Mr Aurélien Dumont

Ms Beatriz Mayor

Dr Elena López-Gunn

IS THE REBOUND EFFECT OR JEVONS PARADOX A USEFUL CONCEPT FOR BETTER MANAGEMENT OF WATER RESOURCES? INSIGHTS FROM THE IRRIGATION MODERNISATION PROCESS IN SPAIN

Keywords: rebound effect, Jevons paradox, efficiency, return flows, modernisation, Spain

Improving the efficiency of water use is usually presented as an opportunity for large water savings, particularly in the agricultural sector. Warnings that this may not translate into reduced consumption is sometimes associated with the rebound effect or Jevons paradox, an appealing concept that can be used to analyse and prevent undesired consequences in the rush for efficiency gains. This article, based on the energy sector, shows that the concept helps to identify possible unintended consequences of increasing efficiency and shows how efficiency gains are shared in society. However, it might be conceptually misleading when applied to water since it reinforces a myth on the consideration of water savings and efficiency, and may be also too restrictive. The recent modernisation of irrigation practices in Spain highlights that the rebound effect is only one of many possible consequences of efficiency improvements.

Introduction

The availability of water has been identified as a fundamental issue concerning the future of food production. The efficient management of water resourcesthe idea of producing more with lessis currently high on the list of strategies required to meet current and future development needs, while reducing the pressure on the environment. The path towards efficiency requires the mobilisation of a series of technical, economic and institutional measures. This article, discusses efficiency from a technical point of view, which is efficiency associated with the physical application of water. This issue has received much attention in recent decades, with the call to reduce the amount of water that is 'wasted' by traditional agricultural practices and to switch to more efficient systems (e.g. sprinkler and drip irrigation). In Spain, in addition to general objectives on resource availability and rural development, efficiency policies undertaken over the past 10 years have been considered a key measure for the implementation of the EU Water Framework Directive (López-Gunn et al., 2012a, b)

(Table 1).

There is increased evidence, however, that investments may not translate into a reduction in the amount of water used. For instance, the paradigm of water savings from efficiency gains has, in some cases, been challenged by referring to the rebound effect or Jevons paradox. This concept refers to efficiency improvement in the technical process of the use of a resource that ultimately defeats the original purpose through a higher overall use by society. This effect and its explanation were first described in relation to the use of coal in England by the economist William Stanley Jevons at the end of the 19th century [START_REF] Alcott | Jevons' paradox[END_REF]. There are many examples of the application of this paradox, principally in relation to energy, an area where the concept has taken hold. The application of this concept also aims to ensure that energy conservations programmes are effective, by devising rules that should be promoted together with planned efficiency improvements ( Van den Bergh, 2010). Therefore, it might appear useful and attractive to apply this framework to water, to evaluate the actual extent of water savings.

The main objective of this article is to analyse whether the rebound effect concept aids the understanding of the consequences of improvements in water efficiency. This is illustrated by considering the modernisation plans in Spain and specifically a case study in the Ebro basin in the north-east of the country. The first section of the article presents the concept of the rebound effect, using energy as an example. The second section highlights the particularities of water when compared to energy, in relation to the concepts of efficiency and losses. The third section focuses on the main explanations for the rebound effect and accompanying policies designed to counteract it. Finally, an alternative framework of analysis is proposed, based on two independent pillarswater resources accounting and the effect on the whole system. This enables the value of applying the rebound effect in the case of water to be recognised, while preventing the misleading aspects that may be implied.

Rationale for the rebound effect and three logical steps in its formulation: from energy to water

Logic of the rebound effect for energy

In general terms, efficiency improvements in a productive process mean a reduction in inputs required per unit of output, thus saving resources. Not all of the projects aimed to increase conveyance or field efficiency (i.e. reducing withdrawals), so the figure for water savings per hectare varies depending on the project.

Figure 1. Estimation of projected savings resulting from improved efficiency in a productive process (2a), and identification and quantification of the rebound effect and Jevons paradox (2b)

The amount of savings can be estimated directly on the basis of the initial and final expected efficiencies and the initial quantity of input, assuming the same level of output (Fig. 1). A typical example for energy is transport: to reach a certain destination, fuel savings will be obtained by switching to a more efficient car or using public transport, instead of using one's own private vehicle. Thus, the first step in the logic of the rebound effect involves identifying the potential savings obtained from efficiency improvements.

In reality, when considering the broader economic system in which technical improvements are nested, economic agents react to changes in efficiency in such a way that the original aim to reduce resources may ultimately trigger an increase in demand. This is the rebound effect (Fig. 1) 1 . A key point in this scenario is that, in parallel to more energy use, more output is also obtained and therefore welfare or final benefits could increase.

Evidence of a higher level of input in comparison to the baseline scenario and/or identifying reasons for this increase to happen forms the second step of the rebound effect. The rationale for identifying a 'rebound' or 'paradox' is related to the difference between the initial projected results at the physical process scale and the observed results within the economic system (Fig. 2).

Various explanations for the rebound effect have been discussed and analysed for energy [START_REF] Alcott | Jevons' paradox[END_REF]Van den Bergh, 2010). These are usually classified as direct and indirect micro-and macroeconomic effects (Table 2).

At the microeconomic level, the basic rule determines that a drop in energy input due to efficiency gains leads to a drop in the marginal cost per unit of product, which potentially implies a lower price for the product and translates into higher demand. The drop in costs can also be accentuated by an associated fall in the input price, which is linked to a lower demand. For this direct rebound effect (the case mainly analysed in the literature), it is essential to consider the elasticity of demand, as consumption would rise only if cost is a limiting factor. Table 2. Main types of rebound effect (adapted from [START_REF] Maxwell | Addressing the rebound effect. European Commission DG Environment[END_REF], using transport as an example

Type of rebound effect Transport example Direct (microeconomic): increased efficiency and associated cost reduction for a product/service results in a higher level of consumption because of an increase in demand Car is used more often or to go further, since the cost for each kilometre travelled has dropped Indirect (microeconomic): savings from cost reductions due to efficiency gains allow more income to be spent on other products and services Money saved due to cheaper public transport is used to go on overseas holiday Economy-wide (macroeconomic): increased efficiency drives overall economic productivity, which results in increased macroeconomic growth and consumption at a macroeconomic level As a general consequence of increased mobility, the economy works better, so that incomes (and hence expenditure on goods and services) rise

Once the rebound effect mechanism is identified, accompanying policy measures are usually explored to ensure that the potential reduction in consumption due to efficiency improvements are secured, or at least that the rebound effect is minimised (Van den Bergh, 2010) (third step in Fig. 2). At this point, the key for designing an effective set of measures lies in the correct identification of the drivers of the rebound effect that need to be targeted. In relation to microeconomic effects, the usual proposal is to increase the price of energy, which offsets the drop in cost. In the case of transport, this may involve introducing a tax on fuel in response to more efficient cars.

Rebound effect for water

The rebound effect or Jevons paradox has recently been considered in an increasing number of reports or scientific papers in relation to water (e.g. EEA, 2012;UNEP, 2012). In many cases it implies that an improvement in efficiency may not materialise. Only a few papers provide a deeper analysis (e.g. Pfeiffer and Lin, 2012;[START_REF] Gómez | Enhancing irrigation efficiency but increasing water use: the Jevons' paradox[END_REF][START_REF] Llop | Economic impact of alternative water policy scenarios in the Spanish production system: an input-output analysis[END_REF]. Use of the three sequential steps described for energy in Fig. 2 provides an opportunity to analyse the unintended consequences of increased water efficiency: 1) Quantification of potential savings. Traditionally, the quantification of water savings is based on the premise that, with more efficient systems, a higher fraction of the water being withdrawn is destined to the desired output, allowing a reduction in initial withdrawals. Efficiency can be considered relative to the final output, such as plant transpiration for biomass production or water delivered to homes, or to intermediary transfers (conveyance efficiency). As with energy, the volume withdrawn initially and the initial and expected efficiencies are the data needed to estimate water savings (Fig. 1).

2) Evidence of, or reason for, higher water demand. In many cases, only the basic direct microeconomic effect is described, as is often the case for energy. For instance, UNEP (2012) states: "This 'rebound effect' occurs because greater efficiency reduces the price of the goods produced, incentivising higher levels of consumption". Equally, [START_REF] Llop | Economic impact of alternative water policy scenarios in the Spanish production system: an input-output analysis[END_REF] identifies that "… any improvement in the technical efficiency of water requirements reduces water demand, and this causes a decrease in the price of water. Price reduction leads to an increase in water use, so the initial efficiency is completely or partially cancelled out". Energy and water are similar within this framework, with demand elasticity being essential in the characterisation of the rebound effect, since the explanation involves supply, demand and price.

The indirect rebound effect, in which monetary savings from efficiency cost reductions enable more income to be spent on other products and services, does not usually occur for water, but it could if, for example, money saved from reduced groundwater pumping costs is used to finance an increase in the irrigated area or for drilling a new well. From a macroeconomic perspective, the issue of potential impacts from efficiency improvements for the whole economy has also been raised, principally through the application of input-output models [START_REF] Llop | Economic impact of alternative water policy scenarios in the Spanish production system: an input-output analysis[END_REF]. Studies that introduce a more detailed analysis (Pfeiffer and Lin, 2012; Gómez and Gutierrez, 2011) also identify other reasons for a rise in demand (to be discussed below).

3) Designing complementary preventive measures. Accompanying measures are also proposed to make savings effective, as for energy. In line with the direct price effect explanation, it can consist of "price interventions in order to produce the desired effect on water resources and water shortages" [START_REF] Llop | Economic impact of alternative water policy scenarios in the Spanish production system: an input-output analysis[END_REF]. Another measure would be to explicitly specify that the savings should not satisfy new demands, e.g. through the revision of withdrawals rights. The important point here is that water savings are considered attainable if correct policy measures are implemented.

Consideration of losses and technical efficiency

Definition of losses and efficiency

Analysis of the rebound effect so far has been based on a similar approach for energy and water. The premise is the identification of savings at the scale of the physical process of water application. Economic or systemic effects then counteract these savings. Nevertheless, by focusing on the systemic consequences, the rebound effect does not question the reality of the savings from a physical point of view and, most importantly, takes them for granted to formulate a paradox. However, water and energy are fundamentally different. Energy that is not converted to the useful output is really lost, e.g. being dissipated as friction and heat (Fig. 1). However, a specific use of water does not necessarily consume all the water withdrawn. Most of water abstracted for domestic or industrial use is returned for treatment and reuse downstream (by usually directly discharging into a river) 2 . In the case of irrigation, some of the water will be consumed, either productively (through crop transpiration) or non-productively (by evaporation from wet soil and foliage), while the remaining fraction will flow back into the irrigation canal or river through run-off and drainage, or percolate into the ground and recharge aquifers. Many irrigation areas have developed on alluvial plains where aquifers are directly connected to rivers.

It is problematic to qualify return flows as losses because these will infiltrate back to the river system or aquifers and can eventually generate value for other downstream users, sustain river flows or recharge aquifers. Methodologies to categorise the different fractions of water withdrawals as a function of destination have been proposed. An approach based on the framework by Molden and Sakthivadivel (1999) and Perry et al. ( 2009) (Fig. 3) identifies recoverable return flows, i.e. the fraction of total water that will be available for reuse downstream. Thus, efficiency cannot be considered a definitive indicator for the performance of the system [START_REF] Frederiksen | A common basis for analysis, evaluation and comparison of offstream water uses[END_REF][START_REF] Jensen | Beyond irrigation efficiency[END_REF]. For example, in the Alto Aragón irrigation district of Spain [START_REF] Lecina | Irrigation modernization and water conservation in Spain: the case of Riegos del Alto Aragón[END_REF], the switch from surface irrigation to 'more efficient' sprinkler irrigation indicated that a higher share of withdrawals was effectively consumed (and therefore not available downstream) (Box 1). The debate has also been presented using the terms 'dry' and 'wet' water savings (Seckler, 1996).

Nevertheless, this critique should not lead to the extreme opposite view that all water applied in excess of crops' evapotranspiration or excessive urban use will necessarily end up downstream for reuse. Return flows might go into saline or contaminated aquifers, percolate into areas where aquifers are so deep that they are not economically viable at the present time or return during floods and go directly to the sea. Excessive evaporation also occurs in poorly managed irrigation systems. Meanwhile, excessive abstractions can lead to serious environmental damage of part of a river, depending on where return flows rejoin it. Thus, the main issue is to insist on an accurate evaluation of local hydrological and hydrogeological settings and conditions of use beforehand, to identify losses and target savings.

Modernisation of irrigation in Spain

In Spain, the reality of the projected savings of the irrigation modernisation plans can be questioned. Many of the savings were to be obtained from areas located in the upper or middle sections of river basins (Fig. 4), meaning that return flows were potentially reused by a series of users downstream. However, there is no evidence of a methodology integrating the reusability of the return flows in the technical reports on the implementation of the plans (e.g. MARM, 2010). In fact, the actual amount of projected savings (Table 1) indicates that quantification may have relied on the 'traditional approach' as shown in

Misleading approach reproduced by the rebound effect framework

The vision of savings based on a 'traditional quantification' (Fig. 3) or that drip irrigation saves water appeals to the common sense, up to the point that it appears unnecessary to introduce a detailed methodology (as happened in Spain) and is almost considered as self-explanatory. It is the same for formulation of the logic of a 'rebound' or 'paradox' (Fig. 2), in which the quantification of savings is an intuitive premise. The possibility to obtain the savings is also reinforced by the proposal of accompanying measures to make them effective (third step in Fig. 2). However, the basic formula, which may be valid for energy, does not apply to water from the physical point of view. Introducing the concept of the rebound diverts attention from the core of the debate on water saving quantification, while helping to create a false expectation.

Can unintended consequences really be prevented?

Traditional explanation based on price

The above observations do not mean that the effects of efficiency improvements should not be considered for water in relation to induced changes in the behaviour of economic agents. In fact, they highlight that these effects should not be referred to as a 'rebound' or 'paradox' as there is no obvious method to quantify water savings that may or may not be real from a physical perspective. Thus, the economic or systemic consequences, potentially based on the case of energy, should be considered. However, possible explanations should be carefully identified, particularly since the proposed accompanying measures depend on an accurate diagnosis of the problem. The principal explanation is linked to the reduction of the relative cost of water per unit of output (UNEP, 2012), potentially accompanied by a reduction in the absolute price of water (EEA, 2012). Accordingly, an increase in demand would occur if water use was initially limited by the price of water. However, price rarely constrains water use, particularly in the agricultural industry [START_REF] Hellegers | Can irrigation water use be guided by market forces? Theory and practice[END_REF]; rather, the limiting factor is water availability or water rights. Therefore, it is problematic to identify a reduction in priceand subsequent increase in demandas the prevalent reason of the rebound effect, since water use is usually inelastic to price change. This means that the related accompanying policy of increasing water price as a disincentive to water

Actual effects at system scale

The absence of price effects does not mean that there is no increase in consumption. In fact, Box 1. Modernisation in the Alto Aragón irrigation project (based on [START_REF] Lecina | Irrigation modernization and water conservation in Spain: the case of Riegos del Alto Aragón[END_REF] The Riego de Alto Aragón irrigation project is located in the Ebro river basin, north-east Spain. In 2003, the irrigated area covered 121,000 ha, with 73% using surface irrigation and 27% sprinkler irrigation. As part of the National Irrigation Modernisation Plan, 52,200 hectares that used 292 Mm 3 of water were upgraded from surface to sprinkler i igatio . The i vest e t of € illio had a ave age a ual a o tisatio ost of a out € /ha fo fa e s, including public subsidies and interest rates. Based on traditional accounting (Fig. 3), 47 Mm 3 of water savings could be expected (Fig. 6). Evidence for such savings can be obtained by considering the destination of water flows at the start and under different scenarios (Fig. 6):

 Analysis of the different fractions at the start reveals that the maximum attainable savings (i.e. with 100% efficiency) would be no more than 10 Mm 3 (the sum of the non-beneficial evapotranspiration [ET] and nonreusable return flows).  Sprinkler 1 scenario is a hypothetical case that assumes the beneficial ET remains unchanged, the usual assumption when calculating savings (see Figs. 1 and3). Even if withdrawals are lower, sprinkler irrigation consumes 23 Mm 3 of water more than surface irrigation. Indeed, most of the non-productive fraction is destined to be reused downstream in the Ebro river basin because the irrigation project is located in the middle section of the basin. Furthermore, overall evaporation (non-beneficial ET) increases because of highwind evaporation that is associated with water application by sprinkler.  Sprinkler 2 depicts the actual result of modernisation when there are no restrictions on water availability.

The modernised system withdraws and consumes more water. Consumptive use (previously available for other uses downstream) increases by 134 Mm 3 . Yet, the higher beneficial ET implies that higher profits are obtained (higher land productivity) because of, among other factors, the increased yields and the shift to more profitable and water-intensive crops.

 Sprinkler 3 considers a constant consumed fraction. The rise in total and consumptive use as illustrated in Sprinkler 2 is only possible if sufficient water is available (e.g. more water is maintained in the reservoirs thanks to higher efficiency) and water rights are not exceeded. However, to ensure there is no increase in consumptive use, a constant consumed fraction could be imposed. Nevertheless, land productivity remains basically the same as before modernisation, and the investment is not amortised. introducing a preconceived explanation of the rebound effect concept causes a distraction from the relevant identification of unintended consequences. The example from Alto Aragón (Box 1) based on [START_REF] Lecina | Irrigation modernization and water conservation in Spain: the case of Riegos del Alto Aragón[END_REF] shows how efficiency improvements can lead to higher consumptive use. This is primarily due to the higher efficacy and reliability of sprinklers: the more an irrigation technology is able to satisfy the water requirements of crops, the more it is efficacious, and efficacy rises with efficiency [START_REF] Dinar | The economics of resource-conservation, pollutionreduction technology selection. The case of irrigation water[END_REF]. Indeed, technologies such as sprinkler or drip irrigation enable the application of water on demand, i.e. when the plant needs it most. This generates increased yields compared to surface irrigation systems, which usually operate by turns [START_REF] Lecina | Irrigation modernization and water conservation in Spain: the case of Riegos del Alto Aragón[END_REF]Ward and Pulido-Velázquez, 2008). Farmers may also switch to more profitable and water-consuming crops as a result of this higher reliability.

An additional effect is that more water is left in reservoirs because of a lower demand in terms of withdrawals. This conserved water may allow farmers to fully use their irrigation quotas, or it can be kept for a subsequent irrigation campaign. Thus, efficiency improvements exert an effect similar to dams: water is kept upstream, potentially affecting downstream users and the environment (Ward and Pulido-Velázquez, 2008;[START_REF] Molle | Demand management in a basin perspective: is the potential for water saving overestimated?[END_REF]. Furthermore, more efficient irrigation technologies allow the irrigation of low-quality land (Pfeiffer and Lin, 2012;[START_REF] Dinar | The economics of resource-conservation, pollutionreduction technology selection. The case of irrigation water[END_REF].

All these consequences seem far more difficult to control than the ones traditionally described in the rebound effect framework (e.g. rising prices). It is often claimed that water rights should be revised according to the efficiency improvement to ensure that consumptive use does not rise. However, if not enough water was initially delivered to farmers to meet their rights (which is common in Spain), the increased upstream availability could be used to respect these rights, leading to more consumption even if they are revised according to the change in efficiency. Moreover, rights are sometimes defined in terms of area. To really prevent any quantitative impact on downstream users, the consumptive use could be conserved on the basis of the initial real situation and not the theoretical situation based on rights. However, the example from the Ebro river (Box 1) shows that modernised systems may not be able to increase profitabilitywith the same level of consumptive use, the investment is not paid back. Similar results have been found by Ward and Pulido-Velázquez (2008) who, apart from questioning the general interest of modernisation, show that farmers might use more water to obtain returns on their investment. It would also be hard to prevent farmers from using the full capacity of the new systems. Nevertheless, an opposite effect should also be considered: the high running costs of modernised (pressurised) systems linked to higher energy input may incentivise farmers to use water more rationally, while establishing the conditions for volumetric pricing (Rodríguez-Díaz et al., 2011).

Conclusions and recommendations

The main reasons the rebound effect is not considered to be the appropriate term to use for the consequences of water efficiency, compared to energy, comes from analysing the three sequential and interrelated steps from the rebound effect and the recommended policy prescriptions. This is summarised in Fig. 7:

 First, the use of the Jevons paradox or rebound effect reinforces the mistaken view that water not consumed is definitively lost, which is not the case. This is because the identification of savings is the basis for the formulation of a 'rebound' or 'paradox'. Furthermore, the common accompanying measures to effectively reach the projected savings assume these are attainable.  Secondly, a unique and rarely effective explanation, based on price mechanisms, is usually given, distracting from the identification of other effects that are much more difficult to control.

An alternative framework addressing the full range of consequences from a shift in irrigation technology should be based on two distinct pillars (Fig. 7):  Water flows accounting: physical implications in terms of the reorganisation of the water flows through changes in the different fractions of destination of water applied to identify real potential water savings.  Systemic assessment: an approach that looks at the entire system, framed into and influenced by economic and institutional conditions, thus enabling a more accurate and contextual identification of the main factors for the change in water use.

The price effect (i.e. what is usually understood as the rebound effect) would be integrated in the second pillar, and should be considered as relevant in some specific situations. Additional insights could be obtained from the rebound effect or Jevons paradox from other types of resources, once it is recognised that its scope is limited to the second pillar.

In Spain, the reusability of return flows has not been assessed in the modernisation plans. In addition, the example from Alto Aragón (Box 1) shows the potential consequences of a more efficient but also more efficacious technologya higher consumptive use. In this context, there are increasing demands, especially from environmental non-governmental organisations, for measures that would ensure real 'savings' (i.e. more water for the environment). However, the same critique as that used for the rebound effect could be applied: by claiming the realisation of savings and the establishment of measures to make them effective, the physical reality of these savings is not questioned.

This article has focused on efficiency from a quantitative perspective, mainly because the rebound effect concept addresses this aspect. However, once the principal justification of efficiency improvements (the estimated large water savings) is demystified, it appears even more necessary to have a detailed analysis on all the potential consequences that result from increasing efficiency. The implications in terms of water quality [START_REF] Gleick | Water-use efficiency and productivity: rethinking the basin approach[END_REF][START_REF] Dinar | The economics of resource-conservation, pollutionreduction technology selection. The case of irrigation water[END_REF], energy requirements (Rodríguez-Diaz et al., 2011) or farmers' quality of life and rural development (CEPS, 2012) should also be assessed. Other benefits and associated costs should be identified, allowing detailed cost-benefit or cost-effectiveness analysis to determine whether the technology change is justified or to compare it with other possible measures.

Meanwhile, the true nature of the investment could be recognised with it being more justified for rural development reasons than for environmental reasons. What was a rebound or even an unintended consequence (e.g. the intensification of irrigation) would be the central objective of the project and the possible savings a potential cobenefit. However, the silver lining is that an increasingly efficient and efficacious use in the context of a semi-arid environment fully optimises the use of water to the point of having possibly low redundancy and thus a less resilient system.

1

The Jevons paradox is a rebound effect higher than 100%, although both terms are usually used interchangeably. In this article, only the concept of rebound effect is used, which is thus a wider term than the Jevons paradox.

Figure 2 .

 2 Figure 2. Three successive steps of the rebound effect concept and identification of the logic of a e ou d o pa adox

Fig. 3 .

 3 The expectation of large water savings due to a technological change was the main evidence for the investment. In a context of political tension around massive water transfers, the win-win opportunity (water savings and rural development) offered by the modernisation of irrigated areas was largely supported (López-Gunn et al., 2012b) (Fig. 5). A real estimation of the potential savings should have been based on a detailed local assessment, as presented by Lecina et al. (2010) (Box 1).

Figure 3 .

 3 Figure 3. Quantification of water savings Within the traditional view, it is based on the change in efficiency. In a detailed accounting framework (Perry et al. 2009; Molden and Sakthivadivel, 1999), the reusability of the return flows must be assessed.

Figure 5 .

 5 Figure 5. Shock Plan: the main justification for the modernisation of irrigation in Spain is to save water fo a sustai a le u al develop e t Sou e: Pla de Cho ue

Figure 4 .

 4 Figure 4. Map of the irrigated areas affected by the Shock Plan (Plan de Choque 2006) within the river basins in Spain (based on MARM, 2010)

Figure 6 .

 6 Figure 6. Traditional accounting of water savings and the real effect of modernisation of 52,200 ha in the Alto Aragón irrigation district (Ebro river basin, Spain). Adapted from: Lecina et al. (2010)

Figure 7 .

 7 Figure 7. From the rebound effect framework to a proposed framework based on two pillars (water flows accounting and systemic assessment)

Table 1 .

 1 Investment, affected area and projected water savings relative to the two Spanish plans of irrigation modernisation[START_REF] Marm | Estrategia nacional para la modernización sostenible de los regadíos H2015. Informe de sostenibilidad ambiental[END_REF]

	Plan	Area (ha)	Investment € 6 )	Total water savings (10 6 m 3 )	Water savings per hectare (m 3 /ha)
	National Irrigation Plan 2000-2008	1,134,900 1,528	1,375	1,212
	Shock Plan 2006-2008	866,900	2,409	1,162	1,340
	Intermediary evaluation relative to Shock Plan 2006 in MARM (2010)	250,000	-	500	2,000

Acknowledgements

The authors would like to thank Enrique Cabrera, Joan Corominas, Emilio Custodio, Carlos Mario Gómez, Sergio Lecina, Chris Perry, Bart Snellen, James Winpenny and two anonymous reviewers, who have provided valuable comments and suggestions on a previous version of this article. This article was motivated by interesting discussions on the Jevons paradox with Laurent Hardy and Jan Lundqvist. What is presented, however, is the sole responsibility of the authors.

2

Return flows should not be identified as new resources, and the race towards wastewater reuse as a way to alleviate water stress is also pernicious in many cases. Perry, Chris J.


Zheng Han

Adrien Allain

Hadi Arjmandi-Tash

Konstantin Tikhonov

Mikhail Feigel'man

Benjamin Sacépé

Vincent Bouchiat

Collapse of superconductivity in a hybrid tin-graphene Josephson junction array

When

a Josephson junction array is built with hybrid superconductor/metal/superconductor junctions, a quantum phase transition from a superconducting to a two-dimensional (2D) metallic ground state is predicted to happen upon increasing the junction normal state resistance. Owing to its surface-exposed 2D electron gas and its gate-tunable charge carrier density, graphene coupled to superconductors is the ideal platform to study the above-mentioned transition between ground states. Here we show that decorating graphene with a sparse and regular array of superconducting nanodisks enables to continuously gate-tune the quantum superconductor-to-metal transition of the Josephson junction array into a zero-temperature metallic state. The suppression of proximity-induced superconductivity is a direct consequence of the emergence of quantum fluctuations of the superconducting phase of the disks. Under perpendicular magnetic field, the competition between quantum fluctuations and disorder is responsible for the resilience at the lowest temperatures of a superconducting glassy state that persists above the upper critical field. Our results provide the entire phase diagram of the disorder and magnetic field-tuned transition and unveil the fundamental impact of quantum phase fluctuations in 2D superconducting systems.

"

The ground state terminating 2D superconductivity in the superconductor-(metal)-insulator quantum phase transition remains an unsolved fundamental problem [START_REF] Goldman | Superconductor-insulator transitions in the two-dimensional limit[END_REF][START_REF] Dobrosavljevic | Conductor-insulator quantum phase transitions[END_REF] . In granular [START_REF] Jaeger | Onset of superconductivity in ultrathin granular metal films[END_REF][START_REF] Allain | Electrical control of the superconducting-to-insulating transition in graphene-metal hybrids[END_REF] and some amorphous superconducting thin films [START_REF] Ephron | Observation of quantum dissispation in the vortex state of a highly disordered superconducting thin film[END_REF][START_REF] Mason | Dissipation Effects on the Superconductor-Insulator Transition in 2D superconductors[END_REF][START_REF] Mason | Superconductor-insulator transition in a capacitively coupled dissipative environment[END_REF][START_REF] Qui | Magnetically induced Metallic Phase in superconducting tantalum films[END_REF][START_REF] Aubin | Magnetic-field-induced quantum superconductor-insulator transition in Nb 0.15 Si 0.85[END_REF][START_REF] Lin | Suppression of the Berezinskii-Kosterlitz-Thouless transition in 2D superconductors by macroscopic quantum tunneling[END_REF] , oxide interfaces [START_REF] Caviglia | Electric field control of the LaAlO3/SrTiO3 interface ground state[END_REF][START_REF] Biscaras | Multiple quantum criticality in a two-dimensional superconductor[END_REF] .and Josephson junction arrays [START_REF] Van Der Zant | Field-Induced Superconductor-to-Insulator Transitions in Josephson-Junction Arrays[END_REF][START_REF] Van Der Zant | Quantum phase transitions in two dimensions: Experiments in Josephson-junction arrays[END_REF] , an intervening metallic state is often observed experimentally at the disorder-tuned or magnetic field-tuned quantum critical point separating the superconducting from the (weak) insulating state. In contrast to usual non-interacting 2D electronic systems in which electron localization yields insulating ground state [START_REF] Abrahams | Scaling theory of localization: Absence of quantum diffusion in two dimensions[END_REF] , this metallic state is characterized by the presence of electron-pairing fluctuations that are indicated in experiments by a partial drop of resistance -a reminiscence of the nearby superconducting phase -preceding the nearly T-independent resistive state.

The possibility that a 2D system can undergo a zero-temperature superconducting to metal transition is rather intriguing [START_REF] Feigel'man | Quantum superconductor-metal transition in a 2D proximity coupled array[END_REF][START_REF] Feigel'man | Quantum Superconductor-Metal Transition in a Proximity Array[END_REF][START_REF] Das | Existence of a Bose metal at T=0[END_REF][START_REF] Kapitulnik | Effects of dissipation on quantum phase transitions[END_REF][START_REF] Spivak | Quantum superconductor-metal transition[END_REF][START_REF] Philips | The Elusive Bose Metal[END_REF][START_REF] Spivak | Theory of quantum metal to superconductor transitions in highly conducting systems[END_REF] . The nature of this metallic state, sometimes termed Bose-metal [START_REF] Das | Existence of a Bose metal at T=0[END_REF][START_REF] Philips | The Elusive Bose Metal[END_REF] , where charge carriers are supposedly Cooper-pairs, is not clear.

Considerable theoretical efforts have been focused on the role of quantum phase-fluctuations of the superconducting order-parameter in the regime where the superconducting phase stiffness breaks down. In particular, models involving specific types of Josephson junction arrays in which superconducting disks are coupled together via a 2D disordered metal through proximity effect have been considered [START_REF] Feigel'man | Quantum superconductor-metal transition in a 2D proximity coupled array[END_REF][START_REF] Feigel'man | Quantum Superconductor-Metal Transition in a Proximity Array[END_REF][START_REF] Spivak | Quantum superconductor-metal transition[END_REF][START_REF] Spivak | Theory of quantum metal to superconductor transitions in highly conducting systems[END_REF] . Such proximity-coupled arrays encompass all the physical ingredients: conventional superconducting electron pairing, embedded dissipation channels in the normal disordered metal, and quantum fluctuations of superconducting phases due to weak Coulomb blockade on superconducting disks [START_REF] Feigelman | Weak charge quantization on a superconducting disk[END_REF] .

The theoretical analysis of these models shows that a quantum phase transition from a superconducting state to a non-superconducting state can be tuned by two different means:

either geometrical by increasing the distance in between disks, or by increasing the metal resistance. Both parameters directly affect the superconducting proximity effect and result in a dramatic enhancement of quantum phase fluctuations. Though proximity effect is expected to extend over infinite distance at zero temperature leading systematically to a superconducting ground state, the main conclusion of recent theories [START_REF] Feigel'man | Quantum superconductor-metal transition in a 2D proximity coupled array[END_REF][START_REF] Feigel'man | Quantum Superconductor-Metal Transition in a Proximity Array[END_REF][START_REF] Spivak | Quantum superconductor-metal transition[END_REF][START_REF] Spivak | Theory of quantum metal to superconductor transitions in highly conducting systems[END_REF] is that quantum phase fluctuations lead to a collapse of superconductivity, that is, of the array phase stiffness. This should arise at a given critical value of the tuning parameter, leading to a zero-temperature 2D metallic state.

On the experimental side, it is hardly conceivable to access this quantum breakdown of superconductivity by tuning continuously the resistance of a metallic thin film that carries the superconducting proximity effect. Varying the inter-disk distance on a large set of samples has thus remained the only reliable approach with thin films. A recent study of proximity arrays made of gold thin-films covered with an array of niobium nano-disks [START_REF] Eley | Approaching zero-temperature metallic states in mesoscopic superconductor-normal-superconductor arrays[END_REF] has demonstrated that the critical temperature of such proximity coupled arrays decreases upon increasing the inter-disk distance and extrapolates to zero for a finite inter-disk distance, suggesting a quantum phase transition to a non-superconducting, metallic ground state.

In this work, we take a different route to tune the quantum phase transition in a controllable fashion. We use graphene [START_REF] Geim | The rise of graphene[END_REF] , a purely 2D crystal of carbon atoms, to serve as the diffusive metal, which we decorate with a triangular array of tin disks (a type-I superconductor). The gate-tunable sheet resistance of graphene, which can approach the quantum of resistance (h/e 2 ), enables us to continuously vary the strength of disorder which affects the inter-disk proximity effect and therefore tune the transition from the superconducting to the metallic state. Furthermore, the charge-carrier density of graphene is always several orders of magnitude below those of classical metals, preventing the weakening of the superconductivity on the disks by inverse proximity effect [START_REF] Gennes | Superconductivity of metals and alloys[END_REF] . As we describe below, these capabilities, along with quantitative comparisons with theory, allow us to unveil both the collapse of superconductivity by quantum phase fluctuations, and the ensuing 2D metallic state. In addition, under perpendicular magnetic field a resilient superconducting glass state is observed at our lowest temperatures, as expected from theory [START_REF] Spivak | Mesoscopic Effects in Disordered Superconductors near H c2[END_REF][START_REF] Galitskii | Disorder and Quantum Fluctuations in Superconducting Films in Strong Magnetic fields[END_REF][START_REF] Feigel'man | Proximity-induced superconductivity in graphene[END_REF] .

Berezinski-Kosterlitz-Thouless transition in superconducting array

Our proximity-induced array consists of a triangular lattice of disk-shaped superconducting islands that are deposited onto a graphene Hall bar equipped with a back-gate electrode (see Fig. 1). The superconducting disks are made of 50 nanometers thick tin films, and have a diameter 2a = 400 nm, with a thickness of 50 nm. Disks are separated by a distance b = 1 µm in between their centres (see Fig. 1). The electron mobility of the graphene extracted from a fit of the field effect and from Hall measurements is about 680 cm 2 V -1 s -1 at high gate voltage (Fig. S1), while the mean free path l can be varied with the backgate within the range of 10 to 30 nm (Fig. S2), corresponding to a diffusion coefficient D = 50 to 140 cm 2 s -1 (see Fig. S2). Our system differs considerably from previous experiments performed on Sn-graphene hybrids [START_REF] Allain | Electrical control of the superconducting-to-insulating transition in graphene-metal hybrids[END_REF][START_REF] Kessler | Tunable superconducting phase transition in metal-decorated graphene sheets[END_REF] for which tin clusters of random shapes self-assembled densely on graphene surface by dewetting. Those experiments were performed in the opposite regime of short junctions (coherence length >>b) and for one of it 4 in the limit of strongly disordered graphene (mean free path << b) for which a superconductor-to-insulator quantum phase transition has been observed upon decreasing the graphene charge carrier density.

We begin the presentation of the data by describing the two-step transition [START_REF] Resnick | Kosterlitz-Thouless transition in proximity-coupled superconducting arrays[END_REF][START_REF] Abraham | Resistive transition in two-dimensional arrays of superconducting weak links[END_REF] that the sample undergoes towards the superconducting state.

Upon cooling from the normal state, a gate-independent resistance drop occurs at a temperature of ~ 3.6 K (Fig. 2a, red and blue curves, also Fig. S3). We attribute this first drop to the superconducting transition of the Sn nanodisks. The resistance drop ΔR at this transition amounts to 20 % on the electron side. This number is in good agreement with the surface filling factor of ~15 % (see Fig. S3) which demonstrates the good transparency of nanodisks/graphene Andreev contacts.

Further cooling enhances the proximity effect via the graphene, which eventually leads to the percolation of superconductivity and the establishment of a 2D

superconducting state (see Figs S3a &S3b). As shown in Fig. 2a, at gate voltages between 0 and 30 V, the resistance decreases when temperature is lowered from 3 K to 0.06K. An opposite trend is seen for gate voltages close to Dirac point (namely, -30 V to 0 V).

Below 1 K (Fig. S3b) and for gate voltage above -3V, the proximity-coupled array develops full superconductivity with a zero resistance state and a gate-dependent critical temperature. The transition to the superconducting state in our 2D sample is captured by a Berezinski-Kosterlitz-Thouless (BKT) mechanism [START_REF] Berezinskii | Violation of long range order in one-dimensional and two-dimensional systems with a continuous symetry group. I. Classical systems[END_REF][START_REF] Kosterlitz | metastability and phase transitions in two-dimensional systems[END_REF] that describes the superconducting transition as a proliferation and unbinding, at For Josephson junction arrays, the value of

BKT c
T is directly proportional to the

Josephson energy (, )

J Eb T of a single junction formed between two superconducting nano-disks. For a triangular lattice, the relation reads [START_REF] Feigel'man | Proximity-induced superconductivity in graphene[END_REF] :

,

where E J (b,T) depends only on the metal conductance, the inter-disk distance b and the diffusion coefficient D (see methods). Using experimentally extracted values of D and of the graphene resistance measured at 4 K for all back-gate values, we calculated

BKT c
T by solving (1) self-consistently with respect to T for the entire back-gate range (see methods).

In Fig. 2b, the resulting

BKT c
T is shown by the dashed line. For high gate voltage, that is,

V g > 10V, this theoretical value of BKT c
T is in excellent agreement with the sample critical temperature without any fitting parameter, thereby validating our theoretical description.

The critical supercurrent at T=0 provides another computable physical quantity on such arrays. As shown in Fig. 3a, the plot of the differential resistance dV/dI versus bias current and V g allows to obtain the full V g dependence on I c at T= 0.06 K. In the zero temperature limit, the critical current of an array of nanodisks can be inferred from the Josephson coupling energy of neighbouring disks [START_REF] Feigel'man | Proximity-induced superconductivity in graphene[END_REF] , which is given by: ,

where is the dimension-less conductance of graphene in the normal state,

T c BKT  1.47E J (b,T c BKT ) E J (b,T = 0) = π 3 4 g D b 2 ln 2 (b / a ) =  2e I 1 g =  / (e 2 R  )
with the sheet resistance; and I 1 is the maximum supercurrent between two neighbouring nanodisks. The total critical current I c can be estimated by summing the critical current I 1 of each nanodisk neighbours (see Fig. 1b) of our array, which are contributing as parallel channels, and by neglecting small contribution to I c from non-nearest pairs of disks, that is, above a given doping level this quantity reaches a constant level of 3.6×10 9 cm -2 which matches the quantity (dashed line). Therefore, at high gate voltages, this experiment follows the theoretical predictions for such proximity-coupled arrays of disk sparsely decorating graphene [START_REF] Feigel'man | Proximity-induced superconductivity in graphene[END_REF] .

Quantum breakdown of superconductivity

The central result of this work is the anomalous reduction of the sample's critical temperature T c with respect to the calculated T BKT upon increasing graphene resistance, leading to a sudden collapse of superconductivity when approaching the charge neutrality point of the graphene layer (dark red region of Fig. 2b). Indeed, the V g -dependence of T c

shown in Fig. 2b first deviates from T c BKT towards lower temperatures for V g < 10V, and then abruptly drops to zero at around -3V. The suppression of superconductivity is also apparent in the V g -dependence of the critical supercurrent I c at 0.06 K. As can be seen in Fig. 3, the zero-resistance state (blue region, marked as "S" in Fig. 3a) disappears when V g = -3V in a similar fashion as for T c . We attribute these significant deviations, culminating in the collapse of superconductivity, to the failure of the BKT model [START_REF] Feigel'man | Proximity-induced superconductivity in graphene[END_REF] , which only take into account thermal phase fluctuations. Therefore our results are a direct consequence of the breakdown of the superconducting phase stiffness due to the

R  eI c R N / D 3π 3 / b 2 ln 2 (b / a )
emergence of quantum phase fluctuations [START_REF] Feigel'man | Quantum superconductor-metal transition in a 2D proximity coupled array[END_REF][START_REF] Feigel'man | Quantum Superconductor-Metal Transition in a Proximity Array[END_REF][START_REF] Feigel'man | Proximity-induced superconductivity in graphene[END_REF] .

The quantum origin of phase fluctuations in such proximity-coupled arrays stems from the particle number-phase uncertainty relation that states that when the number of particles in an isolated superconducting disk is fixed, the phase of the superconducting order parameter will undergo strong non-thermal fluctuations. In tunnelling Josephson junction arrays, such quantum phase fluctuations are expected to happen when the disk charging energy overcomes the Josephson coupling energy [START_REF] Van Der Zant | Field-Induced Superconductor-to-Insulator Transitions in Josephson-Junction Arrays[END_REF] , a situation that leads to charge quantization and the Coulomb blockade of the tunnelling into the disks.

Proximity-coupled arrays behave differently since their superconducting disks are well-coupled to a metal layer without tunnel barrier. When the normal-state resistance of the array approaches a few kΩ, which is achievable with graphene, the Andreev conductance, that is, the conductance for electron pairs, can be such that a weak charge quantization [START_REF] Feigelman | Weak charge quantization on a superconducting disk[END_REF] of charge 2e prohibits the charge transfer into the disks. As the transfer of quasiparticles is also hindered at c TT = by the superconducting gap, the total charge of the disks tends to become fixed and thus promotes quantum fluctuations of the phase of the superconducting order parameter (see Supp. Info). As a result, theory predicts that the presence of quantum phase fluctuations leads to an effective decoupling of superconducting phases between disks, and eventually to the collapse of the array phase stiffness for a given critical resistance of the 2D metal [START_REF] Feigel'man | Quantum superconductor-metal transition in a 2D proximity coupled array[END_REF][START_REF] Feigel'man | Quantum Superconductor-Metal Transition in a Proximity Array[END_REF] . The wide range of tunability of graphene sheet resistance is the key ingredient that allows to access to this regime of weak charge quantization and the ensuing superconducting-metal transition.

We next investigate the nature of the state terminating superconductivity for V g < -3V. We systematically measured the T-dependence of the resistance, that we term R-T curve, for gate-voltage close to the critical V g = -3V. Shown in Fig. 4a, the R-T curves in the non-superconducting regime at V g < -3V exhibit a resistance drop below 0.5 K reminiscent of the superconducting transition of those at V g > -3V, which indicates that superconducting fluctuations have developed into the graphene. However, instead of falling into a zero-resistance superconducting state, the R-T curves level off into finite resistance plateaus of gate-dependent values that can reach up to 28 kΩ. Importantly we can exclude the possibility of electron heating as the origin of this metallic tail at low temperature, since in a magnetic field such that the tin nanodisks turn metallic the resistance shows a clear T-dependence till the base temperature (see Fig. S4).

To get more insights on this critical regime we show in Fig. 3b a set of differential resistance versus bias current curves extracted from the data of Fig. 2a for few values of V g . In the superconducting state, zero resistance is observed up to the critical current as discussed before (see Fig. 3b, curve at V g = 0 V). However, for V g < -3 V where the R-T curves saturates at finite values upon cooling, the differential resistance around zero current bias transform into a zero-bias dips which persist even close to the charge neutrality point at V g = -13 V. These dips give a clear indication that superconducting fluctuations exist for all gate voltages. Therefore we are led to conclude that the ground state of our array, which exhibits finite resistance plateaus, is superconductivity-related.

Moreover, the fact that the R-T curves conspicuously extrapolate to a constant resistance at zero-T suggests that a 2D quantum metallic state is approached. We believe that the nature of this collapse of superconductivity and the resulting quantum metallic state arises from the enhancement of quantum phase fluctuations upon increasing graphene resistance, which eventually leads to the effective decoupling of the nanodisks superconducting phases. For critical temperatures below the typical Thouless energy of a single junction, , the impact of quantum fluctuations on T c is expected [START_REF] Feigel'man | Quantum Superconductor-Metal Transition in a Proximity Array[END_REF] to be dramatic with an exponential suppression in the form , where g c is the critical conductance of the graphene and c a constant, which qualitatively explains the observed collapse of superconductivity. The variation of the resistance plateau in the metallic state (Vg < -3V) with gate voltage may be partially attributed to superconducting fluctuations of Aslamazov-Larkin [START_REF] Aslamazov | Effect of Fluctuations on the Properties of a Superconductor Above the Critical Temperature[END_REF] and Maki-Thompson [START_REF] Maki | Critical Fluctuation of the OrderParameter in a Superconductor[END_REF][START_REF] Thomson | Flux Flow, and Fluctuation Resistance of Dirty Type-II Superconductors[END_REF] types which would include the phase fluctuations, leading to a positive correction to the conductance. A non-perturbative theory of these fluctuations is required to quantitatively explain the very strong contribution of superconducting fluctuations to conductivity near critical Vg = -3 V.

E Th = D b 2 T c ~ETh exp - c g -g c

Superconducting glassy state

Another route to induce a quantum phase transition to the normal state in our superconducting array consists in applying a perpendicular magnetic field, H. The destruction of the array superconductivity by the magnetic field -we consider here a magnetic field smaller than the critical field of tin nanodisks -results from the interplay between three different effects. First the magnetic field imposes phase frustration between the nanodisks. This frustration, which is given by 00 / fH S =Φ (

2 0 3/ 2 Sb =
is the area of the array elementary cell in the triangular lattice and 0 /2 he Φ= the flux quantum), impedes to reach a set of phase differences between nano-disks which simultaneously minimises the Josephson energy of all junctions. Second, the magnitude of each individual Josephson coupling J E between two neighbouring disks i and j decreases upon increasing magnetic field in a way such that J E drops exponentially beyond the magnetic length 0 H LH =Φ due to random magnetic field-induced phase shifts, whereas the typical modulus of E J is suppressed by a factor ~1/g only [START_REF] Al'tshuler | Mesoscopic fluctuations in a superconductor-normal metal-superconductor junction[END_REF] . Thereby both these effects tend to suppress superconductivity in the array. The second effect however leads to the enhancement of quantum phase fluctuations by the mitigation of the proximity effect [START_REF] Feigel'man | Quantum Superconductor-Metal Transition in a Proximity Array[END_REF] , whereas frustration may yield non-monotonic behaviour of the resistance upon increasing H as observed in tunnelling Josephson junction arrays [START_REF] Van Der Zant | Field-Induced Superconductor-to-Insulator Transitions in Josephson-Junction Arrays[END_REF][START_REF] Van Der Zant | Quantum phase transitions in two dimensions: Experiments in Josephson-junction arrays[END_REF] . The third effect to consider here comes from the phase coherent nature of the disordered graphene electron gas and the resulting mesoscopic fluctuations. As a result of the multiple interferences of electron wave functions, the graphene conductance undergoes mesoscopic fluctuations that lead, in the normal state, to universal conductance fluctuations. In our proximity-coupled array, such mesoscopic fluctuations are present together with superconducting fluctuations, which result in resistance fluctuations upon varying electron density, as apparent in the gate evolution of the resistance measured at 0.06 K shown in Fig. 2a. In a similar fashion, small variations of magnetic field can randomly modulate the Andreev conductance of each Josephson junction by dephasing the electron interference pattern [START_REF] Spivak | Mesoscopic Effects in Disordered Superconductors near H c2[END_REF][START_REF] Al'tshuler | Mesoscopic fluctuations in a superconductor-normal metal-superconductor junction[END_REF][START_REF] Denhertog | Sample-Specific Conductance Fluctuations Modulated by the Superconducting Phase[END_REF] and therefore play an important role in the setting of global phase coherence, especially when the mean critical current is strongly suppressed by magnetic field.

The conjunction of these three effects has led to the theoretical predictions that, at very low temperature, superconductivity may persist above the upper critical field, H c2 , in the form of an upturn [START_REF] Galitskii | Disorder and Quantum Fluctuations in Superconducting Films in Strong Magnetic fields[END_REF][START_REF] Feigel'man | Proximity-induced superconductivity in graphene[END_REF] of the T-dependence of H c2 , or a possible re-entrant superconducting phase [START_REF] Spivak | Mesoscopic Effects in Disordered Superconductors near H c2[END_REF] . The resulting state is expected to be a superconducting phase glass due to the phase frustration between nanodisks and to the mesoscopic fluctuations that arbitrarily affect each Josephson coupling.

To address these fundamental aspects we systematically measured the T-evolution of the magneto-resistance of the sample in the superconducting phase far from the charge neutrality point (V g = 30 V). Fig. 5 presents the whole set of data in the form of a plot

showing the colour-coded magneto-resistance as a function of T and H. The dark blue area indicates vanishingly small resistance, that is, superconducting state, and the other colours indicate the resistive normal state. Interestingly we observe that the superconducting phase extends over a wide and continuous range of magnetic field, which resembles the mixed phase of bulk superconductors. The T-evolution of H c2 from 0.8 K down to 0.2 K indeed follows closely the theoretical behaviour of type-II superconductor [START_REF] Gennes | Superconductivity of metals and alloys[END_REF] that we have traced in black line in Fig. 5. At this gate voltage, the zero-T extrapolation of this theoretical fit gives 2.2 mT which is close to full frustration f=1 reached at the field value

00 0 /2 . 3 c HS m T =Φ =
, that is, one flux quantum per unit cell.

However, inspecting Fig. 5 for T below 0.2 K, we observe that the experimental H c2 (T) curve deviates upward and draws a resilient superconducting pocket persisting up to 3.3 mT. Replicas of different sizes, though of finite resistivity, are even visible at higher magnetic field. Importantly, in this re-entrant superconducting phase, the supercurrent is strongly suppressed in comparison with the main superconducting region at the same temperature (see Fig. S5).These observations comply with the theoretically expected superconducting phase glass state [START_REF] Spivak | Mesoscopic Effects in Disordered Superconductors near H c2[END_REF][START_REF] Galitskii | Disorder and Quantum Fluctuations in Superconducting Films in Strong Magnetic fields[END_REF][START_REF] Feigel'man | Proximity-induced superconductivity in graphene[END_REF] . Our measurements therefore constitute a clear demonstration of reentrant superconductivity in a inhomogeneous 2D superconducting system and provide another fundamental aspect of such proximity coupled arrays.

Phase diagram of the quantum superconductor-to-metal transitions

The use of graphene has provided here a unique opportunity to explore in a single device the phase diagram in phase-space variables T, H and V g , where V g is equivalent to disorder, which drive the thermal and the quantum phase transitions to the normal state.

Due to the continuous tuning of disorder through the gate-tuneable graphene electron density, we can therefore construct the entire phase diagram of our proximity-coupled JJ array. The result shown in Fig. 6 provides the complete picture of the experimental temperature, disorder and magnetic field-driven superconductor-to-metal transitions addressed in this work.

The similarities with the original phase diagram based on the dirty-boson model [START_REF] Fisher | Quantum phase transitions in disordered two-dimensional superconductors[END_REF] are striking and show that our proximity coupled array behaves in many respects as disordered superconducting thin films. However, the new physical ingredient here relies on the inhomogeneous nature of superconductivity, which presents built-in superconducting disks subject to phase fluctuations. Such inhomogeneities also emerge close to the quantum critical point in amorphous thin films [START_REF] Finkelstein | Superconducting transition temperature in amorphous films[END_REF] in the form of self-induced fluctuations of the superconducting state induced by disorder [START_REF] Skvortsov | Superconductivity in disordered thin films: giant mesoscopic fluctuations[END_REF] .

To conclude, our work demonstrates that by increasing the normal metal resistance in a proximity-coupled array, a metallic state can terminate the superconducting state due to a strong enhancement of quantum phase fluctuations, which dramatically suppress the critical temperature T c . The use of graphene as 2D electron gas platform has enabled us to continuously map the entire disorder and magnetic field-driven superconductor-to-metal transition. This plot has let us to observe the appearance of a reentrant superconducting phase glass state under magnetic field above the predicted critical field. This system offers new insights to unveil the dramatic impact of quantum fluctuations induced by weak charge quantization in 2D superconducting systems.

Methods

We used CVD-grown graphene sheets transferred onto 285 nm oxidized silicon wafer as 2D diffusive metal. As shown in Fig. 1, the sample was patterned by standard e-beam lithography into Hall bar geometry (central square area 6 µm 2 ) and contacted with normal leads (50nm Au/5nm Ti). The entire graphene surface was then decorated with a second lithography step by an array of 50-nm-thick Sn nanodisks.

Samples were anchored to the mixing chamber of a 3 He- [START_REF] Allain | Electrical control of the superconducting-to-insulating transition in graphene-metal hybrids[END_REF] He dilution refrigerator placed inside a shielded cryostat and connected to highly filtered lines. Lossy coaxial lines, capacitive filters, and π-filters were installed at the mixing chamber stage to attenuate the electronic noise from about 10 kHz. A low-frequency lock-in measurement setup based on current biased method (2 to 5nA input current) was used to measure the differential resistance while voltage across the sample was measured with low-noise preamplifiers.

To fit the T BKT in the measured data in Fig. 2b, we replaced out Eq.( 1) with the E J in the left side of the Eq. ( 5) in Ref.

= ∑ . (a)
The above equation, together with (1), can be reduced to

n Tn ωπ =+ is the n th Matsubara frequency at temperature T. The function Color-scaled map of magnetoresistance plotted as a function of temperature and magnetic field measured in the superconducting phase (V g = 30 V). The upper critical field of the array is indicated by the white contour that borders the superconducting and resistive state. The solid line is a fit of the upper critical field by the BCS theory for dirty superconductors. A clear deviation happens below 0.2 K where a re-entrant pocket of superconductivity persists up to 3.3 mT and is followed at higher field by not fully superconducting pockets, non periodic in magnetic field.

0 1 0 () (c o s h) (c o s h) Pz z K z tK z td t ∞ = ∫ , with (

  -antivortex pairs, that is, thermal phase fluctuations of the superconducting order parameter. A colour-scaled map showing resistance versus gate-voltage and temperature is shown in Fig.2b. T c is indicated by the white contour. One can see that higher positive gate-voltage leads to higher T c , whereas T c vanishes in the region near to the Dirac point, between -30 V and -3 V.

1 6 c

 16 II ≈ . According to the above relations, at c TT = , the product of I c R N (R N being the graphene sheet resistance in the normal state) should only depend on diffusion coefficient D for a given geometry. This can be directly tested since the use of graphene as a 2D disordered metal enables to gate-tune the diffusion coefficient while keeping all geometrical aspects of the array constant. By extracting the experimental critical current I c from Fig.3a, and measuring R N at 4 K (Fig.2a), we plot the quantity as a function of the gate voltage in Fig.3b. One can see that

  energy of the pair of disks separated by distance b, a the diameter of superconducting disks, (

  function. Eq. (b) contains only one variable, which is the critical temperature T at the BKT transition, since the parameters of g and D can be obtained from measured data. By summing up (b) (high orders of n do not affect the sum significantly since the function P decays rapidly with increasing n), one gets the dashed line as the unique solution of Eq. (b) from experimentally extracted g and D. E Th = D b 2

Figure 1 .

 1 Figure 1. Proximity-coupled array of superconducting nanodisks on graphene. (a) Schematics of the device involving a triangular array of superconducting tin disks decorating the bare graphene surface. The device connected to a current source and measured using a 4-probe measurement setup. (b) Optical micrograph of the sample. The black lines underline the graphene sample plasma etched hall bar, which is connected to gold electrodes seen on the edge of the image. The graphene surface is decorated with a regular array of tin nanodisks of diameter 400nm separated by 1µm between centres. (c) Scanning electron micrograph showing the triangular lattice of tin disks. (Scale bars in both images are 1 micrometer).

Figure 2 .

 2 Figure 2. Collapse of superconductivity in the proximity coupled array.(a) Gate voltage dependence of the four-probe resistance for three different temperatures. The curve at T = 4K gives the gate-dependence of the device resistance above the tin nanodisks superconducting transition. At T=3K, the resistance drops over the full gate range due to the superconducting transition of the tin nanodisks. At the base temperature of 0.06 K, the array undergoes a transition from superconducting to resistive state for -29 V<V g <-3 V and the resistance exhibits a sharp peak at the charge neutrality point. (b) Color-scaled map of resistance versus temperature and gate voltage. The superconducting phase (S) on the electron-doped side is shown by the blue area and the superconducting critical temperature bordering the resistive state is indicated by the white contour. The dashed line is calculated from Eq. (1) without fitting parameter. The experimental T c deviates around V g = 5 V towards a sharp breakdown of the superconducting state at V g = -3 V.

Figure 3 .

 3 Figure 3. Critical current in the proximity coupled array at 0.06 K. (a) Color-scaled map of the differential resistance dV/dI measured at 0.06 K versus gate voltage and bias current. The border between the zero-resistance state (blue area) and the resistive state (red area) indicates the critical current I c . (b) The quantity calculated from data of (a) and from Fig. S2 and plotted as a function of gate voltage. The red dotted line corresponds to the theoretical value calculated with eq. (2).

Figure 4 .Figure 5 .

 45 Figure 4. Zero-temperature metallic state at criticality. (a) Temperature dependence of array resistance measured at different gate voltages in a temperature range of 0.07 K -0.5 K. While the resistance drop into a full superconducting state for V g > -5V, the superconducting transition is incomplete for -29 V<V g < -5V and a T-independent resistance plateau extends till the base temperature of 0.06 K. (b) Individual current-biased differential resistance measured at 0.06 K for a set of gate voltages ranging from -13 V to 0 V.

Figure 6 .

 6 Figure 6. Phase diagram of the superconductor-to-metal transitionA 3D-phase diagram showing the superconducting state reconstructed from three color-scaled maps of the array resistance in (V g ,H,T) space. The resistance in (V g -H) space is measured at 0.06 K. As the gate voltage V g has a direct action on the effective disorder of the graphene mediating superconductivity, this axis has been relabelled accordingly. Notice the superconducting reentrance above the first critical field which can be seen in both (V g ,H) and (H,T) planes.

Figure 4

 4 Figure 4 Han et al.

Figure 6

 6 Figure 6 Han et al.

  29, which is the Josephson energy derived from the collective Josephson coupling in a 2D array using Matsubara-space Usadel equation, with

	the result:
	{ Jn ( , ) 4 /[2 ln( / 2 / )] Eb T g T D a ππ ω } 2 0 n ω >	P	(	ω	n	/ 2 ) T h E

& ' '

Acknowledgements. Samples were fabricated at the NANOFAB facility of the Néel Institute, the technical team of which has been of critical help for this work. We gratefully thank D. Shahar for valuable discussions and comments on the manuscript. We thank N. Bendiab, H. Bouchiat, C. Chapelier, J. Coraux, C.O. Girit, B. M. Kessler, L. Marty, A. Reserbat-Plantey, and A. Zettl for stimulating discussions. This work is financially supported by ANR-BLANC projects SuperGraph, TRICO and Cleangraph. DEFI Nano ERC Advanced Grant MolNanoSpin. Z.H. and H.A-T acknowledge grant support from Cible program of Région Rhone-Alpes and from Nanosciences foundation respectively. Research of M.F. was partially supported by the RFBR grant #13-02-00963.