Maxime Federico

Philippe Grangier

A contextually objective approach to the extended Wigner's friend thought experiment

. We show by using various arguments, including textbook quantum mechanics and the ontological approach of Contexts, Systems, Modalities (CSM), that no contradiction arises if one admits that agents must agree on what is considered as a system and what is not. In such a contextually objective approach of quantum mechanics, the apparent contradiction is automatically removed. We also discuss why this mutual agreement between agents is already implicit in the standard formulations of quantum mechanics, and why removing it leads to inconsistencies.

I. INTRODUCTION

Wigner's friend thought experiment has been proposed by Wigner in 1967 [START_REF] Wigner | Symmetries and Reflections[END_REF]. Recently, an extended version was proposed [START_REF] Frauchiger | Quantum theory cannot consistently describe the use of itself[END_REF] in order to test how quantum mechanics describes agents who themselves use that theory to predict results of experiments, that may include other agents. This article triggered a large number of reactions [3] and it is not our purpose here to analyse all of them. We will rather summarize the argument, and show that the claimed contradiction between three arguments is removed not by giving up one of these three arguments, but rather by adding a fourth one. We will justify this additional argument by different approaches, including e.g. textbook quantum mechanics (QM), and the ontological approach of Contexts, Systems, Modalities (CSM). It may be concluded that the theory presented as QM by Frauchiger and Renner is not the actual QM, but a different theory that is not consistent indeed.

In Sections II and III we present the argument and the contradiction as introduced in [START_REF] Frauchiger | Quantum theory cannot consistently describe the use of itself[END_REF], and in Sec. IV how to remove it by adding a simple fourth requirement. Then we discuss how to relate this argument to Hardy's paradox [START_REF] Bub | In a defense of a "Single World" interpretation of quantum mechanics[END_REF][START_REF] Hardy | Quantum Mechanics, Local Realistic Theories, and Lorentz-Invariant Realistic Theories[END_REF] (Sec. V), to textbook QM as discussed by Laloë [START_REF] Laloë | Can quantum mechanics be considered consistent? a discussion of Frauchiger and Renner's argument[END_REF] (Sec. VI), and finally to the CSM point of view (Sec. VII). We conclude in Sec. VIII, and present some additional issues in Appendices 1 and 2.

II. THE FRAUCHIGER-RENNER THOUGHT EXPERIMENT

The experimental protocol is the following (see Fig. 1): an agent F uses a quantum random generator (quantum coin) with output r ∈ {heads, tails}. The coin quantum state is

|r = 1 √ 3 |h + 2 3 |t , (1)
FIG. 1. Sketch of the thought experiment. Image adapted from [START_REF] Frauchiger | Quantum theory cannot consistently describe the use of itself[END_REF].

where |h and |t are the states of heads and tails respectively. Probabilities are therefore 1/3 for heads and 2/3 for tails. During her measurement F gets entangled with the coin, so the state of the total system including F (and her lab) becomes

|Ψ = 1 √ 3 |h F : h + 2 3 |t F : t , (2)
using the notations [START_REF] Laloë | Can quantum mechanics be considered consistent? a discussion of Frauchiger and Renner's argument[END_REF] where (3)

F
Then, she sends this electron to a second agent F , located in an other lab isolated from F one (except during arXiv:2301.03016v1 [quant-ph] 8 Jan 2023

the exchange of the electron assumed very short) who can measure the spin projection of that electron in the basis {|↓ , |↑ }. At this point, the total system including agents themselves is in the overall entangled state

|Ψ = 1 √ 3 |h F : h |↓ |F : ↓ + |t F : t |↓ |F : ↓ + |t F : t |↑ |F : ↑ , (4)
where |F : ↑ and |F : ↓ are the states of F (and her lab) observing spin up or down. Two other separate agents W and W can moreover perform measurements on the systems F and F respectively with respect to the basis states

OK = 1 √ 2 |h F : h -|t F : t , (5)
f ail = 1 √ 2 |h F : h + |t F : t , (6)
|OK = 1 √ 2 |↓ |F : ↓ -|↑ |F : ↑ , (7)
|f ail = 1 √ 2 |↓ |F : ↓ + |↑ |F : ↑ . ( 8
)
Here we name the "sub-experiments" that composed the thought experiment by the name of agents corresponding to the different measurements F , F , W and W . However, it should be clear that F and F , who were agents in the initial step, are then considered as systems from the point of view of W and W . Therefore the states ( 5)-( 8) written above contain agents, labs and measured systems on which W and W are supposed to make quantum measurements, so that the whole ensemble of F and F agents, labs and measured systems are projected into superposition states. This is clearly not feasible in practice, but the whole idea is to consider this operation feasible as a thought experiment, and to examine which consequences can be drawn from it. Formally, considering the four situations where the active agents are either (F , F ), (W , F ), (F , W ), or (W , W ), by using and combining appropriate bases, |Ψ can be written equivalently as

|Ψ = 1 √ 3 |h F : h |↓ |F : ↓ + |t F : t |↓ |F : ↓ + |t F : t |↑ |F : ↑ (9a) = 2 3 f ail |↓ |F : ↓ + 1 √ 6 f ail -OK |↑ |F : ↑ (9b) = 1 √ 6 |h F : h |OK + |f ail + 2 3 |t F : t |f ail (9c) = 1 √ 12 OK |OK - 1 √ 12 OK |f ail + 1 √ 12 f ail |OK + √ 3 2 f ail |f ail . (9d)
Using these four expressions for the state vector |Ψ , one can deduce the following statements, which are close to the ones listed in [START_REF] Frauchiger | Quantum theory cannot consistently describe the use of itself[END_REF]: However this contradiction may likely be attributed to the undefined status of F and F , who switch between being agents (able to make quantum measurements) and systems (being acted upon by quantum measurements). It is therefore required to clarify the definition and the role of agents. From here, several points of view exist and we are going to explore some of them.

III. FRAUCHIGER & RENNER: GETTING

A CONTRADICTION.
This point of view is based on three assumptions which allow authors of Ref [START_REF] Frauchiger | Quantum theory cannot consistently describe the use of itself[END_REF] to show a "no-go theorem".

First assumption (Q) : This defines what an agent must do to interpret and predict measurements. Suppose that a system S (external to the agent) is in state |ψ of a Hilbert space at time t 0 ; an outcome x can be measured on S at time t with respect to a family of projectors {π x } in Heisenberg representation. If ψ| π ξ |ψ = 1, then the agent can conclude that x = ξ at time t. From the point of view on this individual agent, this theory looks like quantum mechanics, and it will be used successively from the "subjective" point of view of different agents, in order to get a serie of statements.

First, if r = heads, the spin is in state |↓ , and a measurement made by agent F with respect to the basis

{π ↓ = |↓ ↓| , π ↑ = |↑ ↑|} gives ↓| π ↓ |↓ = 1.
Therefore agents F and F can conclude that if F gets heads, F will get spin down; which is logically equivalent to: if F gets spin up, F got tails, in agreement with eq. (9a). By considering successively the different pairs of agents, other statements can be obtained and lead to Additional assumptions (C) and (S): These two hypotheses stipulate that agents can trust predictions made by other agents, irrespective of their status, and that all predictions apply in the same universe, avoiding Everett's multi world interpretation. It is under those three assumptions that a contradiction arises.

Indeed, if statement [2.D] is valid, [2.B] implies that F gets spin up, [2.A] that F gets tails, and [2.C] that W gets f ail. The last statement clearly contradicts the starting point that W gets OK. Using their assumptions as just shown, the authors of [START_REF] Frauchiger | Quantum theory cannot consistently describe the use of itself[END_REF] can thus formulate the following no-go theorem:

(Q) + (C) + (S) ⇒ Contradiction ( 10
)
which has been formulated as the statement "quantum theory cannot consistently describe the use of itself" [START_REF] Frauchiger | Quantum theory cannot consistently describe the use of itself[END_REF].

IV. REMOVING THE CONTRADICTION BY AN AGREEMENT BETWEEN AGENTS.

A major remark is that, though it is presented as equivalent to quantum mechanics (QM), the assumption (Q) of Frauchiger and Renner implies a very unusual use of the quantum formalism, where "self-proclaimed" agents may become systems for other agents. It means that the usual measurement postulate can hardly be applied in a consistent way [START_REF] Laloë | Can quantum mechanics be considered consistent? a discussion of Frauchiger and Renner's argument[END_REF]. A simple way to remove this problem, and also the contradiction, is to consider the additional assumption (A) All agents must agree on the definition of the quantum system to which they apply assumption (Q); as a consequence, no agent should be included in what another agent considers as the measured system.

The second part of (A), i.e. that no agent should be included in what another agent considers as the measured system, is a joint consequence of (Q), stating that the agents apply QM to a system external to themselves, and of the first part of (A), i.e. the agents agreement on the definition of the system. Then the quantum system is the same for all agents, and contains no agent.

Given (A), it is clear that the previous contradiction vanishes since the four statements cannot be true together:

(i) if F and F are agents, the first statement is true, but then the other three make no sense since F and F cannot behave as (superposable) systems from the point of view of W and W ;

(ii) if F and F are systems, then the four lines correspond to the agents W and W making four incompatible measurements on the global system composed of F , F , the coin and the spin. Incompatible measurements cannot be true together as it is usual in QM, such that there is no contradiction.

Therefore, with all of the four assumptions, we obtain [START_REF] Grangier | Revisiting Quantum Mysteries[END_REF] which means that assumption (A) restricts (Q) to some safe range where no contradiction arises. This could be summarized as : "Objective quantum mechanics can describe the use of itself", where "objective" is meant as requiring a mutual agreement between all the agents.

(Q) + (C) + (S) + (A) ⇒ No contradiction
In the following we will develop these arguments, and in particular give more justifications and illustrations of the points (i) and (ii) above, as well as some intermediary situations. We will also see that assumption (A) can take different forms depending on which point of view we adopt, but always removes the contradiction.

V. LOOKING AT W AND W AS AGENTS: LINK TO HARDY'S PARADOX

The Frauchiger and Renner (F&R) paradox is, as it was pointed out by Bub [START_REF] Bub | In a defense of a "Single World" interpretation of quantum mechanics[END_REF], not only a new formulation of Wigner's friend experiment but also a twisted version of Hardy's paradox originally presented to show a similar result as Bell's theorem [START_REF] Bell | On the Einstein-Podolski-Rosen paradox[END_REF]. To do that, Hardy [START_REF] Hardy | Quantum Mechanics, Local Realistic Theories, and Lorentz-Invariant Realistic Theories[END_REF] used a Mach-Zehnder interferometer for electrons and positrons in order to get four entangled states which are similar to those of the F&R Gedankenexperiment. The difference arises only with the way one imposes that these four statements must be true together. In the following, we will assume that a hidden variable exists to describe the various states of eq. ( 9), in agreement with the formulation proposed in [START_REF] Hardy | Quantum Mechanics, Local Realistic Theories, and Lorentz-Invariant Realistic Theories[END_REF].

Let λ be the hidden variable which describes the states before measurements and assume that QM is a local realist theory. It means that results of measurements do not depend on the choice of measurement done by other agents. We introduce notations F (h, λ) = 1 if F measures the state |h , and F (h, λ) = 0 otherwise. The same notations will be applied to other agents and their respective measurable states. The statements obtained in the preceding sections can be recovered using the expressions of |Ψ in the different bases. Indeed, from (9a) and for every experiment described by a given value of λ,

F (h, λ)F (↑, λ) = 0. ( 12
)
From (9b) we additionally have

if W (OK, λ) = 1 then F (↑, λ) = 1, (13) from (9c) if W (OK, λ) = 1 then F (h, λ) = 1, (14)
and from (9d) W (OK, λ)W (OK, λ) = 1 for 1/12th of experiments, (15) which can be summed up in the four following statements completely equivalent to [1.A]-[1.D]:

• F (h, λ) and F (↑, λ) cannot occur at the same time

• W (OK, λ) is true ⇒ F (↑, λ) is true • W (OK, λ) is true ⇒ F (h, λ) is true • W (OK, λ) and W (OK, λ) occur one time out of 12
One can see that even if the assumptions and consequently the formalism are not exactly the same, the contradiction arises in the same way as when the four states of eq. ( 9) are considered simultaneously true. In that case inspired by Hardy's paradox, this is due to the hidden variable hypothesis. Therefore, it appears that in both situation, the contradiction is not due to QM itself but to the assumption that all conclusions drawn from eq. ( 9) are simultaneously true. This explains why the use of assumption (A) removes the contradiction, by making clear that they are incompatible in a quantum sense.

VI. TEXTBOOK QM: PROJECTIVE MEASUREMENTS DEFINE AGENTS

Another approach to the F&R contradiction can be found in [START_REF] Laloë | Can quantum mechanics be considered consistent? a discussion of Frauchiger and Renner's argument[END_REF]. This point of view stipulates that defining agents is equivalent to define where the projective measurements are done, i.e. where the state is projected. For instance, without any projection, the state of the total system after "measurements" by F , F , W and W should be written as

|Ψ = 1 √ 12 OK W : OK |OK |W : OK - 1 √ 12 OK W : OK |f ail |W : f ail + 1 √ 12 f ail W : f ail |OK |W : OK + √ 3 2 f ail W : f ail |f ail |W : f ail . (16)
In this state, all observers are entangled as if we had considered Everett's multi-world interpretation. Then all possibilities can be investigated by choosing where the projections are made. The calculations are detailed in [START_REF] Laloë | Can quantum mechanics be considered consistent? a discussion of Frauchiger and Renner's argument[END_REF] and Appendix 2, here we simply describe qualitatively the results. First, we consider F and F as agent (point (i) of Section IV), and project states in agreement with their measurements. The result (see Appendix 2) is that whatever W and W measure, they cannot obtain informations on spin state or coin state. Therefore, the statements [2.B] and [2.C] of Section III cannot be formulated and the contradiction does not arise.

In the second option which considers F and F as systems, and only W and W as agents, W cannot say anything about the spin state, therefore using this way of defining agents and writing states after measurements, also prevent from any contradiction.

It is also instructive to consider the slightly modified situation where F can "protect" her result by using an other qubit stored in her lab, in such a way that it escapes to W 's measurement (refered as "hidden qubit" [START_REF] Laloë | Can quantum mechanics be considered consistent? a discussion of Frauchiger and Renner's argument[END_REF]). If F observes "heads", the hidden qubit is in state G : h and in state G : t if she observes "tails". The overall state can thus be written as

|Ψ = 1 √ 3 |h F : h G : h |↓ |F : ↓ + |t F : t G : t |↓ |F : ↓ + |t F : t G : t |↑ |F : ↑ . ( 17
)
In

|Ψ = 1 √ 3 |h |h G + |t |t G |↓ + 1 √ 3 |t |t G |↑ , (18)
and the bases used by W and W to perform their respective measurements are

OK = 1 √ 2 (|h -|t ), f ail = 1 √ 2 (|h + |t ), (19)
|OK = 1 √ 2 (|↓ -|↑ ), |f ail = 1 √ 2 (|↓ + |↑ ). ( 20
)
The global state can then be put in the form

|Ψ = OK |OK 1 √ 12 |h G + OK |f ail 1 √ 12 |h G - 1 √ 3 |t G + f ail |OK 1 √ 12 |h G + f ail |f ail 1 √ 12 |h G + 1 √ 3 |t G . (21)
As it was pointed out in [START_REF] Laloë | Can quantum mechanics be considered consistent? a discussion of Frauchiger and Renner's argument[END_REF], the state |t G is only correlated with the result |f ail which is not surprising since when F sends a spin in the state |→ , it is orthogonal to |OK . It means that the hidden qubit keeps a memory of F 's measurement and therefore all statements derived before do not hold anymore. We emphasize that there is no need to read out the value of the hidden qubit: as in an interference experiment, it is enough that the "which path information" is stored somewhere to forbid the quantum superposition.

It is also interesting to investigate different scenarii for the two accessible states of the hidden qubit e.g. if h G |t G = 1 i.e. |h G and |t G are the same state, denoted for instance |G , then the overall system is in the following state

|Ψ = 1 √ 12 OK |OK |G - 1 √ 12 OK |f ail |G + 1 √ 12 f ail |OK |G + √ 3 2 f ail |f ail |G , (22)
which is the same as state (9d) without the hidden qubit.

|G can be factored out and the hidden qubit plays no role since it is no more entangled with F 's results. Otherwise, if h G |t G = 0 i.e. the qubit states are orthogonal and the overall state is identical to (21). If we project this global state onto each state of the hidden qubit

h G |Ψ = 1 √ 12 OK |OK + OK |f ail + f ail |OK + f ail |f ail = 1 √ 3 |h |F : h |OK + |f ail √ 2 , ( 23
)
t G |Ψ = 1 √ 3 f ail |f ail -OK |f ail = 2 3 |t F : t |f ail , (24)
we recover the situation where F is an agent, and W cannot project her in a superposition.

This calculation has two consequences: first, it gives a way to make a transition between the different options, by allowing h G |t G to take any value between 0 and 1. We emphasize that this calculation makes sense in a framework where the only agents are W and W . If h G |t G = 0 then F and F should be considered as decohered systems, as they appear in the usual theory of decoherence after tracing out the "ancilla" qubits. Second, it provides a "toy model" for a situation where a system actually does not behave as such: if W tries to make a projective measurement on F , including her lab and all the surrounding environment, it is enough that a single qubit escapes from the action of W to make the projective measurement failed. When F is a macroscopic system, it can be considered impossible for W to get full control on every single qubit entangled with F ; then considering F as a system makes no sense.

Following the idea of Section IV, this can be seen as a reformulation of assumption (A): (A) All agents must agree on the definition of the quantum system to which they apply assumption (Q), that is to say where the state projection is done; as a consequence, no agent should be included in what another agent considers as the measured system.

VII. CONTEXTS, SYSTEMS AND MODALITIES: THE CSM APPROACH.

Here, we want to show that in the framework of the CSM (Contexts, Systems and Modalities) approach, developped by A. Auffèves and P. Grangier [START_REF] Grangier | Contextual objectivity: a realistic interpretation of quantum mechanics[END_REF][START_REF] Auffèves | Contexts, Systems and Modalities: a new ontology for quantum mechanics[END_REF][START_REF] Auffèves | Deriving Born's rule from an Inference to the Best Explanation[END_REF][START_REF] Grangier | Revisiting Quantum Mysteries[END_REF], the assumption (A) introduced and discussed above is no more an assumption but a theorem. We recall some basics of CSM which starts with a new physical ontology (i.e. a definition of all objects that are described by the theory), by considering three entities:

• System = subpart of the world, isolated well enough to have physical properties that can be studied.

• Context = all other physical systems external and possibly in contact with the studied one (ex: measurement apparatus, labs, etc); in CSM, the context corresponds to a classically defined measurement setup, and it acts like an interface between the system and the observer, see Fig. 2.

• Modalities = the set of answers (or values) that can be predicted with certainty and obtained repeatedly for a given system in a given context.

These definitions are bound together by the following rule (or axiom): in quantum mechanics, modalities are attributed jointly to the system and the context. It means that the quantum properties of a system do not "belong" to the system alone, but to the system within a context.

In order to give an operational content to CSM, we need a second axiom called Quantization Principle:

(i) For each well-defined system and context, there is a discrete number N of mutually exclusive modalities. This number N depends on the system, but does not depend on any particular context.

(ii) Modalities, when defined in different contexts, are generally not mutually exclusive, and they are said to be "incompatible".

CSM is based on a physically realistic point of view, i.e. physics is applied to objects which exist independently from any observer. However, the "object" for CSM is a system within a context, which makes it highly nonclassical. As it can be shown on Figure 2(c), in CSM the observer is outside of the context, illustrating again that the (objective) state of the context must be taken into account in defining the modality. Such a "contextual objectivity" [START_REF] Grangier | Contextual objectivity: a realistic interpretation of quantum mechanics[END_REF] is quite different from the classical "absolute objectivity" of Figure 2(a), but it fully agrees with quantum reality as deduced from empirical evidence. Now, coming back to the original problem, CSM was hidden so far but in fact already there. In order to define systems and contexts according to CSM, we have to avoid mixing notations between agents, labs and experimental set ups. First, as it was pointed out before, FIG. 2. Graphic representations of different ontologies: (a) Classical ontology: the observer can know the "real" physical properties of the system, and the context is only used as an auxiliary tool for measurements. (b) Usual quantum ontology: through successive "entangling" interactions and unitary evolution, the system will include the context, and also (ultimately) the observer or "agent", whose status may be problematic. (c) CSM ontology: the context appears always between the system and the observer, and definite values of the relevant physical properties (modalities) are attributed jointly to the system and the context. there is only one ultimate reality which imposes that a context cannot be included in a system and vice versa (see Figure 3 (o)). For consistency, different agents must agree on the definition of systems and contexts which is exactly the purpose of assumption (A), formulated according to CSM as:

(A) All agents must agree on the definition of the system and context to which they apply assumption (Q); as a consequence, no agent should be included in what another agent considers as the measured system. Again, several cases are possible, corresponding e.g. to point (i) and (ii) of Section IV, and are drawn in Figure 3. In case (i), F and F perform quantum measurements on their systems within their respective contexts. Then W and W cannot make quantum measurements on them (composite systems in yellow and orange dash lines of Figure 3), and the F&R reasoning does not hold. In case (ii), F and F are considered as systems so W and W can perform quantum measurements on those systems in two different contexts on each side (i.e. projecting F or F in superposed or non-superposed states). The modalities defined in these different contexts are incompatible and thus not simultaneously true, no contradiction can arise.

To conclude, we have shown that within the CSM interpretation, the F&R contradiction is removed without additional assumption because CSM is already an "contextually objective" theory [START_REF] Grangier | Contextual objectivity: a realistic interpretation of quantum mechanics[END_REF], where agents cannot become systems for other agents.

VIII. CONCLUSION

To conclude, we have presented several ways to escape the "no-go theorem" introduced by F&R. Our arguments can be summed up by "Contextually objective quantum mechanics can describe the use of itself".

This statement does not allow every single world interpretation of quantum mechanics to be self-consistent, but only those which rely on some form of objectivity. A consequence is that the (single world) quantum rules must introduce a clear distinction between agents and systems, embedded in the additional assumption (A). Then we have shown that in the framework of CSM approach, no F&R contradiction can arise since CSM already contains (A) as a theorem, deduced from the central concept of contextual objectivity [START_REF] Grangier | Contextual objectivity: a realistic interpretation of quantum mechanics[END_REF] which governs quantum measurements in the CSM framework. In their article [START_REF] Frauchiger | Quantum theory cannot consistently describe the use of itself[END_REF], F&R insist on the temporal order of measurements made by successive agents, associated to changing the status of F and F between system and context. In the CSM interpretation, changing the agents status in a same experiment cannot occur, therefore the chronology is not relevant. It could be if we had adopted Everett's point of view also called "multi-world"; but this interpretation is in conflict with assumption (S), so the F&R no-go theorem also does not apply.

More precisely, through the usual quantum or CSM point of view, an ideal measurement takes place as follows: (a) entanglement of the system with the pointer (also quantum) and then (b) reading of the pointer, which brings the result at the single classical context level. From the Everett's point of view, there is no single classical context level but many possible results which are associated to differents universe branches. Then it is, in principle, possible to reverse the evolution of these branches, rewriting the history as if no measurement were done, and to make another different measurement, creating again new branches. So in Everett's point of view, temporal order makes sense to describe first a measurement by F , and then its "erasure" by W , followed by a new measurement in the (OK; f ail) basis.

In CSM, the measurement can be reversed only if it has not reached the context level, i.e. between steps (a) and (b). When the measurement is over (step b), the result is macroscopic and unique and the modalities corresponding to other results in other contexts cannot coexist. Explicitly, if F and F are superposable systems, we can measure them in superposed bases The four situations we have discussed correspond to four combined measurements (N , N ), (N , S), (S, N ), (S, S).

In that case, temporal order is not relevant since only one of these four measurements can be made, as in Hardy's paradox, such that there is a unique objective macroscopic world in which QM is consistent.

We note also that the CSM point of view makes a clear distinction between the usual "pre-measurement" stage, that is entangling the system with a probe in a reversible way, and the actual irreversible measurement that brings the result to the context level. It can be told that the F&R paradox fails because it mixes these two stages in an inappropriate way, as explained in details in [START_REF] Zukowski | Physics and Metaphysics of Wigner's Friends: Even Performed Premeasurements Have No Results[END_REF]. Another approach to the F&R contradiction can be found in [START_REF] Laloë | Can quantum mechanics be considered consistent? a discussion of Frauchiger and Renner's argument[END_REF]. This point of view stipulates that defining agents is equivalent to define where projective measurements are done, that is to say, where mathematically the state is projected. For instance, without any projection, the state of the total system after "measurements" by F ,

F , W and W is |Ψ = 1 √ 12 OK W : OK |OK |W : OK - 1 √ 12 OK W : OK |f ail |W : f ail + 1 √ 12 f ail W : f ail |OK |W : OK + √ 3 2 f ail W : f ail |f ail |W : f ail . (25)
In this state, all observers are entangled as if we had considered Everett's multi-world interpretation.

Then all possibilities can be investigated for the projections. First, we consider F and F as agent (point (i) of Section IV). Therefore, they have to project states in agreement with their measurements. According to F , two states are available

|Ψ tails = |t F : t 1 √ 2 |↓ + |↑ , (26)
|Ψ heads = |h F : h |↓ . ( 27
)
If then we consider F 's projections which lead to many cases depending of the result she gets but also of F 's results, we have the states

|Ψ ↓ tails = |t F : t |↓ |F : ↓ , (28)
|Ψ ↑ tails = |t F : t |↑ |F : ↑ , (29)
|Ψ ↓ heads = |h F : h |↓ |F : ↓ , (30)
or equivalently written in the other bases The second option is to consider F and F as systems, and W and W as agents (point (ii) of Section IV). Before they perform measurements, the system is in state (4). Then, two outcomes are possible for W and give the states

|Ψ ↓
|Ψ OK = -|↑ |F : ↑ OK W : OK = 1 √ 2

FIG. 3 .

 3 FIG. 3. (o)Naive way to see Wigner's friends thought experiment. Such a picture is impossible for CSM, because a (quantum) system cannot include a (classical) context. (i) Case where F and F are agents making quantum measurements on systems with their "own" classical contexts. (ii) Second option where F and F are considered as systems.

APPENDIX 1 :

 1 ABOUT THE TIME EVOLUTION OF THE EXPERIMENT

S

  = {OK; f ail}, S = {OK; f ail} or in non superposed ones N = {heads; tails}, N = {up; down}.

APPENDIX 2 :

 2 ADDING A HIDDEN QUBIT

  : h is the state of F (and her lab) if she observes heads and F : t if she observes tails. Depending on the result, she prepares the spin of an electron in state |↓ if r = heads or in state |→ = 1

			√	2 (|↓ + |↑ ) if r = tails. Therefore the
	total state becomes
	|Ψ =	1 √ 3	|h F : h |↓ +	2 3	|t F : t	1 √ 2	|↓ + |↑ .

  the following, to simplify notations, we will replace |h F : h by |h , |t F : t by |t , |↓ |F : ↓ by |↓ and |↑ |F : ↑ by |↑ since agents (and their labs) are always in the state corresponding to the outcome they have measured. We also condense the hidden qubit notation in |h G and |t G . The state becomes

  Here we see that all these states are products; whatever W and W measure, they cannot obtain informations on spin state or coin state. Therefore, statements [2.B] and [2.C] of Section III are wrong and no contradiction can appear.

	tails =	1 2	f ail -OK	|f ail + |OK , (31)
	|Ψ ↑ tails =	1 2	f ail -OK	|f ail -|OK , (32)
	|Ψ ↓ heads =	1 2	f ail + OK	|f ail -|OK . (33)

  We see again, that the result OK is only correlated to spin up, and that W cannot predict anything on W 's measurement since both results OK and f ail are still available. In the second option where W gets f ail, no predictions can be obtained about the spin state. If we furthemore add the projection made by W we obtain|Ψ OK OK = |OK |W : OK OK W : OK ,(36)|Ψ f ail OK = -|f ail |W : f ail OK W : OK , (37)|Ψ OK f ail = |OK |W : OK f ail W : f ail ,(38)|Ψ f ail f ail = |f ail |W : f ail f ail W : f ail , (39)or equivalently expressed in F and F basis Again, since every state is a product state, statements [2.B] and [2.C] cannot be true and the contradiction disappears. Therefore this way to define agents and to write states after measurements, also erases the contradiction.

	|Ψ OK OK =	1 2	|h -|t	|↓ -|↑ ,	(40)
	|Ψ f ail OK =	1 2	|h + |t	|↑ -|↓ ,	(41)
	|Ψ OK f ail =	1 2	|h -|t	|↓ + |↑ ,	(42)
	|Ψ f ail f ail =	1 2	|h + |t	|↓ + |↑ .	(43)

|OK -|f ail OK W : OK , (34)

|Ψ f ail = 2 |↓ |F : ↓ + |↑ |F : ↑ f ail W : f ail = 3 √ 2 |f ail |W : f ail + 1 √ 2 |OK |W : OK ⊗ f ail W : f ail .

(35)


Yoshihiko Nishikawa

Ludovic Berthier

Collective relaxation dynamics in a three-dimensional lattice glass model

We numerically elucidate the microscopic mechanisms controlling the relaxation dynamics of a three-dimensional lattice glass model that has static properties compatible with the approach to a random first-order transition. At low temperatures, the relaxation is triggered by a small population of particles with low-energy barriers forming mobile clusters. These emerging quasiparticles act as facilitating defects responsible for the spatially heterogeneous dynamics of the system, whose characteristic lengthscales remain strongly coupled to thermodynamic fluctuations. We compare our findings both with existing theoretical models and atomistic simulations of glass-formers.

The glass transition of supercooled liquids is characterized by a drastic dynamical slowing down, with timescales that increase much faster than the Arrhenius law in fragile glass-formers [START_REF] Debenedetti | Supercooled liquids and the glass transition[END_REF][START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF]. These dynamics become increasingly spatially heterogeneous upon approaching the glass transition [START_REF] Ediger | Spatially Heterogeneous Dynamics in Supercooled Liquids[END_REF][START_REF] Berthier | Dynamical Heterogeneities in Glasses, Colloids, and Granular Media[END_REF], in a manner revealed by four-point dynamic susceptibilities and correlation functions [START_REF] Berthier | Dynamical Heterogeneities in Glasses, Colloids, and Granular Media[END_REF][START_REF] Lačević | Spatially heterogeneous dynamics investigated via a time-dependent four-point density correlation func-tion[END_REF][START_REF] Toninelli | Dynamical susceptibility of glass formers: Contrasting the predictions of theoretical scenarios[END_REF][START_REF] Karmakar | Growing Length Scales and Their Relation to Timescales in Glass-Forming Liquids[END_REF]. Over the last decade, massive efforts were also deployed to establish and characterize the growth of relevant thermodynamic fluctuations and static correlation lengthscales accompanying this slowing down [START_REF] Biroli | Thermodynamic signature of growing amorphous order in glass-forming liquids[END_REF][START_REF] Karmakar | Growing length and time scales in glass-forming liquids[END_REF][START_REF] Kob | Nonmonotonic temperature evolution of dynamic correlations in glass-forming liquids[END_REF][START_REF] Berthier | Overlap fluctuations in glass-forming liquids[END_REF], with results compatible with an approach to a random-first order transition (RFOT) [START_REF] Berthier | Configurational entropy measurements in extremely supercooled liquids that break the glass ceiling[END_REF][START_REF] Berthier | Zero-temperature glass transition in two dimensions[END_REF][START_REF] Guiselin | Random-field ising model criticality in a glass-forming liquid[END_REF][START_REF] Guiselin | Statistical mechanics of coupled supercooled liquids in finite dimensions[END_REF][START_REF] Guiselin | Is glass a state of matter?[END_REF].

The central open question that remains, then, concerns the existence of a causal link between static and dynamic fluctuations, to go beyond known correlations [START_REF] Richert | Dynamics of glass-forming liquids. V. On the link between molecular dynamics and configurational entropy[END_REF]. While RFOT theory suggests that static fluctuations directly control the evolution of dynamic heterogeneity and explain the slowing down of the dynamics [START_REF] Kirkpatrick | Scaling concepts for the dynamics of viscous liquids near an ideal glassy state[END_REF][START_REF] Bouchaud | On the Adam-Gibbs-Kirkpatrick-Thirumalai-Wolynes scenario for the viscosity increase in glasses[END_REF][START_REF] Biroli | The RFOT Theory of Glasses: Recent Progress and Open Issues[END_REF], alternative approaches use local energy barriers and dynamic facilitation [START_REF] Garrahan | Geometrical Explanation and Scaling of Dynamical Heterogeneities in Glass Forming Systems[END_REF][START_REF] Garrahan | Coarse-grained microscopic model of glass formers[END_REF][START_REF] Chandler | Dynamics on the Way to Forming Glass: Bubbles in Space-Time[END_REF][START_REF] Keys | Excitations Are Localized and Relaxation Is Hierarchical in Glass-Forming Liquids[END_REF][START_REF] Dyre | Colloquium: The glass transition and elastic models of glass-forming liquids[END_REF], making no reference to the underlying complex free energy landscape.

Experimental measurements do not easily discriminate these views [START_REF] Albert | Fifth-order susceptibility unveils growth of thermodynamic amorphous order in glassformers[END_REF], and original strategies are needed [START_REF] Berthier | Self-induced heterogeneity in deeply supercooled liquids[END_REF]. Recent developments in computational studies of atomistic glass-formers [START_REF] Berthier | Modern computational studies of the glass transition[END_REF] helped reveal the importance of dynamic facilitation at very low temperatures [START_REF] Chacko | Elastoplasticity Mediates Dynamical Heterogeneity Below the Mode Coupling Temperature[END_REF][START_REF] Guiselin | Microscopic origin of excess wings in relaxation spectra of supercooled liquids[END_REF][START_REF] Nishikawa | Collective dynamics in a glass-former with Mari-Kurchan interactions[END_REF][START_REF] Scalliet | Thirty Milliseconds in the Life of a Supercooled Liquid[END_REF], strengthening earlier results [START_REF] Keys | Excitations Are Localized and Relaxation Is Hierarchical in Glass-Forming Liquids[END_REF][START_REF] Vogel | Spatially heterogeneous dynamics and dynamic facilitation in a model of viscous silica[END_REF]. The reported decoupling between static and dynamic lengthscales [START_REF] Scalliet | Thirty Milliseconds in the Life of a Supercooled Liquid[END_REF][START_REF] Charbonneau | Decorrelation of the static and dynamic length scales in hard-sphere glass formers[END_REF] weakens the idea that statics fully controls dynamics, as does the observation that local Monte Carlo algorithms strongly impact the equilibrium dynamics [START_REF] Wyart | Does a Growing Static Length Scale Control the Glass Transition?[END_REF]. While RFOT theory accounts for all these observations [START_REF] Biroli | The RFOT Theory of Glasses: Recent Progress and Open Issues[END_REF][START_REF] Xia | Microscopic theory of heterogeneity and nonexponential relaxations in supercooled liquids[END_REF][START_REF] Berthier | Can the glass transition be explained without a growing static length scale?[END_REF], the current situation is confused.

Simplified lattice models with minimal (but not too minimal) ingredients are ideally suited to clarify these questions [START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF]. Kinetically constrained lattice glass models have no complex thermodynamics but display dynamic heterogeneity stemming instead from dynamic facilitation and kinetic constraints [START_REF] Fredrickson | Kinetic Ising Model of the Glass Transition[END_REF][START_REF] Jäckle | A hierarchically constrained kinetic Ising model[END_REF][START_REF] Kob | Kinetic lattice-gas model of cage effects in high-density liquids and a test of modecoupling theory of the ideal-glass transition[END_REF][START_REF] Toninelli | Jamming Percolation and Glass Transitions in Lattice Models[END_REF][START_REF] Ritort | Glassy dynamics of kinetically constrained models[END_REF]. Interacting plaquette spin models form another class, in which simi-lar kinetic constraints emerge from interacting degrees of freedom [START_REF] Garrahan | Glassiness through the emergence of effective dynamical constraints in interacting systems[END_REF][START_REF] Jack | Caging and mosaic length scales in plaquette spin models of glasses[END_REF][START_REF] Jack | Phase Transition for Quenched Coupled Replicas in a Plaquette Spin Model of Glasses[END_REF]. Finally, lattice glass models with frustrated interactions also exist [START_REF] Biroli | Lattice Glass Models[END_REF][START_REF] Ciamarra | Lattice glass model with no tendency to crystallize[END_REF][START_REF] Rivoire | Glass models on Bethe lattices[END_REF][START_REF] Darst | Dynamical heterogeneity in lattice glass models[END_REF][START_REF] Foini | Quantum Biroli-Mézard model: Glass transition and superfluidity in a quantum lattice glass model[END_REF]. Differently from the other two families, they display, just like atomistic models, a random first-order transition in the mean-field limit. They are therefore ideal candidates to study the interplay between static and dynamic fluctuations in finite dimensions. This is the central motivation for our work.

Recently [START_REF] Nishikawa | Lattice Glass Model in Three Spatial Dimensions[END_REF], a lattice glass model with good glassforming ability in d = 3 was numerically shown to display thermodynamic fluctuations consistent with both RFOT theory and finite-d glass-formers. Here, we study its dynamics to shed light on how structural relaxation occurs in a many-body particle system approaching a RFOT. We find that the dynamics becomes glassy and spatially heterogeneous, with a strong coupling between static and dynamic fluctuations. However, structural relaxation is triggered by emerging quasiparticles composed of localized clusters of particles with low-energy barriers, which gradually relax the correlated slow regions in a manner reminiscent of dynamic facilitation. While statics plays a role in the dynamics, this differs strongly from the predictions of RFOT theory.

We study a binary mixture of particles [START_REF] Nishikawa | Lattice Glass Model in Three Spatial Dimensions[END_REF] on a periodic cubic lattice of linear dimension L and Hamiltonian

H = i j δ(|r i -r j |, 1) -ℓ σi 2 , (1)
with δ(•, •) the Kronecker delta, r i the position of particle i taking values r i,α = 1, 2, • • • L (α = x, y, z), and σ i ∈ {1, 2} specifying the particle type. Each particle type has a preferred number of neighbors, ℓ σ , and quadratic deviations from this number provide the local energy cost, see SM [START_REF]See supplemental material at [URL will be inserted by publisher] for additional descriptions of the model, detailed methods for calculating the relaxation spectrum χ ′′ (ω), and the physical[END_REF]. We set ℓ 1 = 3 and ℓ 2 = 5, fix the density to ϕ = N/L 3 = 0.75, where N is the number of particles, and the concentrations ρ 1 = N 1 /N = 0.4 and ρ 2 = N 2 /N = 0.6 [START_REF] Nishikawa | Lattice Glass Model in Three Spatial Dimensions[END_REF]. We use L = 20, apart from the temperature T = 0.26, where we set L = 10 as the simulations take much longer to converge. To obtain equilibrium configuration of the system defined by Eq. (1) we use a non-local swap dynamics, where two lattice sites with different occupation numbers or particle types are randomly chosen and swapped with Metropolis probability. To study the physical dynamics, we restrict the above rule to pairs of nearest neighbors [START_REF] Nishikawa | Lattice Glass Model in Three Spatial Dimensions[END_REF]. This local dynamics allows us to explore structural relaxation down to T ≈ 0.3, while Ref. [START_REF] Nishikawa | Lattice Glass Model in Three Spatial Dimensions[END_REF] locates the putative Kauzmann transition near T ≈ 0.24. At T = 0.3, the non-local swap provides a comfortable speedup of about 3 , allowing us to explore the equilibrium dynamics of the model at temperature where the physical dynamics is extremely slow. The unit time represents L 3 attempted moves.

The structural relaxation of the system can be followed using the collective overlap In this expression the local overlap q i (t) remains 1 only when the site occupied by particle i at time 0 is again occupied by any particle of the same type at time t later, where q i (t) = j δ(r i (0), r j (t))δ(σ i (0), σ j (t)) and

⟨Q(t)⟩ = 1 1 -Q 0 1 N i q i (t) -Q 0 . (2)
Q 0 = ϕ(ρ 2 1 + ρ 2 2
). The brackets ⟨•⟩ stand for an average over initial equilibrium configurations. The collective overlap in Eq. ( 2) is the lattice analog of the overlap defined for particle systems [START_REF] Kob | Nonmonotonic temperature evolution of dynamic correlations in glass-forming liquids[END_REF]. It is normalized to decay from 1 to 0 when the structure at t = 0 has fully relaxed. Numerical results are shown in Fig. 1(a). At high temperatures T ≫ 1, the system rapidly decorrelates. When T ≲ 0.55, ⟨Q(t)⟩ starts to have a plateau [see zoom in Fig. 1(b)] and the corresponding relaxation time τ α departs from an Arrhenius behavior, see Fig. 1(c).

With decreasing T further, the dynamics keeps slowing down, and at very low temperatures T ≲ 0.325, a more complex short time dynamics emerges. This is better appreciated in the Fourier relaxation spectrum [START_REF] Berthier | Numerical Study of a Fragile Three-Dimensional Kinetically Constrained Model[END_REF]

χ ′′ (ω) = d log τ d ⟨Q(τ )⟩ d log(τ ) ωτ 1 + (ωτ ) 2 , (3)
as shown in Fig. 1(d) (see details in SM [START_REF]See supplemental material at [URL will be inserted by publisher] for additional descriptions of the model, detailed methods for calculating the relaxation spectrum χ ′′ (ω), and the physical[END_REF]). This representation clearly reveals additional relaxation processes taking place at intermediate frequencies between the microscopic peak and the main peak corresponding to the structural relaxation, in qualitative agreement with recent observations in particle models [START_REF] Guiselin | Microscopic origin of excess wings in relaxation spectra of supercooled liquids[END_REF][START_REF] Nishikawa | Collective dynamics in a glass-former with Mari-Kurchan interactions[END_REF].

We turn to the mean-squared displacement (MSD)

∆ 2 (t) = 1 N i r i (t) -r i (0) 2 , (4)
to understand how particle motion leads to structural relaxation, see Fig. 2(a). We find, as usual, that the MSD only becomes diffusive at long enough times, after a transient plateau that becomes longer at lower temperatures and a decreasing diffusion constant. However, the MSD slows down much less than the overlap. For instance, at the lowest temperature T = 0.26 and at t = 3 × 10 8 , ⟨Q(t)⟩ ≃ 0.98 while ∆ 2 (t) ≃ 10 2 . It is a priori surprising that such large particle displacements lead to so little relaxation in the structure.

To better understand this finding, we determine the corresponding van Hove distribution

P t (∆x) = 1 3N i,α δ (∆x -(r i,α (t) -r i,α (0))) , ( 5
)
see Fig. 2(b,c). At high temperatures, the particle dynamics is diffusive and P t (∆x) very close to a Gaussian at any time. In contrast, P t (∆x) is non-Gaussian at short times with nearly-exponential tails [START_REF] Chaudhuri | Universal Nature of Particle Displacements close to Glass and Jamming Transitions[END_REF] when decreasing temperature, and only becomes Gaussian when t ≫ τ α , see T = 0.4 in Fig. 2(b). At very low temperature, e.g., T = 0.26 in Fig. 2(c), P t (∆x) develops extremely extended tails coexisting with a large peak at ∆x = 0. This peak corresponds to a large fraction of particles which have not moved at all since t = 0, coexisting with a population of particles which have covered distances up to tens of lattice sites. Clearly, dynamics is highly heterogeneous, and different particles can exhibit different behavior. Also, the large decoupling between the overlap and the MSD arises because a small fraction of particles travels large distances while many others have not yet moved. The emergence of a small population of fast moving particles in an otherwise nearly frozen backbone is unexpected as it is not directly included in the Hamiltonian in Eq. ( 1), contrary to kinetically constrained models. To understand this feature, we measure for lattice site i in an equilibrium configuration a local energy barrier, ∆e i , defined as the minimum energy cost to swap with one of its nearest neighbors. Such analysis is not possible in offlattice models, but is very easy here. We show in Fig. 3 In Fig. 3(b-d), we show the lattice sites with ∆e ≤ 0 at T = 0.3. A significant fraction of these are dimers and trimers, which result in reversible local particle exchanges. More rarely, we observe a larger cluster, as highlighted in red. Our visualizations indicate that these localized clusters can move very rapidly throughout the system. In the example of Fig. 3, the cluster travels more than L/2 over about 400 time steps. The identity of the particles that belong to the cluster also changes rapidly. Therefore, these fast moving localized clusters are emerging quasiparticles which can move large distances. Particles advected by clusters are responsible for the extended tails in the van Hove distributions. After many quasiparticles have been observed, the sites that have relaxed are not homogeneously distributed, as shown in Fig. 3(e). This suggests the existence of specific paths along which the motion of quasiparticles occurs preferentially. As a corollary, there exist large compact domains in which fast motion cannot occur and can only relax by the repeated motion of many quasiparticles at their boundaries. This represents, for the lattice glass model under study, the analog of the dynamic facilitation reported for off-lattice simulations [START_REF] Keys | Excitations Are Localized and Relaxation Is Hierarchical in Glass-Forming Liquids[END_REF][START_REF] Scalliet | Thirty Milliseconds in the Life of a Supercooled Liquid[END_REF].

To describe more quantitatively the emerging dynamic heterogeneity, we measure the four-point correlation function G 4 (r, t), defined as

G 4 (r, t) = 1 4πr 2 1 N i,j δq i (t)δq j (t)δ(r -|r i -r j |) , (6)
with δq i (t) = q i (t)j ⟨q j (t)⟩ /N the local fluctuation of the overlap. We extract the dynamic lengthscale ξ 4 (t) using the definition G 4 (ξ 4 , t)/G 4 (0, t) = 10 -2 , as well as the dynamical susceptibility χ 4 (t) = dr4πr 2 G 4 (r, t). At high temperatures, G 4 (r, t) decays rapidly over a few lattice sites even near τ α , showing that relaxation is purely local. Spatial correlations revealed by a much slower decay in G 4 (r, t) appear at lower temperatures, see Fig. 4(a) for T = 0.325 where ξ 4 grows up to ξ 4 ≈ 6 near τ α ≈ 10 8 . This evolution is observed for both ξ 4 and χ 4 , whose time dependencies are shown in Fig. 4. These data are familiar [START_REF] Toninelli | Dynamical susceptibility of glass formers: Contrasting the predictions of theoretical scenarios[END_REF], with a slow growth for t ≪ τ α followed by a maximum near τ α . Interestingly, at very low temperatures, the slow growth of ξ 4 towards its peak is compatible with a power law, ξ 4 ∼ t 1/z , with a small exponent 1/z ≈ 0.15, in line with recent off-lattice findings [START_REF] Scalliet | Thirty Milliseconds in the Life of a Supercooled Liquid[END_REF].

We collect all relevant lengthscales in Fig. 5 where we show both ξ 4 and χ 4 measured at their maximum near τ α . Our data indicate that the simple relation χ 4 ∼ ξ holds (not shown), confirming that slow domains have a compact geometry and thus we report the quantity χ

1/3 4
in Fig. 5. We normalize these quantities by their values at temperature T = 0.45, below which important thermodynamic quantities have been determined before [START_REF] Nishikawa | Lattice Glass Model in Three Spatial Dimensions[END_REF]. Two estimates of the configurational entropy [START_REF] Berthier | Configurational entropy of glass-forming liquids[END_REF] were measured. First, the total entropy s(T ) itself was measured using thermodynamic integration. It should be close to the configurational entropy since vibrational contributions are negligible on the lattice. Second, the inverse of the critical coupling field ε c [START_REF] Nishikawa | Lattice Glass Model in Three Spatial Dimensions[END_REF] in the Franz-Parisi scheme [START_REF] Franz | Phase Diagram of Coupled Glassy Systems: A Mean-Field Study[END_REF] also represents a solid estimate of the configurational entropy [START_REF] Berthier | Novel approach to numerical measurements of the configurational entropy in supercooled liquids[END_REF], which is itself inversely proportional to the point-to-set correlation lengthscale quantifying static correlations [START_REF] Kirkpatrick | Scaling concepts for the dynamics of viscous liquids near an ideal glassy state[END_REF][START_REF] Bouchaud | On the Adam-Gibbs-Kirkpatrick-Thirumalai-Wolynes scenario for the viscosity increase in glasses[END_REF][START_REF] Berthier | Configurational entropy of glass-forming liquids[END_REF] (see SM [START_REF]See supplemental material at [URL will be inserted by publisher] for additional descriptions of the model, detailed methods for calculating the relaxation spectrum χ ′′ (ω), and the physical[END_REF] for a review of these connections). These two indirect estimates of a static lengthscale are included in Fig. 5, also rescaled at T = 0.45. They appear to grow at least as significantly (if not more strongly) than dynamic lengthscales. The coupling between static and dynamic lengthscales is therefore stronger here than in atomistic models [START_REF] Kob | Nonmonotonic temperature evolution of dynamic correlations in glass-forming liquids[END_REF][START_REF] Scalliet | Thirty Milliseconds in the Life of a Supercooled Liquid[END_REF][START_REF] Charbonneau | Decorrelation of the static and dynamic length scales in hard-sphere glass formers[END_REF][START_REF] Berthier | Static point-to-set correlations in glass-forming liquids[END_REF]. This finding encourages us to directly test the relation between timescales and lengthscales predicted in RFOT theory, see Fig. 5, where the relation log τ α ∼ ξ ψ 4 /T with ψ ≈ 0.84 is followed by the data. While a larger exponent ψ (between d/2 and d) is expected from cooperative relaxation events, it is not surprising that a lower apparent value is found here, as dynamic facilitation via mobile quasiparticles provides more efficient dynamic pathways. This result indicates that the RFOT theory picture of cooperative activated dynamics is not dominant for the studied model. Interestingly, values ψ < 1 were also inferred from analysis of experimental data [START_REF] Capaccioli | Dynamically Correlated Regions and Configurational Entropy in Supercooled Liquids[END_REF][START_REF] Ozawa | Does the Adam-Gibbs relation hold in simulated supercooled liquids?[END_REF] and numerically [START_REF] Karmakar | Growing length and time scales in glass-forming liquids[END_REF][START_REF] Cammarota | Numerical determination of the exponents controlling the relationship between time, length, and temperature in glass-forming liquids[END_REF].

In conclusion, our results establish that, differently from all other classes of lattice models for glassy dynamics, the 3d lattice glass model studied here displays thermodynamic and dynamic properties that compare fa-vorably with atomistic glass-formers, with growing static point-to-set and four-point dynamic lengthscales. While static and dynamic fluctuations appear to be strongly coupled, the relaxation dynamics is nevertheless different from the RFOT theory description invoking cooperative relaxation events. Instead, while the system genuinely approaches a random first-order transition, it also leaves behind a small population of weakly constrained particles forming emergent quasiparticles whose fast propagation slowly relaxes the entire system eventually at large times. These quasiparticles thus resemble the dynamic defects postulated in kinetically constrained models, or those emerging from the specific interactions of plaquette models. They also lead to signatures at intermediate timescales that are reminiscent of recent findings in atomistic models [START_REF] Guiselin | Microscopic origin of excess wings in relaxation spectra of supercooled liquids[END_REF][START_REF] Scalliet | Thirty Milliseconds in the Life of a Supercooled Liquid[END_REF].

While details of the emergent quasiparticles and dynamical defects may be specific to our model, the physical picture emerging from our study appears generic. It is consistent with the recent body of results regarding glassy dynamics at very low temperatures and it supports the generic conclusion that cooperative events from statically correlated domains are preempted by faster relaxation events involving localized regions acting as facilitating defects [START_REF] Nishikawa | Collective dynamics in a glass-former with Mari-Kurchan interactions[END_REF][START_REF] Scalliet | Thirty Milliseconds in the Life of a Supercooled Liquid[END_REF][START_REF] Mishra | Dynamical facilitation governs glassy dynamics in suspensions of colloidal ellipsoids[END_REF]. Our results provide a unified picture for glassy dynamics near the glass transition where facilitated dynamics takes place near localized regions, whose temperature evolution is strongly connected to emerging static correlations. We hope that future studies will confirm the validity of these conclusions across a broader range of models (including molecular systems) and spatial dimensions, and study the fate of models even closer to the putative ideal glass transition where cooperative events could finally become dynamically relevant. spectrum, and we can then safely say that the systematic error due to a limited simulation time is negligible there, see Fig. S1 for the spectra at T = 0.3. In Fig. 1(d), we show the relaxation spectrum only for the ω regime where we have confirmed |χ ′′ (ω) -χ′′ (ω)| /χ ′′ (ω) < 2 × 10 -2 , which implies that truncation of the Fourier integral contributes very little to the results.

The Franz-Parisi potential and the ε-coupling construction We consider the system coupled to a fixed configuration r ′ sampled at inverse temperature β, as originally proposed in Ref. [S4]. The total Hamiltonian of this coupled system is H ε (r|r ′ ) = H(r) -εQ(r, r ′ ). (S4) where Q(r, r ′ ) is the instantaneous collective overlap between two configurations r and r ′ ,

Q(r, r ′ ) = 1 1 -Q 0   1 N i,j δ(r i , r ′ j )δ(σ i , σ ′ j ) -Q 0   . ( S5
)
Coupling to a fixed configuration introduces quenched disorder into the system, and the system can have a finite overlap with the fixed configuration. We measure the overlap distribution averaged over fixed configurations,

[P ε (q)] = [⟨δ(q -Q(r, r ′ ))⟩ ε ], (S6)
where the brackets [•] stands for an average over fixed configurations and ⟨•⟩ ε is the thermal average of the coupled system. We then calculate the overlap susceptibility

χ SG = N q 2 ε -⟨q⟩ 2 ε (S7)
as a function of the coupling ε. At low temperature, Ref. [S5] shows that the coupling induces a first-order phase transition at ε c , where the susceptibility χ SG diverges in the limit N → ∞. We thus estimate the transition point ε c as the peak position in χ SG of the system with L = 12. The Franz-Parisi potential V (Q) is related to the probability distribution of the overlap, V (Q) ∼ -(βN ) -1 log[P (Q)]] and represents the free energy cost to observe the value Q of the overlap. The free energy difference between large and small Q values is proportional to ε c . It is also an estimate of the configurational entropy s conf [S6] and the point-to-set static lengthscale ξ PTS [S7] with a nontrivial surface exponent θ < d as

ε c ∼ s conf ∼ ξ θ-d
PTS .

(S8)

FIG. 1 .

 1 FIG. 1. (a, b) Temperature evolution of the collective overlap ⟨Q(t)⟩. (c) Arrhenius plot of the relaxation time τα; the broken line is an Arrhenius fit valid at high temperatures. (d) Relaxation spectrum from Eq. (3); dashed curves represent the spectrum corresponding to the final stretched exponential decay of the overlap.

FIG. 2 .

 2 FIG. 2. (a) Time dependence of the MSD; diffusive behavior is indicated with dashed lines. Van Hove distribution functions at (b) T = 0.4 where τα ≃ 2 × 10 5 and (c) T = 0.26 where τα ≫ 10 10 .

  (a) the cumulative probability distribution of ∆e/T . Lattice sites with ∆e/T ≥ 10, for instance, are only swapped with probability ≤ 10 -4 . At high temperature, a large fraction of lattice sites have ∆e/T ≤ 0 and can move with no rejection. With decreasing temperature, local energy barriers become very large. At T = 0.3, a majority of sites have ∆e/T ≥ 26, yielding a local timescale ≥ 10 11 . Strikingly, however, a fraction of about 0.1% of the lattice sites remains totally free to swap.

FIG. 3 .

 3 FIG. 3. (a) Cumulative distribution of local energy barriers. The vertical line indicates ∆e = 0. (b-d) Snapshots showing lattice sites with ∆e ≤ 0 at T = 0.3 at three different times. The time intervals between (b) and (c) and between (c) and (d) are 50 and 350, respectively. Sites highlighted in red belong to a quasiparticle moving rapidly. (e) Lattice sites with states that differ between times 0 and 7 × 10 4 , showing the superposition of multiple quasiparticle paths.

FIG. 4 .

 4 FIG. 4. (a) Four-point correlation function G4(r, t) at T = 0.325 where τα ≈ 10 8 . (b) Dynamical susceptibility χ4(t) and (c) dynamical lengthscale ξ4(t) demonstrating increasing dynamic heterogeneity.

FIG. 5 .

 5 FIG.5. Evolution of two estimates of a dynamic lengthscale, (χ4) 1/3 and ξ4, and of two estimates of a static correlation lengthscale 1/s(T ) and 1/εc(T ). Each quantity is normalized by its value at T = 0.45. Inset shows T log(τα) as a function of ξ4, the dashed line is a fit to log(τα) ∼ ξ ψ 4 /T with ψ = 0.84.

  FIG.S1. Relaxation spectra of the lattice glass model at T = 0.3 from the real and synthetic data. At ω ≳ 10 -6 , two spectra are almost identical while at smaller ω, they significantly deviate from each other due to the limited simulation time. In the main figure, we only show the real data when they coincide with the synthetic one. The broken line represents the spectrum of the stretched exponential decay fitted to the observed overlap ⟨Q(t)⟩ and the arrow indicates the smallest ω for χ ′′ (ω) we show in the main figure.

Proceedings of the National Academy of Sciences 111,

We thank P. Charbonneau, A. Ikeda, H. Ikeda, J. Takahashi, H. Yoshino, and F. Zamponi for useful discussions. Y.N. acknowledges support from JSPS KAKENHI (Grant No. 22K13968), L.B.'s work is supported by a grant from the Simons Foundation (#454933).

Supplemental Material for "Collective relaxation dynamics in a three-dimensional lattice glass model"

Yoshihiko Nishikawa 1 and Ludovic Berthier 2, 3

1 Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan 2 Laboratoire Charles Coulomb (L2C), Université de Montpellier, CNRS, 34095 Montpellier, France 3 Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK

Model

We describe the lattice glass model in more detail. The Hamiltonian of the model is

where δ(•) is the Kronecker delta. Each particle lies on a lattice site, and each site cannot be occupied by more than one particle. The energy of particle i depends on the number of neighboring particles, which is counted in the term j δ(|r ir j |, 1), and on the parameter ℓ σi with σ i ∈ {1, 2} denoting the particle type. The energy for particle i is zero only when j δ(|r ir j |, 1) = ℓ i , and it is positive when the equality is not satisfied. Therefore it is energetically favorable that particles surround themselves by exactly ℓ i neighbors, but simultaneously satisfying this constraint for all particles is not possible. In our study, we use a binary mixture of particles with ℓ 1 = 3 or ℓ 2 = 5, and we can think of particles with a smaller ℓ as being "larger" than particles with a larger ℓ which can possess more neighbors.

We mention the relation with another lattice glass model proposed by Biroli and Mézard (BM) [S1] and its generalization [S2]. The original BM model has a hard-core interaction, where, for particle i with a parameter ℓ i , the number of neighboring particles cannot exceed ℓ i : The energy cost becomes infinity when the number of neighbors becomes larger than ℓ i , and it is zero otherwise. Ref. [S2] then introduced a generalized BM model with the Hamiltonian

where θ(•) is the Heaviside step function. In the zero-temperature limit, this generalized model converges to the original, hard-core BM model. Comparing the generalized BM model and our lattice glass model, we notice that the number of neighbors achieving the minimum energy differs in the models: In our lattice glass model, only when j δ(|r ir j |, 1) = ℓ i the interaction energy is zero while, in the generalized BM model, any number of neighbors smaller than or equal to ℓ i yields zero energy. While all three models only differ by the specific choice of the constraints, it results in very different dynamic and static behaviors at low temperature and high density.

The relaxation spectrum

The relaxation spectrum χ ′′ (ω) is given by

To compute the spectrum [S3], we first interpolate ⟨Q(t)⟩ on a very fine temporal grid, and we then numerically differentiate the interpolated ⟨Q(t)⟩, to perform the numerical integration of d⟨Q(τ )⟩ d log(τ ) ωτ 1+(ωτ ) 2 . However, at low temperatures, e.g., T = 0.3, 0.325, and 0.35, we have not reached a timescale where ⟨Q(t)⟩ ≃ 0 due to extremely long relaxation times, which could result in systematic errors in χ ′′ (ω) at small ω. To avoid the errors, we compare χ ′′ (ω) from ⟨Q(t)⟩ and the one from an extrapolated ⟨Q(t)⟩ as follows. We first fit a stretched exponential function ∼ exp(-(t/τ ) β ) to the observed collective overlap using only data points where ⟨Q(t)⟩ < 0.9.

We then construct an extrapolated overlap Q(t) that decays to 0 at a very long time and compute the relaxation spectrum χ′′ (ω) from it. For a given ω, if χ′′ (ω) ≃ χ ′′ (ω), then the extrapolation does not affect the relaxation arXiv:2307.08110v2 [cond-mat.dis-nn] 8 Feb 2024