left panel of Fig.3. Notice the analogy with the quartet of mutually orthogonal degenerate eigenstates in a centrosymmetric non-magnetic crystal, with a doublet of |p and Iθ|p at momentum p and the doublet of I|p and θ|p at -p.[START_REF]The momentum labels of the three states I|p , θT a |p and IθT a |p can be found as it was done for a non-magnetic crystal, by acting on the state with a translation operator T b , that is a symmetry of the Néel state. As an example[END_REF]

FIG. 2 :

 2 FIG.2:(color online). Two points in space, r and r + a, separated by half a period a of the Néel order -and the exchange field ∆ r and ∆ r+a at these two points, upon applying magnetic field H transversely to the initial direction n of the staggered magnetization. The figure also shows how the exchange field changes upon various transformations, such as (i) U n (π) -spin rotation by π around n, (ii) θU n (π) -combination of U n (π) with time reversal θ, and (iii) θU n (π) combined with translation T a . To simplify notations, the transformations are shown as if they were applied directly to ∆ r rather than to ∆ r • σ. Notice that, in a finite transverse field H, the triple product θT a U n (π) is a symmetry of the tilted Néel state.

  Firstly, as outlined above, antiferromagnetic order couples to the electron spin σ via the Néel exchange coupling H N = ∆ r • σ with collinear field ∆ r , such that ∆ r+a = -∆ r . Secondly, an intrinsic spin-orbit coupling of the form H SO = (d p • σ) involves the field d p that is, generally, non-collinear, and transforms as a vector representation of the crystal point group. Finally, magnetic field H gives rise to the Zeeman term H Z = (H • σ).Without magnetic field, the double degeneracy at a given momentum is protected by the very same symmetry θT a as in an antiferromagnet without intrinsic spin-orbit coupling: the H SO = (d p • σ) respects both θ and T a separately, while the Néel coupling H N = ∆ r • σ is symmetric only under the product θT a , which protects the double degeneracy at those momenta p in the Brillouin zone, that are equivalent to their opposite (-p = p + Q) in the antiferromagnetic Brillouin zone. While the set of such momenta in the antiferromagnetic state is different from its paramagnetic counterpart, the protecting symmetry θT a remains the same. This simple picture changes once magnetic field is turned on. The reason can be traced back to a Néel antiferromagnet without intrinsic spin-orbit coupling, where Kramers degeneracy in a transverse magnetic field hinges on the combined symmetry θT a U n (π), with U n (π) being the spin rotation by π around the unit vector n of the Néel magnetization.Generally, intrinsic spin-orbit coupling H SO = (d p • σ) is not invariant under U n (π), let alone U n (φ) with an arbitrary rotation angle φ. Thus, with the H SO present, θT a U n (π) is no longer a symmetry of the problem.To begin with, in the presence of H SO we must specify the orientation of the collinear field ∆ r with respect to the d p ; the latter is generally non-collinear and realizes a vector representation of the crystal point group. My goal here is not to provide a complete classification, but to demonstrate an interesting possibility which is, at the same time, general enough. Not surprisingly, such a possibility appears in a symmetric configuration, where the ∆ r points along a high-symmetry direction of the d p . For the Bernevig-Hughes-Zhang (BHZ) model[37] with d BHZ p ∝ (p x , p y , d z [p 2

  in a two-dimensional crystal of square symmetry, with d BHZ p ∝ (p x , p y , d z [p 2 x + p 2 y ]) near the point Γ of the Brillouin zone. What is the unitary symmetry of the BHZ spin-orbit coupling H BHZ SO = d BHZ p • σ? Upon spin rotation by angle φ, the d BHZ p rotates by the same angle. To make a symmetry, this spin rotation must be compensated by orbital rotation R n (-φ) by the opposite angle. The H BHZ SO is thus symmetric under the product U n (φ)R n (-φ) with an angle φ, respecting the point symmetry of the d BHZ p .

4 .

 4 It is convenient to illustrate the arguments above by the following scheme, showing the symmetries of the various terms, and how they are broken as new terms are included in the Hamiltonian. The arrows show how the broken symmetries combine to form a surviving symmetry. From left to right, the columns represent the Néel exchange H N = ∆ r • σ, the intrinsic spin-orbit coupling H SO = d p • σ, and the combination of the two with the transverse Zeeman term H Z = H ⊥ • σ.

FIG. 4 :

 4 FIG. 4: (color online). The Brillouin zone of a three-dimensional Néel antiferromagnet with intrinsic spin-orbit coupling. The momentum plane p • a = ±π marks the paramagnetic Brillouin zone boundary in the z direction, and is shown in darker gray. As explained in the main text, both the Néel half-period a and the staggered magnetization ∆ r point along the high-symmetry axis n of the spin-orbit coupling, that is along the z axis of the Brillouin zone, shown by the red arrow. The blue arrow shows the magnetic field, normal to z. The momentum plane p • a = ±π/2, shown in lighter gray, marks the antiferromagnetic Brillouin zone boundary in the z direction, and hosts doubly degenerate Bloch states.

FIG. 5 :

 5 FIG. 5: (color online). Typical variation g ⊥ (p) as a function of momentum along the local normal to the degeneracy manifold g ⊥ (p) = 0. Within a momentum range of the order of /ξ, the g ⊥ (p) varies essentially linearly: g ⊥ (p) ∼ pξ/ . Beyond this range, the g ⊥ (p) becomes nearly constant.

137 2 .

 2 Relative to the speed of light c, this yields α R /c ∼

FIG. 6 :

 6 FIG.6:(color online). Zeeman splitting of small carrier pockets, notionally centered at the points X and Σ in the first quadrant of the Brillouin zone of a two-dimensional Néel antiferromagnet on a square lattice. The splitting is induced by magnetic field, transverse to the staggered magnetization.The dashed line, passing through the points X and Σ, is the magnetic Brillouin zone boundary, where g ⊥ (p) = 0. The pocket sizes and the splitting are exaggerated.

FIG. 7 :

 7 FIG. 7: (color online). Band structure of GaAs, sketched near fundamental gap E 0 in the vicinity of the Γ-point. As explained in the main text, the conduction band is formed by s-orbitals while the valence band is formed by p-orbitals. The valence band (J = 3/2) splits into J z = ±3/2 heavy hole (HH) band and J z = ±1/2 light hole (LH) band, touching at the Γ-point. The HH and LH bands are separated from the J = 1/2 band by the gap ∆ 0 .

and g 2

 2 (φ) and g 4 (φ) are functions of the momentum direction in the plane of the well.[START_REF] Winkler | Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems[END_REF][51][52][53][54] Hole carriers in a GaAs quantum well are thus subject to Zeeman spin-orbit coupling. Let us compare the latter with its counterpart in a Néel antiferromagnet:Firstly, in semiconducting quantum wells Zeeman spin-orbit coupling appears due to anisotropy, enhanced by reduced dimensionality of the well. By contrast, in a Néel antiferromagnet, Zeeman spin-orbit coupling emerges due to conspiracy of the anti-unitary symmetry of the Néel order with the symmetry of the crystal lattice. As shown in Section III.c, in an antiferromagnet intrinsic spin-orbit coupling and reduced dimensionality may play a subsidiary role.

Acknowledgements

It is my pleasure to thank Profs. M. M. Glazov, Y. B. Lyanda-Geller and E. Ya. Sherman for helpful discussions of the physics of quantum wells.

APPENDIX A: ORTHOGONALITY RELATION AND KRAMERS THEOREM

This Appendix proves the relation

and points out some of its consequences. Here |φ and |ψ are arbitrary states, O is an arbitrary unitary operator, and θ is time reversal. In the main text, this relation is used for |φ = Oθ|ψ ; in this case, when read right to left, Eqn. ( 9) yields

Whenever |ψ is an eigenvector of the linear operator [Oθ] 2 with an eigenvalue different from unity, the Eqn. (10) proves orthogonality of |ψ and Oθ|ψ .

The proof of Eqn. ( 9) is based on the obvious relation (Cφ, Cψ) = (ψ, φ) for arbitrary complex vectors φ and ψ, where (ψ, φ) ≡ i ψ * i φ i denotes scalar product, and C is complex conjugation. Hence, for an arbitrary unitary operator O, one finds (OCφ, OCψ) = (ψ, φ), due to invariance of scalar product under unitary transformation. Time reversal θ can be presented as a product of C and a unitary operator [START_REF] Wigner | Group Theory and its Application to the Quantum Mechanics of Atomic Spectra[END_REF]: θ = VC , thus C = V -1 θ and, therefore, (Oθφ, Oθψ) = (ψ, φ). As a result, for arbitrary states |ψ and |φ , one finds φ| [Oθ] + | [Oθ] |ψ = ψ|φ , which indeed amounts to (9).

Electron being a spin-1/2 particle leads one to define time reversal θ for a single-electron wave function as per θ = -iσ y C, where the Pauli matrix σ y acts on the electron spinor, and