Elvire Bestion
email: e.bestion@outlook.com

Julien Cote

Staffan Jacob

Laurane Winandy

Delphine Legrand

Habitat fragmentation experiments on arthropods: what to do next?

Keywords: habitat fragmentation, arthropod, insect, review, experiment, eco-evolutionary dynamics, species traits

Habitat fragmentation has the potential to influence ecological and evolutionary dynamics in various ways. Fragmentation experiments explore these multiple influences and the underlying mechanisms. We review experiments used in arthropods and highlight gaps in biological focus, methodology and questions addressed. While the consequences on community structure were often reported, fewer studies focused on ecosystem functions and evolutionary processes, with striking gaps on genetic and eco-evolutionary dynamics. Regarding fragmentation components, matrix quality was often overlooked while inter-patch (and source-patch) distance was the most studied component. The identified gaps outlined our need to study fragmentation at different time-scales, and on teasing apart the respective roles of each fragmentation component on each eco-evolutionary process.

INTRODUCTION

Habitat loss and fragmentation profoundly alter biodiversity [1], although some debated data recently suggested potential benefits of fragmentation per se [2][3][4]. Habitat fragmentation involves the transformation of large expanses of habitat into a number of smaller patches of smaller total area, isolated from each other by a matrix of altered habitat [START_REF] Wilcove | Habitat fragmentation in the temperate zone[END_REF]. It involves the conjunction of four components: (a) reduction in habitat amount, (b) increase in the number of habitat patches, (c) decrease in patch size, and (d) increase in patch isolation [START_REF] Fahrig | Effects of habitat fragmentation on biodiversity[END_REF]. Consequently, the edge-to-core habitat ratio also increases with fragmentation [START_REF] Laurance | Ecosystem Decay of Amazonian Forest Fragments: a 22-Year Investigation[END_REF]. The relative importance of these components on biodiversity may vary between landscapes, making the investigation of fragmentation impacts complex. This investigation is further hampered by the confounding effects on biodiversity between fragmentation per se (i.e., the breaking apart of habitat, controlling for changes in habitat amount, [START_REF] Fahrig | Effects of habitat fragmentation on biodiversity[END_REF]) and reduction in habitat amount. As habitat fragmentation is inherently linked to habitat loss in most landscapes, there is a correlational structure between the effect on biodiversity attributable to habitat loss and to fragmentation per se [START_REF] Didham | Rethinking the conceptual foundations of habitat fragmentation research[END_REF]. Further, fragmentation may have confounding, synergetic and/or antagonist effects with other global change aspects such as urbanisation, climate change or biological invasions [START_REF] Travis | Climate change and habitat destruction: a deadly anthropogenic cocktail[END_REF][START_REF] Brook | Synergies among extinction drivers under global change[END_REF][START_REF] Cs | Interactions between climate and habitat loss effects on biodiversity: a systematic review and meta-analysis[END_REF]. Fragmentation effects might also be nonlinear but increase exponentially after a threshold of habitat loss (e.g. [START_REF] With | Threshold effects of landscape structure on biological control in agroecosystems[END_REF][START_REF] Starzomski | Landscape geometry determines community response to disturbance[END_REF] but see [START_REF] Parker | Habitat loss and the habitat fragmentation threshold: an experimental evaluation of impacts on richness and total abundances using grassland invertebrates[END_REF]), which might be especially relevant given the dire predictions for future habitat degradation [START_REF] Pereira | Scenarios for Global Biodiversity in the 21st Century[END_REF]. Such complexity pleads for using experimental approaches to better estimate the relative consequences of different fragmentation components [START_REF] Diamond | Overview: Laboratory Experiments, Field Experiments, and Natural Experiments[END_REF], the interactive effects with other abiotic and biotic drivers and the effects at different spatio-temporal scales, potential sources of discordance in results [3]. While "natural experiments" (sensu Diamond 1986 [16], i.e. field observations) escape spatio-temporal scale issues and benefit from higher realism and applicability, laboratory and field manipulative experiments (Hurlbert 1984 [17]) allow to disentangle between the effects of correlated components fragmentation, test for interactive effects of other ecological factors, and tackle mechanisms behind biodiversity changes. Here, we aim at providing an overview of the current experimental approaches testing for habitat fragmentation consequences in arthropods.

Arthropods constitute a major part of biodiversity [START_REF] Scheffers | What we know and don't know about Earth's missing biodiversity[END_REF] and provide fundamental ecosystem services [START_REF] Seastedt | The Role of Microarthropods in Decomposition and Mineralization Processes[END_REF]. As small ectotherms, arthropods might be especially sensitive to the accumulation of physical, biotic and climatic dispersal barriers created by fragmentation [START_REF] Caplat | Looking beyond the mountain: dispersal barriers in a changing world[END_REF]. Further, many arthropods depend on multiple habitats (aquatic, terrestrial or aerial) during their ontogeny, imposing distinct constraints on movement. Unfortunately, the number of fragmentation studies on arthropods is not proportional to their biological importance and sensitivity to fragmentation [2,[START_REF] Fardila | A systematic review reveals changes in where and how we have studied habitat loss and fragmentation over 20years[END_REF] .

We created a data base of existing experiments on habitat fragmentation on arthropods using a systematic review of the literature. From this database, we aimed to provide a full picture of how fragmentation has been manipulated in arthropod experiment by classifying the taxa, the biological level, the fragmentation components manipulated and the response variables. We also aimed to identify gaps in the questions addressed and potential shortcomings in experimental approaches. We searched Web of Sciences with experiment* AND fragment* OR main fragmentation components OR metasystem type (see Supplement for exact search), yielding 5865 articles, of which 212 were finally included (Fig S1). Criteria for inclusion comprised (a) study involves arthropods, (b) is a manipulative experiment (sensu [START_REF] Hurlbert | Pseudoreplication and the Design of Ecological Field Experiments[END_REF][START_REF] Mcgarigal | Comparative Evaluation of Experimental Approaches to the Study of Habitat Fragmentation Effects[END_REF]) performed through landscape manipulation in the field or in the laboratory, and (c) focuses on one or more of the above defined fragmentation components, irrespectively of their distinction of fragmentation per se. Natural experiments (sensu [START_REF] Diamond | Overview: Laboratory Experiments, Field Experiments, and Natural Experiments[END_REF]) without landscape manipulation per se (e.g. translocation between landscapes) were therefore excluded. Our aim was to provide a full picture of how fragmentation has been manipulated in arthropod experiments. The identified gaps were used to propose guidelines to improve our understanding of fragmentation impacts.

BIOLOGICAL FOCUS OF STUDIES

A third of the studies focused on large arthropod communities, and a quarter on a single species (Fig 1A). The most investigated insect orders were Coleoptera, Hymenoptera, Lepidoptera and Hemiptera, and the most investigated non-insect classes where Arachnida and Malacostraca (Fig S2A). Formicidae, Apidae, Aphididae and Delphacidae were well represented families (Fig S2B ); Prokelisia crocea (planthopper), Anagrus columbi (fairyfly) and Junonia coenia (butterfly) were the most studied species. Surprisingly, very few studies used model species like Drosophila, Culex or Bombyx (but see [START_REF] Shirley | Metapopulation Dynamics of Fruit Flies Undergoing Evolutionary Change in Patchy Environments[END_REF][START_REF] Bakker | Consequences of fragmentation for the ability to adapt to novel environments in experimental Drosophila metapopulations[END_REF]), at the exception of bees and, to a lesser extent, Daphnia.

Aquatic systems were particularly neglected as well as their interface with terrestrial environments (Fig 1, but see [START_REF] Lecraw | Metacommunity size influences aquatic community composition in a natural mesocosm landscape[END_REF]). Insects with complex lifecycles (e.g. dragonflies) might undergo different ecoevolutionary pressures induced by fragmentation during ontogeny, potentially affecting ecological dynamics at a regional scale even when fragmentation occurs at very local scales. More than half of the studies focused on the community level, and drew general inferences mostly on fragmentation effects on species richness and abundance. While understanding how fragmentation modifies interaction strengths is crucial to assess its impacts on community dynamics, very few studies tackled species interactions, in particular competitive strengths (Fig S3B). Within species, roughly equal number of studies focused on population abundance and dispersal or movement. Fewer studied focused on other phenotypic traits, and only one study on genetics (Fig S3C). This points out a lack of evolutionary consideration, with 1% of studies explicitly testing evolutionary processes. (a) Biological focus of the study. Small communities are <10 species, large communities <100 species and very large communities >100 species, the ecosystem category is for studies focusing on ecosystem functions. (b) Type of experimental setting. Microcosms (<1 m²) are divided into indoor microcosms (16 studies), outdoor microcosms (1 study), and undefined (5 studies). Mesocosms (>1m²) are divided into indoor mesocosms (3 studies) and outdoor mesocosms (26 studies). Small experimental landscapes are <1ha, and large >1ha (see Supplement for discussion of area). The last category involves studies manipulating landscape features in natura. (c) Duration of the study. (d) Type of fragmentation components investigated. Notice that the total number of studies for this plot is more than 212 as several studies focus on more than one fragmentation component. Also notice that "fragmentation" defines studies where all components vary together, for instance "continuous vs fragmented".

While arthropods are crucial for many ecosystem functions (e.g., nutrient cycling [START_REF] Seastedt | The Role of Microarthropods in Decomposition and Mineralization Processes[END_REF]) themselves affected by fragmentation [START_REF] Resasco | The contribution of theory and experiments to conservation in fragmented landscapes[END_REF], only 5% tackled ecosystem functions or related relevant traits like biomass and decomposition rate (Fig 1A,Fig S3C).

EXPERIMENTAL SETTINGS AND FRAGMENTATION COMPONENTS STUDIED

Forty-four percent of the studies made use of small or large experimental landscapes, 30% manipulated landscape features in natura and only 24% used micro-or mesocosms (Fig 1B). Study duration varied from minutes to decades, but was predominantly months (Fig 1C, see supplement for discussion on generation times). Having decade-long studies such as [START_REF] Evans | Short-and long-term effects of habitat fragmentation differ but are predicted by response to the matrix[END_REF][28][29] is an asset as time lags are crucial in understanding the effects of fragmentation, with potentials for extinction debts arising [30].

Half of the studies were designed to isolate the effect of one component (e.g. patch number, interpatch distance, matrix composition), the others investigated 2 or more components (Fig 1D) with more than half manipulating all fragmentation components at the same time. The latter mainly contrasted two fragmentation levels (continuous/fragmented) while fewer used 3 levels (continuous/slightly fragmented/highly fragmented) or more (Fig S4A), despite the fact that fragmentation processes are non-linear [START_REF] With | Threshold effects of landscape structure on biological control in agroecosystems[END_REF][START_REF] Starzomski | Landscape geometry determines community response to disturbance[END_REF]. Patch size and edge effect were frequently studied, in comparison to the number of patches. Connectivity was studied through the manipulation of interpatch distance, corridor presence or dispersal manipulation, but rarely through matrix composition modification (Fig 1D).

Interactions between fragmentation and other aspects of global changes can represent deadly cocktails for biodiversity [START_REF] Travis | Climate change and habitat destruction: a deadly anthropogenic cocktail[END_REF]. However only 6% of experiments also manipulated other aspects of global change (temperature, pollution, invasive species). More generally, only 33% and 13% of the studies manipulated or quantified the biotic quality (mainly vegetation composition) and the abiotic quality (e.g. salinity, nutrients) of patches respectively.

As previously outlined [31], we observed a trade-off between ecological realism and the number of experimental replicates within studies (Fig S4B). This trade-off is further compounded when multiple studies use the same experimental landscape, thus decreasing the ecological replication among ecosystems. Indeed, half of the studies on large experimental landscapes hinged on four projects only (Savannah River Site, Miami research center, Wog Wog or BDFFP project). On the one hand, the detailed consequences of fragmentation on well-known experimental systems provide unique comparative power that deserves special attention. On the other hand, the low ecological replication might lead to over-generalising results potentially idiosyncratic and dependent on specific features of the studied systems. Particularly, most of the studies focus on forests or grasslands [31] and are located in temperate zones [START_REF] Fardila | A systematic review reveals changes in where and how we have studied habitat loss and fragmentation over 20years[END_REF]. Results that apply to these ecosystems might not be extendible to other regions and/or ecological biomes. We urge to increase the ecological replication of experimental settings, especially adding new large/small experimental systems in diverse regions and ecological biomes including at the terrestrial/aquatic interface.

WHAT TO DO NEXT?

Despite a wide breadth of experimental studies either directly-or indirectly-focused (e.g. metasystem's literature) on habitat fragmentation, we confirmed previous identified gaps, and highlighted unexpected ones in the biological focus, questions, fragmentation components studied (Fig 2, Table S1A-C) and methodology used (Table S1D). We hereafter suggest improvements of experimental studies. As our goal was not to compare experimental and non-experimental studies, identified gaps may also (and often do) apply to non-experimental studies. S1 for a more detailed analysis of the gaps), with below 10 studies tackling them. Concepts in orange were tackled by 10-29 studies, in light blue by 30-59 studies and in dark blue by >60 studies. Notice that for each biological level, we underline only a few aspects (e.g. under phenotypic traits, we detail morphology but not behaviour).

Figure 2: Summary of the potential impacts of fragmentation components on different levels of biodiversity, and gaps in our understanding of these impacts.

Concepts are underlined in different colours depending on the number of studies in which they are investigated. Concepts underlined in red represent strong gaps in our understanding (see Table

Our main takeaway is that a large portion of experiments usefully describe changes in community structure, but rarely examine the underlying changes in species interaction and the subsequent changes in community dynamics. Such gaps on species interactions were highlighted 20 years ago [31], with some improvements in the recent years for arthropods (Fig S5). We also lack studies on ecosystem functions such as decomposition rate, which is key to nutrient cycling. Future fragmentation experiments on arthropods should therefore shift from community descriptive investigation to the study of species interactions and consequences on ecosystem functions. Dedicated fragmentation platforms should help to achieve these goals [32,33]. Although a significant number of studies focused on the species level, the evolutionary consequences of fragmentation were largely ignored, with the exception of dispersal-related studies, which were pointed out as a gap 20 years ago [31] that have been partly filled since (Fig S5). Therefore, the respective roles of adaptation, drift, plasticity or mutation in the response to fragmentation are little known in arthropods. This knowledge is however crucial to correctly interpret patterns and predict biodiversity changes. This limited interest for the evolutionary consequences of fragmentation explains why theoretical predictions about the role of fragmentation in eco-evolutionary feedbacks remain untested [34]. Future studies should thrive to understand how species traits, but also genomes, evolve with fragmentation in complex biological settings where interacting species can jointly coevolve. The development of omics and/or use of arthropod models such as Drosophila or Daphnia should help to achieve such mechanistic goals. Coupling these experimental studies with both theoretical models and validation in the field should help better understanding how eco-evolutionary processes affect arthropod biodiversity.

Regarding fragmentation components, matrix composition remains poorly studied since Debinski and Holt's review [31]. While matrix is at the core of fragmented landscapes [START_REF] Resasco | The contribution of theory and experiments to conservation in fragmented landscapes[END_REF]35], its resistance, a fundamental parameter to assess functional connectivity [36], is understudied, and has been called for. We also call for an increase in the type of ecological biomes studied with large replication within landscapes, and for more numerous levels of habitat fragmentation with continuous landscapes as controls. Finally, we urge for the integration of other components of global changes in fragmentation experiments. Climate change, pollution and the spread of invasive species can often co-occur with habitat fragmentation. Experimenters should aim at assessing interactions and eco-evolutionary consequences of these multiple global change drivers.

To conclude, we hopefully provided an objective synthesis of what is known, and what is unknown from fragmentation experiments in arthropods. To this end, we searched for all experiments, irrespective of their integration of only patch-scale or landscape-scale processes, and their control for habitat amount. While such information is obviously crucial to better capture the complexity of fragmentation (and is available from our database), we are convinced that among scientists' general interests, one crucial goal is to describe patterns and mechanistically explain them, avoiding judgement on potential beneficial and deleterious effects. It is hazardous to decide, for example, if an increase in species abundance or diversity is beneficial given that any change in ecological networks due to fragmentation can have detrimental consequences for ecosystem functioning as a whole. For instance, an increase in Orthoptera abundance with fragmentation [37] can negatively affect plant biomass [38], with potential cascading effects on belowground species and ecosystem function [39]. It might even occur that these fragmentation-induced changes feedback to habitat fragmentation itself through eco-evolutionary loops [34]. In the future, we hope that long-term experiments will bring elements to feed such questioning and to inform on long-term stability and recovery of anthropized systems.

Supplementary Methods

We created a data base of existing experiments on habitat fragmentation on arthropods using a systematic review of the literature. From this database, we aimed to provide a full picture of how fragmentation has been manipulated in arthropod experiment by classifying the taxa, the biological level, the fragmentation components manipulated and the response variables, and to identify gaps in the questions addressed and potential shortcomings in experimental approaches. Note that our database was focused on experimental studies, and we did not provide a comparison with nonexperimental studies because we did want to opt for point-scoring between the two valuable experimental and non-experimental approaches. We also did not examine whether fragmentation effects were beneficial or detrimental as it would require a proper meta-analysis and we consider that any change, increase or decrease of a biological metric (e.g. species abundance) can be detrimental at wider ecological scales (see discussion).

We conducted the literature search in two steps: a first step in which we used only the keywords fragmentation AND experiment* (see below), from which we got 2496 results that we then filtered through a two-step exclusion process to get to a set of 159 articles from which we extracted data for our quantitative overview, and a second step in which we used more keywords that we directly derived from the 166 article database, yielding 3369 more articles of which 53 were included. This two search steps procedure allows to avoid forgetting certain types of keywords by searching for keywords that were often present in our initial search and completion of the database. In total, this two-step procedure led to the building of a database of 212 articles. Figure S1 shows the process of study identification, selection and the data extracted from each relevant article.

We first searched Web of Knowledge for the keywords fragmentation AND experiment*, refining for Web of Science Categories biodiversity conservation OR biology OR ecology OR entomology OR environmental sciences OR evolutionary biology OR forestry OR marine freshwater biology OR microbiology OR plant sciences OR soil science OR zoology. The search, undergone in November 2018, revealed 2496 results. We did not include taxonomic information in the search terms to avoid excluding articles where the keywords and abstract did not include taxonomic information, or too specific taxonomic information that would be impossible to search for, but chose to exclude nontaxonomically relevant studies during the first filter. Notice that by focusing on experiment*, we certainly miss studies that were experimental by nature but did not state it explicitly. We used twostep procedure to reduce the 2496 articles to a list of 159 articles fitting our inclusion criteria, first examining the title and abstract, and second examining the full text of the remaining studies and extract information (see below). Examining the full text allowed to identify more keywords that we then used in a second step of search to complete our database. In this step, we focused on studies that would not have fragmentation as a topic but a shorter version of the word (e.g. fragmented), searching for TOPIC: (fragment* AND experiment* NOT fragmentation) Refined by: Web of Science Categories: (Ecology OR Environmental Sciences OR Entomology OR Evolutionary Biology OR Marine freshwater biology OR Plant Sciences or Zoology). Timespan: 1975-2018. We also searched for studies that did not have the fragmentation keyword in the topic but could have keywords linked to one of the fragmentation components that we identified in our first search and analysis of the database, and keywords linked to meta-systems, with the search TOPIC: ((patch number) OR (patch size) OR (edge effect) OR (patch distance) OR (patch isolation) OR (corridor) OR (matrix composition) OR (metapopulation*) OR (metacommunit*) OR (metaecosystem*) OR (meta-population*) OR (meta-communit*) OR (meta-ecosystem*)) AND (experiment*) NOT (fragmentation) Refined by: Web of Science Categories: (Ecology OR Evolutionary Biology. Timespan: 1975-2018. This second step of search yielded 3369 more articles, which were then filtered through our two-step exclusion process in the same way as the articles issued from the first step of search.

From the list of 2496 original articles, and from the list of 3369 articles from the second search, we examined each title and abstract to determine whether articles met the criteria for inclusion in our overview of the literature. Criteria for inclusion comprised (a) the taxonomic identity of the species or group of species studied, keeping only studies that involved arthropoda, (b) the fact that the study was a manipulative experiment (sensu [START_REF] Hurlbert | Pseudoreplication and the Design of Ecological Field Experiments[END_REF]) by manipulating landscape features, and (c) the fact that the study focused on one or more of the four aspects of habitat fragmentation defined by [START_REF] Fahrig | Effects of habitat fragmentation on biodiversity[END_REF], that is reduction in habitat amount, increase in the number of habitat patches, decrease in the size of habitat patches, and increase in the isolation of habitat patches. We included papers on habitat fragmentation per se (without habitat loss) and papers where the manipulation of fragmentation was linked to a loss of habitat, but not papers that manipulate only habitat loss (exclusion of papers manipulating only patch size). We excluded review papers, purely theoretical papers, and papers that did not fit our definition of experimental, that is a manipulative experiment sensu [START_REF] Hurlbert | Pseudoreplication and the Design of Ecological Field Experiments[END_REF] instead of a mensurative experiment. For instance, we excluded capture-mark-recapture studies that did not explicitly manipulate fragmentation, but only captured individuals present in habitats with various levels of fragmentation, which corresponds more to a mensurative experiment. We also excluded studies that e.g. used pitfall traps to recover insects present in naturally fragmented habitat, if there was no manipulation of the habitat. However, we included all manipulative experiments irrespective of the strength or statistical significance of the results. At times, the title and abstract were too vague to positively assess these three criteria of inclusion, and the articles were kept for further detailed reading of the text; they could be thus excluded in a second filtering session. The first filtering session led to a selection of 345 studies from the original search and 199 studies from the second step of search that fitted the scope of this overview (Fig S1).

The remaining 345 original articles and 199 articles from the second search were then reviewed in full to determine whether they fitted our inclusion criteria, contained relevant data and whether the results were presented with sufficient clarity (Fig S1). This step led to the further exclusion of 186 articles from the first search and 146 articles from the second search. From the final set of 212 studies, we then extracted data relevant to several questions:

1) In which kind of ecological system was fragmentation studied (terrestrial, freshwater, marine)?

2) Which was the biological focus of the study: species level, two-or three-species level including e.g. pollination or predator-prey interactions, community level (small community, <10 species, large community, <100 species, very large community, >100 species), ecosystem level? Which taxa where considered? When studies focused on less than four species, we noted the three species studied.

When studies focused on more species, we noted the main taxa studied (e.g. Aphidae and Coccinellidae, maximum three taxa at the lower level to avoid listing each and every taxon in very large communities).

3) Which type of experimental setting was used: artificial habitat in nature/natural landscape, small experimental landscape (<1ha) large experimental landscape (>1ha), indoor or outdoor mesocosms (>1m), indoor or outdoor microcosms (<1m)? Which kind of experiment was done: landscape manipulation, translocation, dispersal manipulation? Notice that the size of experimental landscapes was arbitrarily divided into small and large based on the 1ha threshold to simplify analyses, but the scale of habitat fragmentation might depend on the size of the organism studied and the question posed. For instance, 100 m² is a very large landscape for a mite, but quite small in terms of home range for a dragonfly. Further, studies on dispersal might need very big landscape sizes when focusing for example on dispersal distances, but might be more small-scaled when focusing for example on emigration decision [32,[START_REF] Legrand | Ranking the ecological causes of dispersal in a butterfly[END_REF]. Notice also that while moss systems are correctly considered as natural microcosms, we decided to classify them as artificial habitat in nature/natural landscape in our study, at the rare exception of studies enclosing mosses in incubators (classified here as microcosms).

4) Which was the focus of the fragmentation study: e.g. patch number or size, inter-patch distance, patch-source distance, edge effect, corridor, matrix composition? Which other features non-directly related to the fragmentation were studied (e.g. biotic or abiotic patch quality)? As habitat amount is inherently linked with many components of the general fragmentation process (e.g. patch size, patch number), and we did not aim to separate habitat fragmentation per se from more general fragmentation, some components may covary with habitat loss.

5) What was the duration of the study: minutes (up to 59), hours (up to 23), days (up to 6), weeks (up to 3), months (up to 11), years (up to 9), decades? Because extremely few studies reported time in generations, we only report calendar time. However, studies within the same time scale might mean completely different things for long-lived, long generation time species such as cicadas and shortlived, short generation time species like acari.

6) What where the metrics studied: e.g. community structure, population abundance, survival, interspecific interactions, dispersal, phenotypic traits? Please note that studies that involved more than 3 taxa (resp. order or family) were not taken into account when counting the number of studies using a specific taxon, as it would be complicated to track each and every taxon used in studies on large communities. Also note that the sum of the number of studies investigating each taxon can be superior to the total number of studies as each study can investigate several taxa. Need for freshwater and marine experimental platforms. Need for work at the interface between aquatic and terrestrial realms and how varying fragmentation between the two realms affects arthropods persistence [START_REF] Lecraw | Metacommunity size influences aquatic community composition in a natural mesocosm landscape[END_REF] Studies mainly focused on certain taxa Need for studies on lesser studied insects, e.g. Ephemeroptera, Odonata and Trichoptera which have complex aquatic/terrestrial life cycles. Need for investigations in forgotten arthropods like e.g. Myriapoda, Pycnogonida [START_REF] Mccauley | Foraging Rates of Larval Dragonfly Colonists Are Positively Related to Habitat Isolation: Results from a Landscape-Level Experiment[END_REF] Lack of studies on species interactions and cascading effects More approaches in semi-natural systems with controlled communities can allow to better understand how e.g. effects of fragmentation on one species affect other species through cascading effects [START_REF] Harvey | Trophic island biogeography drives spatial divergence of community establishment[END_REF] Studies mainly focused on adults Need for understanding of effects at different life stages, particularly for species with complex life cycles [START_REF] Lecraw | Metacommunity size influences aquatic community composition in a natural mesocosm landscape[END_REF] Lack of studies on model organisms Need for building on long-lasting models like Drosophila species to access mechanistic understanding of fragmentation effects [START_REF] Shirley | Metapopulation Dynamics of Fruit Flies Undergoing Evolutionary Change in Patchy Environments[END_REF][START_REF] Bakker | Consequences of fragmentation for the ability to adapt to novel environments in experimental Drosophila metapopulations[END_REF][START_REF] Cranmer | Landscape structure influences pollinator movements and directly affects plant reproductive success[END_REF] (b) Issues of eco-evolutionary questions addressed [START_REF] Staddon | Connectivity, non-random extinction and ecosystem function in experimental metacommunities[END_REF] Lack of studies on species interactions Need for more small community studies where researchers track the effect of fragmentation on interaction strength. Particularly, need for studies on species competition [START_REF] Braschler | Experimental small-scale grassland fragmentation alters competitive interactions among ant species[END_REF] Lack of studies on descriptors other than abundance and community structure Studies often focus on species richness while species respond differently to fragmentation, need for more studies on species traits as predictors to sensitivity to fragmentation [START_REF] Davies | Which traits of species predict population declines in experimental forest fragments[END_REF] Lack of studies on species traits Need for studies on phenotypic trade-offs or syndromes to assess the constraints of trait response to fragmentation [START_REF] Cote | Evolution of dispersal strategies and dispersal syndromes in fragmented landscapes[END_REF] Lack of studies at the (epi)genetic level Need for developing monitoring of (epi)genetic diversity change in experiments. Need for omics to unravel the (epi)genetic bases of response to fragmentation [34] Lack of studies on (meta)population and community dynamics (often temporal snapshots)

Need for temporal monitoring of populations and communities, ideally with individual monitoring and information on survival/reproduction/dispersal rates [START_REF] Evans | Short-and long-term effects of habitat fragmentation differ but are predicted by response to the matrix[END_REF]29] No studies on ecoevolutionary dynamics Need for experimental designs deciphering the respective effects of evolutionary processes. Include common gardens and control without fragmentation in designs The quality of the matrix is mainly tackled through the presence of corridors, need for more studies on matrix quality gradients [START_REF] Evans | Short-and long-term effects of habitat fragmentation differ but are predicted by response to the matrix[END_REF] Lack of studies on patch number

The mere of patch number remains poorly tackled. Develop designs where patch number varies independently of habitat amount [START_REF] Goodwin | How does landscape structure influence landscape connectivity[END_REF] Lack of interactions with other components of global change Need for studies combining e.g. climate change and pollution with fragmentation [START_REF] Shirley | Metapopulation Dynamics of Fruit Flies Undergoing Evolutionary Change in Patchy Environments[END_REF][START_REF] Tuff | A framework for integrating thermal biology into fragmentation research[END_REF] Few levels of habitat fragmentation Need for studies with higher number of levels than just continuous vs fragmented, i.e. continuous number of fragmentation levels [START_REF] Brudvig | Evaluating conceptual models of landscape change[END_REF] Not enough ecological replication Need for more replicates at the landscape level in the design. Need for more biological replicates when relevant and for more long-term systems in different biomes Micro/mesocosms offer more control and replication particularly for complex ecoevolutionary designs, relevant for studies requiring biological reshuffling [START_REF] Gianuca | Effects of dispersal and environmental heterogeneity on the replacement and nestedness components of β-diversity[END_REF] Lack of comparability between study results -Use effect sizes -report time in calendar days as well as in generation time when possible -use the same protocols to calculate traits linked to morphology, feeding, life history, physiology or behaviour -report numerical abundance as well as density per m² [START_REF] Moretti | Handbook of protocols for standardized measurement of terrestrial invertebrate functional traits[END_REF] Not enough information on taxa -always specify order and family of the described species -follow the ARRIVE guidelines for reporting studies on animals [START_REF] Kilkenny | experimental habitat fragmentation on ground beetles (Coleoptera, Carabidae) in a boreal spruce forest[END_REF] Not link with theoretical models Build fragmentation experiments to validate theoretical predictions.

Develop mixed approaches where model can help interpreting results. Feed existing models with measured parameters [START_REF] Goodwin | How does landscape structure influence landscape connectivity[END_REF] Macreadie Pi, Geraldi Nr, Peterson Ch: Preference for feeding at habitat edges declines among juvenile blue crabs as oyster reef patchiness increases and predation risk grows. Marine Ecology Progress Series 2012 doi: 10.3354/meps09986 Macreadie Pi,Connolly Rm,Keough Mj,Jenkins Gp,Hindell Js: Short-term differences in animal assemblages in patches formed by loss and growth of habitat. Austral Ecology 2009 doi: 10.1111/j.1442-9993.2009.02060.x Margules Cr, Milkovits Ga, Smith

Figure 1 :

 1 Figure 1: Number of studies by biological focus, type and duration of experiment and fragmentation components.

Figure S1 :

 S1 Figure S1: Flow chart detailing the process of study identification, selection and data extraction

Figure S3 :

 S3 Figure S3: Design of the fragmentation studies in terms of secondary measures and eco-evolutionary processes investigatedNote that the total number of studies mapped is superior to the number of studies assessed, as one study can investigate several fragmentation processes and several metrics at once. Black bars: studies that tackled only one measure/feature/metric, grey bars: studies that tackled multiple measures/features/metrics at once (dark grey: 2 measures, light grey: 3 measures)

Figure

  Figure S5 Cumulative number of studies over the years investigating several topics of importanceDotted lines and empty circles represent the gaps highlighted byDebinski and Holt in 2000 [31], while dashed lines and empty triangles represent gaps highlighted in this review and full lines and filled circles represents topics that are not considered as gaps.

  long-term experiment, and more time series to have the dynamics of the metapopulations, -communities, and -ecosystems on the long term[3] Lack of studies on random vs non-random dispersal Studies manipulating dispersal as a proxy of connectivity often do not take into account the potential for dispersal to be non-

  Gt: CONTRASTING EFFECTS OF HABITAT FRAGMENTATION ON THE SCORPION CERCOPHONIUS-SQUAMA AND AN AMPHIPOD. Ecology 1994 doi: 10.2307/1941608 Marshall Sd, Walker Se, Rypstra Al: Two ecologically-divergent generalist predators have different responses to landscape fragmentation. Oikos 2006 doi: 10.1111/j.2006.0030-1299.14115.x Martinko Ea, Hagen Rh, Griffith Ja: Successional change in the insect community of a fragmented landscape. Landscape Ecology 2006 doi: 10.1007/s10980-005-5322-0 Martins Gm, Matias Mg, Moniz I, Rius C, Sanderson J, Neto Ai, Jenkins Sr: Post-settlement dispersal ability determines structure of marine benthic metacommunities. Marine Ecology Progress Series 2017 doi: 10.3354/meps12108 Matias Mg: Macrofaunal responses to structural complexity are mediated by environmental variability and surrounding habitats. Marine Biology 2013 doi: 10.1007/s00227-012-2155-7 Mccauley E, Kendall Be, Janssen A, Wood S, Murdoch Ww, Hosseini P, Briggs Cj, Ellner Sp, Nisbet Rm, Sabelis Mw, Turchin P: Inferring colonization processes from population dynamics in Brodin T, Hammond J: Foraging rates of larval dragonfly colonists are positively related to habitat isolation: results from a landscape-level experiment. The American Naturalist 2010 doi: 10.1086/650444 Mcintyre Ne, Wiens Ja: How does habitat patch size affect animal movement? An experiment with darkling beetles. Ecology 1999 doi: 10.1890/0012-9658(1999)080[2261:HDHPSA]2.0.CO;2 Mcintyre Ne, Wiens Ja: Interactions between habitat abundance and configuration: experimental validation of some predictions from percolation theory. Oikos 1999 doi: 10.2307/3546577 Mestre L, Jansson N, Ranius T: Saproxylic biodiversity and decomposition rate decrease with small/j.biocon.2018.09.023 Minello Tj, Zimmerman Rj, Medina R: The importance of edge for natant macrofauna in a created salt marsh. Wetlands 1994 doi: 10.1007/BF03160655 Moore Ec,Hovel Ka: Relative influence of habitat complexity and proximity to patch edges on seagrass epifaunal communities. Oikos 2010 doi: 10.1111/j.1600-0706.2009.17909.x Morel-Journel T, Hautier M, Vercken E, Mailleret L: Clustered or scattered? The impact of habitat quality clustering on establishment and early spread. Ecography 2018 doi: 10.1111/ecog.03397

long-term habitat fragmentation experiment leads to morphological change in a species of carabid beetle: Fragmentation changes carabid morphology

  28. Evans MJ, Banks SC, Barton PS, Davies KF, Driscoll DA: A *29. Haddad NM, Gonzalez A, Brudvig LA, Burt MA, Levey DJ, Damschen EI:

. Ecological Entomology 2018, 43:282-293. *

Experimental evidence does not support the Habitat Amount Hypothesis. Ecography 2017, 40:48-55. Experimental results refute the Habitat Amount Hypothesis. Not only the total level of habitat in landscape is important, but also the configuration of patches. 30

  . Haddad NM, Brudvig LA, Clobert J, Davies KF, Gonzalez A, Holt RD, Lovejoy TE, Sexton JO, Austin MP,Collins CD, et al.:

Habitat fragmentation and its lasting impact on Earth's ecosystems.

	Science Advances 2015, 1:e1500052.
	31. Debinski DM, Holt RD: A Survey and Overview of Habitat Fragmentation Experiments.
	Conservation Biology 2000, 14:342-355.

32. Legrand D, Guillaume O, Baguette M, Cote J, Trochet A, Calvez O, Zajitschek S, Zajitschek F, Lecomte J, Bénard Q, et al.: The Metatron: an experimental system to study dispersal and metaecosystems for terrestrial organisms. Nature Methods 2012, 9:828-833. 33. Haddad NM: Connecting ecology and conservation through experiment. Nat Meth 2012, 9:794-795. **34. Legrand D, Cote J, Fronhofer EA, Holt RD, Ronce O, Schtickzelle N, Travis JMJ, Clobert J: Ecoevolutionary dynamics in fragmented landscapes. Ecography 2017, 40:9-

25. The consequences of habitat fragmentation on eco-evolutionary dynamics have been intensively studied theoretically, but not experimentally. The authors propose some experimental developments to fill this gap. 35

. Prugh LR, Hodges KE, Sinclair ARE, Brashares JS: Effect of habitat area and isolation on fragmented animal populations. PNAS 2008, 105:20770-20775. 36. Baguette M, Blanchet S, Legrand D, Stevens VM, Turlure C: Individual

dispersal, landscape connectivity and ecological networks. Biol Rev 2013, 88:310-326. 37

. Braschler B, Marini L, Thommen GH, Baur B: Effects

of small-scale grassland fragmentation and frequent mowing on population density and species diversity of orthopterans: a long-term study

. Ecological Entomology 2009, 34:321-329. 38. Bestion E, Cucherousset J, Teyssier A, Cote J: Non-

consumptive effects of a top-predator decrease the strength of the trophic cascade in a four-level terrestrial food web.

	Oikos 2015,
	124:1597-1603.
	39. Wardle DA, Bardgett RD, Klironomos JN, Setälä H, Putten WH van der, Wall DH: Ecological
	Linkages Between Aboveground and Belowground Biota. Science 2004, 304:1629-1633.

Table S1 :

 S1 Identified gaps and suggestions for closing them(a) identified gaps on the biological focus of the study. (b) Identified gaps on the type of ecological and evolutionary questions addressed. (c) identified gaps in the fragmentation components studied. (d) Other methodological issues.

	(a) Issues of biological focus
	Identified Gap	Suggestions	Example of
			study or
			conceptual
			reference
	Lack of studies on aquatic
	systems

ACKNOWLEDGEMENTS

EB was supported by a Young Researchers Award from the Bettencourt-Schueller Foundation. This work was supported by the French Laboratory of Excellence project TULIP (ANR-10-LABX-41), and by an Investissements d'avenir programme from the Agence Nationale de la recherche (no. ANR-11-INBS-0001AnaEE-Services).

Supplementary Material for


O Vaccarelli

A Honecker

P Giura

K Béneut

B Fåk

G Rousse

G Radtke

Triplet excitations in the frustrated spin ladder Li 2 Cu 2 O(SO 4 ) 2

Magnetic excitations of the recently discovered frustrated spin-1/2 two-leg ladder system Li2Cu2O(SO4)2 are investigated using inelastic neutron scattering, magnetic susceptibility and infrared absorption measurements. Despite the presence of a magnetic dimerization concomitant with the tetragonal-to-triclinic structural distortion occurring below 125 K, neutron scattering experiments reveal the presence of dispersive triplet excitations above a spin gap of ∆ = 10.6 meV at 1.5 K, a value consistent with the estimates extracted from magnetic susceptibility. The likely detection of these spin excitations in infrared spectroscopy is explained by invoking a dynamic Dzyaloshinskii-Moriya mechanism in which light is coupled to the dimer singlet-to-triplet transition through an optical phonon. These results are qualitatively explained by exact diagonalization and higher-order perturbation calculations carried out on the basis of the dimerized spin Hamiltonian derived from first-principles.

I. INTRODUCTION

Frustrated spin-1/2 two-leg ladders form a class of models which has been the subject of intense theoretical interest during the last decades as it fulfills all the requirements favoring the emergence of exotic phenomena [START_REF] Lacroix | Introduction to Frustrated Magnetism: Materials, Experiments, Theory[END_REF][2][3][4][5][6][7] . True material realizations of these models, however, are very scarce in the literature as the stringent conditions defining the geometry of frustrated ladders are rarely met in natural or synthetic compounds.

A noticeable exception is BiCu 2 PO 6 , a system built from coupled spin-1/2 two-leg ladders where frustration arises from competing antiferromagnetic nearest and next-nearest neighbor interactions along the legs 8 . The rich physics emerging from this particular geometry has triggered many experimental investigations of the magnetic properties of this compound over the past decade. These studies reveal a spin-gap behavior 8,9 , frustrationinduced incommensurate dispersion of triplet quasiparticle excitations 10,11 , triplet dispersion renormalization resulting from a repulsion with the multi-quasiparticle continuum, 12 or a cascade of field-induced phase transitions 13 including, presumably, the formation and collapse of a quantum soliton lattice 14,15 .

A new material realization of a frustrated spin-1/2 twoleg ladder system, Li 2 Cu 2 O(SO 4 ) 2 , has recently been discovered [16][17][18] . At high temperature, this compound crystallizes in a tetragonal structure where [Cu 2 O(SO 4 ) 2 ] 2- chains running along the c axis are well separated by Li + ions, thus forming quasi-1D structural units (see Fig. 1(a) and (b)). A first-principle investigation of the electronic structure of this compound 18 clearly identified these structural units as magnetically equivalent to frustrated two-leg ladders where, as in BiCu 2 PO 6 , frustration arises from competing interactions along the legs.

The geometry of this system, however, differs substantially from that of BiCu 2 PO 6 because (i) inter-ladder interactions are very weak, (ii) the rung coupling (J ⊥ in Fig. 1(c)) is presumably ferromagnetic and (iii) additional antiferromagnetic diagonal interactions (J × in Fig. 1(c)) occur.

The phenomenology of this compound is also markedly different as a structural transition to a distorted triclinic phase occurs near 125 K 17 (see Fig. 1(d)). Although lattice parameters are barely affected by this transition, large variations of the interplatelet Cu-O-Cu superexchange angles drastically impact the amplitudes of leg and diagonal interactions J and J × . These couplings, equivalent by symmetry in the high-temperature phase, are largely split by the triclinic distortion, leading to the emergence of a dominant antiferromagnetic interaction J d . The resulting structure, while maintaining the global geometry of a ladder, displays a strong magnetic dimerization, where dimers are staggered on the legs of the ladder, as shown in Fig. 1(e). This scenario, describing the evolution of a complex geometry with the temperature, is however essentially derived from first-principle calculations and a definite assessment of the model requires further experimental investigations.

In this work, we report the first detailed investigation of the low-temperature magnetic excitations of Li 2 Cu 2 O(SO 4 ) 2 combining inelastic neutron scattering (INS), magnetic susceptibility and infrared (IR) spectroscopy measurements carried out on powder samples. Dispersive triplet excitations are observed in INS above a spin gap ∆ = 10.6 meV at 1.5 K, a value consistent with the estimates extracted from magnetic susceptibility. Moreover, an absorption band visible only in the dimerized phase below 70 K is observed in IR spectroscopy and attributed to a triplet excitation arising at ⊥ in green the couplings between the chains along the rungs and the NNN coupling along the legs J2 in red.

14.3 meV. The dynamic Dzyaloshinskii-Moriya mechanism is invoked in this case to explain the absorption of light by this low-dimensional spin system. All these observations are qualitatively explained by higher-order perturbation and exact diagonalization calculations of triplet quasi-particle excitations carried out on the basis of the dimerized geometry derived from previous firstprinciple calculations 18 .

II. EXPERIMENTS

A. Time-of-flight neutron spectroscopy

Inelastic neutron scattering experiments were performed at the Institut Laue-Langevin in Grenoble on the thermal time-of-flight spectrometer IN4 using incoming energies of 16.6, 32.0, and 66.6 meV. Most measurements were done with an incoming energy of 32.0 meV from the (004) reflection of a pyrolytic graphite monochromator. A Fermi chopper was used with a 2 • straight slit package spinning at 15000 rpm. The energy resolution (full width at half maximum) at elastic energy transfer was 2.1 meV. A total of 6.8 g of powder sample of Li 2 Cu 2 O(SO 4 ) 2 was synthesized according to the procedure described in Ref. 16, put in a flat aluminum plate holder of dimensions of 40 × 28 × 2 mm 3 , and mounted in a standard helium cryostat. Measurements were carried out at temperatures T = 1.5, 40, 60, 80 and 100 K for a typical acquisition time of 12 h per temperature. The scattering from the empty Al-holder was measured and subtracted from the data, which was subsequently corrected for self-shielding and absorption using standard data reduction routines. Data treatments were carried out using the LAMP software 19 .

Figure 2(a)-(e) shows the maps of the dynamic susceptibility χ (|Q|, E), obtained by normalizing the background-subtracted neutron scattering intensity S(|Q|, E) by the thermal occupancy factor 1-e -E/kBT at 1.5, 40, 60, 80, and 100 K, respectively. The temperature dependence of the dynamic susceptibility clearly reveals the presence of two dominant contributions arising from phonon and magnetic excitations. Whereas the scattering cross-section of the former scales as |Q| 2 , that arising from magnetism scales with the square of the form factor associated with the magnetic ions, and falls off with increasing |Q| [START_REF] Lovesey | Theory of Neutron Scattering from Condensed Matter[END_REF][START_REF] Squires | Introduction to the Theory of Thermal Neutron Scattering[END_REF] . The high-temperature dynamic susceptibility is therefore largely dominated by phonon scattering (see Fig. 2(d)-(e)) whereas the weight of the magnetic contribution progressively increases with decreasing temperatures, as it can be observed in the low-|Q| region of Fig. 2(a)-(c). Assuming, in first approximation, that the intensity is entirely associated with phonons scattering for the highest temperature measurement (T = 100 K in our case), the magnetic contribution to the dynamic susceptibility at low-temperature can simply be isolated by plotting the difference

22 ∆χ (|Q|, E) = χ (|Q|, E)| 1.5K -χ (|Q|, E)| 100K . (1)
The corresponding difference map is shown in Fig. 2(f) and clearly reveals the presence of dispersive magnetic excitations of bandwidth ≈ 5 meV above a large spin gap ∆ = 10.6 meV, measured as the inflection point of the first peak (A). Constant-Q cuts extracted for a |Q| value of 2 ± 0.1 Å-1 for each temperature measured experimentally are shown in Fig. 3. The broad band of magnetic excitations, extending over 5 meV above the spin gap, includes the first low-intensity peak (A) cen- tered at 11.2 meV followed by the most intense structure (C) at 14.7 meV. A low-energy shoulder (B) of this dominant structure is also present at about 13.5 meV, particularly visible on the data obtained at 1.5 K. Inelastic neutron scattering experiments therefore clearly reveal the presence of dispersive magnetic excitations, mostly localized at low-|Q| < 3.5 Å-1 and vanishing at temperatures above ∼ 80 K, consistent with spin triplet excitations out of the reported singlet ground state of this compound 17,18 .

B. Static magnetic susceptibility

These results can be further confirmed by analyzing the low-temperature behavior of the experimental static magnetic susceptibility. Fig. 4 reproduces the experimental data corrected from paramagnetic impurities and temperature-independent contributions, already reported in Ref. 17. The susceptibility exhibits the typical behavior of a gapped low-dimensional antiferromagnet with a high-temperature paramagnetic regime reaching a broad maximum at about 125 K and an exponential decay at low-temperature. Note, however, that the temperature dependence of the magnetic susceptibility is largely affected by the structural transition occurring at the same temperature as the maximum (125 K) since it is accompanied by a substantial magnetic dimerization within the ladders. As reported in Ref. 18, although this transition extends over a large temperature range, the low-temperature triclinic phase is already mostly stabilized at about 80 K, and the 2-80 K range can therefore be used to estimate the corresponding spin gap. A rough estimate can be obtained by fitting the experimental data using the general expression for thermally activated processes

χ(T ) ∝ e -∆/T (2)
leading to a value of ∆ ≈ 9.1 meV. It should however be noted that (2) would be valid for non-interacting particles, but magnons are not free particles. A suitable expression that takes the hard-core repulsion between the magnons into account is given by 23

χ lad. (T ) ∝ T -1/2 e -∆/T (3)
for a quadratic band minimum. Equation (3) has also been employed (see Fig. 4). The resulting estimate for the spin gap, ∆ ≈ 11.6 meV, is slightly larger than the value obtained with the previous expression. Despite these small variations, essentially related to the rather low accuracy of this approach and to the limited applicability of the simple ladder model to Li 2 Cu 2 O(SO 4 ) 2 , these estimates are however fully consistent with the spin gap value obtained from inelastic neutron scattering.

C. IR spectroscopy

Infrared absorption spectroscopy was finally employed as a complementary technique to investigate the lowenergy excitations of Li 2 Cu 2 O(SO 4 ) 2 . These experiments were carried out using a Fourier transform Bruker IFS66 v/s spectrometer at the IMPMC-Sorbonne University spectroscopy platform. The instrument was aligned in transmission geometry. Isotropic pellets of ∼ 13 mm diameter were prepared by mixing the original powder samples with transparent matrix materials. Pure KBr powder was used as a matrix for the pellets employed in the middle-infrared (MIR), whereas polyethylene (PE) was employed to prepare pellets for the far infrared (FIR) measurements. The former were obtained by mixing 2.5 mg of sample and 80 mg of pure PE; for the latter, 1.4 mg of sample was mixed with 200 mg of KBr powder, placed in an oven at T = 150 • C in order to remove water contamination from the KBr powder, and then pressed to obtain high quality pellets. Transmission spectra were taken as a function of temperature from 10 to 300 K using a continuous Janis liquid helium cryostat working in vacuum. Each spectrum was acquired in the frequency region 60 -640 cm -1 for the FIR measurements and 580 -4400 cm -1 for the spectrum in the MIR, with a spectral resolution of about 2 cm -1 . The FIR spectra were recorded with a DTGS-Pe detector and a multilayer mylar beamsplitter. The MIR data were obtained with a HgCdTe (MCT) detector and a Ge coated KBr beamsplitter. For both region we used a Globar (SiC) source.

The temperature dependence of the transmission IR powder spectrum of Li 2 Cu 2 O(SO 4 ) 2 measured between 10 and 300 K is shown in Fig. 5. Most of the absorption bands visible between 100 and 1300 cm -1 are associated with the electric dipole excitation of optical phonons. The group of high-energy modes located around 1100 cm -1 is exclusively associated with internal [SO 4 ] 2-bond stretching [START_REF] Farmer | of Great Britain and Ireland[END_REF] . The 500 -700 cm -1 range is dominated by [SO 4 ] 2-tetrahedra bending modes involving progressively the displacement of Cu and O atoms forming the chain backbone, as the frequency decreases. Symmetry can be employed to determine the maximum number of infrared active optical phonons in both phases of this compound. In its high temperature phase, Li 2 Cu 2 O(SO 4 ) 2 belongs to the P 4 2 /m space group. A factor group analysis indicates that the vibrational degrees of freedom decompose as

Γ HT = 11A g + 13B g + 9E g + 10A u + 8B u + 15E u (4)
on the irreducible representations of the D 4h point group. Subtracting the A u + E u acoustic modes, we find 37 IR-active modes, decomposed in 23 potentially distinct 9A u +14E u bands. The triclinic distortion, occurring below 125 K, further reduces the crystal symmetry to P 1. A similar analysis leads to the following decomposition on the only two irreducible representations of C i

Γ LT = 42A g + 48A u (5)
leading, after subtraction of the 3A u acoustic modes, to 45A u IR-active modes. As it can be observed in Fig. 5, the exact number of bands detected in these experiments is difficult to assess, due to the broad and asymmetric profile of certain peaks. The above group theoretical analysis therefore only provides an upper bound for the number of bands distinguishable in the experimental spectra. Qualitatively, however, the large increase of active modes due to the symmetry lowering triggered by the triclinic distortion is clearly visible on the experimental spectra when decreasing the temperature below the transition (∼ 125 K) and therefore consistent with the structural data 17 . Besides this increase in the number of phonon lines, the main effects of decreasing the temperature consist in a slight hardening and narrowing of most of the bands, usually attributed with anharmonic effects and in particular, for the former, with the overall unit cell volume contraction 17 . However, a few bands display a softening in the temperature range of the structural transition, characteristic of magneto-elastic effects associated in our case to the rise of the dimerization [START_REF] Rõõm | [END_REF] . At low energy, however, a pronounced transfer of spectral weight toward low frequencies reveals the rise of a weak excitation at 115 cm -1 (14.3 meV). This band is indicated by a blue arrow in Fig. 6(a).

In order to quantify these spectral changes, a leastsquare fit of the low-frequency range of the spectra based on a superposition of Lorentzians has been carried out. The temperature dependence of the energy of the different modes observed in the 105-152 cm -1 range resulting from this fit is displayed in Fig. 6(b). A weak hardening of the modes identified as polar phonons (grey filled diamonds) is visible with decreasing temperature except in the transition temperature range (80 -130 K) where a sizable jump is observed. Concomitantly, the band shown in blue in Fig. 6(b) has an energy of 14.3 meV, that falls precisely in the broad band of magnetic excitations observed by INS, in a region characterized by a large spectral weight. Moreover, this excitation is only visible at temperatures well below the structural transition, i.e. in the magnetic dimerized phase. These observations therefore suggest that this excitation might involve, to a certain extent, the spin degrees of freedom of this system.

III. THEORY

In order to explain this set of experimental findings and explore the landscape of low-energy magnetic excitations in Li 2 Cu 2 O(SO 4 ) 2 , both exact diagonalization and higher-order perturbation theory calculations have been carried out. The Q-dependence of the dynamic susceptibility χ should be dominated by that of the J d dimers 26 , or more generally be related to the exchange constants via a first-moment sum rule [27][28][29][30] . However, any such analysis is complicated in Li 2 Cu 2 O(SO 4 ) 2 by the large phonon background. We have therefore decided to focus rather on the clear magnetic features A, B, and C visible in Fig. 3 by comparing to both exact diagonalization and higher-order perturbation theory calculations.

A realistic spin Hamiltonian (see Appendix A) capable of describing the magnetism of this compound in the triclinic phase has been derived previously 18 from firstprinciple calculations and requires seven distinct couplings to account for the low symmetry of the crystal. The geometry of this model is depicted in Fig. 1(e) where J d = 1 and J a = 20/330 = 2/33 are antiferromagnetic and alternate along the legs of the ladder so as to form a staggered dimer structure, J a ⊥ ≈ J b ⊥ = -110/330 = -1/3 are the ferromagnetic couplings along the rungs of the ladder, J b × = 78/330 = 13/55 and J c × = 133/330 are antiferromagnetic diagonal couplings between the legs and, finally, J 2 = 112/330 = 56/165 is the antiferromagnetic NNN interaction along the legs. This model therefore neglects the supposedly very weak inter-ladder couplings 18 as well as any other term beyond the bilinear, Heisenberg like, interactions.

Exact diagonalization (ED) calculations have been performed using finite lattices of N = 12, 16, 20, 24, 28, and 32 sites with periodic boundary conditions along the legs, as the one-dimensional magnetic unit cell contains two dimers, i.e. four spins. For system sizes exceeding N = 20 we have used the Lanczos algorithm 31,32 in order to compute low-lying eigenvalues. Furthermore, perturbation expansion of the one-triplet dispersion relation up to the fifth order has been carried out around the limit of isolated dimers. In this approach, the unperturbed ground state corresponds to a product of singlets, |s = (|↑↓ -|↓↑ )/ √ 2, on the leg dimers defined by the dominant antiferromagnetic coupling J d . Lowenergy magnetic excitations of this system are obtained by promoting one dimer into a triplet state, |t -1 = |↓↓ ,

|t 0 = (|↑↓ +|↓↑ )/ √ 2 or |t 1 = |↑↑ .
Since the unit cell contains two dimers, we find two separate single-particle bands, as shown in Fig. 7 (see Appendix A for details).

Numerical results are summarized in Fig. 7. Fig. 7(a) and (b) show the convergence of the perturbation expansion by comparing, respectively, the one-triplet dispersion relations obtained at different expansion orders and the highest order perturbation theory with exact diagonalization results. As it clearly appears, a remarkable convergence towards the exact results is achieved for the higher-order expansions. These calculations indicate the presence of two dispersive and slightly overlapping triplet bands above a large spin gap. The lowest band displays a behavior characteristic of antiferromagnetically coupled dimers with a maximum at the zone center and a minimum close the Brillouin zone edge. The actual minimum arises at an incommensurate wave vector and results from the presence of frustrating couplings. In addition to the one-triplet excitation bands, the lower boundary of the two-triplet continuum is calculated as

E 2 (k) = min q∈1BZ m,n=1,2 [ω m (k -q) + w n (q)] ( 6
)
where ω 1 (k) and ω 2 (k) represent the two one-triplet bands, is also shown in Fig. 7(b). The large value of the spin gap compared to the modest triplet excitation bandwidth, pushes this continuum lower bound well above the maximum of the highest one-triplet band. The ED results are close to the fifth-order expansion, i.e., both of them can be considered accurate. The exception is the top of the upper band where proximity to the continuum leads to larger finite-size effects and slower convergence of the series (see Appendix B for details). Exact diagonalization, furthermore, reveals the presence of lower-lying singlets above and below the continuum, which do not interfere with the upper triplet band, as it can be seen in Fig. 7(b). Similar excitations have already been reported in ladder systems where they can be understood as bound states of two triplets 8,33,34 .

IV. DISCUSSION

The theoretical results presented in the previous section provide solid ground for analyzing the experimental results obtained on Li 2 Cu 2 O(SO 4 ) 2 . It should be noted, however, that the global energy scale obtained from firstprinciple calculations 18 is not consistent with our experimental observations. Indeed, as already reported, a straightforward use of the magnetic couplings provided by density functional theory (DFT) calculations leads to a substantial overestimation of the experimental spin gap 18 . Although the amplitudes of these couplings are often overestimated and strongly depend on the approximate treatment of exchange and correlation employed in the calculations 35,36 , their ratios are expected to be subject to smaller errors 37 . In this framework, the ratios between the seven couplings involved in the spin Hamiltonian were considered as fixed. The global energy scale was thus taken as the only variable parameter, adjusted so as to reproduce the experimental value of the spin gap. This led to an approximate 50 % downscaling of the DFT coupling amplitudes. The resulting energy scale in millivolts is shown on the vertical axes of Under these assumptions, powder INS can be qualitatively discussed in terms of one-triplet excitation densityof-states (DOS) shown in Fig. 7(c). In first approximation, the experimental peaks A, B and C shown in Fig. 3 can indeed be interpreted as arising from the DOS singularities, at the bottom of the lowest band for peak A and in the overlapping region of the two bands for peaks B (bottom of the upper band) and C (top of the lower band). Although this qualitative analysis provides a satisfying explanation regarding the origins of the lowenergy part of the INS data, it also predicts the presence of higher-lying features corresponding to the top of the highest one-triplet band, i.e. at ∼ 20 meV, which was not observed experimentally. This could simply be a matrix element effect. Alternatively, although our model locates the two-triplet continuum lower bound above the highest one-triplet branch over the entire first Brillouin zone (see Fig. 7(b)), they remain close in energy. Therefore, only minor modifications of the model employed in this work would be necessary to change this picture and, in particular, restore a significant overlap between the highest triplet quasi-particle mode and the two-particle continuum. This overlap will provide spontaneous decay channels 38,39 leading to significant damping of these quasi-particles and therefore to the absence of visible signatures in INS data.

A second important question arises from the likely detection of triplet excitations in IR spectroscopy presented in Sec. II C. Indeed, dominant electric dipole transitions induced by light are strictly confined to spin-conserving excitations (∆S = 0) and are therefore, in principle, unable to reveal singlet-to-triplet transitions. However, it has been shown that, in a number of low-dimensional quantum magnets, this selection rule can be circumvented through essentially two mechanisms relying on the presence of spin-phonon coupling and involving one or multiple magnetic excitations.

A successful and now well-established model employed to describe the infrared optical absorption of one and two-dimensional undoped cuprates is based on phononassisted bi-magnon absorption 34,[40][41][42][43] . The excitation of singlet bound states, resulting from the coupling of two spin-carrying modes (triplets, in our case) in such a way that the total spin amounts to zero, indeed obeys the imposed spin selection rule. Lorenzana and Sawatzky further showed that, when a center of inversion is present, dipole-allowed absorption is only possible if a symmetrybreaking phonon is also involved in the process 40,41 . In our case, an attribution of the IR band observed at 14.3 meV for T < 70 K to the absorption of phononassisted bi-magnons is very unlikely as the typical en- ergy of these excitations, already of the order of ∼ 2∆ = 21.2 meV when neglecting the phonon energy, are much larger.

An alternative mechanism, arising from the spin-orbit coupling, has been proposed to explain the detection of singlet-to-triplet excitations in dimerized quantum magnets using IR absorption [START_REF] Rõõm | [END_REF]44 . It can be described qualitatively as a process where light excites the system into a virtual spin-singlet one-phonon state coupled, through a dynamic Dzyaloshinskii-Moriya (DM) interaction, to a spin-triplet zero-phonon state 45 . This mechanism thus relies on the assumption that the virtual polar phonon involved in the process is associated with atomic displacements able to induce an instantaneous variation of the DM vector. Assuming that such a mechanism is effective in the low-temperature phase of Li 2 Cu 2 O(SO 4 ) 2 , the 14.3 meV IR absorption band would, quite accurately, match the zone center maximum of the lowest one-triplet excitation and the corresponding Van Hove singularity in the DOS.

V. CONCLUSION

In conclusion, we report the first experimental investigation of magnetic excitations in the low-temperature, dimerized phase of the recently discovered frustrated spin-1/2 two-leg ladder Li 2 Cu 2 O(SO 4 ) 2 . Through a combined analysis of inelastic neutron scattering, magnetic susceptibility and infrared absorption spectroscopy data obtained on powder samples, dispersive excitations of bandwidth of the order of 5 meV have been clearly identified above a large spin gap of 10.6 meV. Exact diagonalization and higher-order perturbation theory calcu-lations allowed for an overall consistent interpretation of these results in terms of one-triplet quasi-particle excitations above the singlet ground state. While experiments and theory show an overall good agreement, the only exception lies in the high-energy part of the triplet excitation spectrum, where a possible coupling between the quasi-particles and the high-lying many-particle continuum may be responsible for the absence of high-energy structure in the INS spectra. This calls for further experimental and theoretical investigations of this very rare example of frustrated spin-1/2 ladder, which will heavily rely on the future availability of single crystals. expansion in the spirit of textbooks such as chapter 11 of Ref. 46. Following Ref. 18, the triclinic phase can be described by the staggered S = 1/2 dimer structure schematized in Fig. 1(e). In this structure one of the couplings along the legs, J d , is much stronger than all the others. The triclinic Hamiltonian H can thus be written as the sum of an unperturbed part H 0 , for decoupled dimers along the legs (bold blue lines in Fig. 1

FIG. 1 .

 1 FIG. 1. (a) Structure of tetragonal Li2Cu2O(SO4)2 viewed along the [001] direction. (b) Structure of the [Cu2O(SO4)2] 2-chains running along the c axis in the high-temperature (T > 125 K) phase. (c) Schematic representation of the corresponding spin system. Intra [Cu2O6] 8-platelet rung coupling J ⊥ is represented in green, leg and diagonal couplings J = J× are represented in blue and next-nearest neighbor (NNN) couplings along the legs J2 are in red. (d) Structure of the [Cu2O(SO4)2] 2-chains in the distorted low-temperature (T < 125 K) phase. (e) Schematic representation of the corresponding staggered-dimer structure: J a and J d in blue represent the alternating couplings along the legs, J b × and J c × in light blue the diagonal inter-chain couplings, J a ⊥ and J b ⊥ in green the couplings between the chains along the rungs and the NNN coupling along the legs J2 in red.

FIG. 2 .

 2 FIG. 2. Experimental dynamic susceptibility χ (|Q|, E) plots for Li2Cu2O(SO4)2 measured at different temperatures: (a) T = 1.5 K, (b) T = 40 K, (c) T = 60 K, (d) T = 80 K, and (e) T = 100 K. Magnetic contributions are isolated in (f) by displaying the difference ∆χ (|Q|, E) from Eq. (1).

FIG. 3 .

 3 FIG. 3. Constant-Q cuts of the dynamic susceptibility χ (|Q|, E) extracted for a |Q| value of 2 ± 0.1 Å-1 from the plots shown in Fig. 2(a)-(e).

6 FIG. 4 .

 64 FIG.4. Temperature dependence of the static magnetic susceptibility of Li2Cu2O(SO4)2 corrected from paramagnetic impurities and temperature-independent contributions. Grey circles correspond to the experimental points, the solid blue and red lines correspond, respectively, to the best fit obtained using the general expression (2) or a spin-1/2 two-leg ladder Heisenberg model(3). The corresponding values of the extracted spin gap are indicated.

FIG. 5 .

 5 FIG. 5. Temperature dependence of the powder transmission infrared spectra of Li2Cu2O(SO4)2 in the FIR region. The blue arrow indicates the spectral feature associated to the magnetic excitation. Top inset: The lowest and highest temperature spectra are reported in logaritmic scale up to 1800 cm -1 to show the high energy modes associated with the internal [SO4] 2-bond stretching.

FIG. 6 .

 6 FIG. 6. (a) Enlarged view of the powder transmission infrared spectra in the 80 -155 cm -1 range. (b) Temperaturedependence of the mode frequencies extracted from the Lorentzian fit. Experimental data are represented with filled diamond symbols, lines are only guides for the eye. The phonon bands are represented in black whereas the magnetic band is in blue.

  Fig 7(a) to (c).

FIG. 7 .

 7 FIG. 7. (a) One-triplet dispersion relation calculated from second to fifth order perturbation theory. (b) Comparison of the one-triplet dispersion relation obtained for the fifth order perturbation theory with exact diagonalization results obtained on finite lattices up to N = 32 sites. First few lowest-lying singlets (in red) and triplet (in blue) obtained from ED are shown. The blue filled area corresponds to the free two-triplet continuum. (c) One-triplet density of states obtained from fifth order perturbation theory dispersion relations shown in (b). Energies given in meV have been obtained by downscaling the DFT isotropic magnetic couplings by 50 %.

2 √

 2 (e)) and a perturbation, W , accounting for the coupling between the dimers withH 0 = m J d S m 1,a • S m 2,a + S m 1,b • S m • S m 2,b + J b ⊥ S m-1 2,a • S m 1,b + J 2 S m 1,a • S m+1 1,a + S m 2,a • S m+1 2,a + S m 1,b • S m+1 1,b + S m 2,b • S m+1 2,b + J a S m 1,a • S m-1 2,a + S m 1,b • S m-1 2,b + J b × S m-1 1,a • S m 1,b + S m-1 2,a • S m 2,b + J c × S m 1,a • S m 1,b + S m 2,a • S mwhere m is the cell index, a and b denote the two legs of the ladder, the number 1 or 2 distinguishes the upper and the lower spin-site of a dimer and S m i,α with α = {a, b} and i = {1, 2}, are the spin-1/2 operators.At W = 0, the system consists of isolated dimers and the unperturbed ground state corresponds to a product of singlets on the leg dimers. The first excited state is the one-triplet state |t m α , a state with a single triplet on a dimer (m, α) and singlets on all the other dimers. As Li 2 Cu 2 O(SO 4 ) 2 contains two dimers per unit cell, a 2 × 2 effective Hamiltonian, W has to be computed for each value of k in Fourier space. This leads to two separate bands of triplets. The dispersion relation is obtained by diagonalizing the effective Hamiltonian up to a given order starting from the Bloch states |T α = Nc m e ikm |t m α , where α = {a, b} and N c is the number of unit cells.

1 . 2 +

 12 FIG. 8. Exact diagonalization results obtained on finite lattices up to N = 32 sites. The triplet (S = 1) energies for k = 0 and k = π are plotted as a function of 1/N .

ACKNOWLEDGMENTS

This work was supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02, and more specifically within the framework of the Cluster of Excellence MATISSE led by Sorbonne Universités. The authors thank Dr Meiling Sun for providing the sample used in this study. Inelastic neutron scattering experiments were performed at the Institut Laue-Langevin (ILL) in Grenoble.

Appendix A: Series expansions

The evaluation of the one-triplet dispersion relation has been carried out solving the spin Hamiltonian parametrized from first-principles around the limit of isolated dimer (low-temperature phase), implementing a high-order perturbative approach in the strong coupling

Fifth order

To proceed beyond second order, we have adapted a C-code that was used previously, e.g., in Refs. 48 and 49 to the present situation. The main challenges here include the large number of variables (6 coupling ratios) as well as the rather long-range nature of some couplings (see Fig. 1(e)). This limits us to fifth order, the resulting expressions are too cumbersome to be presented here (the full expression of the perturbation matrix W can be found in the Mathematica notebook provided in the Supplemental Material 50 that also contains a seventhorder series for the ground-state energy). Nevertheless, we note that there is a freedom of the choice of basis and implementation of the perturbation theory that starts to impact the effective Hamiltonian W at order three. For efficiency reasons, we decided to implement the perturbative corrections to the basis vectors starting from |T α such that orthonormality is lost at higher order. As a consequence, the matrix representation of W no longer needs to be Hermitian and, indeed, we find complex diagonal matrix elements starting at order 3. Nevertheless, we have checked that the eigenvalues of this 2 × 2 matrix are always real, as they should be since they are independent of the choice of basis.

We have also double-checked that we recover known limiting cases from our 6-variable expressions: for J a ⊥ = J b ⊥ = J a = J 2 = 0 and J b × = J c × we recover the conventional two-leg ladder, and our results for the eigenvalues are indeed consistent with Ref. 51; for

(A7)

For reasons of compactness of presentation, we have rounded the numerical coefficients to three significant digits. Expression (A7) has been used in Fig. 7 for order five in panel (a) and systematically in panel (b).

Appendix B: Finite-size effects

Here we take a closer look at the finite-size effects present in the numerical results for the one-triplet dispersion shown in section III. Indeed, Fig. 7(b) exhibits a remarkable convergence of the numerical results towards the fifth-order perturbative expansion with increasing N . Figure 8 shows the ED energies of the triplet excitations for two different values of k, k = 0 and k = π, as a function of the inverse of the size, 1/N , including the smaller system sizes N = 12 and 16 not shown previously. One observes that the values of the energies for k = π and for the lower band of the triplet at k = 0 converge rapidly. Larger finite-size effects are only observed at the top of the upper band at k = π. This corresponds to the region where the series also show a slow convergence (see Fig. 7(a)) and we speculate that this is again due to the proximity with the continuum. Still, for systems with N > 20, the data can also be considered to converge to the thermodynamic limit. Even in this least favorable case, finite-size corrections to the N = 32 data are presumably negligible for both bands and all values of k.