Hicham Ezzat
email: hicham.ezzat@mines-paristech.fr

Pascal Le Masson
email: pascal.le_masson@mines-paristech.fr

Weil Benoit
email: benoit.weil@mines-paristech.fr

Leading in the Unknown with Imperfect Knowledge: Situational Creative Leadership Strategies for Ideation Management

In a fast-changing world, constantly innovating remains one of the principal challenges most organizations are facing nowadays. Survival of organizations became principally linked to the creative generation capacity of their staffs. Nonetheless, fixation imposes a key constraint to the aptitude of individuals to constantly come up with innovative ideas.

Numerous studies have highlighted the significant role that could be played by leadership in this regard. Nevertheless, most of these works studied leadership's role from a social perspective, reducing the function of creative leaders as facilitators. From a more cognitive perspective, very few works have shed the light on the role of creative leaders during ideation processes. However, very recent studies showed that leaders could efficiently play the role of de-fixators, by preparing carefully their interventions (instructions, feedbacks, etc.) within the ideation process, according to their capacity to recognize the frontier between fixation and de-fixation of a project. In the present paper, we have furthered these findings, by exploring the effect of feedbacks, in specific cases in which leaders lead their teams in the unknown with imperfect knowledge. Based on varying levels of knowledge (leaders' ability to recognize if a particular idea generated by his team is inside or outside fixation), we implemented a theoretical model for ideation management using design and probability theories. Using a theory-driven experimental procedure, we showed in this paper that leadership strategies for ideation management should adopt less generic and universal tactics (such as brainstorming rules for example), but rather more situational approaches depending on followers' capacity to think out of the dominant design.

INTRODUCTION

Research context: innovation economy

Creativity has become a fundamental constituent of the so-called "Innovation Economy". Innovation economy, also called creative economy has come to replace knowledge-dependent economy. In fact, the more innovations generated in markets, the more job opportunities created in societies. Therefore, creativity and innovations stand today with no doubt as one of the principal motors that drive the world economy. To illustrate this new phenomena, the world's best selling magazine Business Week has published a special issue entitled "innovation economy" in honor of its 75 th birthday, highlighting in it the achievements of great innovators such as Apple late CEO Steve Jobs or the World Wide Web inventor Tim Berners-Lee, and presenting in its edition a variety of inspiring strategic innovation articles such as "Building an Idea Factory".

Within this new economy, it is worth mentioning that innovation is seen as a fundamental philosophy and culture that should be encouraged by governments. Indeed, this new innovation economy came to shift the focus of governments and societies to encourage creativity, to develop novel and useful business models, as well as innovative products and services.

Innovation is a long process comprising several stages, such as idea generation, evaluation, development and implementation (as illustrated in the figure 1

below).

According to Oxford dictionary, innovation means the introduction of something new, such as a new product, service or process into the market place. [START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF] The very early stage of innovation is usually considered to be ideation. Ideation or idea generation is the process of coming up or generating ideas, while creative ideation consists of coming up with creative ideas, i.e. ideas that are both novel and useful [START_REF] Amabile | The social psychology of creativity: A componential conceptualization[END_REF]. In the problem-solving literature, ideation is the initial stage in which individuals are required to come up with as many possible alternatives to a particular problem.

The most famous ideation method is considered brainstorming. Brainstorming is a group creativity technique popularized by Alex Faickney Osborn in his book Applied Imagination [START_REF] Osborn | Applied imagination[END_REF]. It consists of holding a group discussion to produce ideas and solutions to problems. According to the rules of Osborn [START_REF] Osborn | Applied imagination[END_REF], individuals performing brainstorming sessions should: generate as many ideas as possible, withhold criticism of ideas, freewheeling (welcome wild ideas), and combine on ideas of others.

Research problematic: fixation, a major cognitive bias against creativity

However, the process of idea generation is not an easy task at all. In fact, despite the favourable conditions and necessary resources available in organizations to innovate, most firms have been facing serious difficulties precisely in the early innovation process, when their employees are required to ideate, and more specifically to generate creative ideas and solutions to problems.

One of the origins of these blocs against innovation is well known in the domain of cognitive psychology. Indeed, numerous studies have highlighted the negative role of cognitive biases against creativity, which occurs during the early stages of the innovation process, namely ideation. One of the most common and major obstacle against creative ideation is recognized as the fixation effect, also called functional fixedness [START_REF] Jansson | Design fixation[END_REF].

Fixation has always been a central topic of study in specific domains, such as design sciences or psychology [START_REF] Jansson | Design fixation[END_REF]). Fixation's definition differs lightly according to different domains' perspectives. From a psychological point of view, it is a "mental block against using an object in a new way that is required to solve a problem" [START_REF] Duncker | On problem-solving[END_REF]. In creativity, it is frequently described as the fact that some existing knowledge and obvious solutions are spontaneously activated in individuals' mind during a creativity context, consequently constraining their ability to generate more creative ideas and solutions [START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF][START_REF] Cassotti | Fixation effect in creative ideas generation: Opposite impacts of example in children and adults[END_REF].

One of the first tasks (but not the most famous one) that illustrated fixation is called the "two cord problem" [START_REF] Maier | Reasoning and learning[END_REF]. Participants were given two cords tied to the ceiling, as well as a pair of pliers. Individuals were asked to tie together the free ends of these two cords, knowing that the cords are short and cannot be held in the hands at the same time so that one can easily tie them together (see figure 2). One solution of this problem was simply to tie the pliers to one of the cord to form a pendulum that will swing to reach the second one. In this "two cord problem" experiment, most participants were "fixated" to their existing knowledge of pliers and their conventional uses in their mind, and were unable to look at it alternatively as a pendulum to solve the problem. [START_REF] Maier | Reasoning and learning[END_REF] Due to its growing importance, overcoming fixation became a key topic of study for several researchers from different disciplines such as design, engineering, cognitive science, and management. Among all these above-mentioned disciplines, it is worth mentioning that leadership has a central role to play. In fact, in all these different disciplines, a leader is implicitly or explicitly present in the ideation process, and plays a direct or indirect role to help individuals (designers, engineers, experts, etc…) bypass these cognitive obstacles against creativity. For these reasons, overcoming fixation (also called the process of defixation) becomes with no doubt a principal challenge for the field of leadership.

LITERATURE REVIEW: LEADING FOR DEFIXATION

What is Creative Leadership?

To better understand the role of creative leaders to help followers bypass the principal cognitive biases against creativity (and more specifically fixation), we must go through a brief review of what is leadership, who are creative leaders, and how do they drive their teams for innovation.

Leadership is with no doubt a very ambiguous concept that has as many definitions as who have tried to define it, according to one of the most prominent leadership scholars Richard Stodgill [START_REF] Bass | Stogdill's handbook of leadership: A survey of theory and research[END_REF]. Most people know what leadership is, and are capable to recognize the characteristics of a leader, but would be hardly able to define it. It is generally described as "a process of social influence in which an individual (the leader) is able to enlist the support of others (subordinates or followers) in the accomplishment of a common task" [START_REF] Chemers | An integrative theory of leadership[END_REF]. In other words, a leader is someone who can influence his/her team to achieve certain organizational goals, and at the same time has some kind of managerial authorities on them.

Studies on leadership have produced numerous theories all sorted according to a varying number of leadership study eras [START_REF] Bolden | A review of leadership theory and competency frameworks: Centre for Leadership Studies[END_REF][START_REF] Brungardt | The making of leaders: A review of the research in leadership development and education[END_REF][START_REF] Daft | The leadership experience: Cengage Learning[END_REF][START_REF] Van Seters | The evolution of leadership theory[END_REF][START_REF] Yukl | Leadership in organizations: Pearson Education India[END_REF]. Among these multiple eras, we could today identify most important leadership study eras as: personality and traits, power and influence leadership, behavioral, situational, contingency, transactional, and transformational leadership eras.

However, in most of these leadership study eras, leadership scholars seem to have directly integrated creativity in the broader and larger concept of efficiency. But recently, being a subject of interest among scholars from different disciplines ranging from neurosciences to management, and as a direct consequence of its growing importance, leadership theorists and academics have finally began to add the variable 'creativity' as an imperative constituent of leadership.

Creative leadership could be considered as a certain form of mixture between creativity (defined as the ability to generate ideas that are both novel and useful according to its definition) and leadership (defined as the ability to influence others to accomplish this generation process). In other words, we could say that a creative leader is someone who leads others toward the attainment of a creative outcome.

Creative leaders generally respond to challenges differently than traditional leaders. They typically avoid incremental improvements of products or services, and rather favor the implementation of radical and disruptive innovation. Creative leaders are frequently associated with the figures of great innovators such as Thomas Edison, Steve Jobs, Mark Zuckerberg, Elon Musk, Jeff Bezos, Larry Page, Bill Gates, etc..

What are the principal roles of creative leaders?

Literature review on creative leadership has underlined the role played by leaders for creativity [START_REF] Mumford | Leading creative people: Orchestrating expertise and relationships[END_REF]. Prior works have majorly reduced and concentrated leaders' role as facilitators, mentors, or mediators to organizational creativity [START_REF] Woodman | Toward a theory of organizational creativity[END_REF]. Indeed, numerous studies have examined varied factors that can either foster or hinder employees' creativity at individual, group, and organizational levels, and have then introduced the role of creative leaders in this regard. To foster followers' creativity, leaders could for instance create creativity-supportive climates for their teams, impact their extrinsic motivation (transactional leadership) [START_REF] Jung | Transformational and transactional leadership and their effects on creativity in groups[END_REF][START_REF] Wofford | A cognitive interpretation of transactional and transformational leadership theories[END_REF], or even intrinsic motivation (transformational leadership) [START_REF] Bass | From transactional to transformational leadership: Learning to share the vision[END_REF] Literature review on creative leadership has shown that leaders can enhance subordinates' creativity by directly or indirectly adapting appropriate contextual factors like group climate, group composition, resources, knowledge management, or even human resources issues [START_REF] Hemlin | Creative knowledge environments[END_REF][START_REF] Shalley | What leaders need to know: A review of social and contextual factors that can foster or hinder creativity[END_REF][START_REF] Zhou | Awakening employee creativity: The role of leader emotional intelligence[END_REF].

These various contextual factor could be managed and operated using certain types of leadership tools, such as: goal-settings [START_REF] Carson | Managing Creativity Enhancement Through Goal-Setting and Feedback*[END_REF][START_REF] Locke | A theory of goal setting & task performance[END_REF], instructions [START_REF] Ezzat | Solution-oriented versus Novelty-oriented Leadership Instructions: Cognitive Effect on Creative Ideation[END_REF][START_REF] Paulus | Effects of quantity and quality instructions on brainstorming[END_REF][START_REF] Runco | Creativity, originality, and appropriateness: What do explicit instructions tell us about their relationships?[END_REF][START_REF] Runco | Explicit Instructions to Be Creative and Original[END_REF][START_REF] Runco | The instructional enhancement of the flexibility and originality scores of divergent thinking tests[END_REF], feedbacks [START_REF] De Stobbeleir | Feedback-Seeking Behavior as a Self-Regulation Strategy for Creative Performance[END_REF][START_REF] Ezzat | Leadership-driven Ideation: The Cognitive Effects of Directive Feedbacks on Creativity[END_REF][START_REF] Zhou | Feedback valence, feedback style, task autonomy, and achievement orientation: Interactive effects on creative performance[END_REF][START_REF] Zhou | When the presence of creative coworkers is related to creativity: role of supervisor close monitoring, developmental feedback, and creative personality[END_REF][START_REF] Zhou | Can supervisor feedback always promote creativity? The moderating role of employee self-monitoring[END_REF], and even unusual leadership tools like non-verbal devices [START_REF] Brun | Managing the impacts of non-verbal devices on idea generation: a new challenge for creative leaders[END_REF][START_REF] Cardoso | The influence of different pictorial representations during idea generation[END_REF][START_REF] Chrysikou | Following the wrong footsteps: fixation effects of pictorial examples in a design problem-solving task[END_REF].

Do creative leaders play "defixator" roles for their teams?

When we look at leadership literature, we could note that numeral researches have paid attention to the role of leadership for creativity in general, without particularly paying attention to the cognitive biases against creativity. As an indication of this fact, we could easily find researches linking leadership to creativity in famous leadership journals such as the leadership quarterly [START_REF] Lowe | Ten years of the leadership quarterly: Contributions and challenges for the future[END_REF], but no studies relating leadership to fixation.

Moreover, todays' existing figures of creative leadership seem at first sight not to be appropriate enough to help their teams overcome the critical cognitive biases they face during creative ideation. In other words, current standard models of creative leadership do not play an explicit and clear role of "defixator" for their teams, most probably because the cognitive dimension behind these creative leadership styles is not well highlighted, or seems to be hidden among multiple traditional variables of the creative leadership equation (such as motivation for instance).

Nevertheless, there is clear evidence in the literature review of leadership that the existing forms of creative leadership incorporate certain cognitive dimensions that seem to play an indirect role of defixator since they are able to stimulate the creative performance of followers (the element 'intellectual stimulation' of transformational leadership is the perfect example). However, these underlying defixation cognitive processes are not well emphasized, and not clearly demonstrated in any study to the best of our knowledge.

What can creative leaders learn from defixation methods?

When we look at the literature review related to defixation processes, we clearly see that developing defixation tools, methods and managerial procedures has always been a priority for researchers in diverse areas. Over the past decades, various methods from diverse disciplines (ranging from psychology to design science) have been proposed to help individuals overcome fixation, in order to generate creative ideas [START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF][START_REF] Agogue | Resisting Classical solutions: the creative mind of engineers and designers[END_REF][START_REF] Hatchuel | A new approach of innovative Design: an introduction to CK theory[END_REF][START_REF] Kowaltowski | Methods that may stimulate creativity and their use in architectural design education[END_REF].

For instance, recent findings demonstrated that overcoming fixation could be made through expansive examples, i.e. examples of ideas and solutions that are outside the fixation effect, and consequently helps increase the creative generation capacities of individuals [START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF]. Other studies showed that we could overcome fixation by reframing problems [START_REF] Linsey | A study of design fixation, its mitigation and perception in engineering design faculty[END_REF][START_REF] Zahner | A fix for fixation? Rerepresenting and Abstracting as Creative Processes in the Design of Information Systems[END_REF], providing analogies [START_REF] Moreno | A step beyond to overcome design fixation: a design-by-analogy approach[END_REF][START_REF] Smith | A three-pronged approach for overcoming design fixation[END_REF][START_REF] Tseng | Overcoming blocks in conceptual design: the effects of open goals and analogical similarity on idea generation[END_REF], given de-fixation instructions [START_REF] Ezzat | Solution-oriented versus Novelty-oriented Leadership Instructions: Cognitive Effect on Creative Ideation[END_REF], or even enabling incubation [START_REF] Smith | Incubation and the persistence of fixation in problem solving[END_REF].

In the field of design sciences, several researchers have identified creativity stimulation techniques to overcome cognitive biases against creativity [START_REF] Kowaltowski | Methods that may stimulate creativity and their use in architectural design education[END_REF]. For instance innovative design methods like the Concept-Knowledge theory [START_REF] Hatchuel | A new approach of innovative Design: an introduction to CK theory[END_REF], and KCP method [START_REF] Hatchuel | Design theory and collective creativity: a theoretical framework to evaluate KCP process[END_REF] are well recognized to help individuals and teams not only identify, but also bypass fixation in contexts of innovative design.

At first sight, and by going through all these diverse creativity stimulation practices, we could inaccurately consider that they do not require any type of leadership, and that these "leaderless" defixation methodologies seem to be simply reduced to simple instructional processes. Moreover, this would be an excellent indication for management and creativity theorists claiming that creativity does not necessarily require "leadership" control [START_REF] Amabile | How to kill creativity[END_REF]. In other words, it would be perfectly consistent with the idea that creativity flourishes from certain levels of freedom and autonomy.

However, by looking much deeper into these defixation procedures; we could model a very particular type of leader (with very specific managerial actions and competences) hidden behind these instructional means. For instance, if we consider the role of expansive examples as a defixation methodology [START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF], we could model figures of leader that have the competence not only to identify fixation, but also to provide their teams with examples of solutions outside fixation, as well as a certain level of legitimacy to impose the given expansive example on their teams. Therefore, it is worth mentioning that although most defixation methodologies seem at first glance to be leaderless, they implicitly hide an important and inevitable role of leadership with specific means of actions, competences and strategies.

HOW CAN CREATIVE LEADERS

Research method: A theory-driven experimental procedure

To answer this research question, we adopted an atypical research method in management science called theory-driven experiments, which consists of building a theoretical model, simulating it, and then test it using laboratory experiments (see figure 4 below). This method is usually applied to design experimental procedures in cognitive psychology. Theory-driven experiments' methodology relies on random assignment and laboratory control to ensure the most valid and reliable results. We chose this particular research method among others because answering research questions required having highly controlled settings, in order to neutralize most of the typical variables of creative leadership literature that could interfere with the defixation process (such as leaders motivating followers or creating creativity-supportive environment, leaders giving creative ideas to followers, leaders providing creativityrelevant resources to followers, leaders recruiting creative members, etc..). Moreover, it enables having accurate results given the fact that studying creative ideation requires some theoretical framework to measure creativity.

Given the above, we first built a theoretical model of "leadership-driven ideation", based on both design and probability theories. We then implemented an experimental protocol, listed the theoretical predictions of our model before testing it experimentally.

Finally, we used some case-control study of historical cases to illustrate the managerial implications of our experimental results.

A LEADERSHIP-DRIVEN IDEATION MODEL

Modeling ideation using Concept-Knowledge theory: fixation versus expansion

Modeling the process of ideation is important to facilitate the understanding of the cognitive processes behind creativity. Moreover, modeling ideation is highly useful for leadership to better understand how defixator leaders could manage these cognitive processes, and more precisely how they could help followers to overcome cognitive traps occurring during these processes (such as fixation).

To model ideation, we chose to use the C-K theory [START_REF] Hatchuel | A new approach of innovative Design: an introduction to CK theory[END_REF], because it enables both the precise identification of fixation within creative design processes, as well as the managerial operations and mechanisms that could enable defixator leaders to overcome it. C-K theory is well renowned as a tool to distinguish between a fixation path that is based on the spontaneous activation of existing knowledge (inside fixation), and an expansive path that is based on the activation of less accessible knowledge (outside fixation).

Using C-K theory, we first modeled the generation of an idea as a two-stage cognitive process involving two operations: CàK and KàC:

• Stage 1: Conjunction (CàK)
Beginning from an initial concept C0 in C-Space, the ideator activates in his K-Space a specific knowledge basis.

• Stage 2: Disjunction (KàC)

The ideator then generates an idea (lowest branch of the concept tree), associated with the activated knowledge in stage 1. From a creative ideation perspective, fixation is described as the fact that some existing knowledge and obvious solution are spontaneously activated in individuals' mind (inside the fixation path), constraining individuals to generate creative ideas (outside fixation, i.e. inside the expansion path). As illustrated in the figure 5 above, we could then say that an idea in fixation is generated using spontaneously activated knowledge (knowledge basis inside the fixation path). This knowledge basis (F for fixation) is easier to retrieve from memory, and is effortless. On the contrary, an idea in expansion is generated using non-spontaneously activated knowledge basis (knowledge basis in expansion). This knowledge basis (E for expansion) is harder to retrieve from memory, and demands much more efforts from individuals.

Modeling "sequential" ideation using Bayesian theories: a Markov model However, simply modeling ideation as a two-stage cognitive process wouldn't be accurate enough, since it doesn't take into consideration the dependency between the current idea and the later idea. In fact, ideation could be considered a stochastic process if we assume that the generation of an idea could be influenced in certain ways by the previous generated idea.

To better understand this fact, Wang gave the problem-solving example of: "how to improve university parking" [START_REF] Wang | Modeling idea generation sequences using Hidden Markov Models[END_REF]. The author explains that when individuals generate ideas to this creative ideation task, the idea of "encouraging students to walk" is more likely to occur with the idea of "riding bicycles" than the idea of "increasing the parking fee".

For these reasons, modeling the process of sequential ideation (sequence or chain of ideas) would be much more precise since it would clarify the dependency and link between successive ideas. Moreover, by modeling sequential ideation, we could enable defixator leaders to intervene during the creative ideation process itself, to stimulate followers and deviate their ideation paths towards certain directions (either in the fixation path, or in the expansion path) depending on followers' performance.

To do that, we modeled sequence of ideas as a two-state Markov chain. Markov chains are stochastic processes (named after its founder Andrey Markov) that satisfy the Markov property [START_REF] Markov | Extension of the limit theorems of probability theory to a sum of variables connected in a chain[END_REF]. Frequently described as memory-less systems, a process satisfies the Markov property if predictions of the future states depend uniquely on its present state. In other words, in a Markov process, what happens next depends only on the current state of the system.

Consider the example of figure 6. An ideator generates a sequence of five ideas during an ideation task. Each generated idea is either in fixation (F) or in expansion (E).

The sequence of ideas the ideator generated is: FFEEF. The Markov chain presents two states: fixation (F) and expansion (E), with four transitional probabilities:

• The probability of maintaining the state of fixation (α): it refers to the probability that ideators generate an idea in fixation after having generated an idea in fixation.

This probability refers to the incapacity of ideators to defixate. In figure 6, it is illustrated by the transition (in red) from the first idea to the second one.

• The probability of de-fixation (1-α): it refers to the probability that ideators generate an idea in expansion after having generated an idea in fixation. In other words, it is the probability that ideators defixate. In the figure below, it is illustrated by the transition (in green) from the second idea to the third one.

• The probability of maintaining the state expansion (β): it refers to the probability that ideators generate an idea in expansion after having generated an idea in expansion. It is the probability that ideators maintain the flow of creative ideas. In the figure below, it is illustrated by the transition (in orange) from the third idea to the fourth one.

• The probability of re-fixation (1-β): it refers to the probability that ideators generate an idea in fixation after having generated an idea in expansion. In other words, it is the probability that ideators become re-fixated again. In the figure below, it is illustrated by the transition (in purple) from the fourth idea to the last one. Using the sequence of ideas, we could then estimate a transition matrix (I), which gives us a good indication of the creative performance of an ideator. A creative ideator should have a low probability of maintaining fixation (α) and a high probability of maintaining expansion (β). However, this transition matrix (I) parameters' estimation is significant only if the ideator exhibits a high ideational fluency (which means come up with many ideas). It is worth mentioning that if the ideator generates few ideas, the Markov model could be inaccurate to determine his/her creative performance.

The four types of Ideators

Based on the Markov model presented above, we could say that the vector composed of the fixation rate (F) and the expansion rate (E) at sequence t+1 (of the later state), corresponds to the multiplication of the transition matrix (I) by the vector composed of the fixation rate (F) and the expansion rate (E) at sequence t (of the current state)

F t+1 E t+1 ( ) = I × F t E t ( ) F t+1 E t+1 ( ) = α 1-α 1-β β # $ % % & ' ( ( × F t E t ( )
Based on the above, we then obtain the following equations:

F t+1 = α × F t + (1-β) × E t E t+1 = (1-α) × F t + β × E t F t+1 + E t+1 = 1
At steady state, we could then deduce this interesting relationship between refixation (1-β) and defixation (1-α) as follows:

(

1-β) = 1-E E (1-α)
This formula leads us to plot the relationship between re-fixation (1-β) and defixation (1-α) as shown in the figure 7 below: From the figure above, we could notice the following:

• Good ideator creative performance means having high score of E (high expansion rate), and means moving clockwise.

• Different couples of α and β could give the same creative performance.

However, depending on ideator's fluency, we could have fast (large sequence of ideas) or slow (short sequence of ideas) ideation performance.

• There are four different types of ideators' creative performance o Ideator type A: this ideator is low at defixation, and high at re-fixation.

o Ideator type B: this ideator is high at defixation, and low at re-fixation.

o Ideator type C: this ideator is low at defixation, and low at re-fixation.

o Ideator type D: this ideator is high at defixation, and high at re-fixation.

Integrating leadership in the ideation process

Now, the intuition behind our model is that an external factor (the leader "L") could play a pivotal role in "triggering" some of followers' (ideator "I") generated ideas at certain instants (see figure 8). This could then force followers' ideation paths to take unexpected tracks they wouldn't have take if they weren't confronted to this external factor. A "defixator" leader should help followers to decrease the probability of maintaining fixation (α), i.e. help to defixate his/her followers, and to increase the probability of maintaining expansion (β), i.e. help his/her team to persist in generating creative ideas. mentioning that in previous cognitive psychology studies, this triggering phenomenon was called attention. In the work of Janis [START_REF] Janis | Victims of groupthink: a psychological study of foreign-policy decisions and fiascoes[END_REF] for instance, triggering referred to paying attention of individuals to a range of category of ideas in group ideation interactions [START_REF] Brown | Modeling cognitive interactions during group brainstorming[END_REF].

It is important to note that depending on the type of ideator (type A, B, C or D), leaders could have different contingent strategies to defixate and maintain defixation. For instance if ideators are very good at defixation (thinking outside the dominant design), they would need a leader to help them maintain their creative performance (persisting in the generation of creative ideas). Otherwise, if the ideators are poor at defixation, leaders should mainly concentrate on managing the process of defixation (deviating their ideation paths outside the dominant design).

FEEDBACKS: A TOOL FOR LEADERSHIP-DRIVEN IDEATION

State of the art of the role of feedbacks for creativity

We have seen that leaders could manipulate the contextual factors for creativity through various tools, mostly based on instructional processes. In the present study, we chose leadership feedbacks as a tool to manage followers' ideation. We chose feedbacks, since it is the convenient tool to manage followers' ideation paths by paying attention to (or triggering) certain ideas during the process of ideation.

Few studies have been devoted to the relationship between feedback and creativity. Most researchers have examined feedback from a very broad perspective.

These researchers have investigated the influence of evaluative information on creative performance and argued that it could have a strong influence on enhancing creative processes [START_REF] De Stobbeleir | Self-regulation of creativity at work: The role of feedback-seeking behavior in creative performance[END_REF]. Indeed, these studies have underscored the importance of being exposed to others' ideas and perspectives in the stimulation of the generation of creative ideas.

Other studies have noted that feedback could considerably help to control individuals' creative performances [START_REF] Zhou | Can supervisor feedback always promote creativity? The moderating role of employee self-monitoring[END_REF]. Moreover, other findings have argued that delivering negative and controlling feedback to individuals can damage their creative performance, and in contrast, the delivery of constructive or developmental feedback can exert a positive influence on creativity [START_REF] Carson | Managing Creativity Enhancement Through Goal-Setting and Feedback*[END_REF][START_REF] De Stobbeleir | Feedback-Seeking Behavior as a Self-Regulation Strategy for Creative Performance[END_REF][START_REF] Zhou | Feedback valence, feedback style, task autonomy, and achievement orientation: Interactive effects on creative performance[END_REF][START_REF] Zhou | When the presence of creative coworkers is related to creativity: role of supervisor close monitoring, developmental feedback, and creative personality[END_REF][START_REF] Zhou | Can supervisor feedback always promote creativity? The moderating role of employee self-monitoring[END_REF].

Recent studies by Ezzat et al. analyzed the impact of directive feedbacks during an interactive creative ideation task, and demonstrated that leaders could drive followers' ideation paths in distinct directions according to their expertise [START_REF] Ezzat | Leadership-driven Ideation: The Cognitive Effects of Directive Feedbacks on Creativity[END_REF]. They showed that the process of de-fixation is in itself important but not enough, and that creative leaders must equally pay attention to strategies to de-fixate their teams, as well as to maintain their ideation in a state of de-fixation.

In this paper, we have furthered these previous findings, by exploring the effect of feedbacks, in specific cases in which leaders lead followers in the unknown with imperfect knowledge. We have implemented an experimental protocol, in which leaders had varying levels of knowledge (leaders' ability to recognize if a particular idea generated by his team is inside or outside fixation), and interacted with their teams to defixate them.

Modeling three types of leaders' feedbacks

We then modeled three types of leaders' feedbacks:

• Neutral Feedback: "I confirm receipt of your idea, and await for the next one".

This feedback was considered to be neutral and to have no impact on ideators' sequence of ideas.

• Positive Feedback: "continue in this path".

This feedback was considered to encourage ideator to generate ideas that would be similar to a generated idea in expansion. According to C-K theory, this positive feedback should force ideators to activate the knowledge basis associated with the idea in expansion (see figure 9 below).

• Negative Feedback: "search for another path":

This feedback was considered to force ideator to search for ideas that would be dissimilar from a generated idea in fixation. This negative feedback should help individuals to inhibit the knowledge basis associated with the idea in fixation (as shown in the figure below). • State F-: Leaders give negative feedback given the idea is in fixation. In this case, the leader has the capacity to perfectly recognize that follower's idea is in fixation (γ), and consequently gives follower a negative feedback. This negative feedback will help decreasing the probability of maintaining the state of fixation (α decreases to α').

• State F+: Leaders give positive feedback given the idea is in fixation. In this case, the leader fail to recognize that follower's idea is in fixation (1γ), and consequently gives follower a positive feedback. This positive feedback will help increasing the probability of maintaining the state of fixation (α increases to α'').

• State E+: Leaders give positive feedback given the idea is in expansion. In this case, the leader has the capacity to perfectly recognize that follower's idea is in expansion (σ), and consequently gives follower a positive feedback. This positive feedback will help increasing the probability of maintaining the state of expansion (β increases to β'').

• State E-: Leaders give negative feedback given the idea is in expansion. In this case, the leader fail to recognize that follower's idea is in expansion

(1-σ), and consequently gives follower a negative feedback. This negative feedback will help decreasing the probability of maintaining the state of expansion (β decreases to β').

α * = γα '+ (1-γ )α '' β * = σβ ''+ (1-σ )β '
If leaders have a high capacity to detect fixation (high probability γ), the new α* after the intervention of leaders tend to decrease the probability of maintaining the state of fixation (α transforms to α'). On the contrary, if leaders have a low capacity to detect fixation (low probability γ), the new α* after the intervention of leaders tend to increase the probability of maintaining the state of fixation (α transforms to α'').

Similarly, if leaders have a high capacity to detect expansion (high probability σ), the new β* after the intervention of leaders tend to increase the probability of maintaining the state of expansion (β transforms to β''). On the contrary, if leaders have a low capacity to detect expansion (low probability σ), the new β* after the intervention of leaders tend to decrease the probability of maintaining the state of expansion (β transforms to β').

These transformation of parameters α to α*, and β to β* depending on leaders' knowledge, leads us now to an interesting relationship between the new probabilities of re-fixation (1-β*) and defixation (1-α*) as follows:

(1-

β * ) = 1-E + E + (1-α * )
Then, according to these equations we could plot the relationship between the new probabilities of re-fixation (in y-axis) and defixation (in x-axis) as shown in figure 11 below. From the figure above, we could notice the following:

• A creative leader should manage followers to force defixation (pushing right-side)

and not to re-fixation (maintain expansion by pushing downside).

• We could note interesting situational strategies to manage leadership-driven ideation:

o If leaders have perfect knowledge (high γ, and high σ), they would be able to push followers towards defixating them and maintaining expansion (see red arrow in the figure). We named this leader "expert".

o If leaders have imperfect knowledge (low γ, and low σ), they won't be able to push followers towards defixating them, and they won't be able as well to maintain expansion if followers are defixated (see orange arrow in the figure). We named this leader "inexpert".

o If leaders have perfect knowledge of expansion (low γ, and high σ), they won't be able to push followers towards defixating them, but they would be able to maintain expansion if followers are defixated (see pink arrow in the figure). We named this leader "expert in expansion".

o If leaders have perfect knowledge of fixation (high γ, and low σ), they would be able to push followers towards defixating them, but won't necessarily help them to maintain expansion if followers are defixated (see blue arrow in the figure). We named this leader "expert in fixation".

Theoretical predictions: a situational case

According to the Markov model above, we then listed three hypotheses regarding leadership-driven ideation with various levels of knowledge as follows:

• Hypothesis 1: Ideators leaded by "expert" leaders will have the highest expansion rate.

• Hypothesis 2: Ideators leaded by "inexpert" leaders will have the lowest expansion rate.

• Hypothesis 3: Ideators leaded by "expert in fixation" leaders will have a higher expansion rate than ideators leaded by "expert in expansion" leaders, if the group is more or less fixated.

To verify and test these theoretical predictions, we constructed an experimental protocol in which leaders (the experimenters) intervene during the ideation process. The experimenters directed the ideators (the participants of the experiment) through feedbacks with varying levels of knowledge (capacity to recognize fixation and expansion).

A THEORY-DRIVEN EXPERIMENT: LEADING FOR DEFIXATION USING

FEEDBACKS

A creativity task: the hen's egg task

We chose to perform our experiment using a creative ideation task called the hen's egg task. This creativity task consist of proposing the maximum number of original solutions to ensure that a hen's egg dropped from a distance of ten meters does not break.

We chose this particular creativity task since we have an existing database of ideas and solutions of more than thousands subjects that have performed this task within the past years. This database reveals that more than 80% of participants generate ideas around three main categories of obvious "restrictive" solutions (which are damping the shock, slowing the fall, and protecting the egg). However, only 20% of participants usually generate non-obvious "expansive" solutions (for instance: before and after the fall, with a living device, using the intrinsic properties of the environment, etc..). Using CK theory, we were able to differentiate between restrictive paths of ideas and solutions (dark partitions) and expansive ones (light partitions) as shown in figure 12 above. By doing this, two distinct sub-spaces could be identified: fixation and expansion zones [START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF].

Participants of the experiment

Participants (N=100) of the Faculty of Psychology of Paris Descartes University have participated in this study. Subjects were between 19 and 29 years old, with a mean age of 22. 68% of the participants were females. All subjects recruited to perform this task haven't previously heard about the hen's egg task.

Experimental procedure

Participants were recruited from the faculty of psychology and were asked to perform a creative task via an online text conversation with their experimenter. The participant and the experimenter were in two different rooms (see figure 13 below). We chose to perform the complete task online via a written chat conversation on Skype to avoid any type of social biases that could appear and affect our experimentation, as a give positive feedback, while the other half the leader would fail to recognize it and give negative feedback.

Experimental Results

We computed the mean expansion rate for each group. The expansion rate was calculated for each participant, by dividing the number of ideas in expansion by the total number of ideas generated (sum of the ideas in fixation and the ideas in expansion). The expansion rate gives us an excellent indication of participants' creative performance for this creativity task. We compared each group's expansion rate with the control group as shown in the figure 14 below. Moreover, we can see that results confirmed the hypothesis 1. In fact, ideators confronted to "expert" leaders obtained the highest expansion rate (in red color). Expert leaders were able to identify fixation and deviate their teams from it, as well as identify expansion and maintain the creative ideation flow of their teams. They increased the expansion rate to 46%. Nonetheless, hypothesis 2 was not fully confirmed. In fact, ideators confronted to "inexpert" leaders obtained a low expansion rate (in orange color) compared to the control group, since the expansion rate was decreased to 27%. However, they performed slightly better than the group confronted to the expert in expansion leaders. Indeed, inexpert leaders were unable to identify fixation and expansion, and consequently disturbed the ideation process of their teams.

Finally, results confirmed the hypothesis 3. Since the sample was more or less fixated, ideators leaded by "expert in fixation" leaders obtained a higher expansion rate (in pink color) than ideators leaded by "expert in expansion" leaders (in light blue color).

Indeed, ideators leaded by "expert in fixation" leaders highly increased the expansion rate to 45% (which is quite similar to the expansion rate obtained for the ideators leaded by expert leaders), while ideators leaded by "expert in expansion" leaders highly decreased the expansion rate to 23% (which was the lowest expansion rate in all groups).

CONCLUSION AND MANAGERIAL IMPLICATIONS

In the present study, we demonstrated that leaders could efficiently play the role of de-fixators, by preparing carefully their interventions (through feedbacks) within the ideation process, according to their capacity to recognize the frontier between fixation and de-fixation of a project.

We have explored the effect of feedbacks, in specific cases in which leaders lead their teams in the unknown with imperfect knowledge. We designated varying levels of knowledge by leaders' aptitude to recognize if a particular idea generated by his team is inside or outside fixation.

In this study, we implemented a theoretical model for leadership-driven ideation management using design (C-K theory) and probability theories (Markov chains). Using a theory-driven experimental procedure, in a creativity task where the aim was to propose the maximum number of original solutions to ensure that a hen's egg dropped from a distance of ten meters does not break, we showed that leadership strategies for ideation management should adopt more situational and contingent approaches depending on followers' capacity to think out of the dominant design. The major managerial implications in the present study could be summarized as follows:

• Leaders can in certain specific cases obstruct followers' creative ideation capacities if they are not capable of recognizing the frontier between fixation (the dominant design of a project) and expansion (outside the dominant design).

These findings clearly confirm the point of view in management science arguing that leaders shouldn't control followers and should rather grant them more freedom and autonomy for creativity to emerge [START_REF] Amabile | How to kill creativity[END_REF].

For these reasons, it is worth mentioning that leadership and creativity scholars must pay more attention to the importance of leaders' idea evaluation, specifically based on fixation recognition.

• However, leaders could as well highly improve followers' creative ideation capacities, if they are capable of recognizing the frontier nearby the dominant design of a project.

These findings show that leadership control does not always kill creativity, provided that leaders have good knowledge of the project, and a good vision of creativity.

For these reasons, it is important to note that leadership's role should be explicitly incorporated in followers' creativity process in general, and during the ideation processes more specifically. It highlights the implicit cognitive dimension behind creative leadership (such as overcoming the cognitive biases against creativity), rather than the social perspective of creative leadership (such as motivation). It enables us to slightly redefine leadership for creativity as a process of "cognitive influence" rather than "social influence", toward the attainment of a creative outcome.

• Moreover, the present findings demonstrated that less knowledgeable leaders (having the ability to recognize either fixation or expansion) could also highly increase followers' creative ideation capacities, if they are confronted to appropriate teams.

In fact, our leadership-driven ideation model presents a situational approach to ideation management as shown in figure 15 above. In the present paper, we demonstrated that if teams are more or less fixated, leaders who could recognize the dominant design of the project should preferably lead them.

These findings could interestingly link to situational leadership theories [START_REF] Hersey | Situational leadership, perception, and the impact of power[END_REF], and give new sights to managerial implications towards a situational approach to creative leadership.

Finally, future studies will consist of linking these experimental findings (more specifically the situational aspects of the results) with well-documented historical cases of great innovative leaders (such as Gustave Eiffel, Steve Jobs, Thomas Edison, etc..), using a case-control study method.

Fig. 1 :

 1 Fig.1: Identifying Fixation/Expansion in the Egg's Task[START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF]

Fig. 2 :

 2 Fig. 2: Two Cord Problem[START_REF] Maier | Reasoning and learning[END_REF]

  DEFIXATE FOLLOWERS? Research question: leadership versus fixation Therefore, the research question becomes more precise and well defined now: How can leadership play an explicit role of "defixator" for their teams? In other words, how can creative leaders lead their teams towards overcoming fixation effects occurring during creative ideation contexts? The research question could be illustrated as shown in the figure 3 below.

Fig. 3 :

 3 Fig. 3: An Illustration of the Research Context, Problematic and Question

Fig. 4 :

 4 Fig. 4: Theory-driven Experiments Research Method

Fig. 5 :

 5 Fig. 5: Idea Generation in Fixation/Expansion using Concept-Knowledge Theory

Fig. 6 :

 6 Fig. 6: Markov chains of Sequential Ideation

Fig. 7 :

 7 Fig. 7: Markov chains of Sequential Ideation

Fig. 8 :

 8 Fig. 8: Leadership-driven Ideation Model

Fig. 9 :

 9 Fig. 9: The Impact of Leader's Positive versus Negative Feedback According to C-K Theory

Fig. 10 :

 10 Fig. 10: Markov Model Representing the Effects of Leaders' Feedbacks depending on their Knowledge

Fig. 11 :

 11 Fig. 11: Re-fixation versus Defixation Ideators' Performance after Leaders' Intervention

Fig. 12 :

 12 Fig. 12: Categories of Ideas in Fixation/Expansion of the Hen's Egg Task[START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF]

Fig. 14 :

 14 Fig. 14: Score of expansion rate for each group

Fig. 15 :

 15 Fig. 15: A Situational Approach to Creative Leadership for Ideation Management

Leaders' feedbacks in case of various level of knowledge

Leaders' knowledge is a crucial factor of the evaluation of an idea. In this paper, we considered leaders' knowledge as the capacity to recognize that followers' ideas are in fixation (uncreative) or in expansion (creative). Leaders could occasionally give their followers wrong feedbacks due to their incapacity to recognize if an idea is in fixation or outside fixation.

People usually categorize or evaluate an idea as creative if it is both useful and novel [START_REF] Amabile | The social psychology of creativity: A componential conceptualization[END_REF]. However, leaders could differ on the judgment of what is creative and what it is not. Prior studies proposed that domain knowledge could determine the recognition of creative ideas, arguing that ideas that could be categorized as creative if it is considered new according to one's domain knowledge [START_REF] Elsbach | Assessing creativity in Hollywood pitch meetings: Evidence for a dual-process model of creativity judgments[END_REF].

Studies have also shown that leaders could simply fail to recognize creative ideas, and consequently reject them [START_REF] Mueller | Recognizing creative leadership: Can creative idea expression negatively relate to perceptions of leadership potential[END_REF]. Sometimes the rejection of creative ideas could be done due to risk or uncertainty. Otherwise, number of errors and cognitive biases could also occur leading to incorrect evaluation of the creativity of an idea [START_REF] Licuanan | Idea evaluation: Error in evaluating highly original ideas[END_REF].

However, in the present paper, we argue that recognizing creative ideas does not necessarily mean recognizing uncreative ideas. Leaders could be very good at recognizing the dominant design of a project, and have the strategy to deviate their teams from the dominant design. Other leaders could be excellent at identifying what is creative, and outside the dominant design, but hesitant vis-à-vis of uncreative ideas.

To integrate this knowledge dimension in our leadership-driven ideation model, we considered two supplementary probabilities:

• Probability γ: it refers to the probability that leaders recognize that an idea is in fixation. If γ tends to one, it means that the leader could perfectly recognize if an idea is in fixation.

• Probability σ: it refers to the probability that leaders recognize that an idea is in expansion. If σ tends to one, it means that the leader could perfectly recognize if an idea is in expansion.

Based on the above, we added four new states in our Markov model as shown in figure 10 below: consequence of the physical presence of experimenters face to face in front of participants [START_REF] Belletier | Choking under monitoring pressure: being watched by the experimenter reduces executive attention[END_REF]. The task duration was set to 10 minutes for each participant. The experimenter (playing the role of leader) initially instructed the participant (playing the role of follower) with the task's objective ("I am your team leader for this task, and the aim is to propose the maximum number of original solutions to ensure that a hen's egg dropped from a distance of ten meters does not break."), specifying to them to "be as creative as possible in this task". Then, at each idea generated by the participant, the experimenter was giving a feedback.

Participants of the first group were chatting with a neutral leader, which simply acknowledges reception of each ideas generated by subordinates, awaiting the next one.

We considered group 1 as a control case.

Participants of the second group were chatting with an expert in the recognition of both ideas in fixation and expansion. If the idea generated by participants was in fixation, the leader would recognize it perfectly and give negative feedback ("search for another path"). Similarly, if the idea generated by participants was in expansion, the leader would also recognize it perfectly and give positive feedback ("continue in this path").

Participants of the third group were chatting with an inexpert in the recognition of both ideas in fixation and expansion. If the idea generated by participants was in fixation, the leader would half of the time recognize it correctly and give negative feedback, while the other half the leader would fail to recognize it and give positive feedback. Similarly, if the idea generated by participants was in expansion, the leader would half of the time recognize it correctly and give positive feedback, while the other half the leader would fail to recognize it and give negative feedback.

Participants of the fourth group were chatting with an expert in the recognition of ideas in expansion. If the idea generated by participants was in fixation, the leader would half of the time recognize it correctly and give negative feedback, and the other half the leader would fail to recognize it and give positive feedback. However, if the idea generated by participants was in expansion, the leader would recognize it perfectly and give positive feedback.

Participants of the fifth group were chatting with an expert in the recognition of ideas in fixation. If the idea generated by participants was in fixation, the leader would recognize it perfectly and give negative feedback. However, if the idea generated by participants was in expansion, the leader would half of the time recognize it correctly and


L D C Jaubert

M J Harris

T Fennell

R G Melko

S T Bramwell

P C W Holdsworth

Topological-Sector Fluctuations and Curie-Law Crossover in Spin Ice

Keywords: numbers: 05, 50, +q 05, 70, Jk 75, 10, Hk 75, 10, Kt 75, 25, -j

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

In spin liquid systems, where an absence of symmetry breaking leads to a disordered phase, system spanning correlations that are hidden from conventional probes can be manifest in topological properties [1]. These extended correlations typically requires measurement of non-local quantities, such as the entanglement entropy [2,3], which may not immediately be accessible by experiment. The quest for spin-charge separation [4] in models for high temperature superconductivity has led to extensive studies of model quantum spin liquids [5,6] and quantum critical phenomena [7,8], where topological order can be identified. However, the interest in topology is not limited to quantum systems, as the constraints inherent to geometrically frustrated magnets or classical dimer systems also lead to extremely rich many body behaviour. "Topological order" may also be defined heuristically in classical spin liquids with an absence of conventional order, by a lack of ergodicity between sectors in configuration space with different topological invariants (see e.g. Ref. [9]). The most familiar invariant is the "winding number" which can be used to define Z 2 or integer topological invariants in a variety of 2D and 3D systems. Interestingly, novel topological invariants unique to three-dimensional have recently been identified [10], which could have broad implications for classical spin liquids in higher dimensions, with certain underlying emergent gauge structures arising due to local constraints [11].

The correlations induced by local constraints have been discussed in systems on the kagome lattice with both discrete [12] and continuous symmetry [13], and for frustrated antiferromagnets with a pyrochlore structure [14][15][16]. In these theoretical models, the physics of the "Coulomb phase", with dipolar correlations showing up as pinch point singularities in reciprocal space [17][18][19], is universally present. Among systems showing Coulomb phase physics [20], spin ice models [21] have proved particularly fruitful testing grounds for the collective behaviour associated with topological constraints as the associated spin ice materials are the only experimental magnetic systems for which sharp pinch points have been observed [18,19,22,23]. An extensive manifold of low energy states can be constructed by ensuring that a local constraint -the ice rule -is obeyed on every tetrahedron of the underlying pyrochlore lattice. The ice rule requires that at low temperature, two spins point into and two out of every tetrahedron, as shown in Fig. 1. This local divergence free condition also generates the long range dipolar correlations in the Coulomb phase, even within the confines of the simplest model with nearest neighbour spin interactions only.

In this paper, we show that whilst a spin ice is topologically constrained, it is not topologically ordered as it is able to fluctuate between topological sectors. We show how the susceptibility can be used as an indicator of these topological sector fluctuations (TSF), and present a detailed comparison of our results with both bulk magnetometry and neutron scattering measurements on Ho 2 Ti 2 O 7 [24]. We compare the experimental data to analytical and numerical expressions of the susceptibility, where the emergence of TSF at low temperature appears as a crossover between two Curie laws at specific wave vectors Q in the structure factor S(Q). This study was motivated in part by the experimental observation of an unusual temperature dependence in the wave vector dependent susceptibility [25] (see Fig. 4), one that is fully explained by the theory presented here.

Topological sectors in spin ice: Let us first characterize a model system, of size L, with periodic boundaries and only afterwards relate its properties to experimental observables. We study the nearest neighbour spin ice model (NNSI) [21] with vector spins of unit length, S i placed on the vertices of a pyrochlore lattice, constrained to lie along the body centred crystal field directions of the tetrahedra: S i = ± d i (see Fig. 1) and with ferromagnetic exchange interaction J > 0. The model maps onto Anderson's Ising antiferromagnet with exchange constant J = -J/3 [26] through the definition of Ising pseudospin variables σ i = S i . d i . However it is the ferromagnetic spin ice model that is physically realisable: the Ising pyrochlore antiferromagnet with a single easy axis cannot represent any real magnetic system as its Hamiltonian breaks the cubic lattice symmetry [21]. The ensemble of ground states satisfying the ice rules of two spins in and two out, or i=1,4 σ i = 0, on each tetrahedron is a Coulomb phase [14,15] and leads to the Pauling zero point entropy [27,28] (see Fig. 1). Here we consider cubic systems with L 3 unit cells and N = 16L 3 spins. Throughout the paper, as we have in mind comparisons with Ho 2 Ti 2 O 7 , we take J = 1.8 K, the value found to best parameterize thermodynamics measurements of this spin ice material [24].

A micro state of the Coulomb phase can be classified by a set of strings of alternating out, in, out ... spins which wind through the system along each of the cubic axes. That is, any spin S i with a downward (upward) projection along a given cubic axis, always has at least one nearest neighbour above and below it with the same downward (upward) projection (see Fig. 1). Connecting these spins draws a map of strings spanning the system from top to bottom in the negative (positive) direction. We define an individual string, which does not necessarily close on itself, as an object spanning the system once along one cubic axis, so that each string is composed of 4L spins. Hence each microstate has n - k (n + k ) strings spanning in the negative (positive) direction along cubic axis k, such that the total number, n 0 = n - k + n + k = 4L 2 equals the number of spins on a plane of the pyrochlore lattice perpendicular to k. Each spin belongs to three strings threading along x, ŷ and ẑ. Two strings threading through a tetrahedron in the same direction k are indistinguishable and can in fact be mapped onto the world lines of bosons living in the d-1 dimensional space perpendicular to the string direction [29][30][31]. One can define a topological sector for each configuration through a winding vector w = (w k = n + k -n - k ) k=x,y,z , whose components are even integers taking values between ±4L 2 .

Excitations within the Coulomb phase are non-local and limited to the flipping of a closed loop of spins identified in two categories (see Fig. 2): firstly a non-winding loop of spins which closes within the system. This kind of excitation moves the system between microstates of a given topological sector, re-arranging the string network without changing the winding number. Secondly, a winding loop which closes on itself after passing one or more times through the periodic boundaries. Each passage flips a string of spins and changes the topological sector through a change in one component of the winding vector by two. We define a topologically ordered system as one that is restricted to a single topological sector.

In model systems the Coulomb phase space can be sampled directly in simulations by using a non-local worm algorithm [69]. The worm is a virtual sequence of spins that burrows through the system until it closes on itself becoming a loop excitation that maintains the ice rules. In real materials, deconfined topological defects provide local dynamics [32] and their creation and annihilation allows for the sampling of different constrained states. However, a finite concentration of such defects destroys the Coulomb phase above length scales fixed by their mean separation and the balance between maintaining ergodicity and imposing the constraints is a fine one.

The winding vector is a direct measure of the difference between upward and downward projections along each cubic axis, making it proportional to the magnetization: M = (4L/ √ 3) w. Magnetic fluctuations of a system in the Coulomb phase are therefore a direct measure of topological sector fluctuations: non-winding loop excitations carry no magnetic moment, while winding ones carry a magnetic fingerprint of the change in topological sector. The phase space of constrained Pauling states making up the Coulomb phase is therefore compatible with an ensemble of gauge invariant topological sectors with U (1) symmetry. Each sector is associated with an extensive subset of states with constant magnetization connected by non-winding loop excitations, while magnetic fluctuations correspond to a change in topological sector. Susceptibility measurements could therefore be used as a diagnostic of both the crossover from the high temperature paramagnetic phase into the constrained phase and of the extent of topological sector fluctuations at low temperature. Applying a magnetic field along a cubic axis, ẑ, breaks the symmetry and the system approaches the maximum winding state, w = (0, 0, 4L 2 ) singularly via a Kasteleyn transition [18,29,30,[33][34][35]]. This is an example of a topological ordering transition, as the system enters a state with fixed sector and constant winding vector. The susceptibility diverges on the high temperature side but is strictly zero in the topologically ordered phase. In the rest of the paper we develop the notion of susceptibility as a diagnostic for TSF around w = 0, in zero field. FIG. 1: Top right: A portion of the pyrochlore lattice, made of corner-sharing tetrahedra. All spins respect the ice-rules -"two in two out". The red spins form a string or worm in the negative direction, while the green hexagon represents a non-winding worm of 6 spins: flipping the former changes the topological sector, the latter is a fluctuation within a sector.

Top right: The Husimi tree construction for the pyrochlore lattice. Middle right: Mapping between spins with easy-axis anisotropy respecting the symmetry constraints of the lattice (right), and the corresponding pseudo-spins (left). Bottom: Scattering function S(Q, T ) of the NNSI model at T = 1 K for a system of 4000 spins, simulating Ho2Ti2O7 with J eff = 1.8 K. The pinch points are evident at (0, 0, 2), (1, 1, 1) and (2, 2, 2).

Analytics: We calculate both the magnetic susceptibility and the pseudo-spin susceptibility on a Husimi tree of corner sharing tetrahedra, which preserves the coordinations of the pyrochlore lattice but neglects nonwinding loops (see Fig. 1). Although this is an approxi- mate method, previous work [29] suggests that it should provide an excellent basis for describing magnetic fluctuations, as it allows system spanning strings of flipped spins and hence winding number fluctuations. The approach is reminiscent to the cluster variation method developed in [36]. Spins on the (n + 1) th shell (green in Fig. 1) have three equivalent neighbours and one neighbour on the n th shell (blue in Fig. 1). The total partition function is built up recursively by summing over the degrees of freedom of the spins from the (n + 1) th shell, while holding the spin on the n th shell fixed, with an external field h along the z-axis breaking the up down symmetry [37,38]. In order to neglect boundary effects, thermodynamic quantities such as the longitudinal susceptibility χ, are extracted from the centre of the tree:

χ ≡ ∂m z ∂h h→0 = 2β 3 1 + e 2βJ 2 + e 2βJ + e -6βJ , (1)
where m z = i S z i /N , ... is a thermal average. Here and throughout our susceptibility has dimensions of inverse temperature. To compare with experiment it must be multiplied by 3C where C is the normal Curie constant in the SI system :

C = µ 0 µ 2 N Ho /3V ≈ 4 K for Ho 2 Ti 2 O 7
, where µ 0 is the vacuum permeability, µ the magnetic moment operator for Ho 3+ , N Ho the number of Ho ions and V the system volume.

The asymptotic limits of χ(T ) [39,40] reveal a crossover between unconstrained and constrained, collective paramagnetic regimes, with the Curie constant scaled by a factor of two.

χ(T → ∞) ∼ 1/3T χ(T → 0) ∼ 2/3T. ( 2
)
The factor of 1/3 at high temperature is a necessary property of a system with cubic space symmetry, familiar in the case of a Heisenberg paramagnet where an applied field couples to fluctuations in only one of the three Cartesian components of the magnetization. Although spin ice has local easy-axis anisotropy, it is isotopic in linear response once the full symmetry of the system is accounted for. This symmetry is a key property of spin ice and is related to the almost perfect screening of the long range interactions in the Coulomb phase of the dipolar spin ice model [41]. Further insight into this low temperature crossover can be gained from the pseudo-spin susceptibility (cf. figure 1)

χ 0 ≡ ∂m 0 ∂h 0 h=0 = 4β 1 + e -6βJ 4 + 6e 2βJ -2e -6βJ , (3)
where m 0 = i σ i /N and where h 0 is parallel to the global pseudo-spin axis and thus conjugate to m 0 . The asymptotic limits of χ 0 are

χ 0 (T → ∞) ∼ 1/T χ 0 (T → 0 + ) ∼ 2e -2βJ 3T (4)
without the factor of 1/3 as all pseudo-spins are parallel to the field. As the constraints are imposed, the pseudospin moment vanishes on each tetrahedron and χ 0 falls to zero on the same temperature scale than the crossover of χ between the two Curie laws for the system with real spins (cf. figure 4.A). This reinforces our claim that this crossover can be used as a signal of the system entering the Coulomb phase, and as an indicator of topological sector fluctuations at low temperature.

Comparison with Monte Carlo simulations gives excellent agreement between analytics and numerics as shown in Fig. 4.B, where we show both T χ and T χ 0 from simulations of the NNSI using the worm algorithm. The data agree with the analytic prediction, within numerical error, over the entire range of temperature from 1000 K to 0.5 K, at which point the system enters into the asymptotic regime characterized by equations ( 2) and (4). A closer look at T χ as T → 0 taken from Ref. [39] gives an estimate of T χ = 0.66735 ± 0.0003, very close (but not equal) to the 2/3 predicted by the Husimi tree. One can conclude that as the non-winding loops, which are absent in the tree calculation, carry no magnetization this is of little consequence for magnetic fluctuations, allowing for an extremely accurate estimate of the magnetic fluctuations between different topological sectors.

Moving from the model to real systems, there are no periodic boundaries, so no winding of loops but any finite window will have strings running through it much as in the periodic system. Although boundary effects may change the string statistics [12], the same picture of strings and closed internal loops should hold, motivating detailed comparison between model systems and experiment. In Fig. 3 we superimpose previously unpublished experimental data from susceptibility measurements for a powder sample of Ho 2 Ti 2 O 7 taken between 3 and 15K.

The experimental moment was scaled to 96% of its full value of µ = 10µ B to get the best fit. Even when taking into account this scale factor (which is close to unity) the agreement between experiment, theory and simulation is remarkably good, indicating that the bulk susceptibility does indeed approach the TSF regime as the temperature becomes of order J.

On the Curie-Weiss law: Unfrustrated ferro-or antiferromagnets order on a temperature scale set by the Curie-Weiss temperature, |Θ CW |, estimated from a high temperature expansion for the magnetic susceptibility

(3χT ) -1 ∝ 1 -βΘ CW . (5)
Expanding equation ( 1) to order O(β) one finds Θ CW = 2J for the NNSI. A standard picture of frustrated compounds proposed by Ramirez [42] is that frustration will hinder ordering down to a lower temperature T * Θ CW , with T * → 0 for a spin liquid, or cooperative paramagnet. This is the case for spin ice, although unlike an antiferromagnetic spin liquid, the susceptibility approaches what we call the spin liquid Curie law, characteristic of topological sector fluctuations, rather than a constant value, as T approaches zero. The crossover occurs over a very wide range of temperature, so that while the TSF regime is reached around 1 K, the paramagnetic Curie law is only reached above 100 K. This is extremely important for comparison with experiment, as 100 K is far outside the spin ice temperature range for real materials making an estimate of Θ CW by traditional methods a difficult task. For example, putting J = 1.8 K, the canonical value Ho 2 Ti 2 O 7 gives Θ CW = 3.8 K, which is noticeably bigger than the estimate of Θ CW = 1.9 ± 0.1 K from the bulk magnetometry measurements for Ho 2 Ti 2 O 7 shown in Fig.

3 [24]. The reason is that the rare-earth single ion Ho 3+ starts to lose its Ising nature and becomes more and more Heisenberg-like with easyaxis anisotropy above ∼ 30K [43]. Although well above the scale set by J, it is still in the crossover region and far from the high temperature limit. In this crossover region, the effective measured value of the Curie-Weiss temperature thus depends on the temperature window used for the fit. Of course this does not prevent an estimate of the Curie-Weiss temperature, or the use of Ramirez's criterion mentioned above. We illustrate this in Fig ( 3) where we plot 1/χ against temperature for the bulk susceptibility data together with theory and simulation. The data, now scaled by an effective moment which is 96% of the full moment, compares extremely well with a Curie-Weiss law with Θ CW ∼ 1.9 K, but lies far from the true Curie-Weiss law with Θ CW = 2J = 3.8 K. The dotted lines show Curie-Weiss laws for ΘCW ≡ 2J = 3.8 K (thin) and ΘCW = 1.9 K (thick). The susceptibility is expressed in CGS units cm 3 .mol -1 .

Simulations

imation is defined as

S(Q, T ) ≡ N i=1 S i ⊥ e iQ•ri 2 , (6)
where S i ⊥ is the component of a spin at r i orthogonal to the scattering vector Q(q x , q y , q z ), and where ... represents a thermal average at temperature T . Being the Fourier transform of the thermally averaged two-spin correlation function, S(Q, T ) is related to the wave vector dependent susceptibility, and in certain instances becomes an explicit function of the bulk susceptibility χ(T ). For example in the paramagnetic phase, equation (6) becomes

S(Q, T → ∞) = N i=1 S 2 i x + S 2 i,y = 2N 3 , (7)
where x and y are the axis of the plane orthogonal to Q. Hence, since χ = 1/3T in this regime, S(Q, T ) = 2N T χ(T ) = 2N/3 for all Q as T → ∞: that is, while χ is coupled to only one spin component parallel to the field, S(Q) is coupled to two components orthogonal to the wave vector Q [70].

As one moves into the Coulomb phase and correlations build up, S(Q, T ) develops a strong Q dependence with, in particular, the appearance of the pinch points [15,17,46] that are characteristic of the local divergence free constraint imposed by the ice rules. A map of S(Q, T ) generated from equation (6), in the (h, h, l) plane of reciprocal space, for the NNSI as the Coulomb phase is approached, is shown in Fig. (1). The wave vectors are in units of 2π/a, where a is the side of a 16 site cubic unit cell. The pinch points, narrow regions of intense diffuse scattering can be seen at the reciprocal lattice vectors (0, 0, 2); (1, 1, 1); and (2, 2, 2), that is, at Brillouin zone centres for the face centred cubic lattice of the pyrochlore structure. Near the (0, 0, 2) pinch point the scattering is expected to take the form

S(Q, T ) = C(T ) q2 z + ξ -2 ice (T ) q2 z + q2 x + q2 y + ξ -2 ice (T ) , (8)
where qx = q x , qy = q y , qz = q z -2 × (2π/a) and where, following Youngblood and Axe [46], ξ ice (T ) is a coherence length for the Coulomb phase [15,19]. The amplitude C(T ), which is the value of S(Q) as one traverses the singular point at qx = qy = 0, comes from transverse magnetic fluctuations: spin fluctuations in the plane perpendicular to the wave vector qz that describes them. It is therefore coupled to two spin components and so should constitute two contributions from the bulk susceptibility χ. This expression predicts a ridge of intense diffuse scattering of constant amplitude along the cubic axis (q x = qy = 0), whose width is limited by ξ -2 ice at the pinch point: S(Q = (0, 0, l), T ) = C(T ) = 2N T χ which scales as 4N/3 as T goes to zero, while at higher temperature, as ξ ice (T ) becomes microscopic, the pinch point broadens and one crosses back to isotropic homogeneous paramagnetic scattering, with S(Q, T ) independent of Q and scaling as ∼ 2N/3. As a summary, we expect

S(Q, T ) = 2N T χ(T ) ∀Q T → ∞ ∀T q = 0 (9)
In Fig. 4.C we show simulation results for the NNSI for 3S(Q, T )/2N as a function of temperature for different values of Q. For Q along (0, 0, l) the simulation results confirm the above scenario to a high degree of accuracy, as results for both (0, 0, 1) and (0, 0, 2) follow the theoretical expression given by the tree calculation within numerical precision between T = 0.1 K and T = 1000 K. We expect this argument to hold at any pinch point, as confirmed for (1,1,1) in Fig. 4.C. Away from the line of high symmetry (0, 0, l) and from pinch points, the scattering intensity fails to develop as the constraints are imposed, but remains larger than the high temperature asymptote. As seen from the data at (1, 1, 0) and (2, 2, 0), the topological sector information is not contained in these projections, or at least in a less straightforward way. Pinch point scattering: Although Youngblood and Axe predict constant amplitude along (0, 0, l) it is perhaps surprising to find that this long wavelength expression holds all the way from the zone centre (pinch point) to the zone boundary. The q = (q x , qy , qz ) independence is a consequence of the collective paramagnetism yielding diffuse, rather than either Bragg or critical scattering: at finite q, away from the zone center along the cubic axis (0, 0, l), one observes topological sector fluctuations in an effective system of reduced size ∼ 2π/|q|, which has only small finite size corrections to those in the thermodynamic limit. We show for comparison the Curie-Weiss fit of figure 3 with ΘCW = 1.9 K (green line). The bulk data have been extracted from figure F14.1 in [44] and the error bars are due to digitalization. C: Wave vector dependent susceptibility of the NNSI model at selected Q points, again obtained by Monte Carlo simulation, and compared with the analytical theory. The data for (0, 0, 2) and (0, 0, 1) have been obtained using both single spin flip dynamics and a worm algorithm has been added for (1, 1, 1) to circumvent the low temperature ergodicity loss [45]. The nearest neighbor coupling has been set to J/kB = 1.8 K, as estimated for Ho2Ti2O7, within the nearest neighbour approximation [24]. D: Temperature dependance of S(Q, T ) measured experimentally (see text for details), compared with the analytical theory. The red line is a guide to the eye to illustrate that this crossover could be misinterpreted as a power law within a finite temperature window (χ ∼ T -1.17 here). In each case the intensity is scaled to the value of the susceptibility estimated from the Husimi tree at a single temperature between 30 and 100 K. There are no fitting parameters for the temperature axis.

Specifically, for q along ẑ, we are interested in fluctuations in M x ∝ w x and M y ∝ w y . The strings of alternating "out-in-out-in ..." spins along a given cubic axis behave as random walkers in the plane perpendicular [47,48], so that those oriented along the ŷ axis have a ballistic trajectory in this direction, but make a diffusive random walk in the (x -ẑ) plane. For q along ẑ, string correlations will therefore be lost when the extension of the string in the ẑ direction exceeds λ = 2π/|q|. If the number of steps along the ŷ axis is ˜ y , the diffu-sive orthogonal extension in the (x -ẑ) plane is ∼ ˜ y . For a string oriented along the ŷ axis, each tetrahedron on its way provides 2 possible paths, alternatively along the [101] and [10 1] axes. Hence such string makes an almost isotropic random walk step in the (x -ẑ) plane after spanning 2 tetrahedra, along ±x or ±ẑ approximately. Because there are 4 tetrahedra in a cubic cell, a step length is a/2. Fixing the perpendicular extension (a/2) ˜ y = λ gives ˜ y ∼ (2λ/a) 2 , a number that is always greater than unity, even at the zone bound-ary, where λ = a and ˜ y spans two cubic cells. Hence, even at the zone boundary we are observing strings for a system of large enough effective size to be essentially in the asymptotic regime where one can observe topological sector fluctuations. We are not comparing with a system with periodic boundaries; the soft effective boundaries provided by finite scattering wave vector appear to give similar results, allowing constant scattering amplitude along the entire (0, 0, l) ridge.

To quantify this correlation, we can rewrite the susceptibility as

3χT = 1 N i,j ( S i • S j -S i S j ) = 1 + i =0 S i • S 0 .
Obviously we don't expect any q dependence in the infinite temperature limit. In the Coulomb phase on the Husimi tree lattice, the total correlation between i th nearest neighbors is 37,38]. χ = 2/3T is recovered upon integration of all correlations, and we can estimate that most of the correlations (98 %) are already taken care by the first three nearest neighbors, i.e within a sphere of radius r a.

S i • S 0 = 2/3 i [
In pictorial terms (see Fig.

(2), a neutron of wavelength λ will not see closed loops on that scale, as they do not change the two point correlation function g(r = λ), or magnetic moment calculated over an area incorporating the loop, but they will detect fluctuations on a larger scale (r > λ), which appear as fluctuations of strings as a result of the anisotropic scaling of the string trajectory. Hence the strings should give essentially uniform scattering right up to the zone boundary.

Neutron scattering experiments on Ho 2 Ti 2 O 7 : We now compare our theoretical and numerical findings with experimental measurements of S(Q). In Fig. 4.D we show S(Q) for several Q values in the (h, h, l) plane from two different single crystal neutron scattering experiments of Ho 2 Ti 2 O 7 (that used different samples). Both experiments were performed on the IN14 triple axis spectrometer at the ILL, Grenoble. In the first experiment S(Q) was extracted from the measured unpolarized neutron scattering cross section [49], while in the second it was derived by combining nonequivalent components of the tensor S(Q) measured by polarization analysis [19]. In each case the intensity is scaled to the value of the susceptibility estimated from the Husimi tree at a single temperature between 30 and 100 K. There are no fitting parameters for the temperature axis. One can immediately see that the total intensity at, or near the zone boundaries, (0, 0, 0.9) and (0, 0, 3) is in remarkably good agreement with theory and simulation, and shows the clear signature of a crossover between paramagnetic fluctuations and topological sector fluctuations. However, things are very different at the pinch point, (0, 0, 2). After an initial increase above the Curie law below 30 K, the value of the scattering intensity stagnates and even decreases as the temperature dips below 3 K. This failure to follow the predictions of the NNSI appears despite the fact that, experimentally the pinch points become sharply developed at low temperature, indicating that the topological constraints are imposed to an excellent approximation [19].

A more detailed look at the evolution of the scattering intensity along the ridge is shown in Fig. 5 where S(Q) is plotted as a function of Q(0, 0, l 2π/a) from l ∼ 0.7 to l = 4 and for temperatures between 50 and 1.7 K. This data is extracted from polarized diffuse scattering measurements made using the D7 spectrometer at the ILL, Grenoble, which are described in Ref [19]. Again different polarization channels have been combined so that the total magnetic response is plotted. On the zone boundaries, with l = 1 and l = 3, the crossover between Curie laws is clearly visible, while at the zone centre, l = 2 the crossover fails to develop. Please note that [1.7 K ; 50 K] is not a wide enough temperature window to see the full factor of 2 between the two Curie laws. The variation between both Brillouin zone boundaries (l = 1 and 3) is due to the Q dependence of the atomic form factor, which is not present in the simulations. Without the form factor, the scattering amplitude would be equal for l = 1 and 3, in complete agreement with the numerical simulation.

This difference in behavior at the centre and zone boundaries shows that the Pauling states with strings spanning large distances are partially suppressed in the real material, so that the topological sector fluctuations, defined over large length scales, are less intense than those expected from a pure ice rule system in which all constrained microstates have the same statistical weight. Indeed, as discussed in Ref. [50] the scattering in the closely related spin ice material Dy 2 Ti 2 O 7 tends towards a pattern characteristic of closed 6-membered loops, though retains weak pinch points and the 00l ridge. A first possible explanation is suppression of long loops by further neighbour exchange couplings [50]; another is the closing of field lines -similar to strings -by the magnetic dipole interaction, while a third and more interesting possibility, is condensation into resonating loops stabilized by quantum fluctuations [51][52][53]. Interestingly, the bulk susceptibility measurements from the powder sample are more in line with the topological sector fluctuations of the NNSI than the results for scattering at zero wave vector (Fig. 4). This could be due to a difference between results for single crystals and powder samples [54] but it is an interesting point that requires further investigation. In Fig. 4 we also show the evolution of the scattering intensity at points along the (h, h, 0) axis. The data for h = 3/2 and h = 2 become similar to the numerical scattering data from the NNSI when it is scaled to the theoretical susceptibility at high temperature. As temperature is reduced, the intensity increases slightly before approaching a plateau, well below the theoretical predictions for scattering along the cubic axis [001]. This is to be expected as the spin components perpendicular to this low symmetry direction fail to capture the correlations of the Coulomb phase, as can be read from equation (8).

This data presentation allows us to make some comments regarding the relaxation of the system towards equilibrium, which becomes very slow at low temperature ( 0.6 K), as indicated by field cooled, zero field cooled splitting of the magnetization [55]. This can be understood as a topological ergodicity breaking expected in classical systems [9] where the rarefaction of topological defects, that can be seen as monopoles in presence of dipolar interactions [40,56,57], hinders the magnetic relaxation in both numerical simulations and in experiment [32,55,58,59]. However the unpolarized neutron scattering data points at (0, 0, 0.9) measured on IN14 follow the analytical curve down to ∼ 0.1 -0.2 K, which suggests that equilibrium of large wave vector components of the magnetization can be achieved in spin ice in zero field down to temperatures significantly below the ergodicity temperature of 0.6 K. Less can be said about large length scales as loss of equilibrium does not appear to be the mechanism for the dip in scattering intensity along the (0, 0, l) ridge, at small wave vector. The deviation of S(Q) from the theoretical prediction, at the zone centre occurs on the scale of 10 K, well above the ergodicity temperature. Topological sectors in other systems: Finally, we address the question of generality, as a similar enhancement of paramagnetic fluctuations occurs in other topologically constrained systems if one has access to the appropriate variables. The enhanced fluctuations are a result of the emergent gauge phenomena [60] that characterize the Coulomb phase in magnetic pyrochlore and kagomé systems [12][13][14][15][16]. The emergent field is subject to a local divergence free constraint, leading to dipolar correlations and the structure of topological sectors described above. In spin ice models and materials the emergent gauge field is proportional to the coarse grained field of magnetic moments, or spin configuration. Hence it is directly accessible through both bulk measurement and through scattering experiments. The direct access to the gauge field [19] makes spin ice of particular interest in this context, as we have shown in the present paper. In pyrochlore antiferromagnets the emergent gauge field is a hidden property of the rule of satisfied units [61] and essentially corresponds to an inverse mapping between antiferromagnetically coupled spins and ferromagnetically coupled pseudo-spins that resemble the spin ice degrees of freedom and form an effective magnetic field within a coarse grained description [14]. Hence the physics of the emergent gauge field is common to both classes of system. This is the case for the Heisenberg antiferromagnet: within a Gaussian approximation, the susceptibility for the pseudo degrees of freedom is enhanced by the same factor of two as the susceptibility in spin ice, as one descends into the Coulomb phase [16,62]. The pseudo spin degrees of freedom are not accessible through bulk measurement, but magnetic scattering in Q space is related to that for spin ice, with the topological constraints leading to a pinch point pattern. For nearest neighbour interactions only, one finds lines of constant scattering amplitude for the real spins, this time along the (h, h, 0) axes. As further neighbour interactions are added, the intensity dips at the Brillouin zone centre, even though the lowest energy states remain within the topologically constrained phase space. This result seems in complete analogy with our experimental observations along the (0, 0, l) axis in spin ice, giving weight to our interpretation in terms of topological sector fluctuations of varying intensity, as the effective length scale, ∼ 1/|Q| changes from the zone centre to zone boundary.

The factor of two of this crossover is directly related to the entropic weight of the different topological sectors in the Coulomb phase [16,62]. Hence it is not universal and we can expect the spin liquid Curie laws in other frustrated systems to have a different prefactor.

Conclusion:

The local divergence free constraint of the Coulomb phase in spin ice materials allows a decomposition of the ground state ensemble in terms of topological sectors. Fluctuations between sectors is clearly visible through susceptibility measurements, as the winding numbers characterizing the sectors are directly proportional to the magnetization. This is only true in the Coulomb phase and as the system settles into it, dipolar spin correlations develop, giving a Curie law crossover between paramagnetism and fluctuations characteristic of the topological spin liquid phase. We have compared analytical and numerical results for the nearest neighbour spin ice model with bulk magnetometry measure-ments for a powder and neutron scattering measurements through the structure function S(Q, T ) for a single crystal of Ho 2 Ti 2 O 7 . We find quantitative agreement between theory and experiment for large Q, but near the pinch point, at small wave vector the experimental scattering intensity is suppressed compared with theory. We have discussed how further nearest neighbour exchange and dipole interactions, or quantum fluctuation, may hinder topological sector fluctuations on a large length scale. Because such perturbations can be pertinent in real materials, especially dipolar interactions for spin ice compounds, the success of the nearest neighbour model raises interesting open questions. More experimental and theoretical work is required to understand these mechanisms in details.

We believe this Curie law crossover is in fact a general feature of many frustrated systems [38], apparent as TSF in spin ice materials and encoded into the scattering pattern of related antiferromagnets. As a consequence, the standard Curie-Weiss picture at high temperature appears to be incomplete and should be used with caution. Further work in this direction, and in particular in potential quantum spin liquid compounds -Herbertsmithite [63], Tb 2 Ti 2 O 7 [64,65], Yb 2 Ti 2 O 7 [66,67] would be particularly interesting.

It is remarkable that, in spin ice, a completely local probe readily accessible by experiments is able to identify fluctuations between topological sectors; i.e. the difference between "topological constraints" and "topological order". Until now, this task necessarily fell to non-local probes, such as measures of the winding number, or the topological entanglement entropy [2,3]. In gapless U(1) quantum liquids, the notion of topological entanglement entropy is expected to be ill-defined [68]. In light of the revelation that there may be other, yet undiscovered topological invariants in related systems [10,11], it is interesting to speculate whether similar probes may prove instrumental in characterizing this important class of topological order in the future.

FIG. 2 :

 2 FIG.2: Non-winding loop(blue) and string (red) which can be flipped by winding loops of this length. The winding loop excitation leads to a topological sector fluctuation. The (red) string illustrates the effective finite size system sampled by neutrons of wavelength λ.

FIG. 3 : 1 /

 31 FIG.3:1/χ vs. T for bulk magnetometry measurements of a powder sample of Ho2Ti2O7 (green triangles), Monte Carlo simulation of the nearest neighbour spin ice (red dots) and analytical result from the Husimi tree, equation (1) (blue line). The dotted lines show Curie-Weiss laws for ΘCW ≡ 2J = 3.8 K (thin) and ΘCW = 1.9 K (thick). The susceptibility is expressed in CGS units cm 3 .mol -1 .

FIG. 4 :

 4 FIG.4: A: Temperature dependance of 3χT from Eq. 1 with real spins (top curve) and 3χ0T From Eq. (3) with pseudo-spins (bottom curve) and with J eff = 1.8 K. B: Comparison of the susceptibility of the NNSI model obtained by Monte Carlo simulation with single spin flip dynamics compared with the analytical theory. The blue triangles are bulk susceptibility measurements fitted to theory with a factor of 96% with respect to the expected value of the local magnetic moments of 10 µB. We show for comparison the Curie-Weiss fit of figure3with ΘCW = 1.9 K (green line). The bulk data have been extracted from figure F14.1 in[44] and the error bars are due to digitalization. C: Wave vector dependent susceptibility of the NNSI model at selected Q points, again obtained by Monte Carlo simulation, and compared with the analytical theory. The data for (0, 0, 2) and (0, 0, 1) have been obtained using both single spin flip dynamics and a worm algorithm has been added for (1, 1, 1) to circumvent the low temperature ergodicity loss[45]. The nearest neighbor coupling has been set to J/kB = 1.8 K, as estimated for Ho2Ti2O7, within the nearest neighbour approximation[24]. D: Temperature dependance of S(Q, T ) measured experimentally (see text for details), compared with the analytical theory. The red line is a guide to the eye to illustrate that this crossover could be misinterpreted as a power law within a finite temperature window (χ ∼ T -1.17 here). In each case the intensity is scaled to the value of the susceptibility estimated from the Husimi tree at a single temperature between 30 and 100 K. There are no fitting parameters for the temperature axis.

FIG. 5 :

 5 FIG.5: Neutron scattering amplitude along the ridge (0, 0, l) taken from the second experiment[19], for temperatures between 50 K and 1.7 K. The lines are a guide to the eye.

Acknowledgements It is a pleasure to thank A. Harman-Clarke, M. Gingras, M. Hastings, L.-P. Henry, G. Misguich, R. Moessner, T. Roscilde and N. Shannon for useful discussions; A.R. Wildes and M. Boehm for contributions to the second IN14 experiments; O.

Losserand for cryogenic support during the second experiment and D. Prabhakaran for a sample. We are grateful for financial support from the European Science Foundation PESC/RNP/HFM (LDCJ and PCWH), the Ecole Normale Suprieure de Lyon (RGM) and the Institut Universitaire de France (PCWH).