Natsuda Klongvessa

Christophe Ybert

Cécile Cottin-Bizonne

Takeshi Kawasaki

Mathieu Leocmach

Aging or DEAD: origin of the non-monotonic response

come

I. INTRODUCTION

Half a century ago, Goldstein [START_REF] Goldstein | Viscous Liquids and the Glass Transition: A Potential Energy Barrier Picture[END_REF] proposed that the relaxation of glassy systems is dominated by free energy barriers high compared to thermal energy. At low enough temperatures or high enough densities some barriers are so high that the system cannot explore its whole energy landscape and becomes nonergodic [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] , with the available phase-space region reducing in a non trivial way with temperature or density [START_REF] Charbonneau | Fractal free energy landscapes in structural glasses[END_REF][START_REF] Charbonneau | Glass and jamming transitions: From exact results to finite-dimensional descriptions[END_REF] . Nevertheless partial relaxation still occurs in the nonergodic glass state and typically slows down with the 'age' of the system, i.e. the time after its preparation in the glass state [START_REF] Struik | Physical aging in amorphous polymers and other materials[END_REF][START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF] . More generally, the system behavior depends on its whole history and preparation protocol. Therefore probing the glass state is an out-of-equilibrium issue, and has been performed by studying the response to external driving, for instance temperature cycles [START_REF] Vincent | Slow dynamics and aging in spin glasses[END_REF][START_REF] Scalliet | Rejuvenation and Memory Effects in a Structural Glass[END_REF] or global mechanical driving [START_REF] Viasnoff | Rejuvenation and Overaging in a Colloidal Glass under Shear[END_REF][START_REF] Ikeda | Disentangling glass and jamming physics in the rheology of soft materials[END_REF] .

More recently has emerged a novel class of systems whereby the additional nonequilibrium condition are brought by the self-propulsive properties of each constitutive particles. The physics of such active glasses [START_REF] Janssen | Aging and rejuvenation of active matter under topological constraints[END_REF][START_REF] Janssen | Active glasses[END_REF] is expected to be relevant to describe biological tissues [START_REF] Garcia | Physics of active jamming during collective cellular motion in a monolayer[END_REF][START_REF] Bi | Motilitydriven glass and jamming transitions in biological tissues[END_REF] . Starting from a zoological curiosity such as flocks of birds and schools of fish, active matter has received more attention and is established as a novel area to study new physics in a scope of nonequilibrium systems [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF][START_REF] Vicsek | Collective motion[END_REF][START_REF] Marchetti | Hydrodynamics of soft active matter[END_REF][START_REF] Bechinger | Active particles in complex and crowded environments[END_REF] . In a) Now in: Faculty of Medicine, Bangkokthonburi University, 10170 Bangkok, Thailand rather dilute conditions, several studies have shown that the state of many active systems can be described by an effective temperature [START_REF] Howse | Self-motile colloidal particles: From directed propulsion to random walk[END_REF][START_REF] Tailleur | Sedimentation, trapping, and rectification of dilute bacteria[END_REF][START_REF] Palacci | Sedimentation and effective temperature of active colloidal suspensions[END_REF][START_REF] Ginot | Nonequilibrium equation of state in suspensions of active colloids[END_REF] . In previous experimental studies [START_REF] Klongvessa | Active glass: Ergodicity breaking dramatically affects response to self-propulsion[END_REF][START_REF] Klongvessa | Nonmonotonic behavior in dense assemblies of active colloids[END_REF] , some of us have shown that this mapping holds up close to ergodicity breaking but not beyond. Our main observation is that the relaxation time displays nonmonotonic response with the self-propulsion force. Starting from the nonergodic passive system, the relaxation time increases when the system becomes weakly self-propelled and then decreases when the activity level is high enough to restore ergodicity.

Other nonmonotonic behaviors have been reported from theoretical [START_REF] Liluashvili | Mode-coupling theory for active Brownian particles[END_REF] and numerical [START_REF] Debets | Cage Length Controls the Nonmonotonic Dynamics of Active Glassy Matter[END_REF] studies of ergodic glassy systems. Indeed, close to the glass transition the dynamics does not depend on a single active parameter but on at least two parameters characterizing the activity, e.g. the persistence time and the Peclet number. For instance, increasing the persistence time can shift the glass transition towards higher or lower densities depending on the effective temperature [START_REF] Berthier | How active forces influence nonequilibrium glass transitions[END_REF] . Ultimately, the direction of the shift of glass transition depends on the microscopic details of the activity [START_REF] Nandi | A random first-order transition theory for an active glass[END_REF] .

Beyond the glass transition, some recent studies reveal that the phenomenology of passive glass is qualitatively altered by activity: plasticity and turbulence [START_REF] Mandal | Extreme active matter at high densities[END_REF] , aging behavior [START_REF] Mandal | Multiple types of aging in active glasses[END_REF][START_REF] Janzen | Aging in thermal active glasses[END_REF] , and dynamics heterogeneity [START_REF] Paul | Dynamic heterogeneity in active glass-forming liquids is qualitatively different com-pared to its equilibrium behaviour[END_REF] . However, most of the above effects occur at relatively strong selfpropulsion force, and more scarce are the studies that focus on the perturbation of the thermal motion by a weak propulsion force; precisely the regime where we observed experimentally a nonmonotonic response to activity [START_REF] Klongvessa | Active glass: Ergodicity breaking dramatically affects response to self-propulsion[END_REF][START_REF] Klongvessa | Nonmonotonic behavior in dense assemblies of active colloids[END_REF] .

In Ref. [START_REF] Klongvessa | Active glass: Ergodicity breaking dramatically affects response to self-propulsion[END_REF] , we proposed a model based on a Deadlock due to the Emergence of Active Directionality (DEAD).

Briefly, this model is based on a Goldstein 1 -like picture with an (effective) single particle trapped in a cage of high free energy barriers. We consider that a small amount of self-propulsion does not affect the height of the energy barriers. However, the nature of the motion within the cage affects the frequency at which the particle could attempt to hop out of its cage. A Brownian particle randomly explores its cage and can often test weak points of the cage (low energy barriers). A particle with persistent directed motion will be able to test a single direction until it reorients. At the scale of the cage, persistent motion can be less efficient than Brownian motion, thus the attempt frequency lower and the structural rearrangement at low activities slower than in the equivalent passive system.

Several experimental issues prevented us to test further the DEAD model. (i) The system density was inhomogeneous. (ii) We had no access to the orientation of the self-propulsion force, which prevents testing the cage exploration scenario. (iii) We had a poor control on the system age and history, which prevents ensemble averaging and a systematic study on aging behavior.

Furthermore, in 2D systems like in our experiments, long wavelength sound modes [START_REF] Shiba | Unveiling Dimensionality Dependence of Glassy Dynamics: 2D Infinite Fluctuation Eclipses Inherent Structural Relaxation[END_REF] , also called Mermin-Wagner fluctuations [START_REF] Illing | Mermin-Wagner fluctuations in 2D amorphous solids[END_REF] , are expected and may blur our interpretation in term of a local cage. Indeed, in Ref. [START_REF] Klongvessa | Nonmonotonic behavior in dense assemblies of active colloids[END_REF] we found that most of the observed relaxation was not due to structural relaxation but to motion at a scale of a few particles that we interpreted as collective motion but may have been long wavelength sound modes.

In this work, we use numerical simulations to (i) reproduce the nonmonotonic behavior in dense, nonergodic assembly of soft 2D active Brownian particles while controlling for long wavelength sound modes; (ii) overcome experimental limitations, further characterize this phenomenon; (iii) test the DEAD model and alternative explanations. We provide the simulation details in Section II, confirm the nonmonotonic behavior in Section III, but refute the cage exploration postulated by the DEAD model in Section IV. In Section V, we rather link the nonmonotonic behavior to aging, which is a hallmark of glassy dynamics. Finally, we discuss our results and conclude in Section VI.

II. SIMULATION DETAILS

A. Equations of motion

To simulate our active system, we use 2D Active Brownian Particle (ABP) model [START_REF] Fily | Athermal phase separation of self-propelled particles with no alignment[END_REF][START_REF] Volpe | Simulation of the active brownian motion of a microswimmer[END_REF] , which consists in the overdamped Langevin equation for Brownian particles with an additional active force. The equations of motion of each particle i of diameter σ i are given by

ζ ṙi = ξ i + F I i + f ûi , (1) θi = η i , (2)
where ζ is a drag coefficient. r i = (x i , y i ) and θ i are the particle position and orientation, respectively. The term ξ i stands for the thermal noise of zero-mean and it obeys ξ i (t)ξ j (t ) = 2k B T ζ δ ij δ(t -t )1, where δ ij and δ(t -t ) are the Kronecker and Dirac delta function, respectively, k B is Boltzmann's constant and 1 is the identity matrix. The interaction term between particles is

F I i (t) = -Σ j ∂U (rij )
∂ri , where U (r ij ) is the interaction potential between particle i and j separated by the distance r ij . In this work, we use Harmonic sphere potential [START_REF] O'hern | Random Packings of Frictionless Particles[END_REF][START_REF] Berthier | Compressing nearly hard sphere fluids increases glass fragility[END_REF] , U (r ij ) = (1 -r ij /σ ij ) 2 /2, with the cutoff range σ ij = (σ i + σ j )/2. The active force is represented by a self-propulsion term f ûi (t), where f is a self-propulsion force of a constant magnitude. ûi = (cos θ i , sin θ i ) is a unit vector subjected to rotational diffusion equation (Eq. 2). η i is a zero-mean thermal noise that follows

η i (t)η j (t ) = 2D R δ ij δ(t -t ), and D R is a rotational diffusion coefficient.
By this definition, our particles always have both translational and rotational diffusion originating from Brownian thermal noise. The orientation of the particles θ i is the self-propulsion direction in the case of active particles, f > 0. For passive particles, f = 0, despite their orientation also changing diffusively, there is no influence from this change on the displacement of the particles.

We simulate N = 1000 particles with periodic boundary condition. To avoid crystallization, particle diameter has 10% polydispersity with Gaussian distribution centered at σ 0 = 1. We detect medium-range crystalline order typical of polydisperse glassy systems [START_REF] Kawasaki | Structural evolution in the aging process of supercooled colloidal liquids[END_REF] , but neither long-range crystalline order nor well defined grains and boundaries. The units of length, energy and time in our simulation are σ 0 , and σ 2 0 ζ/ , respectively.

B. Model parameters

Translational diffusion and self-propulsion are the two mechanisms that govern motion for an active Brownian particle. They are characterised by translational (D T = k B T /ζ) and rotational (D R ) diffusion coefficient, respectively. D R sets the rotational time τ R = D -1 R that marks for an individual particle the crossover between ballistic motion at short times and effective diffusion at long time scales [START_REF] Howse | Self-motile colloidal particles: From directed propulsion to random walk[END_REF][START_REF] Volpe | Simulation of the active brownian motion of a microswimmer[END_REF] . In experimental realisation of ABP, D R is linked to D T through a geometric coefficient. For instance for spherical particles of radius σ 0 rotating in 3D, we have D R σ 2 0 /D T = 3. However, the presence of a solid wall can shift this coefficient toward larger values [START_REF] Gomez-Solano | Tuning the motility and directionality of self-propelled colloids[END_REF] . Although some numerical studies keep D R σ 2 0 /D T = 3 [START_REF] Volpe | Simulation of the active brownian motion of a microswimmer[END_REF][START_REF] Digregorio | Full phase diagram of active brownian disks: From melting to motility-induced phase separation[END_REF] , most treat D R as an independent parameter [START_REF] Mandal | Extreme active matter at high densities[END_REF][START_REF] Mandal | Multiple types of aging in active glasses[END_REF] . In the present study, we fix the temperature at k B T = 0.004 and set the rotational diffusion coefficient such that D R σ 2 0 /D T = 10, corresponding to a rotational time τ R = D -1 R = 25. We will discuss further the significance of the value of D R in Section. IV B.

We define the Péclet number Pe = f σ 0 /k B T to describe the activity level of the system. In this work, we will investigate the system dynamics at various activity levels Pe and densities φ = π N i σ 2 i /4L 2 , where L is the length of the simulation domain.

C. Simulation procedure

We solve the equation of motion, Eq. 1-2, using Heun's method with a step size δt = 3 × 10 -4 . Initially, particles are randomly put in space with no self-propulsion force, f = 0. The system quickly relaxes over a short period of time during which originally overlapping particles separate. This rapid evolution is then followed by a much slower one corresponding for instance to aging effects in the glassy state (see further details in Section V). The cross-over between the initialization stage and the later evolution can be monitored through the system potential energy. For all densities this initialization stage is set to a fixed duration t ini = 20 (except in Fig 7b andd, as will be explained in Sec. V B). After that, we turn on the self-propulsion, f = Pe × k B T /σ 0 , which defines the origin of waiting times t w = 0.

Since this preparation protocol involves infinitely fast annealing from random configuration, we expect strong aging effects [START_REF] O'hern | Random Packings of Frictionless Particles[END_REF][START_REF] Berthier | Compressing nearly hard sphere fluids increases glass fragility[END_REF] that will be addressed in Section V. The production run starts at the waiting time t w = 1.5 × 10 4 (unless stated otherwise) over the time interval ∆t = 3 × 10 3 . If the system has completely relaxed during this time interval (see results in Section III A), the simulation results are a time-average over six successive production runs. Otherwise, they are an ensemble-average over 50 independent runs at the same waiting time.

III. ACTIVE GLASSY BEHAVIOR

A. Dynamics before glass transition

In general, the absolute motion of a particle in a glassy system can be split between the motion relative to the cage and the motion of the cage [START_REF] Mazoyer | Dynamics of particles and cages in an experimental 2D glass former[END_REF] . In 2D, it is well known that the contribution of the motion of the cage cannot be neglected due to long wavelength sound modes [START_REF] Shiba | Unveiling Dimensionality Dependence of Glassy Dynamics: 2D Infinite Fluctuation Eclipses Inherent Structural Relaxation[END_REF] , also called Mermin-Wagner fluctuations [START_REF] Illing | Mermin-Wagner fluctuations in 2D amorphous solids[END_REF] which are systemsize dependent. In order to focus on structural rearrangement and get rid of finite-size effects, we define the cagerelative motion of particle i between time t 0 and time t 1 ,

∆r CR i (t 0 , t 1 ) = ∆r i (t 0 , t 1 ) - 1 6 j∈NNi ∆r j (t 0 , t 1 ), (3)
where ∆r i (t 0 , t 1 ) = r i (t 1 )-r i (t 0 ) is the absolute displacement of particle i and NN i is the set of the six-nearest neighbors of particle i at time t 0 . To quantify the system relaxation, we use the cagerelative self-intermediate scattering function F s (q, ∆t), which is expressed as

F s (q, ∆t, t w ) = 1 N N i=1 e iq•∆r CR i (tw,tw+∆t) , (4)
where . . . denotes the average among different runs, either time average or ensemble average as described in Section II C. We choose |q| = 2π, and we also average among four perpendicular directions in Fourier space.

For the sake of readability, in the following, we will use the simpler notation F s (∆t), and t w will be specified separately.

In Fig. 1a, we show F s (∆t) obtained from the passive system, Pe = 0, at various densities ranging from φ = 0.17 to 1.03. For φ ≤ 0.81, F s (∆t) fully decays within the time interval of interest. This means that these systems have completely relaxed and we can per-form a time-average from longer production runs. As density increases, the relaxation proceeds in two steps with a well-defined plateau in between. The system takes longer time to relax and for φ > 0.81 F s (∆t) cannot fully decay within our measurement time. Such a system is not in a steady state and the results shown are obtained from the ensemble averaging at t w = 1.5 × 10 4 . The relaxation of absolute positions (not relative to the cage) display a slopped plateau due to long-range sound modes (not shown). At very high densities, harmonic particles are known to exhibit counter intuitive behavior: due to the bounded potential particles can fully overlap [START_REF] Jacquin | Anomalous structural evolution of soft particles: Equibrium liquid state theory[END_REF] , thus relax faster than at more moderate densities [START_REF] Berthier | Increasing the density melts ultrasoft colloidal glasses[END_REF] . We confirm this reentrant behavior in our passive simulations for φ > 1.1 (not shown). In the following, we will remain at densities below this limit.

We show in Fig. 1b how activity influences the system relaxation. At one fixed density φ = 0.91, the passive system is not relaxed whereas the highly active system with Pe = 4.0 manages to relax. We can also see that the system relaxation is enhanced by activity: the higher the activity level, the faster relaxation.

When F s (∆t) fully relaxes, we can define the relaxation time τ from the time such that F s (∆t) has reached the threshold 1/e. However, when we deal with correlation functions that do not completely relax, we cannot characterize the full relaxation and have to focus on the exit from the plateau. For convenience, we fit this exit with a stretched exponential A exp -(t/T ) β , where both A = 0.89 and β = 0.45 are fitted once at φ = 0.98 in the passive case and then kept constant. T is thus the only fitting parameter. In order to ensure a smooth connection between fully and poorly relaxing states, we intersect the stretched exponential resulting from the fit with the threshold 1/e to obtain a characteristic time τ , as shown on Fig 1a . Although we will call this characteristic time a 'relaxation time' it only quantifies the exit time from the plateau and not the full relaxation. In particular, we make no claim on the shape of F s (∆t) beyond our measurements. Now that we have characterized the exit from the plateau with a single time scale, we can show its dependence on density in Fig. 1c. Starting from the lowest density, the relaxation time τ increases as the system is denser, and this is true for both passive and active systems. The first part of the increase, is well-described by the Vogel-Tamman-Fulcher (VTF) form,

τ α = τ 0 exp Bφ * φ * -φ , (5)
where τ 0 is associated to the relaxation time obtained from the limit φ → 0, B is a constant quantifying the steepness of τ (φ) dependence, and φ * is the density at which (Eq. 5 diverges. At each activity level we perform the VTF fit only using the densities that reach a steady state. We fix the parameter τ 0 , which can be obtained directly from the simulation in the very dilute limit φ = 0.01, and B = 0.75 obtained from the pas-sive case. This leaves φ * as the only fitting parameter for the other activity levels. Our results show that activity pushes glass transition φ * towards a higher density (inset) as also reported in previous simulation [START_REF] Ni | Pushing the glass transition towards random close packing using self-propelled hard spheres[END_REF] and experimental [START_REF] Klongvessa | Nonmonotonic behavior in dense assemblies of active colloids[END_REF] works. We note that in Ref. 46 the value of B was reported to be decreasing with Pe. However, in this work, we are focusing on the very low activity limit where this decrease is negligible. Even if we leave B as a free parameter, the difference in B between the passive case and the highest activity level of interest (Pe = 4) is still smaller than the uncertainty from the fitting, which is about 0.1.

The system relaxation is well-described by the VTF relation for φ/φ * (Pe) < 0.85, at which the system is in an ergodic supercooled state. As it has been shown in the experimental work [START_REF] Klongvessa | Nonmonotonic behavior in dense assemblies of active colloids[END_REF] , our simulation verifies that the density-dependence relaxation of the passive and active systems in the supercooled state can be collapsed into one master curve. That is to say, the mapping between the passive (equilibrium) and active (nonequilibrium) system in the supercooled state is valid.

B. Dynamics in the glass state

At high densities, we observe on Fig 1a that the increase of relaxation time τ is slower than the VTF trend and we notice the beginning of a saturation of τ at φ/φ * ≈ 1, which is typical of soft particles [START_REF] Philippe | Glass transition of soft colloids[END_REF] and in the case of bounded potential is the signature of a reentrant melting [START_REF] Berthier | Increasing the density melts ultrasoft colloidal glasses[END_REF] . Although in the passive case the maximum of τ (φ) is located at a higher density than all explored densities, at Pe ≤ 2 we observe this maximum within our explored density range. The position of the maximum shifts toward lower densities with activity. This behavior is reminiscent of the temperature response of harmonic spheres [START_REF] Jacquin | Anomalous structural evolution of soft particles: Equibrium liquid state theory[END_REF] . However, the previous mapping between passive and active states fails in the glass state: for increasing activity the deviations from VTF occurs at lower reduced density and the curves do not collapse. In other words, beyond the glass transition the behavior of the active system cannot be described by an effective passive system. We actually observe that the maximum of τ (φ)/τ 0 exhibits a nonmonotonic behavior with the activity level. All curves at 0.2 ≤ Pe ≤ 1 are higher than the passive case, whereas all curves at Pe > 1 are lower than the passive case. Although a shift in reentrant melting might explain the decrease of rescaled relaxation time at high Pe, such a monotonic shift cannot be the cause of the slowing down at low Pe.

To confirm that the increase in relaxation time is not an artifact of our fitting procedure, we display directly the dependence of F s (∆t) on activity level at high density (φ = 1.03) in Fig. 2a. In particular, we focus on the exit from the plateau. The inset of Fig. 2a curve), F s (∆t) is significantly above the passive case (black curve), indicating a delayed exit from the plateau. However, when activity is high enough (Pe ≥ 2, purple to magenta curves), the exit from the plateau is faster than the reference passive case and the system relaxes all the more fast than Pe increases.

In Fig. 2b, we quantify the system relaxation using the relaxation time τ extracted from F s (∆t), normalized by its low density value τ 0 (Pe). Even once the trivial enhancement of the dynamics by self-propulsion taken into account, the relaxation time drops by two orders of magnitude between the passive (Pe = 0) and the most active system (Pe = 4). However at low activities (0 < Pe ≤ 1) the relaxation time is slower than the passive case by a factor ≈ 2. Qualitatively similar results are obtained without the normalization by τ 0 (not shown).

This nonmonotonic behavior is the key observation of our previous experiments [START_REF] Klongvessa | Active glass: Ergodicity breaking dramatically affects response to self-propulsion[END_REF] , and we show here that it can be recovered within a simple numerical glass model. Beyond this first achievement, we describe in the next section the numerical phase diagram to demonstrate that the in silico system is fully consistent with experiments to date. Then we are in a position for carrying new detailed characterizations aiming at deciphering the origins of this nonmonotonic behavior.

C. State diagram

The system behavior, which depends on both density φ and activity level Pe, is summarised in the state diagram shown in Fig. 3. The rescaled relaxation time τ /τ 0 is represented by colors. The fitted divergence φ * (Pe) of (Eq. 5) are displayed by the black triangles, and we can use them to illustrate the glass region (grey zone). In the passive system, the saturation of relaxation time occurs around τ /τ 0 ≈ 9 × 10 3 . We draw a contour line (blue line) around the region exceeding that value.

The system behavior at one fixed density can be understood such that we travel vertically from zero to nonzero Pe on the state diagram. Within this visualization, we can locate the nonmonotonic behavior (the rise and fall of τ ) occurs at sufficiently high density inside the glass state. Starting from the passive state at about φ > 0.98, τ increases when the system becomes slightly active, and stays higher than the passive τ at low level of activity (0 < Pe ≤ 1). As the activity level increases further, τ drops and we can see that the system is about to leave the glass state in the upwards direction.

This state diagram is consistent with our experimental measurements [START_REF] Klongvessa | Active glass: Ergodicity breaking dramatically affects response to self-propulsion[END_REF] . It shows that the behavior of an active glass at short aging time cannot be mapped on the passive soft sphere state diagram simply by replacing the temperature by Pe. Now that our simulations proved to capture the salient features of experimental active colloidal glasses, we use them to zoom in to the cage level in order to investigate the influence of activity on structural relaxation behavior.

IV. CAGE EXPLORATION AND ESCAPE

Previously, we have explained the slowing down of structural rearrangements at low activities by an inefficient exploration of the cage [START_REF] Klongvessa | Active glass: Ergodicity breaking dramatically affects response to self-propulsion[END_REF] . With the present simulations, we have the opportunity to test the Deadlock by the Emergence of Directionality (DEAD) model. Contrary to experiments, we have access to the orientation of the propulsion of the particles θ i and thus can test our hypothesis on cage exploration.

A. Angular distribution of nearest neighbors

In our previous attempt to understand the experimental non monotonic behavior, we argued that the DEAD occurs when a particle is more confined by its self-propulsion than by the cage [START_REF] Klongvessa | Active glass: Ergodicity breaking dramatically affects response to self-propulsion[END_REF] . Indeed, for times shorter than τ R , the propulsion force can be considered of constant direction and in analogy with sedimentationdiffusion, the confinement length λ P = Pe -1 σ emerges. For Peclet numbers larger than 1, a particle against a hard barrier is confined within less than its own diame- ter. Here, we will check whether this confinement effect exists in our simulations.

For an active particle (Pe > 0), θ i is the direction of self-propulsion. For a passive particle (Pe = 0), we simulated the evolution of θ i although the force attached to the particle is null. Here we investigate how the nearest neighbors of particle i distribute around it with respect to θ i . The azimuthal correlation function c(θ) is defined as,

c(θ) = 2π ∆θ 1 N 1 n N i=1 j∈NNi δ(θ ij ) , ( 6
)
where ∆θ is the bin size. NN i is a set containing n nearest neighbors of particle i. θ ij is the angular position of particle j with respect to the orientation of particle i,

θ ij = θ i -tan -1 y j -y i x j -x i . (7)
δ in Eq. 6 is an rectangular impulse function such that

δ(θ ij ) = 1, if θ < θ ij < θ + ∆θ 0, otherwise. (8)
First, let us focus on the first nearest neighbor. In Fig. 4a, we show the azimuthal correlation c(θ) of the nearest neighbor. The inset illustrates the notation of the direction θ: θ = 0 corresponds to the neighbor being 'in front' of the particle, i.e. in the direction of selfpropulsion for an active particle (arrow). Conversely, the 'rear' corresponds to θ = ±π. For a passive particle, the orientation does not mean anything since there is no self-propulsion. Consistently, c(θ) displays uniform distribution (black line). This means that the probability of finding the nearest neighbor is equal for all directions.

In active systems, the distribution becomes nonuniform. There is a higher chance of finding the nearest neighbor at the front of the particle and lower chance at the rear. The degree of nonuniformity is stronger with the increasing activity level as the peak of c(θ) increases linearly with Pe, as shown on Fig. 4b. This nonuniformity indicates that the closest neighbor mostly locates at the front of an active particle. In other words, an active particle is pushing its cage due to the self-propulsion.

In Fig. 4c we show the value of the peak of the distribution in front of the particle c(θ = 0), function of the selected neighbor, from the first to the sixth nearest. We confirm the effect of self-propulsion, with active particle pushing toward its nearest neighbor and away from the rear of the cage.

This observation is consistent with the hypothesis of our DEAD model. In the following, we will explore whether the escape from the cage is at the origin of the observed nonmonotonic response.

B. Length and timescales of cage exploration

To quantify the length and time scales of cage exploration, we define the cage-relative mean square displacement:

MSD(∆t, t w ) = 1 N N i=1 |∆r CR i (t w + ∆t, t w )| 2 , ( 9
)
where t w = 1.5 × 10 4 . Thus we immediately simplify the notation to MSD(∆t).

In Fig. 5a, we show the cage relative mean square displacement at φ = 1.03 for various activity levels. The passive system (black line) shows the typical behavior of glassy systems: short time diffusion inside the cage, long time diffusion from cage to cage, and a plateau at intermediate times 10 ∆t 10 2 . If a is the typical cage size, the height of this plateau is a 2 . We can thus estimate a ≈ 0.125. The typical time to diffuse across the cage for a passive particle is τ B cage = a 2 /4D T ≈ 1. On Fig. 5a we observe a non monotonic response of the length of the plateau to activity, consistent with Fig. 2a. What the MSD teaches us, is the details of the particle behavior within the cage and at very short times. Low levels of activity do not affect the height of the plateau and thus the size of the cage. Furthermore the short times behaviors are also superimposed, meaning that the nature of the displacement within the cage is not significantly altered by the weak self-propulsion force. This observation goes against the central hypothesis of the DEAD model, i.e. inefficient cage exploration by directed motion.

The DEAD model supposes than the size of the cage is shorter that the persistence length, a condition equivalent to

Pe > a σ 2 D R D T . (10)
With our simulation parameter σ 2 D R /D T = 10 and at φ = 1.03, we expect the DEAD behavior only for Pe > 1.25. Instead, we observe it for Pe < 1 see Fig. 2b. Furthermore, (Eq. 10) predicts that the effect would be visible for a larger range of Peclet number if D R decreases, i.e. the rotational time increases. Consistently, the DEAD model predicts that the magnitude of the slowdown should be τ R /τ B cage and thus increases with the rotational time. Contrary to both predictions, the nonmonotonic behavior is less visible when σ 2 D R /D T = 3, see Fig. 5b where the passive and lowest activity MSD are within the error bars of each other.

Within our simulations, we have thus demonstrated that cage exploration is indeed skewed by activity. However unlike our initial DEAD proposal, the slowdown is more pronounced for shorter rotational time and shorter persistent length, and the mean dynamics within the cage seems to be little affected by the persistent selfpropulsion. Therefore, DEAD does not properly account for the nonmonotonic response with activity, and we need to look for alternative mechanisms. In the following section, we address the link between this response and the hallmark of nonergodicity: aging.

V. AGING

The state of nonergodic systems depends on their history. In particular, the relaxation dynamics of a system brought to a glass state slows down with the time since preparation, a phenomenology called aging. In the inset of Fig. 6a we confirm that phenomenology in our passive system by plotting F s (∆t) for various waiting times ranging from t w = 3.0 × 10 3 (green) to 5.7 × 10 4 (blue) at one fixed density φ = 1.03. As t w increases, the exit from the plateau is more and more delayed. For the longest waiting times t w > 1.5 × 10 4 , the exit time from the plateau is not reliably captured. However, for each waiting time t w ≤ 1.5 × 10 4 , we can fit the exit from the plateau as in Sec. III A to define a characteristic time τ (t w ). This procedure can be generalized at various densities, as shown in Fig. 6a. At low densities, all waiting times follow the same VTF relation between φ and τ . However in the nonergodic regime, the saturation level of the relaxation time occurs at lower values in younger systems (shorter t w ).

In the following, we study how the activity-dependence of the relaxation depends on the preparation history, and how this aging relates to the observed nonmonotonic response of the relaxation to activity.

A. Aging in the active system

We show in Fig. 6b the dependence of τ on t w ≤ 1.5 × 10 4 comparing between different activity levels Pe at the same density φ = 1.03. As expected from previous studies on passive glassy systems [START_REF] Struik | Physical aging in amorphous polymers and other materials[END_REF][START_REF] Berthier | Theoretical perspective on the glass transition and amorphous materials[END_REF] , we observe at Pe = 0 a power-law increase of τ ∝ t δ w , where δ is the aging exponent. By contrast for Pe = 4.0, there is no dependence on waiting time. At this high level of activity aging effects disappear, the system is ergodic even at t w = 3.0 × 10 3 , as self-propulsion promotes system relaxation. For intermediate activity levels, we observe power-law aging of varying exponents and prefactors. On Fig. 6c we show the dependence of the aging exponent on Pe. Overall, the aging exponent drops from δ ≈ 1.6 (Pe < 1) to δ ≈ 0 (Pe = 4). This seems consistent with recent studies on thermal [START_REF] Janzen | Aging in thermal active glasses[END_REF] and athermal 31 active glasses. In particular Mandal and Sollich 31 have described the evolution of the aging exponent as a parabola in an athermal active glass in a regime where the persistence time is very large. Although the persistence time in our system is small (τ R = 25) with respect to relaxation time, we nevertheless observe a decrease of δ compatible with a parabola for Pe > 1 (dotted line on Fig. 6c).

The effect of low amount of activity (Pe < 1) on aging is more complex. In particular, we observe that δ at Pe = 0 lies significantly below the parabola, close to δ = 1 often observed in passive systems [START_REF] Arceri | Glasses and aging: A Statistical Mechanics Perspective[END_REF] . It means that the passive system ages slower than slightly active systems. As a consequence, we can observe a crossing on Fig. 6b: at short waiting times t w = 3.0 × 10 3 , relaxation times monotonically follow Pe, however at longer waiting times the lowest activities can exhibit longer relaxation times than the passive system. From this, we can deduce that the nonmonotonic response of the relaxation develops with waiting time and is an effect of activity-enhanced aging.

We note that the nonmonotonic behavior we investigate here has been observed only with soft interactions (the Yukawa potential of our past experiments and our present harmonic potential simulations). Our results link this behavior with the observation of a saturation or a maximum in τ (φ), which level depends on the age of the system, see Fig. 6a.

Overall, the effect of activity appears quite reminiscent of phenomena associated with passive glasses under mechanical solicitation. There indeed, small shear can trigger a glass over-aging while higher ones can induce its partial to complete system rejuvenation 10 . It is remarkable to note that the same qualitative scenario occurs here from a local energy injection by active particles. Shear-enhanced aging is often understood in terms of potential energy landscape: mechanical energy helps the young system to hop out of relatively shallow energy minima into deeper ones, increasing its age [START_REF] Viasnoff | Rejuvenation and Overaging in a Colloidal Glass under Shear[END_REF] . If this picture applies to our system, we expect activity-enhanced aging to be more pronounced for systems that are prepared in even more shallow minima. We will test this hypothesis below.

B. Protocol dependence

Since aging is by nature protocol dependent, we also checked the relevance of the initialization stage of our simulations. Indeed until now, as explained in Section II, we prepare our system by letting a random configura-tion of harmonic spheres evolve through passive dynamics during t ini = 20 in order to separate overlapping particles. During t ini the potential energy decays by ∆E p (φ). We now introduce a new protocol where we shorten t ini to 7 × 10 -4 that correspond in average to a drop of potential energy of only ∆E p (φ)/4 during initialization at φ = 1.03. This new protocol produces configurations that are less relaxed, at which we set t w = 0 and turn on the self-propulsion.

In Fig. 7 we compare the response to activity of relaxation between these two preparation protocols differing only in the duration of the initialization stage. Fig. 7a and b display F s (q, ∆t) at t w = 3 × 10 3 . For this relatively short waiting time, we have observed until now (t ini = 20) that the exit from the plateau become monotonically earlier with activity, as confirmed in Fig. 7a. By contrast, at the same conditions but with a shorter initialisation time t ini = 7 × 10 -4 , the exit time from the plateau responds nonmonotonically to activity, as shown in Fig. 7b. The response of relaxation times to activity confirm the difference between the two preparation protocols, as show in Fig. 7c andd. Therefore, starting from a poorly relaxed configuration allows to observe the nonmonotonic response of the relaxation in younger systems.

With the new protocol, we have confirmed that the nonmonotonic response was intimately related to nonergodic phenomenology. In particular, a significant amount of aging within the observation window is necessary, which can be enhanced by preparing a poorly relaxed system.

VI. DISCUSSION AND CONCLUSION

To sum up, we have reproduced with numerical simulations the key experimental observations of Refs [START_REF] Klongvessa | Active glass: Ergodicity breaking dramatically affects response to self-propulsion[END_REF][START_REF] Klongvessa | Nonmonotonic behavior in dense assemblies of active colloids[END_REF] . In the ergodic supercooled regime, we confirm the mapping between the passive and active systems via the VTF relation of the density-dependence relaxation. Remarkably, the mapping fails in the glass state and the structural relaxation of the system exhibits a nonmonotonic response to activity level that cannot be explained by the reentrant behavior of soft spheres. Although we verified that the self-propulsion induces an instantaneous anisotropy of cage exploration, as predicted by DEAD model, we refute the predictions of the model regarding the influence of the rotational time. By contrast, we demonstrate by varying the preparation protocol that the nonmonotonic response is linked to activity-enhanced aging. This is consistent with our experimental observations that identified ergodicity breaking and the onset of the non monotonic behavior, both in glass 24 and polycrystals [START_REF] Klongvessa | Nonmonotonic behavior in dense assemblies of active colloids[END_REF] .

From our results, we can conclude that low levels of self-propulsion are not significantly slowing down the way a Brownian particle explores its cage in steady state. By contrast, low levels of self-propulsion during ageing are probably able to enhance the aging process, bring-ing faster the system to deeper minima of the energy landscape. Thus an actively aged system has higher energy barriers to cross than its passively aged counterpart. When the propulsion force is not strong enough to compensate for the higher barrier, the active system is slower than the passive. By contrast, when the propulsion force more than compensate for the higher barriers, the active system is faster. Since the nonmonotonic behavior disappears at long persistence time, it seems that activityenhanced aging needs relatively short persistence time, although this remains to be quantified.

We can relate this activated aging hypothesis with recent simulations showing nonmonotonic enhancement of steady-state relaxation in supercooled liquids [START_REF] Debets | Cage Length Controls the Nonmonotonic Dynamics of Active Glassy Matter[END_REF] . In particular, further simulation works should specifically look at how these enhanced steady-state relaxation translate or not into faster aging beyond ergodicity breaking.

FIG. 1 .

 1 FIG. 1. (a,b) Self-part of the cage-relative intermediate scattering function Fs(∆t) defined in Eq. 4. The relaxation time τ is obtained from the time such that Fs(τ ) = 1/e (horizontal dashed line). (a) Fs(∆t) of the passive system at various densities. The blue dotted curve, which corresponds to φ = 0.91, is a stretched exponential fit to obtain τ (at the red cross) if Fs(∆t) does not reach 1/e. (b) Fs(∆t) at φ = 0.91 of different activity levels represented by Péclet number Pe, and indicated by colors with the same legend in (c). (c) Scaled relaxation time τ /τ0 versus scaled density φ/φ *showing collapse on (Eq. 5, red line). φ * is obtained from the fit of systems able to reach a steady state (empty circles). The relaxation time of systems that do not reach a steady state (filled circles) is obtained from ensemble average of 50 simulations at the waiting time tw = 1.5 × 10 4 . Uncertainty is smaller than the marker size.

FIG. 2 .

 2 FIG. 2. (a) Self-part of the cage-relative intermediate scattering function Fs(∆t) for various activity levels Pe at φ = 1.03 for the waiting time tw = 1.5 × 10 4 . (Inset) Highlight of the nonmonotonic behavior of relaxation time τ with the activity level. (b) Activity dependence of τ /τ0 showing the nonmonotonic behavior.

FIG. 3 .

 3 FIG. 3. Activity level Pe and density φ state diagram showing the dependence of scaled relaxation time τ /τ0, which is shown by colored circles. The triangles represent the activitydependent glass transition density φ * (Pe). Inside the glass state, which is displayed by the grey area, the blue curve is a guide for the eye of a contour line at which τ /τ0 ≈ 9 × 10 3 , that corresponds to the saturation level of τ in the passive system. The results are obtained at the waiting time tw = 1.5 × 10 4 .

FIG. 4 .

 4 FIG. 4. (a) Azimuthal correlation function c(θ) between a particle and its first nearest neighbor with respect to particle's orientation. Activity levels Pe are shown by colors in the same representation as in (c). (Inset) A sketch illustrates how c(θ) is defined for the first nearest neighbors. (b) c(θ = 0) calculated from the first nearest neighbor as a function of activity levels Pe. The black line is a linear fit. (c) c(θ = 0) as a function of the selected nearest neighbor, from the first to the sixth nearest, at various activity levels Pe. Uncertainty is smaller than the marker size.

FIG. 5 .

 5 FIG. 5. Cage-relative mean square displacement MSD at φ = 1.03 and various activity levels Pe. Colored area indicate the error bars from ensemble averaging. Horizontal dotted line indicates the squared cage size a 2 . Vertical dashed lines indicate respectively the time to explore the cage by Brownian motion τ B cage and the rotational time τR. (a) For a rotational diffusion set by σ 2 DR/DT = 10, consistently with other figures. (b) For a longer rotational time such that σ 2 DR/DT = 3, where we notice that the slowing down at low Peclet is less pronounced.

FIG. 6 .

 6 FIG. 6. (a) Density dependence of relaxation time τ of the passive system (Pe = 0). The empty circles are steady state systems and are used for the fit of VTF relation (red line). The waiting time dependence of τ is illustrated by the filled circles colored by waiting time tw. (inset) Fs(∆t) at φ = 1.03 colored by tw from green (3 × 10 3 ) to blue (5.7 × 10 4 ). (b) Scaled relaxation time τ /τ0 as a function of waiting time tw at φ = 1.03 of various activity levels Pe. The solid lines are the power law fit τ /τ0 ∼ t δ w , where δ is an aging exponent. (c) Aging exponent δ function of activity level Pe. The dotted line is a parabolic function eye-guide (see main text) and the dashed line represents the value of δ in the passive case.

6 FIG. 7 .

 67 FIG. 7. Dependence on the initiation time. (a,b) Selfintermediate scattering function Fs(q, ∆t) at φ = 1.03 and waiting time tw = 3 × 10 3 for (a) tini = 20, and (b) tini = 7 × 10 -4 . (c-d) Activity dependence of τ comparing between different initialization time: tini = 20 (c), and tini = 7 × 10 -4 (d).

ACKNOWLEDGMENTS

N.K. was supported by (i) Japan Society for the Promotion of Science (JSPS) (ii) Thailand Science Research and Innovation (TSRI), and (iii) National Science, Research and Innovation Fund (NSRF) of Thailand. T.K. was supported by JSPS KAKENHI Grant Numbers JP20H00128, JP20H05157, JP19K03767, and JP18H01188. We thank David Rodney for pointing to enhanced aging, and Olivier Dauchot for encouraging us in this direction. N.K. and T.K. thank Kunimasa Miyazaki for fruitful discussions.


Michelle Agnes Magalhaes

Benjamin Matuszewski

Frédéric Bevilacqua

Frédéric Norbert Schnell

Adding value to research and technology through integration of artists in projects and synergy creation between creative

Keywords: Collective Interaction, Music Composition, Mobile Technologies, Movement, Music, Performances, Sensors. I

The goal of the Constella(c)tions project was to develop new paradigms for screenless and collective musical interaction, repurposing web and mobile technologies. Constella(c)tions unfolds as a set of collective musical games and interactive pieces involving various gestures, body movements and action of performers and/or public within open musical environments (open work). The project is inscribed in a practice-based methodology with the aim of exploring and developing further technological tools dedicated to play sounds using the motion sensors embedded in smartphones. In this context, the residency allowed the composer to create new musical pieces, involving various performers, from professional music ensemble to pedagogical action in schools, as well as including the general public in performances. Feedback and continued discussions and experimentations allowed us to constantly and reflectively revise both the interaction design and technology. The gained experience ultimately allowed us to design a new version of our technological ecosystem. The results are thus intrinsically of a hybrid nature, beyond strict scientific and artistic boundaries, and allowed us to reconsider our technical setup from a novel epistemic point of view. Several new projects have already been engaged, with dancers, musicians, as well as pedagogical actions. The project allowed us to create novel ways to engage the public, aiming at fostering social interaction and inclusion.

INTRODUCTION

The Constella(c)tions project was proposed by composer Michelle Agnes Magalhaes in response to the third STARTS residency call. The residence proposal was to experiment with collective musical interactions, using gestures and bodily actions. The composer's artistic vision was of shifting towards novel paradigms for musical composition and scores through a series of collective "games" whose rules implied texts, visuals, objects in spaces.

This was matching particularly well with the web-based ecosystem that IRCAM has developed to create embodied collective interaction with media, and particularly with sound and music. In particular, the IRCAM project called BeCoMe was precisely to further develop the technological environment to support artistic use cases of collective 1 Report, February 28, 2020. This work was supported in part by the European Commission through the Coordination and Support Action VERTIGO STARTS Residencies: "Adding socio-economic value to industry through the integration of artists in research and open innovation processes." under EC Grant 732112 We acknowledge additional support from the BeCoMe project (Ircam) and ANR Element (ANR-18-CE33-0002). Michelle Agnes Magalhaes is an invited composer, with IRCAM-Centre Pompidou, Paris France (e-mail: michelleagness@gmail.com). Benjamin Matuszewski is with the Sound Music Movement Interaction team, STMS Lab, IRCAM -CNRS -Sorbonne Université, Paris France (email: Benjamin.Matuszewski @ircam.fr). Frederic Bevilacqua s with the Sound Music Movement Interaction team, STMS Lab, IRCAM -CNRS -Sorbonne Université, Paris France (e-mail: Frederic.Bevilacqua@ircam.fr).

interaction 2,3 . Among our research questions was: how to support and/or question human/human interaction through the mediation of smartphones and web technologies. This use of technology seeks to find an alternative between high-tech and low-tech approaches, by building on top of consumer grade and widely spread technologies, an approach we might call everyday-tech or "mid-tech". As such, the project aims at proposing novel artistic and expressive paths that enable to include the audience by hijacking tools they are familiar with, and expose the bricolage 4 of the scientific, technologic and artistic research to better blend the audience, context and artwork.

Therefore, in Constella(c)tions, the main tool for playing sounds is the motion sensors embedded in smartphones, which are manipulated by the public and by the performers. They permit to trigger and modify sounds according to body gestures and movements. The performance is built from the coexistence between the performers, composer and creators, musician-researcher and audience. The scenography, architecture, stage and musical space are designed as a continuum, as a set of connections between objects and people, that have musical roles through motion detectors.

II. ARTWORK

Constella(c)tions are a set of interactive music pieces. (1987), Ignorant essays involved collective actions through readings, digressions, scenes, movies, songs, in a setting between the scene and the class. Together, they imagined a space for discovery and collective musical creation, which leaded to one of the Constella(c)tion pieces presented at Centquatre-Paris, where, guided by the performers, the public is invited to discover score elements (drawings, instructions, texts, images, actions) intended for all, without prior musical knowledge.

This piece involved SoundInitiative, a new music ensemble dedicated to the interpretation of contemporary repertoire in close collaboration with composers to create new pieces, and in the development of musical events with a strong emphasis on scenography. From the beginning of the Constella(c)tion residency, the SoundInitiative musicians were involved to experiment with the early prototypes, and participated in the development of a series of pieces and public events. The idea of Michelle Agnes Magalhaes to build musical composition as the interaction between score elements built as a set of texts, instructions, objects, considering the different performances space characteristic, fully resonated with the musicians' approach.

In order to fully understand how the Constella(c)tions artworks have been developed, we summarize below the different artistic pieces and events that happen during the residency. Here, the use of objects in the performance has been first experimented: strings were held by the performers, creating visual links between them. The strings roles were both metaphorical and evolving constraints for the performers, enabling them to find original actions and movements.

A different set of sounds were specifically composed for this piece, enriching the available musical environments.

○ TRANSIENTS Festival, New York, September 2019

This performance developed further the use of strings to create original interaction by investigating their potential as a collective instrument. These aspects were developed together with the Soundinitative ensemble through a week-long workshop. The smartphones attached to the string of the performer create an evolving sound environment.

○ Improtech 2019, Athens, September 2019 (ikPA'19)

The performance was part of a festival on musical improvisation in interaction with digital systems, held at the Cultural Center Onassis STEGI, and the University of Athens.

Here, the focus has been put on how the strings, attached to smartphones, can be used for structuring spatially and dynamically the performance space from an "architectural" point of view: the strings were unfold progressively and attached between the piano and the public.

○ Studio 5 en direct, IRCAM, December 14 2019

The concert involved public participation with the help of the Soundinitiatve music ensemble. This performance enabled to blend the different aspects previously developed in a public space at IRCAM normally not devoted to performances, as well as to fully integrate the participation of the public. The performance followed presentations to explain the different aspects of the collaborations of the residency.

○ STARTS Residencies Days, Centquatre-Paris, February 28 and 29, 2020

This performance, conceived as a series of small sound games, was also created with the musical ensemble SoundInitiative and featured Juliana Jardim as writer and theatrical director. It allowed us to further refine and develop the scenography and the audience participation scenarios. Indeed, the public could be close to the performance space and actively participate in the games led by the professional performers.

III. METHODOLOGY

The methodology used in this project can be best described as practice-based research. From a global perspective, the methodology was to develop prototypes and short musical pieces, though rapid iteration cycles. Each cycle allowed us to experiment with specific interaction principles and sonic results. Both technological adjustments of the prototypes and musical ideas were then tested conjointly. The series of public events allowed for assessing the different options, and critically reviewed the system. In addition to the performances described in the previous section, other workshops allowed us to experiment the concepts and technology. In particular, several workshops with musical students were held at Conservatoire of Saint-Brice sous Forêt.

The main objective of the project is to explore further how to control sound synthesis and generate musical content from users' gestures, and more specifically how such an approach can be implemented in collective and participatory settings. This research is based on CoMo-Elements) Adding value to research and technology through integration of artists in projects and synergy creation between creative industries, entrepreneurs, researchers and arts 5

Controller interface of the CoMo application Typical workflow of the CoMo application

The figure above shows the generic workflow proposed by the application:

• A client records a new gesture example.

• The example is sent to the server to be added to the common training set.

• The model is updated according to the new training set.

• The new trained model is sent to all clients of the same project.

• Every client can make use of this new gesture.

The centralized state of the project on the server thus enables the automatic sharing of the gesture models among web clients, allowing for creating a large set of collective interaction scenarios.

Constella(c)tions is built on top of this application and retroactively permitted many improvements, finally leading to its complete refactoring8 . The final work results from a number of smaller musical pieces (cf. Section Artwork) that have all been tested publicly. These public tests had a strong importance in this project in that they served several goals: assessing technical elements and improvements, but also and mainly exploring multiple ways of interacting with the public and with performers.

IV. CO-CREATION PROCESS

In this work, the artistic and techno-scientific endeavors were intimately related, following a process that could be described as: the experimental tasks were performed altogether, leading to specific tasks each different authors performed specifically, such as sound design and composition, specific developments, or testing. Also, while the performances benefited from IRCAM support, all the authors were directly involved in communication and production tasks. Despite that such involvement is beyond what is generally considered for scientific researchers, this was perceived by all stakeholders as beneficial, since it allowed all the authors to test first hand and reflect on all the process. This also contributed to the creation and sharing of a common language and understanding of technological, scientific and creative issues and processes.

V. IMPACT

This project allowed us to develop and evaluate a number of different concrete actions, from music pedagogy to artistic works implying audience participation. Importantly, as described in Section VII, these different actions will be intensively pursued beyond the STARTS residency, which lead us to foresee high societal impact on the long term. The scenarios developed in various music classes (conservatory children class and choirs) offer a complementary approach to current teaching approaches, by implying corporal and collective engagement of the students in a playful manner. This opens opportunities to develop soft skills that are considered as key for personal development beyond music education. Importantly, the pedagogical aspects can also be applied for a large range of situations and people, from amateur to professional, from children to adults. For example, a collaboration with the Ligeti Quartet will allow workshops with deaf children in the UK. Also, a long week workshop with professional dancers will be held with the choreographer Hervé Robbe in the context of the Academy of the Manifeste Festival in Paris.

The more detailed impacts are described below from both the research and artistic perspectives.

A. Research Impact

The collaboration between the ISMM team and the composer stimulated further research and several technological developments. First, it permitted us to test our prototype in real environments, showing some of their flaws and ultimately allowing for making it more robust. Second, it enabled us to test its limits, both technically and in its usability, triggering improvements in form of new software developments and prototyping extensions. Third, it permitted us to reflect upon and enlarge the conceptual background thanks to the confrontation to new conceptual and musical ideas and practices.

These three axes of research and developments were all conducted parallelly in two highly demanding, while very different, contexts. On the one hand, the multiple performance situations allowed us to focus on particular requirements such as robustness, easiness of deployment, controllability. On the other hand, the context of compositional process and elaboration in the studio permitted to focus on aspects such as flexibility, testability and adaptability. Ultimately, the collaboration contributed in large parts to the formalization, design and development of a novel version of our tool (https://github.com/ircam-ismm/como), hopefully unfolding new paths for scientific, design and artistic research.

B.

Artistic Impact From a formal point of view, the collaboration challenged the composer Michelle Agnes Magalhaes to expand her practices of musical composition. This led her to conceive novel musical forms: she oriented the composition process towards open-work, conceiving her compositions as "interactive puzzle". Moreover, the project led the composer to develop a holistic approach of sound-action interaction, avoiding to consider gesture, movement and sound as separate elements as often found in digital practices. This enabled the composer to create a coherent and personal interactive sound environment, through an iterative process of samples recording and testing both in studio and in real-world situations.

Finally, the project already led to new artistic collaborations with different artists, and will be shown internationally in the following years. As already mentioned, the project led (in less than a year) to six public performances in Europe and the US.

VI. ART-SCIENCE INTER-RELATIONSHIPS

Our project was precisely to further develop both artistic, user interactions and technological aspects. Among our research questions was how to support and/or question human/human interaction through the mediation of "screenless" devices, developing novel multimodal interaction paradigms using smartphones and web technologies. We believe that these questions can only be research in implying highly multidisciplinary approaches, with both artists, scientists and technologists.

In Constella(c)tions, the main tool for playing sounds are the motion sensors embedded in smartphones, which are manipulated by the public and by the performers. The musical space, scenography and stage architecture are designed as a continuum of affordances, from objects to sounds, enabling connections between the environment and the people. Therefore, the works are co-created at several levels, built from the coexistence between the artists, performers, researchers and finally, also the audience.

This collaboration is inscribed in the elaboration of a conceptual and epistemological framework we are currently

Adding value to research and technology through integration of artists in projects and synergy creation between creative industries, entrepreneurs, researchers and arts 7 developing. This framework makes use of the concept of experimental system as defined by Rheinberger 9 , to overcome current issues often described in art/science relationships 10 . More precisely, we postulate that such an approach of experimental systems -as systems composed of epistemic things and technical objects in constant evolution and reconfiguration -is a fruitful basis in providing a common and operational ground for pluralistic research approaches and collaborations (scientific, technological and artistic), by allowing to abstract from disciplines and concurrent epistemologies. In this context, we think the Contella(c)tions project offered an interesting contribution and test-case in the development of this framework.

VII. FUTURE DIRECTION AND ACTIONS

Several projects and collaborations are currently pursued beyond the official end of the STARTS residency. In particular:

• Dance and choreographic work with Hervé Robbe, a week workshop at the pompidou (from 15 to 19 June) with a presentation at the end (19 June). The event is part of the programming of the Centre National de la Danse (Camping https://www.cnd.fr/en/page/33-camping) associated with the Ircam Festival (Manifesto). In all we will be a team of 5 musicians + 15 dancers + our team at Ircam. Our device will have a scenographic function with a strong visual-sound and interactive aspect.

VIII. CONCLUSION

The residency was judged by all parties as highly valuable, and raised motivations to pursue the project and collaborations. It validates our web-based tools, expecting then a large range of novel scientific and artistic applications. Moreover, the continuation of this collaborative work should lead to potentially high societal impacts, in creating original events implying audience participation and pedagogical actions with children and adults, and fostering inclusion.

A CKNOWLEDGMENTS

We acknowledge the whole ISMM research team (including Jean Philippe Lambert and Estelle Dupré), Juliana Jardim, and the performers of SoundInitiative, Joshua Hyde, Gwen Rouger, Franco Venturini, Louise Leverd, Szymon Kaça, Shao-Wei Chou, Winnie Huang, as well as performers Yves Candau, Sarah Fdili Alaoui, Bertha Bermudez, Maussane Lerer, Lola Malique, Lucas Struna and Ronan Gil.

Special thanks to the IRCAM production and communications teams, in particular Aline Morel, Audrey Gaspar, Damien Ripoll, Jérémie Bourgogne, Sylvie Benoit, Deborah Lopatin.

○

  Mova Lab Days, Nantes (France), April 2019, (http://www.mova.one/nantes_meeting.html) During this workshop hosted by the Stereolux Arts and Technology Laboratory, two pieces were publicly shown: Extensio and Two Speeds (shown on the left), with Yves Canbdau, Sarah Fdili Alaoui and Bertha Bermudez. These musical pieces are played collectively following a set of textual instructions. The sounds is activated through the movement using smartphones attached to the wrists. These first performances represented an important step, both artistically and technologically. It also validated the technical setup as means to mediate the musical collective interactions. ○ 'Première de couverture" evening , Centre les Recollet, Paris (France), June 2019. Constella(c)tions was performed by Maussane Lerer, Lola Malique, Lucas Struna and Ronan Gil.

  • 2020: Workshops with the students of the Conservatory of Saint-Brice sous Forêt, with a public presentation in a large gymnasium (500 seats) • June 13 2020, Performance with the device and the SoundInitiative ensemble at the Arts Centre in Enghien-les-Bains. • For the second semester the London based Ligeti Quartet is offering workshops with deaf children in the UK. The new version of CoMo will be released in 2020, with various forms: CoMo-Art (directly aimed at Music and Dance performances), CoMo-Education (in collaboration with CRI-Paris, Université de Paris), CoMo-Research for scientific studies.

  Each work is presented as a collective musical game, which unfolds as a set of interactions involving various gestures, body movements and action of performers and/or public within a composed musical environment. Mobile technologies, such as smartphones, are used to capture and play sounds. Constella(c)tions has been developed from different encounters between composer Michelle Agnes Magalhaes and various artists, performers, musicians as well as with researchers from the Interaction Sound Music Movement team at IRCAM-STMS.An important encounter is the one, several years ago, of Michelle Agnes Magalhaes and theater director, actress and teacher Juliana Jardim, where they participated together in the creation of The Accordion Experience, based on Das Badener Lehrstücke von Einverständnis by Bertolt Brecht, as stage and music performers, respectively, with Dedé Pacheco, Maria Tendlau and Magali Biff. From 2008 to 2019, Juliana wrote and directed her project called Ignorant essays (www.ensaiosignorantes.com), which served as conceptual foundation for developing Constella(c)tions. Associated with the idea of emancipating the person as developed by Joseph Jacotot (1770-1840) and reported in Le maître ignorant by Jacques Rancière

  5,6 , a template application designed toward non-developer users for gesture-based and distributed User-Centered Machine Learning 7 . As such, the application is specifically designed to provide an environment where users without expert programming knowledge can create their own instance of the application (e.g. behavior and mappings of different clients) by simply editing a configuration file.

5 Benjamin Matuszewski, Joseph Larralde, Frédéric Bevilacqua. Designing Movement Driven Audio Applications Using a Web-Based Interactive Machine Learning Toolkit. Web Audio Conference (WAC), Sep 2018, Berlin, Germany. 6 https://github.com/ircam-cosima/elements 7 Marco Gillies et al.. (2016, May). Human-centred machine learning. In Proceedings of the 2016 CHI Conference Extended Abstracts on Human Factors in Computing Systems (pp. 3558-3565).

  Thanks to Emmanuel Valette, Claire Bouchard and Anthony Lefaure for video shooting, and Valentin Boulay for photography.

9 Rheinberger, Hans-Jörg. "Consistency from the Perspective of an Experimental Systems Approach to the Sciences and Their Epistemic Objects." Manuscrito 34, no. 1 (June 2011): 307-21. 10 Schwab, M. Experimental Systems: Future Knowledge in Artistic Research. Orpheus Institute Series. Leuven University Press, 2013. p. 214

https://github.com/ircam-ismm/como


X T Pham Phu

V Thanh Ngo

H T Diep
email: diep@u-cergy.fr

CROSSOVER FROM FIRST TO SECOND-ORDER TRANSITION IN FRUSTRATED ISING ANTIFERROMAGNETIC FILMS

Keywords: numbers: 75.10.-b General theory and models of magnetic ordering, 75.40.Mg Numerical simulation studies, 75.70.Rf Surface magnetism

In the bulk state, the Ising FCC antiferromagnet is fully frustrated and is known to have a very strong first-order transition. In this paper, we study the nature of this phase transition in the case of a thin film, as a function of the film thickness. Using Monte Carlo (MC) simulations, we show that the transition remains first order down to a thickness of four FCC cells. It becomes clearly second order at a thickness of two FCC cells, i.e. four atomic layers. It is also interesting to note that the presence of the surface reduces the ground state (GS) degeneracy found in the bulk. For the two-cell thickness, the surface magnetization is larger than the interior one. It undergoes a second-order phase transition at a temperature T C while interior spins become disordered at a lower temperature T D . This loss of order is characterized by a peak of the interior spins susceptibility and a peak of the specific heat which do not depend on the lattice size suggesting that either it is not a real transition or it is a Kosterlitz-Thouless nature. The surface transition, on the other hand, is shown to be of second order with critical exponents deviated from those of pure 2D Ising universality class. We also show results obtained from the Green's function method.

Discussion is given.

I. INTRODUCTION

This paper deals with the question whether or not the phase transition known in the bulk state changes its nature when the system is made as a thin film. In a recent work, we have considered the case of a bulk second-order transition. We have shown that under a thin film shape, i.e. with a finite thickness, the transition shows effective critical exponents whose values are between 2D and 3D universality classes. 1 If we scale these values with a function of thickness as suggested by Fisher 2 we should find, as long as the thickness is finite, the 2D universality class.

In this paper, we study the effect of the film thickness in the case of a bulk first-order transition. The question to which we would like to answer is whether or not the first order becomes a second order when reducing the thickness. For that purpose we consider the face-centered cubic (FCC) Ising antiferromagnet. This system is fully frustrated with a very strong first-order transition.

On the one hand, effects of the frustration in spin systems have been extensively investigated during the last 30 years. In particular, by exact solutions, we have shown that frustrated spin systems have rich and interesting properties such as successive phase transitions with complicated nature, partial disorder, reentrance and disorder lines. 3,4 Frustrated systems still challenge theoretical and experimental methods. For recent reviews, the reader is referred to Ref. 5.

On the other hand, physics of surfaces and objects of nanometric size have also attracted an immense interest. This is due to important applications in industry. [START_REF] Zangwill | Physics at Surfaces[END_REF][START_REF]Ultrathin Magnetic Structures[END_REF][START_REF] Diehl | Phase Transitions and Critical Phenomena[END_REF] In this field, research results are often immediately used for industrial applications, without waiting for a full theoretical understanding. An example is the so-called giant magneto-resistance (GMR) used in data storage devices, magnetic sensors, ... [START_REF] Baibich | [END_REF]10,11,[START_REF]Solid State Physics[END_REF] In parallel to these experimental developments, much theoretical effort has also been devoted to the search of physical mechanisms lying behind new properties found in nanometric objects such as ultrathin films, ultrafine particles, quantum dots, spintronic devices etc. This effort aimed not only at providing explanations for experimental observations but also at predicting new effects for future experiments. [START_REF] Ngo | [END_REF]14 The above-mentioned aim of this paper is thus to investigate the combined effects of frustration and film thickness which are expected to be interesting because of the symmetry reduction. As said above, the bulk FCC Ising antiferromagnet is fully frustrated because it is composed of tetrahedra whose faces are equilateral triangles. The antiferromagnetic (AF) interaction on such triangles causes a full frustration. [START_REF]Frustrated Spin Systems[END_REF] The bulk properties of this material have been largely studied as we will show below. In this paper, we shall use the recent high precision technique called "Wang-Landau" flat histogram Monte Carlo (MC) simulations to identify the order of the transition. We also use the Green's function (GF) method for qualitative comparison.

The paper is organized as follows. Section II is devoted to the description of the model.

We recall there properties of the 3D counterpart model in order to better appreciate properties of thin films obtained in this paper. In section III, we show our results obtained by MC simulations on the order of the transition. A detailed discussion on the nature of the phase transition is given. In the regime of second-order transition, we show in this section the results on the critical exponents obtained by MC flat histogram technique. Section IV is devoted to a study of the quantum version of the same model by the use of the GF method.

Concluding remarks are given in section V.

II. MODEL AND GROUND STATE ANALYSIS

It is known that the AF interaction between nearest-neighbor (NN) spins on the FCC lattice causes a very strong frustration. This is due to the fact that the FCC lattice is composed of tetrahedra each of which has four equilateral triangles. It is well-known 5 that it is impossible to fully satisfy simultaneously the three AF bond interactions on each triangle.

In the case of Ising model, the GS is infinitely degenerate for an infinite system size: on each tetrahedron two spins are up and the other two are down. The FCC system is composed of edge-sharing tetrahedra. Therefore, there is an infinite number of ways to construct the infinite crystal. The minimum number of ways of such a construction is a stacking, in one direction, of uncorrelated AF planes. The minimum GS degeneracy of a L 3 FCC-cell system (L being the number of cells in each directions), is therefore equal to 3 × 2 2L where the factor 3 is the number of choices of the stacking direction, 2 the degeneracy of the AF spin configuration of each plane and 2L the number of atomic planes in one direction of the FCC crystal (the total number of spins is N = 4L 3 ). The GS degeneracy is therefore of the order of 2 N 1/3 . Note that at finite temperature, due to the so-called "order by disorder", 15,16 the spins will choose a long-range ordering. In the case of AF FCC Ising crystal, this ordering is an alternate stacking of up-spin planes and down-spin planes in one of the three direction. This has been observed also in the Heisenberg case, 17 as well as in other frustrated systems. 18 The phase transition of the bulk frustrated FCC Ising antiferromagnet has been found to be of the first order. 19,20,21,22,23 Note that for the Heisenberg model, the transition is also found to be of the first order as in the Ising case. 17,24 Other similar frustrated antiferromagnets such as the HCP XY and Heisenberg antiferromagnets 25 and stacked triangular XY and Heisenberg antiferromagnets 26,27 show the same behavior.

Let us consider a film of FCC lattice structure with (001) surfaces. The Hamiltonian is given by

H = - i,j J i,j σ i • σ j (1)
where σ i is the Ising spin at the lattice site i, i,j indicates the sum over the NN spin pairs σ i and σ j .

In the following, the interaction between two NN on the surface is supposed to be AF and equal to J s . All other interactions are equal to J = -1 for simplicity. Note that in a previous paper, 28 we have studied the case of the Heisenberg model on the same FCC AF film as a function of J s .

For Ising spins, the GS configuration can be determined in a simple way as follows: we calculate the energy of the surface spin in the two configurations shown in Fig. 1 where the film surface contains spins 1 and 2 while the beneath layer spins 3 and 4. In the ordering of type I [Fig. 1(a)], the spins on the surface (xy plane) are antiparallel and in the ordering of type II [Fig. 1(b)] they are parallel. Of course, apart from the overall inversion, for type I there is a degenerate configuration by exchanging the spins 3 and 4. To see which configuration is stable, we write the energy of a surface spin for these two configurations

E I = -4|J s | E II = 4|J s | -4|J| (2)
One sees that E I ≤ E II when J s /J ≥ 0.5. In the following, we study the case J s = J = -1 so that the GS configuration is of type I.

III. MONTE CARLO RESULTS

In this paragraph, we show the results obtained by MC simulations with the Hamiltonian (1) using the high-precision Wang-Landau flat histogram technique. 29 Wang and Landau recently proposed a MC algorithm for classical statistical models. The algorithm uses a random walk in energy space in order to obtained an accurate estimate for the density of states ρ(E) which is defined as the number of spin configurations for any given E. This method is based on the fact that a flat energy histogram H(E) is produced if the probability for the transition to a state of energy E is proportional to ρ(E) -1 . At the beginning of the simulation, the density of states (DOS) is set equal to one for all possible energies, ρ(E) = 1.

We begin a random walk in energy space (E) by choosing a site randomly and flipping its spin with a probability proportional to the inverse of the momentary density of states.

In general, if E and E ′ are the energies before and after a spin is flipped, the transition

probability from E to E ′ is p(E → E ′ ) = min [ρ(E)/ρ(E ′ ), 1] . (3)
Each time an energy level E is visited, the DOS is modified by a modification factor f > 0 whether the spin flipped or not, i.e. ρ(E) → ρ(E)f . At the beginning of the random walk, the modification factor f can be as large as e 1 ≃ 2.7182818. A histogram H(E) records how often a state of energy E is visited. Each time the energy histogram satisfies a certain "flatness" criterion, f is reduced according to f → √ f and H(E) is reset to zero for all energies. The reduction process of the modification factor f is repeated several times until a final value f final which close enough to one. The histogram is considered as flat if

H(E) ≥ x%. H(E) (4)
for all energies, where x% is chosen between 70% and 95% and H(E) is the average histogram.

The thermodynamic quantities 29,30 can be evaluated by

E n = 1 Z E E n ρ(E) exp(-E/k B T ) C v = E 2 -E 2 k B T 2 M n = 1 Z E M n ρ(E) exp(-E/k B T ) χ = M 2 -M 2 k B T
where Z is the partition function defined by

Z = E ρ(E) exp(-E/k B T ) (5)
The canonical distribution at any temperature can be calculated simply by

P (E, T ) = 1 Z ρ(E) exp(-E/k B T ) (6)
In this work, we consider a energy range of interest 31,32 (E min , E max ). We divide this energy range to R subintervals, the minimum energy of each subinterval is E i min for i = 1, 2, ..., R, and maximum of the subinterval i is

E i max = E i+1 min +2∆E
, where ∆E can be chosen large enough for a smooth boundary between two subintervals. The Wang-Landau algorithm is used to calculate the relative DOS of each subinterval (E i min , E i max ) with the modification factor f final = exp(10 -9 ) and flatness criterion x% = 95%. We reject the suggested spin flip and do not update ρ(E) and the energy histogram H(E) of the current energy level E if the spin-flip trial would result in an energy outside the energy segment. The DOS of the whole range is obtained by joining the DOS of each subinterval (E i min + ∆E, E i max -∆E). The film size used in our present work is L × L × N z where L is the number of cells in x and y directions, while N z is that along the z direction (film thickness). We use here belongs to the second layer [see Fig. 1(a)]. In our simulations, we used N z FCC cells, i.e.

2N z atomic layers. We used the symmetry of the two film surfaces.

A. Crossover of the phase transition

As said earlier, the bulk FCC antiferromagnet with Ising spins shows a very strong firstorder transition. This is seen in MC simulation even with a small lattice size as shown in Fig. 3.

Our purpose here is to see whether the transition becomes second order when we decrease the film thickness. As it turns out, the transition remains of first order down to N z = 4 as seen by the double-peak energy histogram displayed in Fig. 4. Note that we do not need to go to larger L, the transition is clearly of first order already at L = 40.

In Fig. 5 we plot the latent heat ∆E as a function of thickness N z . Data points are well fitted with the following formula

∆E = A - B N d-1 z 1 + C N z , (7)
where d = 3 is the dimension, A = 0.3370, B = 3.7068, C = -0.8817. Note that the term

N d-1
z corresponds to the surface separating two domains of ordered and disordered phases at the transition. The second term in the brackets corresponds to a size correction. As seen in Fig. 5, the latent heat vanishes at a thickness between 2 and 3. This is verified by our simulations for N z = 2. For N z = 2 we find a transition with all second-order features: no discontinuity in energy (no double-peak structure) even when we go up to L = 150.

Before showing in the following the results of N z = 2, let us discuss on the crossover. In the case of a film with finite thickness studied here, it appears that the first-order character is lost for very small N z . A possible cause for the loss of the first-order transition is from the role of the correlation in the film. If a transition is of first order in 3D, i. e. the correlation length is finite at the transition temperature, then in thin films the thickness effect may be important: if the thickness is larger than the correlation length at the transition, than the first-order transition should remain. On the other hand, if the thickness is smaller than

T E -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 1.4 1.6 1.8 2 2.2 2.4
FIG. 6: Energy versus temperature T for L = 120 with film thickness N z = 2. that correlation length, the spins then feel an "infinite" correlation length across the film thickness resulting in a second-order transition.

B. Film with 4 atomic layers (N z = 2)

Let us show in Fig. 6 and Fig. 7 the energy and the magnetizations of sublattices 1 and 3 of the first two cells with L = 120 and N z = 2.

It is interesting to note that the surface layer has larger magnetization than that of the second layer. This is not the case for non frustrated films where the surface magnetization is always smaller than the interior ones because of the effects of low-lying energy surfacelocalized magnon modes. 33,34 One explanation can be advanced: due to the lack of neighbors surface spins are less frustrated than the interior spins. As a consequence, the surface spins maintain their ordering up to a higher temperature.

Let us discuss on finite-size effects in the transitions observed in Figs. 8 and9. This is an important question because it is known that some apparent transitions are artifacts of small system sizes. To confirm further the observed transitions, we have made a study of finite-size effects on the layer susceptibilities by using the Wang-Landau technique described above. 29 We observe that there are two peaks in the specific heat: The first peak at T 1 ≃ 1.927, corresponding to the vanishing of the sublattice magnetization 3, does not depend on the lattice size while the second peak at T 2 ≃ 1.959, corresponding to the vanishing of the sublattice magnetization 1, does depend on L. Both histograms taken at these temperatures and the near-by ones show a gaussian form indicating a non first-order transition [see Fig. 10]. The fact that the peak at T 1 does not depend on L suggests two scenarios: i) the peak does not correspond to a real transition, since there exist systems where C v shows a peak but we know that there is no transition just as in the case of 1D Ising chain, ii) the peak corresponds to a Kosterlitz-Thouless transition. To confirm this we need to check carefully several points such as the behavior of the correlation length etc. This is a formidable task which is not the scope of this work.

Whatever the scenario for the origin of the peak at T 1 , we know that the interior layers are disordered between T 1 and T 2 , while the two surface layers are still ordered. Thus, the transition of the surface layers occurs while the disordered interior spins act on them as dynamical random fields. Unlike the true 2D random-field Ising model which does not allow a transition at finite temperature, 35 the random fields acting on the surface layer are correlated. This explains why one has a finite-T transition here. Note that this situation is known in some exactly solved models where partial disorder coexists with order at finite T . The exponent ν can be obtained as follows. We calculate as a function of T the magneti-

zation derivative with respect to β = (k B T ) -1 : V 1 = (ln M) ′ = E -ME / M
where E is the system energy and M the sublattice order parameter. We identify the maximum of V 1

for each size L. From the finite-size scaling we know that V max 1 is proportional to L 1/ν . [START_REF] Ferrenberg | [END_REF] We show in Fig. 11 the maximum of V 1 versus ln L for the first layer. We find ν = 0.887 ± 0.009. Now, using the scaling law χ max ∝ L γ/ν , we plot ln χ max versus ln L in Fig. 12. The ratio of the critical exponents γ/ν is obtained by the slope of the straight line connecting the data points of each layer. From the value of ν we obtain γ = 1.542 ± 0.005. These values do not correspond neither to 2D nor 3D Ising models (γ 2D = 1.75, ν 2D = 1, γ 3D = 1.241, ν 3D = 0.63). We note however that, if we think of the weak universality where only ratios of critical exponents are concerned, 38 then the ratios of these exponents 1/ν = 1.128 and γ/ν = 1.739 are not far from the 2D ones which are 1 and 1.75, respectively.

IV. GREEN'S FUNCTION METHOD

We consider here the same FCC system but with quantum Heisenberg spins. To compare the results with the Ising case, we add in the Hamiltonian an Ising-like anisotropy interaction term. In addition, this term avoids the absence of long-range order of isotropic non Ising spin model at finite temperature (T ) when the film thickness is very small, i.e. quasi twodimensional system. 39 The Hamiltonian is given by

H = - i,j J i,j S i • S j - i,j I i,j S z i S z j (8)
where S i is the Heisenberg spin at the lattice site i, i,j indicates the sum over the NN spin pairs S i and S j . J i,j and I i,j are antiferromagnetic (negative). Note that in the laboratory coordinates, the up spins have S z ≥ 0 while the down spins have S z ≤ 0.

We can rewrite the Hamiltonian (8) in the relative local spin coordinates as

H = - <i,j> J i,j 1 4 (cos θ ij -1) S + i S + j + S - i S - j + 1 4 (cos θ ij + 1) S + i S - j + S - i S + j + 1 2 sin θ ij S + i + S - i S z j - 1 2 sin θ ij S z i S + j + S - j + cos θ ij S z i S z j - <i,j> I i,j cos θ ij S z i S z j (9)
where θ ij is the angle between two NN spins. Note that in the above expression, we have transformed all S z ≥ 0, the relative spin orientation of each spin pair is now expressed by θ ij . In a collinear spin configuration such as those shown in Fig. 1, cos θ ij = -1 and 1 for antiparallel and parallel pairs, respectively, while sin θ ij = 0. In non collinear structures, the calculation is more complicated. The general GF method for non collinear spin configuration has been proposed elsewhere. 40,41 In the present study, one has a collinear spin configuration shown in Fig. 1 because of the Ising-like anisotropy. We define two double-time GF by

G ij (t, t ′ ) = ≪ S + i (t); S - j (t ′ ) ≫, (10)
F mj (t, t ′ ) = ≪ S - m (t); S - j (t ′ ) ≫ . ( 11
)
where i and j belong to the up-spin sublattice, m to the down-spin one. In the case of thin films, the reader is referred to Refs. 28,33,34 for a general formulation. We describe here only the main steps: we first write the equations of motion for G ij (t, t ′ ) and F mj (t, t ′ ) and we next neglect higher-order correlations by using the Tyablikov decoupling scheme 42 which is known to be valid for exchange terms, 43 and then we introduce the following Fourier transforms in the xy plane

G i,j (t, t ′ ) = 1 ∆ dk xy 1 2π +∞ -∞ dωe -iω(t-t ′ ) . g n,n ′ (ω, k xy ) e ikxy•(R i -R j ) , (12)
F m,j (t, t ′ ) = 1 ∆ dk xy 1 2π +∞ -∞ dωe -iω(t-t ′ ) . f n,n ′ (ω, k xy ) e ikxy•(Rm-R j ) , (13)
where ω is the spin-wave frequency, k xy denotes the wave-vector parallel to xy planes, R i is the position of the spin at the site i, n and n ′ are respectively the indices of the layers where the sites i (or m) and j belong to. One has n, n ′ = 1, 2, ..., 2N z . The integral over k xy is performed in the first Brillouin zone in the xy reciprocal plane whose surface is ∆. Finally, one obtains for all layers the following matrix equation

M (ω) g = u, (14)
Note that though n runs from 1 to 2N z , the matrix M has the dimension of 4N z × 4N z because for each n there are two functions g(n, n ′ ) and f (n, n ′ ). In the above equation, g and u are the column matrices of dimension 4N z which are defined as follows

g =              g 1,n ′ f 1,n ′ M (ω) =                     A + 1 B 1 D + 1 D - 1 • • • • • • • • • -B 1 A - 1 -D - 1 -D + 1 • • • • • • • • • C + 2 C - 2 A + 2 B 2 D + 2 D - 2 • • • -C - 2 -C + 2 -B 2 A - 2 -D - 2 -D + 2 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • C + Nz C - Nz A + Nz B Nz • • • • • • • • • -C - Nz -C + Nz -B Nz A - Nz                     (16)
where for the spin configuration of type I [Fig. 1 (a)] one has

A ± n = ω ± 2J n S z n Z + 8I n S z n B n = -2J n S z n (Zγ) (17)
C + n = +4J n,n-1 S z n cos k y a 2 (18)
C - n = -4J n,n-1 S z n cos k x a 2 (19)
D + n = +4J n,n+1 S z n cos k y a 2 (20)
D - n = -4J n,n+1 S z n cos k x a 2 ( 21
)
in which, Z = 4 is the number of in-plane NN, and

γ = 1 Z 4 cos k x a 2 cos k y a 2 .
Here, for compactness we have used the following notations: i) J n and I n are the in-plane interactions. In the present model J n is equal to J = -1.

All I n are set to be I(< 0).

ii) J n,n±1 are the interactions between a spin in the n-th layer and its neighbor in the (n ± 1)-th layer. Here, we take

J n,n±1 = -1. Of course, J n,n-1 = 0 if n = 1, J n,n+1 = 0 if n = 2N z .
Now, solving det(M) ≡ |M| = 0, we obtain the spin-wave spectrum ω of the present system. For each k xy there are 4N z eigenvalues ω, two by two with opposite signs because of the AF symmetry. The solution for the GF g n,n is given by

g n,n = |M| n |M| , (22)
with |M| n is the determinant made by replacing the n-th column of |M| by u in (15).

Writing now

|M| = i (ω -ω i (k xy )) , (23)
one sees that ω i (k xy ) , i = 1, • • • , 4N z , are poles of the GF g n,n . Now, we can express g n,n

as

g n,n = i f n (ω i (k xy )) (ω -ω i (k xy )) , (24)
where

f n (ω i (k xy )) is f n (ω i (k xy )) = |M| n (ω i (k xy )) j =i (ω j (k xy ) -ω i (k xy )) . (25)
Next, using the spectral theorem which relates the correlation function S - i S + j to the GF, 44 one has

S - i S + j = lim ε→0 1 ∆ dk xy +∞ -∞ i 2π (g n,n ′ (ω + iε) -g n,n ′ (ω -iε) ) • dω e βω -1 e ikxy•(R i -R j ) , (26)
where ǫ is an infinitesimal positive constant and β = 1/k B T , k B being the Boltzmann constant.

Using the GF presented above, we can calculate self-consistently various physical quantities as functions of temperature T . Large values of Ising-like interaction I will enhance the ordering. On the contrary, for I → 0 the transition temperature will go to zero according to the Mermin-Wagner theorem. 39 This is seen in the following. For numerical integretaion, we will use 80 2 points in the first Brillouin zone.

Figure 13 shows the sublattice magnetizations of the first two layers for N z = 2 with I = -0.25 and I = -0.01 (upper and lower figures, respectively)). As seen, the surface sublattice magnetization is larger than the sublattice magnetization of the second layer for N z = 2 in qualitative agreement with the MC results shown in Fig. 7, in spite of the fact that due to a finite-size effect, there is a queue of the sublattice magnetization above the transition temperatures for MC results. Note that the AF coupling gives rise to a zero-point spin contraction at T = 0 which is different for the surface spins and the second-layer spins.

We show in Fig. 14 the phase diagram in the space (I, T ) where T 1 and T 3 are transition temperatures of the surface sublattice 1 and the sublattice 3 of the second layer.

V. CONCLUDING REMARKS

We have shown in this paper the crossover of the phase transition from first to second order in the frustrated Ising FCC AF film. This crossover occurs when the film thickness N z = 2 is smaller than a value between 2 and 4 FCC lattice cells. These results are obtained with the highly performing Wang-Landau flat histogram technique which allows to determine the first-order transition with efficiency.

For N z = 2, we found that in a range of temperature the surface spins stay ordered while interior spins are disordered. We interpret this as an effect of the frustration reduction: due to the lack of neighbors, the surface spins are less frustrated than the interior spins. As a consequence, interior spins are disordered at a lower temperature. This has been verified by the Green's function calculation.

The second-order transition for N z = 2 is governed by the surface disordering and is characterized by critical exponents whose values are deviated from those of the 2D Ising universality class. We believe that this deviation results from the effect of the disordered interior spins which act as "correlated" random fields on the surface spins. We do not know if the critical exponents found here belong to a new universality class or they are just "effective critical exponents" which one could scale in some way or another to bring into the 2D Ising universality class. Anyway, these exponents seem to obey a weak universality. An answer to this question is desirable.

One of us (VTN) thanks the University of Cergy-Pontoise for a financial support and hospitality during the final stage of this work.

FIG. 1 :

 1 FIG.1:The ground state spin configuration of the FCC cell at the film surface: a) ordering of type I for J s < -0.5|J| b) ordering of type II for J s > -0.5|J|.

L 12 FIG. 2 :FIG. 3 :

 1223 FIG. 2: Bulk energy vs T for L = N z = 12. ( ) P E

20 FIG. 4 :FIG. 5 :

 2045 FIG. 4: Energy histogram for L = 20, 30, 40 with film thickness N z = 4 (8 atomic layers) at T = 1.8218, 1.8223, 1.8227, respectively.

FIG. 7 :

 7 FIG. 7: Sublattice magnetization for L = 120 with film thickness N z = 2.

FIG. 8 :

 8 FIG. 8: Specific heat are shown for various sizes L as a function of temperature.

FIG. 9 :

 9 FIG. 9: Susceptibilities of sublattice 1 (a) and 3 (b) are shown for various sizes L as a function of temperature.

33 FIG. 10 :

 3310 FIG. 10: Energy histograms for L = 120 with film thickness N z = 2 at the two temperatures (indicated on the figure) corresponding to the peaks observed in the specific heat. See text for comment.

2 FIG. 12 :

 212 FIG. 12: Maximum sublattice susceptibility χ max versus L in the lnln scale. The slope of this straight line gives γ/ν.

8 FIG. 13 :

 813 FIG. 13: Magnetization of sublattices 1 (surface) and 3 (second layer) versus temperature for N z = 2 and I = -0.25 (upper) and I = -0.01 (lower).

6 FIG. 14 :

 614 FIG. 14: Phase diagram obtained by the GF method. T 1 and T 3 are transition temperatures of the sublattice 1 of the surface and of the sublattice 3 of the second layer. The transition temperatures should go to zero as I → 0 (see text).

  FIG. 11: The maximum value of V 1 = E -M E / M versus L in the lnln scale. The slope of this straight line gives 1/ν.at T 2 . The theoretical argument of Capehart and Fisher 2 does not apply in the present situation because one does not have a single transition here, unlike the case of simple cubic ferromagnetic films studied before.1 So, we wish to calculate the critical exponents associated with the transition at T 2 .

		5.8
			1/ν = 1.128(9)
		5.6
		5.4
	Ln( V 1 max )	4.8 5 5.2
		4.6
		4.4
		4.2 3.6	3.8	4	4.2	L Ln( ) 4.4	4.6	4.8	5	5.2

3,4,[START_REF] Diep | see chapter Frustration -Exactly Solved Frustrated Models in Ref[END_REF]

However, it is not obvious to foresee what is the universality class of the transition