domain: shs scco

text:
Ménétrier

Didierjean

Vieillard

Keywords: visual perception, boundary extension, emotions

. This article looks at whether the emotion category expressed (i.e., happiness, pleasure, irritation, or anger) by actors in short films could have an influence on the BE effect. The results showed that positively-valenced emotions (happiness, pleasure) led to an extension effect, while the negatively-valenced ones (anger, irritation) did not produce any significant memory distortion. The arousal dimension of emotions had no significant effect on BE. The current results were discussed in the light of previous studies on the links between BE and emotions.

Is Boundary Extension Emotionally Selective?

When we observe a visual scene, the memory of it that we create often differs in various ways from the scene initially observed. While this frequently shows up as a loss of some informations (e.g., [START_REF] Simons | Change blindness[END_REF], sometimes the reverse is true: the information memorized is richer than in the original scene. One of the manifestations of this potential enrichment of the memory trace with additional information that might have been just outside the scene's actual boundaries is a phenomenon known as boundary extension, or BE [START_REF] Intraub | Wide-angle memories of close-up scenes[END_REF]e.g., Intraub, 2010, for a review). BE refers to a distortion of visual-spatial memory which occurs when observers trying to recall a picture tend to remember scene layout that was not in the original photograph but could be present just beyond the picture's actual boundaries (e.g., [START_REF] Gottesman | Surface construal and the mental representation of scenes[END_REF].

While there is a large body of research demonstrating that the BE phenomenon is robust (e.g., [START_REF] Dickinson | Transsaccadic representation of layout: What is the time course of boundary extension?[END_REF], there are a number of factors that seem likely to modulate this effect (e.g., [START_REF] Intraub | More than meets the eye: The effect of planned fixations on scene representation[END_REF], or even to make it disappear (e.g. [START_REF] Gottesman | Surface construal and the mental representation of scenes[END_REF]. One such factor is the emotional nature of the stimulus (e.g., [START_REF] Candel | How children remember neutral and emotional pictures: Boundary extension in children's scene memories[END_REF][START_REF] Candel | Boundary distortions for neutral and emotional pictures[END_REF][START_REF] Mathews | Take a closer look: Emotion modifies the boundary extension effect[END_REF][START_REF] Safer | Tunnel memory for traumatic events[END_REF], although few studies have dealt with the impact of emotions on BE, and the ones that have been conducted offer discrepant conclusions. Some have shown a BE effect following the presentation of stimuli with a negative valence (Candel et al., 2003, Experiment 1;[START_REF] Candel | How children remember neutral and emotional pictures: Boundary extension in children's scene memories[END_REF], whereas others have demonstrated the opposite effect: a shrinking of the remembered visual impression of the scene for negative-valence elements (Safer et al., 1998 1 ). While these previous researches were limited to examining the effects of negativelyvalenced stimuli on BE, a study by [START_REF] Mathews | Take a closer look: Emotion modifies the boundary extension effect[END_REF] tested the effects of positively-valenced stimuli, too. These authors also attempted to determine the impact of arousal on the phenomenon. Their results indicated a BE effect, regardless of the stimuli's valence. Moreover, the BE effect observed by these authors was weaker for negative and highly arousing stimuli.

Our study explores a different aspect of the impact of emotions on BE. In the current work, we tested participants on facial and body expressions of emotion acted out by another person and not, like on most studies, on effects of emotional induction. Static stimuli depicting expression of facial emotions present a limited ecological validity, we therefore used dynamic stimuli (short films) showing actors expressing emotions with different valences (positive, negative) and arousal levels (low, high): anger, happiness, irritation, and pleasure. All of the filmed sequences we used were taken from the GEMEP corpus [START_REF] Bänziger | Introducing the Geneva Multimodal Expression Corpus for Experimental Research on Emotion Perception[END_REF], which contains stimuli presenting both facial expressions and postural and gestural attitudes. These stimuli have been found to be powerful in terms of social communication [START_REF] Bänziger | Introducing the Geneva Multimodal Expression Corpus for Experimental Research on Emotion Perception[END_REF][START_REF] Vieillard | Children's perception and understanding of (dis)similarities among dynamic bodily/facial expressions of happiness, pleasure, anger and irritation[END_REF]. Their dynamic nature is another aspect that makes them different from the static stimuli (photographs) used in most BE studies (see however [START_REF] Delucia | Visual memory for moving scenes[END_REF]. Since BE is a process thought to be involved in the visual integration of scenes perceived in succession by the eye (e.g., [START_REF] Dickinson | Transsaccadic representation of layout: What is the time course of boundary extension?[END_REF], it seemed useful to test BE with such dynamic stimuli.

We used a standard BE task, called camera distance paradigm (e.g., [START_REF] Intraub | Looking at scenes while searching for numbers: Dividing attention multiplies space[END_REF], that we adapted to filmed stimuli. The participants underwent two experimental phases: a film-memorization phase, followed by a recognition test. In this second phase, they saw the exact same items as seen in the first phase, but they were told, incorrectly, that while some of the items would be exactly the same, others would be shown from farther away or from closer up. The participants used a five-point scale to rate each film as to whether it was a more close-up view, a more wide-angle, or the same view as in the initial film. With this type of task, the BE effect will show up as a consistent tendency to judge the scenes as being filmed from closer up than they were in the first presentation.

Our objective was to find out if the BE effect was likely to depend upon the valence and the arousal level of the emotions the actors' displayed in the films. On the basis of previous data, it seems difficult to predict the nature of the effects likely to be observed, that is, whether there will be boundary extension [START_REF] Mathews | Take a closer look: Emotion modifies the boundary extension effect[END_REF] or conversely, boundary restriction (BR; [START_REF] Safer | Tunnel memory for traumatic events[END_REF]. However, research on memory for central and peripheral details reported that material depicting positive emotions enhances memory both for central and peripheral elements (i.e., memory broadening effect, e.g., [START_REF] Yegiyan | Encoding details: Positive emotion leads to memory broadening[END_REF], whereas material depicting negative emotions promotes only central elements of an event (i.e., memory narrowing effect; e.g., [START_REF] Levine | Emotion and memory narrowing: A review and goalrelevance approach[END_REF], for a review). On the basis of these observations, BE could be expected when the expressed emotions are positively-valenced and BR could be expected in the case of negativelyvalenced emotions. In this sense, BE could be seen as one example of these general memory mechanisms2 . It is also likely that arousal modulates emotional valence effects on BE. [START_REF] Mathews | Take a closer look: Emotion modifies the boundary extension effect[END_REF] observed a significant interaction between these two variables, corresponding to a BE reduction for scenes which were at once very negative and highly arousing. Even if in our study the emotions were acted out by the actors and not induced by an emotional induction paradigm, a similar interaction is expected, with more BR for anger movies than for irritation ones.

Method

Participants

The participants were 40 students from the University of Franche-Comté (32 women, 8 men), with a mean age of 21 years 2 months (SD = 2 years 11 months). All had normal or correctedto-normal vision. None had knowledge of the experimental objectives.

Experimental Device

The experiment was presented on a MacBook Pro portable computer with a 15-inch screen, and was run by Psyscope software. During the test session, the participants were seated at a distance of about 50 cm from the screen.

Stimuli

We used sixteen short (1-second) silent film clips taken from the GEMEP corpus [START_REF] Bänziger | Introducing the Geneva Multimodal Expression Corpus for Experimental Research on Emotion Perception[END_REF] featuring actors expressing emotions controlled on their valence and arousal [START_REF] Vieillard | Children's perception and understanding of (dis)similarities among dynamic bodily/facial expressions of happiness, pleasure, anger and irritation[END_REF]. Four actors expressed each of the following emotions in turn: anger, happiness, irritation, and pleasure. Two of the emotions had a positive valence (happiness, pleasure) and two had a negative valence (anger, irritation). Likewise, two emotions involved a high degree of arousal (anger, happiness) and the other two, a low degree of arousal (irritation, pleasure).

As with the photographs typically used in BE tasks, the actors were located in the approximate center of the screen, filmed against a plain background consisting of a blue curtain. The framing was done so that their bodies would be visible down to the mid-thigh or knee level, with the lower part of the body off-screen (cf., Figure 1). During display, the films were framed with a white border and filled the entire screen. An additional film clip (neutral emotion) from the GEMEP corpus was presented to the participants in the familiarization phase.

Procedure

The participants were tested individually. Prior to the experiment, they were asked to fill out the STAI-Y (State-Trait Anxiety Inventory, Form Y, [START_REF] Spielberger | Inventaire d'anxiété état-trait forme Y[END_REF][START_REF] Spielberger | Inventaire d'anxiété état-trait forme Y[END_REF]. This was aimed at eliminating from the sample those participants who exhibited anxiety levels (state or trait) that were too high at the time of the experiment. The BE task proposed was an adaptation of the camera distance paradigm to film sequences. It was preceded by a familiarization phase during which the experimenter used a sample film that was emotionally neutral in order to explain the general principle of the study. Then the experiment proper began, in two successive phases: a memorization phase, followed immediately by a recognition test.

Memorization phase. The participants were asked to memorize each of the sixteen filmed sequences to the best of their ability, while paying attention both to the actor and to the manner in which he/she was framed. Each stimulus was preceded by a mid-screen fixation point lasting 600 ms that warned the observer of the imminent arrival of the film clip to memorize. The stimuli were displayed in random order, lasted one second each, and were followed by a blank screen.

Recognition phase. The participants were told that each of the films previously memorized was going to reappear on the screen in random order, but that some of them had been modified with regard to the camera perspective from which it had been filmed. The participants' task was to use a five-point scale (ranging from -2 to 2) to rate each film presented as to whether it seemed identical to the original (a value of 0 on the scale), just a little closer (-1), much closer (-2), a little farther away (1), or much farther away (2). Contrary to the experimenter's statement, the film clips shown the second time were all exactly the same as those presented in the first phase of the experiment3 . No time limit was set for responding.

Results

The mean score distributions on the state-trait anxiety measure [START_REF] Spielberger | Inventaire d'anxiété état-trait forme Y[END_REF][START_REF] Spielberger | Inventaire d'anxiété état-trait forme Y[END_REF] did not contain any extreme values (±3 standard deviations), so none of the participants were excluded from the analysis.

The mean boundary ratings for each expression category are shown in Figure 2. To determine whether each mean rating differed significantly from zero, .95 confidence intervals were computed for each one (presented with error bars in Figure 2). If a rating did not differ significantly from zero ("same"), neither BE nor BR occurred. By contrast, a significant deviation from zero indicated a distortion of the remembered size of the film. A negative rating indicated that the viewer remembered the film as having more background than it actually had (a BE effect); a positive rating indicated that the viewer remembered the film as having less background. For all emotions taken together, the confidence intervals revealed an overall BE effect (mean = -0.17, SD = 0.47). However, as the error bars in Figure 2 show, there was a significant BE effect for only two of the emotions tested: happiness and pleasure. Anger and irritation, by contrast, did not produce a significant memory distortion. We conducted an ANOVA on the boundary ratings for each participant, with valence (positive, negative) and arousal (high, low) as within-subject factors. The results indicated a main effect of valence, F (1, 39) = 14.91, p < .001, h 2 p = .277, suggesting that participants extrapolated the spatial structure more from stimuli depicting positively-valenced emotions than from those showing negatively-valenced emotions (mean positive valence = -0.29, SD = 0.56; mean negative valence = -0.04, SD = 0.58). There was no effect of arousal, F (1, 39) < 1, nor a significant interaction between valence and arousal, F (1, 39) < 1.

In order to control that the effect we observed was due to the emotional valence and not to the spatial expanse of the gestures 4 , we compared (1) the average distances in pixels between the actor's arms and the boundaries (horizontal, vertical), and (2) the minimal distances in pixels between the actor's arms and the boundaries (horizontal, vertical) for negative and positive valences stimuli. In all the cases, no significant difference was observed, with (1) t( 14) = 1.049 and t(14) = .505, for horizontal and vertical average distances respectively, and (2) with t(14) = .21, and t(14) = .397, for horizontal and vertical minimal distances respectively. 4 The more an object fills a stimulus, the greater BE we will observe (e.g., [START_REF] Gottesman | Surface construal and the mental representation of scenes[END_REF].

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

Mean boundary rating

Emotion

Pleasure

Arousal

High Low

Anger Irritation Happiness

Valence Negative

Positive

Discussion

The present study was aimed at determining whether emotions such as happiness, pleasure, irritation, and anger expressed by actors in short films were likely to affect the boundary extension phenomenon. The results indicated a BE effect only for positivelyvalenced stimuli (i.e., happiness, pleasure), while negatively-valenced ones (i.e., anger, irritation) did not lead to memory distortion. Moreover, there was no observed effect of arousal, nor an interaction between valence and arousal.

Firstly, these results show that the BE effect, which to date has been observed almost exclusively with photographs, can also be obtained when films are presented [START_REF] Delucia | Visual memory for moving scenes[END_REF]. This finding is important in the sense that BE, so far mainly studied by having participants memorize still photographs, seems to be involved in the visual integration of scenes perceived in succession by the eye (e.g., [START_REF] Dickinson | Transsaccadic representation of layout: What is the time course of boundary extension?[END_REF].

Concerning the influence of the emotions expressed by the actors, our results support only partially previous observations on the links between BE and emotions. Like [START_REF] Mathews | Take a closer look: Emotion modifies the boundary extension effect[END_REF], we found a BE effect for positive-valence stimuli, but contrary to these authors (but see [START_REF] Safer | Tunnel memory for traumatic events[END_REF], we did not observe a BE effect with negativevalence films. Although the previous researches showed discrepant findings on the effects of emotional stimuli on BE, our results support the observations made by researches on the effects of emotion on memory (e.g., [START_REF] Levine | Emotion and memory narrowing: A review and goalrelevance approach[END_REF]. Indeed, it was shown that emotional valence involves a different treatment as stimuli represent a positive or a negative valence. As positive stimuli broaden attentional focus and facilitate the Memory Broadening phenomenon (e.g., [START_REF] Fenske | Modulation of focused attention by faces expressing emotion: Evidence from flanker tasks[END_REF][START_REF] Yegiyan | Encoding details: Positive emotion leads to memory broadening[END_REF], it is likely that these mechanisms could be the cause of the BE phenomenon observed for happiness and pleasure. On the other hand, negative stimuli classically involve narrowing of attentional focus and memory (e.g., [START_REF] Fenske | Modulation of focused attention by faces expressing emotion: Evidence from flanker tasks[END_REF][START_REF] Levine | Emotion and memory narrowing: A review and goalrelevance approach[END_REF], which could be the cause of the absence of significant memory distortion we observed for anger and irritation. One hypothesis, which would need to be tested in the future, is that the results could be due to a differing attention level or amount of attention directed to a photograph, depending on the valence of the emotions expressed in the films.

Whereas arousal is traditionally described as a fundamental component of emotional stimuli (e.g., [START_REF] Levine | Emotion and memory narrowing: A review and goalrelevance approach[END_REF], we observed no significant effect of this variable on BE. If these results seem surprising, in particular as [START_REF] Mathews | Take a closer look: Emotion modifies the boundary extension effect[END_REF] observed an effect of arousal on BE, it is possible that the nature of the presented stimuli had an incidence on our observations. Contrary to previous studies, arousal was not being designed to induce, but to convey emotions through the portrayal of the actors. On this matter, it has been shown that emotion recognition (i.e., perception) and experience (i.e., induction) refer to independent processes [START_REF] Ille | Emotion recognition and experience in Huntington's disease: Is there a differential impairment[END_REF]. Moreover, the sequences presented during our study are characterized by their dynamic aspect and have the particularity to simulate social interactions. Indeed, emotional expressions transmit socio-communicative signals relative to the internal state of an individual, from which the observer will infer behavioral intentions (Scherer & Bänziger, 2010).

In summary, our results showed that positively-valenced emotions led to an extension effect while the negatively-valenced ones did not produce BE. If different hypotheses can account for the discrepant findings observed in the literature on BE and emotions, one must notice that these studies have an high heterogeneity in the methodology used across studies: drawing tasks (e.g., [START_REF] Candel | Boundary distortions for neutral and emotional pictures[END_REF][START_REF] Candel | How children remember neutral and emotional pictures: Boundary extension in children's scene memories[END_REF], camera distance paradigm (e.g., [START_REF] Candel | Boundary distortions for neutral and emotional pictures[END_REF][START_REF] Safer | Tunnel memory for traumatic events[END_REF]our study), forced-recognition choice test (e.g. [START_REF] Mathews | Take a closer look: Emotion modifies the boundary extension effect[END_REF][START_REF] Safer | Tunnel memory for traumatic events[END_REF]. For a better understanding of the influence of emotion on the BE effect, further researches would have to examine if the observed differences are mainly linked to the different methodologies used, and if so, why.

Figure 1 .

 1 Figure 1. Photographs a, b, c, and d show one of the actors expressing one of the four emotions used in the experiment.

Figure 2 .

 2 Figure 2. Mean boundary ratings for each emotion. Error bars indicate the .95 confidence interval around the means.

The conclusions drawn from this study however probably need to be taken with caution. Indeed, the principal objective of this research was not to study BE, and the method employed was unusual for BE studies (e.g. only two pictures tested).

This idea was suggested by an anonymous reviewer in the first version of this article.

The use of a pre-existing corpus of experimentally validated films[START_REF] Vieillard | Children's perception and understanding of (dis)similarities among dynamic bodily/facial expressions of happiness, pleasure, anger and irritation[END_REF] did not allow us to include distractor films shot from closer up or farther away, as is generally done in studies based on this paradigm (e.g.,[START_REF] Intraub | Looking at scenes while searching for numbers: Dividing attention multiplies space[END_REF]. However, the studies which, like ours, did not employ distractors did not fail to observe a BE effect (e.g.,[START_REF] Munger | Are boundary extension and representational momentum related?[END_REF].

Authors' Notes

Emmanuelle Ménétrier, Laboratoire de Psychologie EA 3188, Université de Franche-Comté, Besançon, France; André Didierjean (1) Laboratoire de Psychologie EA 3188, Université de Franche-Comté, Besançon, France and

(2) Institut Universitaire de France ; Sandrine Vieillard, Laboratoire de Psychologie EA 3188, Université de Franche-Comté, Besançon, France. Correspondence concerning this article should be addressed to Emmanuelle Ménétrier or André Didierjean, Laboratoire de psychologie EA 3188, Université de Franche-Comté, 30-32 rue Mégevand, 25000 Besançon, France. Electronic mail may be sent to menetrier_emmanuelle@yahoo.fr or andre.didierjean@univ-fcomte.fr.

We express our gratitude to T. Bänziger, M. Mortillaro, and K.R. Scherer for granting us permission to use the GEMEP corpus.


O Boulle

S Rohart

L D Buda-Prejbeanu

E Jué

I M Miron

Stefania Pizzini

Jan Vogel

G Gaudin

A Thiaville

Domain wall tilting in the presence of the Dzyaloshinskii-Moriya interaction in out-of-plane magnetized magnetic nanotracks

Domain wall tilting in the presence of the Dzyaloshinskii-Moriya interaction in out-of-plane magnetized magnetic nanotracks

1

The effect of inversion asymmetry on the magnetic and electronic transport properties at interfaces of low dimensional magnetic film is currently attracting a growing attention.

In the presence of spin-orbit coupling, inversion asymmetry leads to an additional term in the exchange interaction, namely the Dzyaloshinskii-Moriya interaction (DMI) [1,2], which tends to make the magnetization rotate around a local characteristic vector D. This can destabilize the uniformly magnetized states leading to novel chiral magnetic order, such as spin spiral [3,4]. Novel out-of-equilibrium transport phenomena have also been demonstrated, such as current induced spin orbit torques induced by the Rashba spin orbit coupling and/or the spin Hall effect, leading to current induced magnetization reversal [5][6][7]. A recent striking example of the impact of the inversion asymmetry in ultrathin magnetic films is the current induced domain wall motion (CIDM) in perpendicularly magnetized nanotracks.

This was first outlined by Miron et al. who reported very efficient CIDM in asymmetric

Pt/Co(0.6 nm)/AlOx multilayers whereas symmetric Pt/Co/Pt multilayers showed no effects [8,9]. The high perpendicular anisotropy in this material leads to very narrow DW (∼ 5 nm), so that in typical experiments, the nanotrack width (∼ 100 nm) is much larger than the DW width. It is thus expected that the magnetization rotates parallel to the DW surface (Bloch DWs) to minimize the magnetostatic energy. Whereas these experiments were first interpreted in terms of a high non-adiabatic torque induced by the Rashba spin orbit coupling, we recently proposed that the high efficiency arises from two key features resulting from the inversion asymmetry and the high spin orbit coupling in this material [10]:

First, a change of the DW equilibrium structure from Bloch to Néel induced by the DMI. This leads to chiral DWs where the DW magnetization rotates perpendicular to the DW surface with a unique sense of rotation [11,12]; Second, a large Slonczewski-like spin orbit torque which is maximal in the Néel configuration [5,13]. Recent CIDM experimental results in Pt/Co/Ni [9] and Pt/CoFeB/MgO multilayers [15] seem to support this scheme.

In this Letter, we show that the inversion asymmetry not only affects the DW dynamics through a change of the internal DW structure but also through a modification of its shape.

In perpendicular magnetized nanotracks, the DW surface is expected to be perpendicular to the nanotrack axis to minimize the DW length and thus its energy. However, in the presence of DMI, when driving the DW dynamics, micromagnetic simulations reveal that a large DMI can lead to a sizable tilting of the DW surface which can strongly affect the DW dynamics. This DW tilting is a dynamical effect which occurs whatever the driving mechanism, e.g. an external magnetic field or a spin polarized current, and is thus intrinsically different from the previously reported current induced DW tilting [1,3,5] [30]. The results are well explained using an analytical model based on a Lagrangian approach where the DMI and the DW tilting are included. We also show that the DW tilting can be controlled using a static transverse magnetic field, and this provides a simple way to measure directly the DMI experimentally. Our results shed light on the unexplained current induced DW tilting recently observed in Co/Ni asymmetric multilayer nanotracks [4] and further support the presence of DMI in these materials [9].

We consider a magnetic ultrathin film grown on a substrate with a capping layer in a different material so that the inversion symmetry is broken along the vertical axis (z). The magnetization is supposed oriented out-of-plane with a strong perpendicular anisotropy.

In addition to the standard micromagnetic energy density which includes the exchange, anisotropy, Zeeman and demagnetizing energy, we add the following DMI that reads in a continuous form [10]

E DM = D m z ∂mx ∂x -m x ∂mz ∂x + id.(x → y)
. This form corresponds to a sample isotropic in the plane, where the Dzyaloshinskii vector for any in-plane direction u is Dz × u with D a uniform constant, originating from the symmetry breaking at the z surface. Micromagnetic simulations are based on the Landau-Lifschitz-Gilbert equation:

∂m ∂t = - γ 0 µ 0 M s δE δm × m + αm × ∂m ∂t -γ 0 H SO Jm × (m × u y ) (1)
where γ 0 = µ 0 γ with γ the gyromagnetic ratio, E the energy density and M s the saturation magnetization. We assume that the injection of a current density J in the nanotrack leads to a Slonczewski-like torque -γ 0 H SO Jm × (m × u y ) [5,7,20]. To simplify, we do not consider the effect of the adiabatic and non-adiabatic spin transfer torque as well as the field like part of the spin orbit torque [21]. In the following, we consider sufficiently large values of D (D > 0.12 mJ/m 2 for our simulation parameters) so that the Néel configuration is stable at equilibrium [10]. 2D micromagnetic simulations are performed using modified homemade micromagnetic solvers [10,22]. The following parameters have been used [9]: exchange parameter A=10 -11 J/m, saturation magnetization M s = 1.09×10 6 A/m, uniaxial anisotropy constant K = 1.25 × 10 6 J/m 3 , Gilbert damping parameter α = 0.5, thickness of t m = 0.6 nm.

The DW tilting induced by the DMI can simply be introduced by considering the effect of a static in-plane magnetic field H y transverse to the magnetic track ( To describe the dynamics of tilted DWs induced by the DMI, we consider an extended collective coordinate model (CCM) [START_REF] Malozemoff | Magnetic domain walls in bubble materials[END_REF] where the DW is described by three variables: its position q in the track, the DW magnetization angle ψ and the tilt angle of the DW surface χ

(cf Fig 1(a)
). The DW profile is described by the following Ansatz for the azimuthal θ and polar angle ϕ: ϕ(x, y, t) = ψ(t) -π/2 and θ = 2 arctan[(x cos χ + y sin χq cos χ)/∆]

(∆ = A/(K -µ 0 M 2 s /2)
). The effect of the DMI on the DW profile and dynamics is taken into account by an additional term in the DW energy (see below) [10].

To derive the dynamical equations, a Lagrangian approach is considered [START_REF] Hubert | Theorie der Domänenwände in geordneten Medien[END_REF][START_REF] Thiaville | [END_REF][26]. The LLG equation can be derived by writing the Lagrange-Rayleigh equations for the Lagrangian L = E + (M s /γ)ϕ θ sin θ with E the micromagnetic energy density and m = (sin θ cos ϕ, sin θ sin ϕ, cos θ). The effect of the damping and spin orbit torques is included

in the dissipative function F = αM s /(2γ)[dm/dt -(γ 0 /α)H SO Jm × u y ] 2 .
The Lagrange-Rayleigh equations then leads to the following CCM equations :

ψ + α cos χ ∆ q = γ 0 H z + π 2 γ 0 H SO J sin ψ, (2)
q cos χ ∆ -α ψ = γ 0 H k 2 sin 2(ψ -χ) + πDγ 0 2µ 0 M s ∆ cos(ψ -χ) - π 2 γ 0 H y sin ψ, (3)
χαµ 0 M s ∆π 2 6γ 0 tan 2 χ + w π∆ 2 1 cos 2 χ = -σ tan χ + πD cos(ψ -χ) + µ 0 H k M s ∆ sin 2(ψ -χ) ( 4
)
where σ is the wall energy per unit area with σ = 4 √ AK +πD sin(ψ-χ)+µ 0 H k M s ∆ sin 2 (ψχ) + π∆M s H y cos(ψ), with H k the DW demagnetizing field [32].

Assuming that αw ≫ ∆, these equations lead to a typical time scale for the tilting to settle τ = αµ 0 M s w 2 /(6σγ 0 ∆). The w 2 dependence is explained by the time to reverse the spins in the nanotrack surface swept by the DW when the tilting takes place. On the other hand, the magnetization angle in the DW frame relaxes on a shorter time scale

τ Φ = 1+α 2 αγ 1
πD/(2Ms∆)-H k which does not depend on w. In the steady state regime ( χ = 0, ψ = 0) and H y = 0, the tilt angle is directly related to the DW velocity v as :

tan χ = 2M s γσ v cos χ (5)
with the DW velocity v = γ 0 ∆ α cos χ (H z + π 2 H SO J sin ψ). This points to the dynamical origin of the DW tilting. A more physical picture can be obtained from the expression of the Lagrangian integrated over the nanotrack L DW /(t m w) = σ cos χ -2 Ms(Φ+χ) γ q, where Φ is the magnetization angle in the DW frame (Φ = ψ -χ). The first term is the DW internal energy proportional to the DW surface and thus scaling as 1/ cos χ. The second term can be seen as a kinetic potential [START_REF] Hubert | Theorie der Domänenwände in geordneten Medien[END_REF], which contrary to a kinetic energy, is linear with the DW velocity and the DW angle. For the field driven case in the steady state regime, Φ is defined only by the in-plane torques due to H z , D and H k (see Eq. Hall effect due to the current flowing in the non magnetic layer and/or from the Rashba spin orbit interaction [5,7]. It leads to an effective easy-axis magnetic field on the DW H SO J proportional to sin ψ (see Eq. 2), which is thus maximal for a Néel DW configuration (ψ = ±π/2) obtained for sufficiently high D. The field H SO can be very large ∼ 0.07 T/(10

12
A/m 2 ) in Pt/Co/AlOx [27] (see also Ref. [6,15,28,29]). Similarly to the action of H y , the In this section, we discuss other mechanisms that may lead to current induced DW tilting in magnetic nanotracks.

Anomalous Hall effect

When injecting a current in a perpendicularly magnetized nanotrack, the Anomalous Hall the DW in the direction of the charge carrier. This effect is mostly prevalent in narrow and thick nanotracks. For nm thin magnetic nanotracks, the DW drag effect is generally very small compared to the spin transfer/spin orbit torque [3]. However, the additional current densities ±∆J on the edges translate into different DW velocities v = v 0 + ±∆v which can create DW tilting. Assuming a constant velocity with time, the tilting angle χ increases with the pulse length τ as:

tan χ = 2τ w ∆v ≈ 2τ w dv dJ J ∆J ( 6
)
where w is the track width. Partin et al. [1] calculated the current density distribution in the nanotrack for a non-tilted DW. To first order in the Hall angle, the additional current density ∆J at the right and left edge of the track reads :

∆J J = ± 40 tan θ H 9π ( 7
)
where J is the injected current density.

For a Hall angle of 1% typically observed in perpendicularly magnetized ultrathin multilayers, this leads to an additional current density of about ±1.5% at the right and left edge of the track. A/m 2 , Eq. 6 leads to a DW tilting angle < 3 • for the 100 ns pulses used in their experiments. This is about an order of magnitude lower than the one observed experimentally (∼ 20 • ).

Reciprocally, the DW tilting observed experimentally would need an unrealistically large

Hall angle of about 16%. Thus the anomalous Hall effect cannot account for the reported DW tilting but may add a small contribution.

Current induced DW tilting was also reported by Yamanouchi et al. in (Ga,Mn)As diluted magnetic semiconductor nanotracks [5,6]. The nanotracks were typically 5 µm wide and 30 nm thick. In Ref. [5], the authors report a large DW tilting (up to 60 • ) for J < 1 × 10 10

A/m 2 which increases with the pulse width. In these experiments, the large thickness of the magnetic films makes unlikely the presence of Dzyaloshinskii-Moriya (DMI). However, (Ga,Mn)As may have very large Hall angles (5 to 10 % [5,7]) and the authors used long current pulses (up to 20 µs). Assuming an Hall angle of 0.1, the additional current circulating around the DW due to the anomalous Hall effect has a density ∆J/J = 15%. Using Eq. 6

and for a DW velocity of about 1.75 m/s for J = 4 × 10 9 A/m 2 , one can estimate a DW tilting of about 60 • in agreement with the experimental results. In the case of (Ga,Mn)As, the anomalous Hall effect seems thus to be the main driving force which creates the DW tilting [19].

Oersted field effect

The Oersted field created by the current circulating in the nanotrack has a perpendicular component which is antisymmetric with respect to the center of the track and can reach relatively large values on the edges (up to 5.2 mT at J = 1 × 10 12 A/m 2 for the 10 µm wide To study the effect of the Oersted field on the DW tilting, we carried out micromagnetic simulations taking into account the Oersted field for a 300 nm wide and 50 µm long track.

The following parameters have been used to mimic the experiments of Ryu et al. [4,9] : the Oersted a large impact on the tilt angle, a large decrease of the tilt angle as w increases would be expected [8], which is not observed. We can thus conclude that the Oersted field has little impact on the steady state tilt angle in these experiments.

A = 1 × 10 11 J/m, D = 0.8 mJ/m 2 , M s = 0.6 × 10 6 A/m, K u = 0.59 × 10 6 J/m 3 ,

Domain wall deformation in soft in-plane magnetized nanotracks

The DW deformation of moving DWs in soft in-plane magnetized nanotracks is a well established phenomenon [10][11][12][13][14][15][16]. In the absence of magnetocrystalline anisotropy, the equilibrium internal DW structure in soft nanotrack is defined by the demagnetizing and the exchange energy and thus depends only on the track aspect ratio for a given material. This leads to wide DWs whose internal structure can be easily deformed as compared to DWs in materials with strong magnetocrystalline anisotropy. Two types of DWs are typically found in such tracks, head-to-head vortex DWs (VDW) and transverse DWs (tDW) [15].

When driving a DW dynamics by an external magnetic field or a spin polarized current, complex DW deformations have been predicted [10][11][12]16]. For example, in the viscous regime, the VDW dynamics is characterized by a shift of the transverse position of the vortex core, whereas in the precessional regime, periodic DW transformations from tDWs to vortex/antivortex DWs occur [10,12,16]. Such periodic transformations of tDW into VDW were observed experimentally in permalloy nanotracks, induced by the injection of consecutive current pulses [14]. However, these changes are in fact a modification of the internal DW structure induced by the precessional torque and not a tilting of the DW surface. For a standard Bloch DW, this is physically equivalent to the in-plane rotation of the DW magnetization. In terms of our collective coordinate model, the DW deformations in VDW/tDW is equivalent to a change of the variable ψ which describes the DW magnetization (see Ref. [16,17] for a more detailed discussion) but not a change of the inclination of the DW surface χ. A tilting of the DW surface in soft magnetic nanotracks were predicted in the special case of a DW moving in the presence of a transverse in-plane magnetic field [18],

which may relate closer to the present study as the DMI acts similarly to a constant external magnetic field exerted on the DW.

  Fig.1(c,d). As expected, the DW tilting increases with H y and, for a fixed H y , increases with D. The tilt angle can be roughly estimated from energetic considerations assuming that the DW always stays in a Néel configuration with an energy per surface unit σ 0 . On the one hand, for a DW tilted by an angle χ, the DW surface and thus the total energy is increased by a factor 1/ cos χ; on the other hand, the Zeeman energy per unit surface scales as σ Z sin χ with σ Z = -πµ 0 H y M s ∆ (∆ is the DW width). This leads to a total DW energy E DW ≈ wt m (σ 0 -σ Z sin χ)/ cos χ, where w is the track width. The minimization of this energy leads to sin χ = σ Z /σ 0 . Importantly, the slope of the DW tilting as a function of H y on Fig.1(c) depends directly on the value of D. This provides a direct way to measure D experimentally, from the dependence of the DW equilibrium tilt angle on H y .In the presence of DMI, a tilting of the DW surface can also be induced dynamically by applying an easy axis external magnetic field H z . The magnetization distribution in the track for different magnetic fields (Fig.2(a), D = 2 mJ/m 2 ) reveals that the DW tilts significantly when driven by H z in the steady state regime. As shown on Fig. 2(b), the steady-state tilt angle rapidly increases with H z and D, although a saturation is observed for large H z . Fig 2(d) shows the DW velocity v along the track direction as a function of H z for different values of D. As expected, the DMI leads to an increase of the Walker field [10]. For large values of H z , the DW velocity significantly deviates from the expected linear scaling as D increases. This deviation is the direct result of the DW tilting: the propagation of the tilted DW at a velocity v n normal to its surface leads to a velocity v = v n / cos χ along the track direction. When considering v n instead of v (Fig. 2(d), inset)), the expected linear scaling is recovered and the velocity in the steady state regime does not depend anymore on D. The time dependence of the DW tilt angle is shown on Fig. 2(c) for several w when applying µ 0 H z = 100 mT at t=0 [31].

  [2][3] and does not depend on χ. The tilt angle in the steady state regime can thus be deduced from the minimization of L DW with χ at fixed Φ, which leads to Eq. 5[33]. The tilt angle is thus the result of a balance between the gain in the kinetic potential resulting from the DW tilting and the cost in the increased DW energy due to the larger surface.We now compare the predictions of this model with the results of the micromagnetic simulations. The continuous lines on Fig1(c,d) show the DW tilting induced by H y predicted by the CCM whereas the DW tilt angle, time dependence and DW velocity driven by H z are plotted in continuous lines on Fig.2(b-d) : a general good agreement is obtained with the micromagnetic simulations despite the simplicity of the model. We also plotted the results of the standard (q,ψ) model on Fig2(d) (dashed line). The model does not reproduce the nonlinear increase of the DW velocity, but a good agreement is obtained when considering the DW velocity in the direction perpendicular to its surface v n (inset). The DW tilting thus does not affect the DW velocity perpendicular to its surface.We now consider the current driven DW dynamics induced by the Slonczewski-like spin orbit torque in the presence of a large DMI. This torque is expected for structure with inversion asymmetry such as Pt/Co/AlOx trilayers[5, 9,21]. It may arise from the spin

  spin orbit torque tends to rotate the DW magnetization along the y direction away from the Néel configuration, providing an additional source for the DW tilting.The results of micromagnetic simulations of the DW dynamics driven by the spin orbit torque with µ 0 H SO = 0.1 T/(10 12 A/m 2 ) are shown on Fig.3. When injecting a current in the track, a fast DW motion is observed against the electron flow and the velocity increases with J and D (see Fig.3(d)). At the same time, a significant tilting of the DW occurs (see Fig.3(a) for D = 2 mJ/m 2 ), which increases with J and D (Fig.3(b)). The DW velocity and the tilting predicted by the CCM are shown on Fig.3(b,d), continous lines. An excellent agreement is obtained with the micromagnetic simulations except at higher current densities for the tilt angle due to the onset of a more complex DW structure (see Fig.3(a) for J = 2.5 ×10 12 A/m 2 ). The DW velocity in the direction perpendicular to the DW surface v n for D = 2 mJ/m 2 is plotted Fig.3(d), inset. Contrary to the field driven case, the standard (q,ψ) model strongly overestimates the DW velocity (continous line). As expected, the DW tilting leads to an additional rotation of the DW angle ψ away from π/2 where the torque is maximal. The DW tilting thus leads to a large decrease of the DW velocity. This clearly illustrates the importance of the DW tilting on the CIDWM for large DMIs. Fig.3(c) shows the time dependence of the tilting for a current pulse of J = 0.25 × 10 12 A/m 2 applied at t=0. The CCM (continuous lines) reproduces well the time scale for the tilting to take place which scales as w 2 . Experimentally, Ryu et al. recently reported fast current induced DW motion associated with a significant DW tilting in asymmetric perpendicularly magnetized (Pt/Co/Ni/Co/TaN) nanotracks [4, 9]. By studying the dependence of the current induced DW velocity on an in-plane longitudinal magnetic field, they present evidence of chiral DWs driven by the Slonczewski like spin-orbit torque in agreement with the presence of a DMI. The DW tilting is reversed for up/down and down/up DW which is well explained by Néel DWs pointing in opposite directions due to the DMI. From the longitudinal magnetic field required to suppress the CIDM and using the magnetic and transport parameters of Ref. [4, 9], one can deduce a DMI of D = 0.8 mJ/m 2 for A=1 ×10 -11 J/m. Using this value, micromagnetic simulations predict a steady state tilt angle of about 18 • for J = 1 × 10 12 A/m 2 close to the one measured experimentally (∼ 20 • ) [34]. Smaller additional contributions may also arise from the anomalous Hall effect and the Oersted field [35]. Our model thus accounts for the DW tilting reported by Ryu et al. which further supports the presence of DMI in these inversion asymmetric multilayers. To conclude, we have shown that the DMI can lead to a tilting of the DW surface in perpendicularly magnetized nanotracks when DW dynamics is driven by an easy axis magnetic field or a spin polarized current. The DW tilting is the result of the balance between the gain in the kinetic potential of the moving DW when tilted and the increased DW energy due to the larger DW surface. The DW tilting affects the DW dynamics for large DMI and the tilting relaxation time can be very large as it scales with the square of the track width. We propose a simple way to estimate the DMI in magnetic multilayers by measuring the dependence of the DW tilt angle on a transverse static magnetic field. Our results shed light on the current induced DW tilting observed recently in perpendicularly magnetized Co/Ni multilayers with inversion asymmetry and further support the presence of DMI in these materials. This work was supported by project Agence Nationale de la Recherche, project ANR 11 BS10 008 ESPERADO.

  effect (AHE) leads to charge accumulations on the edges of the track whose sign depends on the out-of-plane orientation of the magnetization. Around a magnetic domain wall (DW), the abrupt change in the charge polarity leads to a deviation of the current lines around the DW which results in an additional current density +∆J (resp. -∆J) on the right (resp. left) edge of the track. This effect was first described by Partin et al. [1] in 1974 and is at the basis of the DW drag effect introduced by Berger [2]: The current loop around the DW induced by the AHE creates an additional out-of-plane magnetic field which tends to move

and 2 .

 2 65 nm thick track used in the experiments of Ryu et al.[4, 5, 8]). It may thus favor a tilting of the DW. However, in the experiments of Ryu et al.[4], the Oersted field should tilt the DW in a direction opposite to what is observed experimentally and thus can not explained the reported DW tilting. It may however decrease the steady state DW angle as it goes against the effect of the DMI.

For larger width (w = 5 ,

 5 10 and 20 µm), tilt angles of around 20 • (J = 1.1 × 10 12 A/m 2 ) have been reported by Ryu et al. and the tilt angle depends little on the track width. Had

FIG. 1 :FIG. 2 :FIG. 3 :

 123 FIG. 1: (a) Schematic of the tilted DW. (b) Micromagnetic configuration of a 100 nm wide track with D=2 mJ/m 2 and a transverse magnetic field µ 0 H y = 100 mT. The color scale is the same as Fig. 2(a). (c) DW tilt angle as a function of µ 0 H y for D = 2 mJ/m 2 and (d) as a function of D for µ 0 H y = 100 mT. Dots are the results of micromagnetic simulations whereas the continuous lines are the results of the CCM.

  Ryu et al. reported in Ref.[4] current induced DW tilting in Pt(1.5)/Co(0.3)/Ni(0.7)/Co(0.15)/TaN(5) (thickness in nm). Assuming an Hall angle < 2%, one obtains ∆J/J < 3%. For a DW velocity of 100 m/s for a current density of 1 × 10 12

  a spin orbit torque effective field µ 0 H SO = 4.8 × 10 -14 T/(A/m 2 ) corresponding to a spin Hall angle of 0.1, a damping constant α = 0.05, a spin polarization P = 0, a buffer layer of Pt of 1.5 nm and a Co/Ni thickness of 1.15 nm. The current density is assumed homogeneous in the track. In the absence of Oersted field, the steady state angle is 18.3 • for J = 1 × 10 12 A/m 2 ,close to the predictions of the collective coordinate model (17.3 • ). For this wire width, the Oersted field leads only to a slight decrease of the steady state angle (17.6 • ). For a given current density, larger effects of the Oersted field are expected for larger width[8]. To study the effect of the track width, we carried out micromagnetic simulations for a 2 µm wide nanotrack (J = 1 × 10 12 A/m 2 ). Due to the much larger numbers of cells and the larger DW tilting relaxation time, the steady state was not fully reach in this case. However, one can estimate that the Oersted field leads to a decrease of the DW tilting by about 14 -15%, which leads to a steady angle of 15 -16 • . Experimentally, Ryu et al. reported tilt angles up to ∼ 13 • in 2 µm wide wires (J = 1.1 × 10 12 A/m 2 ).


Evan Constable
email: evan.constable@neel.cnrs.fr

Rafik Ballou

Julien Robert

Claudia Decorse

Jean Blaise Brubach

Pascale Roy

Elsa Lhotel

Laurent Del-Rey

Virginie Simonet

Sylvain Petit

J.-B Brubach

S Debrion
email: sophie.debrion@neel.cnrs.fr

Double vibronic process in the quantum spin ice candidate Tb2Ti2O7 revealed by terahertz spectroscopy

Keywords: THz spectroscopy, pyrochlore, frustration, spin-lattice coupling

Over the last decade, spin-ice physics has aroused significant attention [1,2]. It emerges in pyrochlore magnets, such as Ho 2 Ti 2 O 7 or Dy 2 Ti 2 O 7 , as a result of effective ferromagnetic interactions coupling the Ising-like rare-earth magnetic moments constrained along the local 111 directions by the crystal field [3][4][5] (Fig. 1 (a,b)). This combination impedes long-range order, favoring a disordered and macroscopically degenerate magnetic state governed by an organizing principle, the socalled "ice rule," where two spins point into and two out of each elementary tetrahedron in the structure [3].

The possibility that quantum fluctuations can melt the spin ice state is a topical issue. It is expected to lead to a wealth of exotic phenomena such as a quantum superposition of "two-in two-out" configurations and emergent electrodynamics with new deconfined particles [6][7][8][9][10][11].

In this context, the puzzling behavior of the spin liquid Tb 2 Ti 2 O 7 has attracted much attention. Despite effective antiferromagnetic interactions, Tb 2 Ti 2 O 7 features no long-range order [12,13], however short-range correlations are present over a broad temperature range [14][15][16]. Remarkably, the ground state supports elastic power law spin correlations [17,18], bearing some resemblance to the pinch point pattern observed in spin ices [19]. Below 300 mK, a dispersing low-energy collective mode is found to emerge from these pinch points, reaching its maximum energy at ∼0.25 meV [20]. This signal likely corresponds to fluctuations between the two |± electronic states of the crystal electric field (CEF) ground state doublet. It was interpreted as a signature of interactions between quadrupoles, highlighting the importance of those degrees of freedom [18,[20][21][22]. This is because the non-Kramers nature of Tb 3+ ions prevents magnetic exchange alone from inducing such fluctuations within the Ising-like ground state [22][23][24].

A property that distinguishes Tb 2 Ti 2 O 7 from the other rare-earth titanates is the CEF energy spectrum of Tb 3+ . It features a first excited doublet at a low energy of ∆ ≈ 1.5 meV above the ground state [25][26][27][28][29], compared to 25 meV for Ho 3+ in the prototype spin ice compound. Furthermore, below ∼20 K, this excited doublet has been shown to couple to a transverse acoustic phonon, forming a hybrid magneto-elastic excitation [30]. It was proposed that a magneto-elastic coupling, linear in the phonon displacement and in the quadrupolar electronic moments, could explain the peculiar ground state of Tb 2 Ti 2 O 7 . To confirm this hypothesis, it is important to show that this magneto-elastic coupling not only occurs for the excited CEF doublet, but also with the ground doublet.

In this article, we provide evidence for such a coupling using terahertz (THz) spectroscopy, which probes the Brillouin zone center in the energy range close to ∆. Our synchrotron-based experiments, performed with an unprecedented accuracy, show a splitting of both the ground and first excited doublets that occurs by coupling with the acoustic phonon at temperatures as high as 50 K. This confirms that the actual ground state of Tb 2 Ti 2 O 7 has a mixed magnetic and phononic nature. Moreover, we show that this low temperature vibronic process is not alone. A second process is observed as high as 200 K and couples the first excited CEF state and associated transition at ∆ with a silent optical phonon mode. This double vibronic process provides a natural path for spin-lattice quadrupolar coupling that amplifies transverse contributions on top of the Ising anisotropy. The measurements were performed using a Tb 2 Ti 2 O 7 single crystal grown by the floating zone technique previously described in [18,20]. We have measured the polarization dependent THz spectrum in the 0.2-1.0 THz (0.82-4.14 meV) energy range using both standard (SSR) [31] and coherent synchrotron radiation (CSR) [32,33] techniques. Focus is given to the "high temperature" regime, from 300 K to 6 K. Four plaquettes, 2 mm in diameter, were chosen to test the magnetoelectric activity of the excited modes according to the main cubic sym-metry axes. Specifically, the (100) wafer (1 mm thick) is perpendicular to the four-fold axis while the (111) wafer (300 µm thick) is perpendicular to the three-fold axis. To probe the local rare earth symmetry with an Ising direction along [111], two additional wafers (1 10), and ( 112) (300 µm thick) are used allowing three mutually orthogonal planes: (111), (1 10), and ( 112). Note then, that the projection of the polarized THz field in the local frame is mixed between the 4 Tb 3+ sites of each tetrahedron (See Supplemental Material [START_REF]Geometry of Tb2Ti2O7 crystal, CEF-phonon coupling symmetry, CEF doublet transitions, detailed experimental methods and far-infrared phonon spectra[END_REF]).

Typical spectra are given in Fig. 1(c) with a 200 K reference. As the temperature is lowered down to 6 K, an absorption peak develops around 0.42 THz (1.7 meV). This feature is consistent with the first excited CEF level previously reported by neutron and optical spectroscopic studies [25][26][27][28][29]. A much weaker and so far unreported peak is visible at 0.67 THz (2.78 meV). Previous experimental studies suggest that the low-temperature magnetic correlations first begin to develop below ∼100 K with short-range correlations strengthening below 20 K [12,[START_REF] Gardner | [END_REF]. We therefore analyze this temperature regime using the spectrum at 60 K (Fig. 1(d)) as a reference. Noticeably, in Fig. 1(d) the peak at 0.67 THz (2.78 meV) is not present indicating that its strength and energy position do not change significantly over this temperature range and its contribution is divided out along with the background. At the same time, close inspection reveals a three peak structure within the CEF excitation. The four observed peaks, with 200 K and 60 K references, are fit by Gaussian profiles shown in the inset of Fig. 1(d) designated P0, P1, P2 and P3. Their energy position and normalized total area are shown in Fig. 2. No significant temperature shifts are observed for their energy position except for a slight hardening below 20 K for P3 and above 100 K for P1. The precise fitted peak energies at 6 K are E0 = 0.332(4) THz (1.37(2) meV), E1 = 0.415(6) THz (1.71(2) meV), E2 = 0.502(9) THz (2.08(3) meV), and E3 = 0.672(6) THz (2.78(3) meV). P0, P1 and P2 mainly appear below 50 K, while P3 appears at the much higher temperature of 200 K, indicating a different origin.

It is tempting to assign the three peak structure (P0-P2) to a splitting of the ground and first excited CEF levels, that share the E g symmetry. The resulting four level scheme will give rise to the 3 observed excitations if the central peak (P1) corresponds to two different transitions of the same energy (see Fig. 3 (a)) within our experimental precision. The E g splitting is then similar for both the ground state doublet and the first excited doublet at a value of 0.085 THz (0.35 meV). Such a splitting corresponds very nicely in energy with the collective mode measured by neutron scattering at a much lower temperature (50 mK) [20]. It should be emphasized that it is experimentally impossible to assign the entire four peak spectrum to transitions between the two doublets with different splittings [START_REF]Geometry of Tb2Ti2O7 crystal, CEF-phonon coupling symmetry, CEF doublet transitions, detailed experimental methods and far-infrared phonon spectra[END_REF].

The temperature profile for the outer peaks (P0 and

P2

) is modeled by a two-level system with an energy barrier of 0.42 THz (∆ ≈ 1.7 meV). It bears similarity to the temperature dependence of the magneto-elastic mode (MEM) measured by Fennell et al. using inelastic neutron scattering [20,30]. This MEM has been shown to arise from hybridization between the T 1u acoustic phonon and first excited CEF level taking on a magneto-elastic component. A zone center extrapolation of the scattering vector dependent neutron data is in good agreement with the energy of P2 at 0.50 THz (2.08 meV). Since the acoustic phonon branch must also cross the CEF ground state at the zone center, hybridization with the ground state is also expected due to its identical E g symmetry. This is exactly what we observe, with similar splitting for both the ground and first excited states. This suggests that the collective mode built on the basis of the |± states in the ground doublet already incorporates lattice degrees of freedom as does the first excited CEF state. Similar vibronic phonon-CEF interactions have also been shown for CeAl 2 and facilitate mixing between the ground and first excited CEF states [37].

T 2u P1 P3 T 1u P0 P1 P1 P2 T < 200 K T < 50 K E g E g CEF 1 CEF 0 (a) (b) (c) (d)
optical phonon In Tb 2 Ti 2 O 7 , the hybridization process between the two E g CEF levels and the T 1u acoustic phonon depicted in Fig. 3 (b) is not alone. A recent report by Ruminy et al. [29] suggests that other hybridized magneto-elastic optical mode (MEOM) excitations occur at much higher energies around 3.9 THz (16 meV) due to the proximity of optical phonons. Their work is corroborated by a complementary study focused on the phonon spectrum in these compounds [36]. Noticeably, their ab inito calculations show that the lowest-energy phonon is at ∼1.1 THz (4.6 mev) and has T 2u symmetry. This phonon is optically silent (both in Raman and infrared (IR) spectroscopy), and it is not observed by inelastic neutron scattering because of a negligible scattering cross section. Since it is quite close in energy to the additional P3 excitation that occurs below 200 K at 0.67 THz (2.78 meV), we assign P3 to a coupling mechanism between this T 2u phonon and the first excited CEF level. It is worth noting that this optical phonon branch does not cross the E g ground state, and therefore is not coupled to it.

Using infrared reflection measurements, we have checked that the 7 expected IR active phonon modes are present in the range 80-800 cm -1 (2.4-24 THz) [START_REF]Geometry of Tb2Ti2O7 crystal, CEF-phonon coupling symmetry, CEF doublet transitions, detailed experimental methods and far-infrared phonon spectra[END_REF]. The observed phonon energies reveal a discrepancy in comparison to the calculated values which suggests that the lowest-energy T 2u phonon may indeed be even closer to the first excited E g CEF level.

Therefore, two different vibronic processes must occur in Tb 2 Ti 2 O 7 at two different temperature scales. A diagram of the two coupling mechanisms and corresponding level scheme is depicted in Fig. 3. The high-temperature vibronic process involves the first excited E g CEF and the lowest energy T 2u optical phonon. The resulting hybridization develops below 200 K as seen with the appearance of peak P3 at an intermediate energy between the E g CEF level and the T 2u phonon energy. This coupling mechanism transfers magnetic THz activity to an otherwise silent phonon. The low temperature vibronic process occurs when magnetic fluctuations set in below 50 K. It involves the ground and first excited CEF levels, which both have E g symmetry, and the T 1u acoustic phonon. It results in the splitting of both the ground and first excited doublets which acquire a mixed electronicphononic character.

Additional valuable information about these CEFphonon processes is obtained by the selection rules observed for each excitation and their angular dependence when the polarization of the THz electric (e) and magnetic (h) fields is rotated. We first confirm that the hybridized features (P2 and P3) are mainly of magnetic origin. When measuring two configurations with the same direction for h but 90 degrees rotation for e, the same features are present [START_REF]Geometry of Tb2Ti2O7 crystal, CEF-phonon coupling symmetry, CEF doublet transitions, detailed experimental methods and far-infrared phonon spectra[END_REF]. Using a rotative sample mount, we were then able to precisely measure the angular dependence of the spectrum for a rotation about the [100] fourfold and [1 10] two fold axes of the cubic structure (Fig. 4). Note that the [100] axis has no particular meaning in the rare earth site symmetry while the [1 10] direction corresponds to a two fold axis for two sites out of four on each tetrahedron [START_REF]Geometry of Tb2Ti2O7 crystal, CEF-phonon coupling symmetry, CEF doublet transitions, detailed experimental methods and far-infrared phonon spectra[END_REF]. Around [100], the spectrum remains unchanged as expected. In contrast, around [1 10], a strong anisotropy appears for all peaks except P1. P2 has a maximum intensity for h [111], while both P0 and P3 behave inversely with maximum intensities when h⊥ [111]. There is also a transfer of spectral weight between P2 and P3 showing that the two vibronic processes are not independent.

This double vibronic mechanism accounts for a mixing between electronic states and propagating phonon modes, which occur in crystals when the excitations are of the same symmetry and are close enough in energy [START_REF]Geometry of Tb2Ti2O7 crystal, CEF-phonon coupling symmetry, CEF doublet transitions, detailed experimental methods and far-infrared phonon spectra[END_REF]. The transverse components of the acoustic T 1u phonon fulfill the symmetry requirements for both the ground and first excited CEF states and their coupling may induce a splitting of these doublet levels. As for the silent optical T 2u phonon, its longitudinal component is also allowed to couple to the CEF states inducing no splitting but a shift in the CEF level. This optical phonon is close in energy to the first excited CEF doublet and will dominantly form vibrons with it at high temperature. Its dispersion can be reasonably assumed to be flatter than that of the acoustic phonon leading to a larger density of states. At lower temperatures, this vibronic process should weaken owing to thermal depopulation. The remaining acoustic-phonon vibronic process will then gradually take hold. While P3 is still present at 6 K, this pro- cess appears evident in Fig. 2 (c) with a slight reduction in the intensity of P3 at the lowest temperatures.

Importantly, the double vibronic process involves quadrupolar operators which behave as spin flip operators in the |± ground state doublet basis. As a result, phonons (and lattice zero point motions) are now the source of spin fluctuations and by their virtual exchange may give rise to new quantum terms in the effective Hamiltonian of Tb 2 Ti 2 O 7 . Depending on their strength, a long-range quadrupolar order or Coulombphase-like state might be stabilized, provided the Ising term dominates [7,10,21]. Furthermore, an interesting feature of the vibronic process is that the acoustic phonon crosses the two doublets, which themselves being close in energy, will also partially mix. We anticipate that the magnetic dipolar operators will then behave as spin flip operators in the basis of the dressed |± electronic states. This mechanism, already proposed in Ref [38], could be the source of additional fluctuations. It may also be present in Tb 2 Sn 2 O 7 but is not allowed in Ho 2 Ti 2 O 7 and Dy 2 Ti 2 O 7 : Although the symmetries of both phonons and CEF states are identical, the energy of the first CEF state is one order of magnitude larger, reducing the vibronic process solely to the ground state. More importantly, the quadrupolar matrix elements in the basis of the ground |± electronic states are essentially zero for Ho 3+ and Dy 3+ .

In summary, using synchrotron-based THz spectroscopy, we show that two temperature regimes are present in the spin ice candidate Tb 2 Ti 2 O 7 revealing two different vibronic processes. The first one, below 50 K, concerns the ground and first excited CEF levels which couple to the acoustic T 1u phonon. This is the temperature scale of magnetic fluctuations [12] as well as elastic softening [39]. The second regime, below 200 K, is associated with the coupling between the first CEF level and the lowest energy optical phonon of symmetry T 2u and is in agreement with the high temperature plateau observed in the muon spin relaxation rate [12] for this compound. This double vibronic process, of quadrupolar origin, affects the electronic state that can no longer be described solely by electronic wave functions. Rather, a collective vibronic state prevails, built on the ground and first excited CEF states mixed with two different phonons. It is activated with dramatic effects in Tb 2 Ti 2 O 7 by virtue of adequate energy matching and strong quadrupolar moments that occur in this compound. This unique combination of vibronic processes revealed by our THz study supports the formation of the Tb 2 Ti 2 O 7 quantum spin liquid state; it may also provide a broadly applicable model in the formation of other exotic quantum ground states.

SUPPLEMENTARY MATERIAL

Geometry of the rare earth stacking Tb 2 Ti 2 O 7 crystallizes in the pyrhochlore structure with space group F d 3m. The Tb 3+ ions are the only ones carrying a magnetic moment and occupy the 16d site with local symmetry described by the point group 3m, symbolised D 3d in the notation of Schoenflies. The network of apex-connected tetrahedra they form is generated from the Tb 3+ ions of a single tetrahedron by applying translations of the face centered cubic lattice (Fig. S5). In each of the four lattices the local 3-fold axis coincides with one of the four diagonals of the cubic cell. There are three local 2-fold axes perpendicular to each 3-fold axis that coincide with three face diagonals of the cubic lattice. Each 2-fold axis is perpendicular to two 3-fold axes. Calling the z-axis the appropriate cube diagonal and the x-axis one of the 2-fold axes perpendicular to it, we define a local frame on which the THz electric and magnetic field components can be projected. The interrelation between the four local frames and the global cubic frame are provided in table T1.

1 √ 2 [ 110] 1 √ 2 [110] 1 √ 2 [ 11 0] yi 1 √ 6 [11 2] 1 √ 6 [ 112 ] 1 √ 6 [1 12] 1 √ 6 [ 112] zi 1 √ 3 [111] 1 √ 3 [ 11 1] 1 √ 3 [1 11 ] 1 √ 3 [ 11 1] t [11 2] [010] [0, 1 3 , -4 3 √ 2 ] [ 1 √ 3 , -2 3 , 2 3 √ 2 ] [ -1 √ 3 , -2 3 , 2 3 √ 2 ] u [1 10] [100] [ 100] [0, 1 √ 3 , 2 √ 6 ] [0, -1 √ 3 , -2 √ 6 ] v [111] [001] [0, -4 3 √ 2 , -1 3 ] [ 2 √ 6 , 2 3 √ 2 , -1 3 ] [ 2 √ 6 , 2 3 √ 2 , 1 3 ] v [001] [0, - √ 2 √ 3 , 1 √ 3 ] [0, - √ 2 √ 3 , 1 √ 3 ] [0, √ 2 √ 3 , -1 √ 3 ] [0, √ 2 √ 3 , -1 √ 3 ]
Crystal electric field -phonon coupling

The CEF spectrum of each Tb 3+ ion is protected from splitting and mixing by the D 3d symmetry of its charge environment. When the crystal vibrations are taken into account, the positions of the surrounding ions change in time and the cage they form about the Tb 3+ ion is dynamically deformed. This results in symmetry breaking of their electrostatic interactions. The latter can be expanded over the spherical harmonics (Y m n ) on the f electrons angular variables (θ i , φ i ) most generally in the form i nm s A m n s r n i Y m n (θ i , φ i ), where r i is the i-th f electron radial variable and A m n s stands for the contribution associated with the surrounding ion s. The expansion complies with the D 3d symmetry if and only if it involves invariants belonging to the trivial representation A 1g of the point group D 3d , in which case it leads to the CEF Hamiltonian:

H CEF = A 0 2 r 2 C 0 2 + A 0 4 r 4 C 0 4 + A 3 4 r 4 (C 3 4 -C -3 4 ) + A 0 6 r 6 C 0 6 + A 3 6 r 6 (C 3 6 -C -3 6 ) + A 6 6 r 6 (C 6 6 + C -6 6 ), (1)
where A m n accounts for the contribution of all the surrounding charges, r n is the expectation value of r n over the radial part of the f-electron wave function and C m n are the Wybourne operators. The oscillations of the ions modify the coefficients A m n s . This can be accounted for by the expansion A m n s + ∇ rs A m n s • δ r s + • • • provided that the displacement δ r s of the surrounding ion s from its equilibrium positions relatively to the Tb 3+ ion is small enough. A single ion magnetoelastic interaction Hamiltonian H M E then emerges to first order. Non linear displacement effects and cooperative multi-ion magnetoelastic interactions occurring at the higher orders will be ignored. H ME will involve Wybourne operators that are not present in H CEF whenever s ∇ rs A m n s • δ r s is symmetry breaking. In order to find these operators it proves convenient to describe the linear displacements by normal modes u Γ -that transform according to the parity odd irreducible representations Γ -of the point group D 3d and take advantage of the orthogonality properties of the irreducible representations to write the single ion magnetoelastic interaction Hamiltonian in the form:

H ME = nΓγγ ζ nΓγγ u Γ - γ r n m f Γ + γ (C m n ) = nΓγγ ζ nΓγγ (a † Γ - γ + a Γ - γ ) r n m f Γ + γ (C m n ), (2)
where ζ nΓγγ is the magnetoelastic coefficient associated with the normal mode u Γ - γ and a † Γ - γ (a Γ - γ ) the creation (annihilation) operator of a phonon in this mode. γ distinguishes the different modes within a multidimensional Γ -.

m f Γ + γ (C m n
) is a linear combination of the Wybourne operators that are allowed to couple to lattice vibrations. To determine them, we look at symmetry consideration including spatial parity conservation. ζ nΓγγ is a linear combination of the components of ∇ rs A m n s . It is therefore parity odd since the quantity ∇ rs A m n s • δ r s is parity even as a scalar quantity. Accordingly, the invariants f Γ + γ (C m n ) must belong to the parity even irreducible representations Γ + of the point group D 3d whose character is the same as the one of Γ -on the group elements that are not combined with the space inversion and opposite otherwise. With ζ nΓ -∼ e 2 /r 2n+2 s the magnetoelastic coupling energy for displacements in the range 0.01 -0.1 Å is numerically estimated in the order of 0.5 -5 meV for n = 2, in the order of 0.01 -0.1 meV for n = 4 and in the order of 0.0005 -0.005 meV for n = 6. It follows that all the Wybourne operators other than the quadrupolar ones (n = 2) can be ignored in a first approximation. These quadrupolar operators transform according to the irreducible representation D + 2 of the rotation group O(3) associated with the angular momentum n = 2. The subduction to the group

D 3d of D + 2 is reduced into D + 2 ↓D 3d = Γ + 1 ⊕ 2 Γ + 3 = A 1g ⊕ 2 E g .
An inspection of the invariants reported in table T2 then informs us that the phonon modes belonging to the irreducible representation E u can couple to the CEF states through the quadrupolar Wybourne operators C m 2 ± C -m 2 (m = 1, 2). Within the ground multiplet |JM this transposes itself to possible coupling of the E u phonon modes with CEF states through the quadrupolar Stevens operators:

O 2 2 = J 2 x -J 2 y O -2 2 = J x J y + J y J x O 1 2 = J z J x + J x J z O -1 2 = J z J y + J y J z
The only other possible coupling is with the phonon mode belonging to the irreducible representation A 1u through the quadrupolar Weybourne operator C 0 2 which transposes to the Stevens operator in the ground multiplet |JM :

O 0 2 = 3J 2 z -J(J + 1)
The acoustic phonons at the Brillouin zone centre in Tb 2 Ti 2 O 7 belong to the irreducible representation T 1u of the factor group m 3m. Its subduction to the point group D 3d is reduced into T 1u↓D 3d = A 2u ⊕ E u , meaning that the triply degenerate acoustic modes separate themselves into a non degenerate longitudinal mode (A 2u ) and doubly degenerate transverse modes (E u ) with respect to the local 3-fold axis. These transverse acoustic phonon modes are those that might induce the splitting and the mixing of the ground and first excited CEF doublets that are separated only by about 1.5 meV. In the subspace of these two doublets the single ion magnetoelastic interaction Hamiltonian H ME takes the form

H ME = γγ ζ 2Eu γγ (a † Euγ + a Euγ ) r 2 m=1,2 s=± x=0,1 s =± x =0,1 |ψ x s ψ x s | f Egγ (C m 2 ) |ψ x s ψ x s |, (3)
where |ψ 0 ± (resp. |ψ 1 ± ) are two orthogonal states of the ground (resp. excited) doublet. The 3-fold symmetry implies that they must expand on the basis of the multiplet states |JM in the form n α M -3n |M -3n whereas the 2-fold symmetry transforms this into (-1) J-M n α M -3n (-1) n | -M + 3n . The two states are parallel if M ≡ 0 (modulo 3), thus forming a singlet, and orthogonal otherwise, forming a doublet. It follows that:

|ψ x + = α x 5 | + 5 + α x 2 | + 2 + α x -1 | -1 + α x -4 | -4 , |ψ x -= α x 5 | -5 -α x 2 | -2 + α x -1 | + 1 -α x -4 | + 4 ,
with M α x M α x M = 0 to achieve orthogonality of the various states. The oscillator strengths ψ x s | f Egγ (C m 2 ) |ψ x s in eq.3 are then evaluated to:

ψ x s | O 1 2 |ψ x s = (1/i) ψ x s | O -1 2 |ψ x s = (1 -δ s,s ) -3 5/2(α x 2 α x -1 + α x 2 α x -1 ) + (9/2) 11/2(α x 5 α x -4 + α x 5 α x -4 ) ψ x s | O 2 2 |ψ x s = (-1/i) ψ x s | O -2 2 |ψ x s = (1 -δ s,s ) 21α x -1 α x -1 -3 √ 30(α x 2 α x -4 + α x 2 α x -4 ) .
Accordingly, the coupling of the transverse phonons to the CEF doublets can occur through both the quadrupolar operators O ±1 2 and O ±2 2 with oscillator strengths depending on the actual structure of the doublet states, that is, on the parameters α x M . Using numerical values published in the literature for the α M coefficients [2], one observes that the strongest coupling comes from the O ±1 2 quadrupoles. Interestingly, the same numerical calculations performed for Ho 2 Ti 2 O 7 show that those quadrupolar matrix elements are essentially zero. This is also the case for Dy 2 Ti 2 O 7 when generalizing the above analysis to Kramers ions.

In Tb 2 Ti 2 O 7 , an optical phonon is also suspected to exist at energy close to the first CEF excited doublet, which would belong to the irreducible representation T 2u of the factor group m 3m (silent in both IR and Raman spectroscopy). Its subduction to the point group D 3d is reduced into T 2u↓D 3d = A 1u ⊕ E u , meaning again that the associated triply degenerate phonon modes give rise to a singlet (A 1u ), the longitudinal mode, and a doublet (E u ), the transverse modes. These transverse modes can couple to the first CEF excited doublet with similar oscillator strengths as calculated above resulting in the splitting of the CEF excited state. The longitudinal mode can also couple to the first CEF state through the quadrupolar operator O 0 2 . No splitting will result but a shift in energy will occur. This is most likely the process that it activated below 200 K in Tb 2 Ti 2 O 7 .

Crystal electric field transitions

Considering a splitting of the ground and first excited CEF doublets, we can expect four possible transitions as seen in Fig. S6. In the main article, we assign three of the observed peaks (P0, P1 and P2) within this transition scheme. P0 corresponds to transition (1), P1 to transitions (2) and (3), and P2 is (4). P3 does not fit into the scheme and is therefore described by the T 2u coupling mechanism. However, this proposed scheme requires that the difference in energy splitting of each doublet is equivalent (ε 1 -ε 0 = ε 3 -ε 2 ). In such a case, transitions (2) and (3) will have the same energy difference and therefore appear as one peak. If we assume different energy splittings (ε 1 -ε 0 = ε 3 -ε 2 ), then it should be possible to observe the four distinct transitions. We may then expect that the four peaks observed in the main article correspond to the four transitions here. While the different temperature profiles of peaks P0, P1

D 3d 3m C1 C2 C3 C1 C2 C3 Basis Vectors -Invariants A1g Γ + 1 1 1 1 1 1 1 Y 0 2 A2g Γ + 2 1 -1 1 1 -1 1 Jz Eg Γ + 3 2 0 -1 2 0 -1 (Y +m 2 ± Y -m 2 ) (m=1,2) (Jx, Jy) A1u Γ - 1 1 1 1 -1 -1 -1 A2u Γ - 2 1 -1 1 -1 1 -1 Z Eu Γ - 3 2 0 -1 -2 0 1 (X, Y ) TABLE T2
. Irreducible representations of the group D 3d , denoted Γ ± i and according to the Mulliken convention [3], C1 stands for the identity, Cn for the class of the n-fold rotations, C1 for the space inversion and Cn for the class of the n-fold rotoinversion. The basis vectors (or the invariants) associated with each irreducible representation Γ ± i can be computed by means of the projectors

P ± i = dim(Γ i ) Card(D 3d )
g∈D 3d χ i (g) g, where χ ± i is the character of Γ ± i , and when dim(Γi) > 1 the exchangers

Q nm i = dim(Γ i ) Card(D 3d ) g∈D 3d Γ ± i mn (g -1 )
g by applying these operators on trial objects (vector components, spherical functions).

E g E g (2) (3) (4)

(1) S6. Four possible transitions between ground, and first excited Eg CEF doublets. and P2 compared to P3 is a strong indication that P3 does not fit into this scheme, here we show arithmetically that such a four peak scheme is unlikely given their observed energies.

ε 0 ε 1 Δ ε 2 ε 3 CEF 1 CEF 0 FIG.
First we define ε 0 = 0. Then the largest transition (4) ε 0 → ε 3 . If we take the highest energy peak P3 = 0.675 THz as this transition, then ε 3 = 0.675 THz. Now, depending on whether ε 1 -ε 0 > ε 3 -ε 2 or ε 1 -ε 0 < ε 3 -ε 2 , the next largest transition will be either (3) ε 0 → ε 2 or (2) ε 1 → ε 3 . Assuming the first case, P2 = 0.50 THz corresponds to ε 0 → ε 2 setting ε 2 = 0.50 THz. So ε 3 -ε 2 = 0.175 THz. Now, (2) is the next largest transition and must correspond to P1 = 0.414 THz. So ε 3 -ε 1 = 0.414 THz. Since we have shown ε 3 = 0.675 THz, then ε 1 = (0.675 -0.414) THz = 0.261 THz. Therefore, transition (1) which is ε 1 → ε 2 must be (0.50 -0.261) THz = 0.239 THz which is not consistent with our measurement of P0 = 0.33 THz. The same outcome occurs when considering the case ε 1 -ε 0 < ε 3 -ε 2 . Therefore, a scheme that attempts to conform the four observed peaks into the four transitions in Fig. S6 cannot reconcile with the energies in our spectra. Furthermore, such a scheme is also inconsistent with previous inelastic neutron scattering studies.

Detailed experimental methods

Sample details

The single crystal samples used in the experiments were cut from a larger crystal used in a previous neutron study [4]. They were then prepared into thin wafers. Each sample was selected with a specific crystallographic face confirmed and orientated by Laue reflected X rays. Three mutually orthogonal planes (111), (1 10) and ( 112) were chosen to test the electro/magneto activity of the excited modes. With polarized radiation incident on the (111) plane, the magnetic (h) and electric (e) field components can probe either the (1 10) or ( 112) planes. Investigating the (1 10) cut sample allows h and e field probing of the (111) and ( 112) planes. In principle, the h field can therefore be aligned with the (11 2) plane in both cases while the e field probes different planes. In this way, it is possible to isolate the effects of h or e fields on specific planes in the crystal. An example of this is shown in from the P3 mode is strongest when h lies in the (11 2) plane. A final sample cut normal to the low index (100) plane was also selected to compare with any observed spectroscopic anisotropies. The (111), (1 10) and ( 112) cut samples had a thickness of 0.2 mm, 0.3 mm and 0.3 mm respectively. The (100) sample had a thickness of 1 mm.

Terahertz synchrotron setup

The THz spectroscopy experiments were performed at the SOLEIL synchrotron in France on the AILES Far-Infrared beamline. Spectra were obtained in the 0.15-1.0 THz (0.62-3.8 meV) energy range with a Brüker IFS125 spectrometer and 1.6 K bolometer using both standard (SSR) and coherent synchrotron radiation (CSR) techniques. In the studied energy range, the THz wave is linearly polarized at more than 95%. For the SSR measurements, a 125 µm mylar beam splitter was used to access the low energies required for the experiment. Multiple reflections in the beam splitter produce a spectral minimum at0.75 THz, therefore, for SSR measurements, we omit data at this frequency due to a high noise level. In this configuration, samples were mounted at fixed orientations with respect to the THz polarization on the cold head of an adapted Cryomech PT 405 pulse tube cryostat. The temperature range accessible was 6-300 K. In the CSR configuration, the SOLEIL accelerator was operating in the low-α coherent mode, with a α/α 0 parameter of 1/25 [5,6]. Under this operation, a much higher flux of low-energy photons is produced. This facilitated the use of a 50 µm mylar beam splitter that removes the spectral minima at 0.75 THz while maintaining strong signal at low energies. In this configuration, samples were mounted on an Attocube ANR240 rotary stage attached to the cold head of the cryostat. The thermalization of the rotary stage limited the base temperature to 12 K. Using the rotary stage, the sample could be rotated through 100 • with respect to the THz polarization with a precision of 0.1 • . During analysis the data were filtered using a Fourier filtering procedure that removes the 2 cm -1 fringe noise due to multiple reflections in the setup [7].

Far-infrared setup

Far-infrared measurements of the optical phonon spectra were performed in reflection geometry at near normal incidence. The measurements were performed on a Brüker Vertex 70v spectrometer with a mercury discharge-lamp source and 4.2 K bolometer detector. A far-infrared silicon beam-splitter was used, with the measurements covering the 50-900 cm -1 (1.5-27 THz, 6-112 meV) spectral range. The sample was mounted on the cold head of an in-houseadapted Janis flow cryostat reaching temperatures from 6 to 300 K

Far-infrared phonon spectra

To complement the terahertz spectroscopy provided in the article, we have also performed far-infrared reflectance spectra of the (100) cut sample. Here we aim to analyse the infrared active optical phonon spectra and their temperature dependence to gain insight into the lattice dynamics. The reflectance spectra at selected temperatures from 6-300 K are shown in Fig. S8 (a). Here, a number of Reststrahlen bands are present due to phonon absorption. The A 2 B 2 O 6 O stoichiometry of Tb 2 Ti 2 O 7 tells us that the A 3+ and B 4+ ions occupy the 16d and 16c Wyckoff positions while the O and O ions are at the 48f and 8b positions respectively. Factor group analysis of the m3 m point group table revels the allowed normal modes at the zone center, which are given by the following representations [8],

Γ = A g1 ⊕ 3A 2u ⊕ 3E u ⊕ E g ⊕ 4T 2u ⊕ 4T 2g ⊕ 8T 1u ⊕ 2T 1g .
Of these modes, only the 8 T 1u modes are infrared active with one being an acoustic mode. We therefore expect 7 phonon modes to appear in our reflectivity data. Because of the numerous crystal field levels known to be mixed within the phonon spectrum and our symmetry analysis allowing coupling between them, the precise form of the dielectric response is non-trivial. Rather than applying the dispersive method of fitting the reflectivity to a simplified dielectric model (as in the analysis of Bi et al. for Dy 2 Ti 2 O 7 [9]), here we analyze the dielectric response by Kramers-Kronig analysis. The real and imaginary components of the dielectric response, determined by a Kramers-Kronig transform of the reflectivity, are shown in Figs. S8 (b) and (c). Peaks in the imaginary response represent absorptions due to phonon excitations. Seven distinct peaks are observed corresponding to the 7 predicted T 1u optical modes. No soft modes are present which may have indicated a structural instability or phase transition. Notably, the lowest energy phonons (T 1 1u , and T 2 1u ) in close proximity to the 2nd and 3rd crystal field levels, feature abnormal peak profiles and temperature dependence. Nevertheless, approximate phonon energies and intensities can be extracted by Lorentzian profile fits to the curves. The temperature dependence of the peak positions are shown in Fig. S8 (d), with dashed gray lines representing the neighboring crystal field levels. The normalized phonon intensities, determined by the area of the fitted peaks, are shown in Fig. S8 (e). While the 5 higher energy phonons follow a similar temperature dependence in their intensity profile, strengthening or weakening as the lattice contracts between room temperature and 200 K, the two lower energy modes are distinct. These two modes, which are suspected of interacting with the crystal field levels, feature rapid strengthening between 200 K and 80 K. Notably this is similar to the temperature dependence of P3 from Fig. 4 (c). A summary of the results at base temperature is presented in Table T3.

FIG. 1 .

 1 FIG. 1. Local symmetries and THz spectra close to the first CEF excitation in Tb2Ti2O7. (a) One tetrahedron showing the "two-in two-out" spin configuration with the four spin directions. (b) Local D 3d environment of each magnetic ion. (c, d) Temperature dependent THz spectra highlighting the two temperature regimes by using the sample as a reference at 200 K (c) and 60 K (d). The spectra are fitted by four Gaussian profiles designated P0, P1, P2 and P3 (inset). Vertical gray lines indicate base temperature positions of fitted peaks.

FIG. 2 .

 2 FIG. 2. (a-c) Temperature dependence of the P0, P1, P2 and P3 peak intensity determined by the area of the Gaussian fits. Peak area is normalized to the value at 6 K. In panel (a) the intensity of the MEM measured by Fennell et al. [30] at 5 meV is also shown, which represents the onset of magneto-elastic correlations. The population profile of a two-level model, n1 -n0 where ni = exp(-∆i/kBT )/ 1 0 exp(-∆j/kBT ) and ∆1 = 1.7 meV, is added in (a) and (b). In panel (c) the intensity of the MEOM measured by Ruminy et al. [29] at 16 meV represents the onset of high temperature CEF-Phonon coupling. (d) Temperature dependence of fitted peak energies. Uncertainties represent the span of the peak's half width at half maximum.

FIG. 3 .

 3 FIG. 3. Sketch of the two CEF-phonon hybridization processes that occur successively. (a) Four level scheme giving rise to the three excitations P0-P2. (b) Low temperature coupling process involving two CEF states and an acoustic T1u phonon. (c) Two level scheme giving rise to excitations P1 and P3. (d) High temperature coupling process involving the first excited CEF state and a low-energy T2u optical phonon.

FIG. 4 .

 4 FIG. 4. Angular dependence of Tb2Ti2O7 spectra around the [110] (a) and [100] (b) rotation axes obtained using CSR configuration. Insets show fitted peak intensities as a function of the THz magnetic field orientation.

  FIG. S5. The pyrochlore network of Tb 3+ tetrahedra connected at their apexes. The 4-fold [100], 3-fold [111] and 2-fold [110] axes of the cubic F d 3m structure are shown. Image generated using VESTA software package [1].

  FIG. S7. (a)-(e) Polarization dependent absorbance spectra recorded at 6 K or 12 K with SSR or CSR configurations. The hatched rectangles mask the beamsplitter artifact at 0.75 THz in the SSR spectra. The peak fits for P0-P3 are shown in each spectra identifying that the multi peak structure is indeed reproducible. Red arrows and crosses indicate orientations where the P3 excitation is present and absent respectively (f) The (11 2) plane within the pyrochlore lattice. Aligning h in this plane produces the strongest absorption from the P3 excitation.

  FIG. S8. (a) Far-infrared reflectivity of (100) cut sample from 6-300 K. (b) Real and (c) imaginary components of dielectric function determined by Kramers-Kronig analysis. Peaks in 2 represent the absorption bands of phonons. (d) Temperature dependence of fitted phonon frequencies. Dashed gray lines indicate energies of known crystal field levels. (e) Temperature dependence of the normalized peak intensities.

TABLE T3 .

 T3 Table of transverse optical phonon energies determined by spectra at base temperature (6 K) compared with previously reported results. Symmetry This work Ruminy et al. [10] Bi et al. (Dy2Ti2O7)[9]

	Phonon	cm -1	cm -1	cm -1
	T1u	90.9	62.1	85

ACKNOWLEDGEMENTS

This work was supported by the French ANR Project No. ANR-13-BS04-0013-01. We thank J. Debray for sample preparation and F. Lévy-Bertrand for assistance with infrared measurements.