E Burzurí

F Luis

O Montero

B Barbara

R Ballou

S Maegawa

Quantum interference oscillations of the superparamagnetic blocking in an Fe 8 molecular nanomagnet

Keywords:

We show that the dynamic magnetic susceptibility and the superparamagnetic blocking temperature of an Fe8 single molecule magnet oscillate as a function of the magnetic field Hx applied along its hard magnetic axis. These oscillations are associated with quantum interferences, tuned by Hx, between different spin tunneling paths linking two excited magnetic states. The oscillation period is determined by the quantum mixing between the ground S = 10 and excited multiplets. These experiments enable us to quantify such mixing. We find that the weight of excited multiplets in the magnetic ground state of Fe8 amounts to approximately 11.6 %.

High-spin molecular clusters [1,2] display superparamagnetic behavior, very much as magnetic nanoparticles typically do.

Below a time-(or frequency-)dependent blocking temperature T b , the linear magnetic response "freezes" [3,4] and magnetization shows hysteresis [5]. The slow magnetic relaxation of these singlemolecule magnets (SMMs) arises from the presence of high anisotropy energy barriers separating spin-up and spin-down states. Because of their small size, the magnetic response shows also evidences for quantum phenomena, such as resonant spin tunneling [3,[6][7][8]. In the case of molecules that, like Fe 8 (cf Fig. 1A and [4]), have a biaxial magnetic anisotropy, tunneling between any pair of quasi-degenerate spin states ±m can proceed via two equivalent trajectories, which, as illustrated in Fig. 1B, cross the hard anisotropy plane close to the medium anisotropy axis. A magnetic field along the hard axis changes the phases of these tunneling paths, leading to either constructive or destructive interferences. This phenomenon is known as Berry phase interference [START_REF] Berry | Proc. R. Soc. Lond. A[END_REF][START_REF] Garg | [END_REF].

Experimental evidences for the ensuing oscillation of the quantum tunnel splitting ∆ m , shown in Fig. 1C, were first observed in Fe 8 [11,12] and then in some other SMMs [13][14][15][16][17][18][19] by means of Landau-Zener magnetization relaxation experiments. Quantum interference patterns measured on Fe 8 at very low temperatures, which correspond to tunneling via the ground state doublet m = ±10, are well reproduced by the following spin Hamiltonian

H = -DS 2 z +E(S 2 x -S 2 y )+C ( S 4 + + S 4 - ) -gµ B -→ S • -→ H (1)
that applies to the lowest lying spin multiplet, with S = 10, and where D/k B = 0.294 K, E/k B = 0.046 K, C/k B = -2.9 × 10 -5 K are magnetic anisotropy parameters, and g = 2. The sizeable fourth-order parameter C reflects not only the intrinsic anisotropy but, mainly, it parameterizes quantum mixing of the S = 10 with excited multiplets (S-mixing) and how it influences quantum tunneling via the ground state [20].

In the present paper, we study the influence of Berry phase interference on the ac magnetic susceptibility χ and T b of Fe 8 , that is, on those quantities that characterize the standard SMM (or superparamagnetic) behavior. Close to T b , magnetic relaxation is dominated by tunneling near the top of the anisotropy energy barrier, thus also near excited multiplets with S ̸ = 10. In this way, we aim also to use the interference pattern to gain quantitative information on the degree of S-mixing in Fe 8 .

The sample employed in these experiments was a 3 × 2 × 1 mm 3 single crystal of Fe 8 . Each molecule has a net spin S = 10 and a strong uniaxial magnetic anisotropy. Equation (1) defines x, y and z as the hard, medium and easy magnetization axes. In the triclinic structure of Fe 8 , x, y, and z axes are common to all molecules [21]. The complex magnetic susceptibility χ = χ ′ (T, ω) -iχ ′′ (T, ω) was measured between 90 mK and 7 K, and in the frequency range 3 Hz ≤ ω/2π ≤ 20 kHz, using a purpose built ac susceptometer thermally anchored to the mixing chamber of a 3 He-4 He dilution refrigerator. A dc magnetic field -→ H was applied with a 9 T×1 T×1 T superconducting vector magnet, which enables rotating -→ H with an accuracy better than 0.001 • . The magnetic easy axis z was parallel to the ac excitation magnetic field of amplitude h ac = 0.01 Oe. The sample was completely covered by a non-magnetic epoxy to prevent it from moving under the action of the applied magnetic field. The alignment of -→ H perpendicular (±0.05 • ) to z and close (±5 • ) to the hard x axis was done at low temperatures (T = 2 K), using the strong dependence of χ ′ (T, ω) on the magnetic field orientation (see [22] for further details). These data were scaled with measurements performed, for T 1.8 K, using a commercial SQUID magnetometer and a physical measurement platform equipped with ac susceptibility options.

The zero field (H = 0) ac susceptibility χ ′ and χ ′′ components of Fe 8 show the typical SMM behavior (Fig. 2). Maxima of χ ′′ measured at different frequencies define T b and occur when τ ≃ 1/ω, where τ is the magnetic relaxation time. As Fig. 2 shows, τ approximately follows Arrhenius' law τ ≃ τ 0 exp (U/k B T ), where U is the activation energy and τ 0 is an attempt time. However, a closer inspection reveals that the slope of the Arrhenius plot increases gradually with temperature, from a low-T value U/k B ≃ 22 K to more than 32 K. While the former U value agrees with tunneling taking place via m = ±5 states, the latter is close to the maximum energy (≃ 32.5 K) of the S = 10 multiplet.

Figure 3 shows χ ′ (T ) and χ ′′ (T ) data measured at ω/2π = 333 Hz and under three different transverse magnetic fields. By increasing µ 0 H x from 0 to 0.19 T, the superparamagnetic blocking shifts towards higher temperatures. Thereafter, T b decreases again with further increasing µ 0 H x to 0.27 T. The right-hand panel of Fig. 3 shows that T b oscillates as a function of H x . This behavior contrasts sharply with the rapid and monotonic decrease of T b that is observed when -→ H is parallel to the medium axis y (see the inset of Fig. 3 and [22,23]).

Oscillations of T b originate from oscillations of the relaxation time τ , which also lead to oscillations of the dynamical susceptibility. Figure 4A shows χ ′ vs H x data measured at ω/2π = 333 Hz and T = 2.6 ≃ T b (H x = 0) K. Under such conditions, small shifts of T b result in large changes of χ ′ , thus allowing us to monitor these changes very precisely. χ ′ shows three minima, at µ 0 H a ≃ 0.20(1) T, µ 0 H b ≃ 0.56(1) T, and µ 0 H c ≃ 0.90(1) T, with an approximate periodicity

T(K) (A) (B) (D) (s) (C) m = ±3 m = ±4 U/k B (K) 1/T (K -1 ) m = ±5
µ 0 ∆H x ≡ µ 0 [2H a + (H b -H a ) + (H c -H b )] /3 ≃ 0.37 T.
Again, this behavior contrasts with the abrupt increase towards equilibrium that is observed when ⃗ H is applied along y [22]. From the susceptibility we estimate also τ ≃ [r/(sin βπ/2 -r cos βπ/2)] (1/β) /ω, where r = χ ′′ /χ ′ and β ≃ 0.92 was determined from Cole-Cole fits performed at H x = 0 (see Fig. 2B). Figure 4B shows that, as one could anticipate, τ also oscillates with H x . Data of Figs. 3 and4 strongly suggest that the oscillation period δH x remains approximately constant in the temperature range between 2 and 3 K covered by present experiments.

The oscillations can be qualitatively accounted for by recalling the dependence of the quantum tunnel splittings ∆ m on H x (Fig. 1). At zero field and close to T = 2.6 K, magnetic relaxation takes place mainly via thermally activated m = ±4 states. By increasing H x , ∆ 4 gets periodically quenched and therefore tunneling is inhibited. This leads to an increase of τ and thus also of T b , as it is observed experimentally.

However, the giant spin model Eq. ( 1) is unable to provide a quantitative description of the interference pattern. The magnetic relaxation time and the frequency dependent-susceptibility have been calculated by solving a Pauli master equation for the populations of all energy levels of ( 1), following the model described in [24]. For simplicity, we have simulated the effect that dipolar interactions between different Fe 8 clusters have on the spin tunneling probabilities [24,25] by introducing an effective bias field µ 0 H dz = 31 mT. This value is the width of the distribution of dipolar fields in a magnetically unpolarized Fe 8 crystal [26,27]. The results are compared to experimental data in Figs. 3 and4. Although they show oscillations, the theoretical period µ 0 ∆H x = 0.28 T is about 20 % smaller than the experimental one. It is important to emphasize that the discrepancy cannot be ascribed to a small uncertainty in angle ϕ. All theoretical curves with distinct oscillations (for ϕ 10 deg.) show ∆H x smaller than observed. The discrepancy originates instead from the fact that µ 0 ∆H x obtained from Eq. ( 1) strongly decreases with m, from 0.4 T for m = ±10 to 0.25 T for m = ±2. This dependence arises from the presence of a strong fourth-order anisotropy term. The same effect occurs for higher-order terms. The giant spin approximation, which neglects all excited multiplets, is therefore unable to simultaneously account for the quantum interference patterns observed at low and high-T .

These results call for a more complete description, able to explicitly incorporate the effects of S-mixing. A schematic diagram of the magnetic structure of the Fe 8 molecular core [28,29] is shown in Fig. 5. The central diamond (or "butterfly") of spins 1-4 strongly couple antiferromagnetically to give a net spin S 1-4 ≃ 0. Couplings via the butterfly generate effective interactions between the remaining spins 5 -8. The resulting spin configuration, with a ground state S = 10 and a first excited S = 9 multiplet lying about δE 9,10 /k B = 44 K above it [28][29][30][31], follows from the fact that |J 4 | > |J 3 | (all couplings are antiferromagnetic). In addition to symmetric exchange interactions, one has to consider also single-ion magnetic anisotropies, dipolar interactions and antisymmetric Dzyaloshinskii-Moriya (DM) interactions, which mix states of different S [20]. Concerning the latter, although it is in principle possible to make DM interactions irrelevant on a specific bond by a gauge transformation, compatibility conditions must be satisfied to extend this over the molecule. To be precise, the product of the gauge transformations associated to each bond along any closed exchange path in the molecule should be equal to (-1) n I (I ≡ identity, n integer) [32]. The low symmetry of Fe 8 ensures that no such global similarity transformation should exist, mainly because the DM interactions between the individual spins are certainly non uniform. Based on the above considerations, we reduce the full spin Hamiltonian of 8 Fe 3+ ions to a simpler and compu- tationally more affordable one, involving only two spins S I = 5 and S II = 5, defined in Fig. 5. This approximation can be justified by the fact that exchange interactions in Fe 8 as well as in other SMMs have been treated by iteratively coupling spins in pairs [28,33]. The twospin Hamiltonian reads as follows

H = ∑ i=I,II H anis,i - ∑ i=I,II g i µ B -→ S i • -→ H -J -→ S I -→ S II - -→ S I Â-→ S II + --→ d I,II -→ S I × -→ S II (2)
where H anis,i = -D i S 2 i,z +E i (S ) accounts for the magnetic anisotropy of each spin, J > 0 is an effective isotropic exchange constant, Â is an anisotropic coupling tensor, and --→ d I,II is a DM interaction vector. We set D I /k B = D II /k B = 0.625 K, E I /k B = E II /k B = 8.94 × 10 -2 K and C I /k B = C II /k B = -5.7 × 10 -5 K, which give, for the S = 10 multiplet, parameters D = 0.47D I , E = 0.47E I and C = 0.128C I that agree with those determined from EPR experiments [29].

Within this model, dominant symmetric exchange interactions (mainly J 2 and J 4 ) contribute to the formation of the two giant spins ⃗ S I and ⃗ S II , which are then coupled by a weaker effective symmetric exchange. We have set J/k B = 3.52 K to fit the energy gap δE 9,10 between the S = 9 and the S = 10 multiplets. The last two terms in Eq. ( 2) parameterize the effects that dipolar and DM interactions, considered as perturbations, have on the energies of the subspace defined by different ⃗ S I and ⃗ S II orientations. Dipolar interactions between spins (5, 7) and (6,8) give predominantly rise to a term -A xx S I,x S II,x , with A xx /k B ≃ 2.8 × 10 -2 K. This term hardly has any noticeable influence on the period of quantum oscillations and, furthermore, it tends to reduce ∆H x . Because of the close to planar molecular structure and its pseudo C 2 symmetry, the same considerations apply to terms arising from dipolar interactions between any of these spins and those forming the central butterfly. In order to account for the observations, S-mixing must therefore predominantly arise from antisymmetric exchange interactions.

DM interactions between individual Fe 3+ spins are generally of the order of ∆g/g ≈ 0.01 times the symmetric interactions [34][35][36], thus about 0.3 -2 K in the case of Fe 8 , see Fig. 5. The oscillation periods of ∆ m vs H x depend on the magnitude and orientation of --→ d I,II : they increase with increasing d I,II when --→ d I,II points along the hard axis x but decrease when --→ d I,II is along the medium axis y. We have therefore set this vector along x and varied its magnitude d I,II to fit the experimental susceptibility oscillations. As with the giant spin model, the dynamical susceptibility has been calculated by solving a Pauli master equation for the energy level populations of Eq.( 2). The best agreement, shown in Figs. 3 and4, is found for d I,II /k B = 1.28 ± 0.05 K, which is compatible with the above estimates. The model accounts for the oscillations observed in the ac susceptibility and related quantities (T b and τ ) measured between 2 K and 3 K. In addition, it describes well the overall temperature dependence of ∆H x between very low temperatures and 3 K (Fig. 4C). Finally, it predicts a ground state tunnel splitting ∆ 10 /k B = 10 -7 K, which agrees with that determined from Landau-Zener experiments [11,12].

It can be concluded that Eq. ( 2), despite its relative simplicity, agrees not only with available spectroscopic and magnetic data, but also provides a much more accurate description of the spin dynamics in Fe 8 than the giant spin model Eq. (1). It also enables one to quantify the degree of S-mixing. The ground state of Eq.(2) contains 88.4 % of S = 10 states, 10.9 % of S = 9 states and 0.7 % of states from other multiplets.

Summarizing, we have shown that the ac linear magnetic response and the superparamagnetic blocking of molecular nanomagnets are governed by quantum interferences, which can be tuned by an external magnetic field. Furthermore, the period of oscillations is directly linked to the nature of the spin states involved in the tunneling processes, i.e. on whether they are pure ground-S states or quantum superpositions with states of other multiplets. These results confirm that an accurate description of quantum phenomena in single molecule magnets should take into account quantum mixing between ground and excited multiplets. They also illustrate the sensitivity of interference phenomena to small changes in the wave function describing a physical system. By contrast, the relaxation time depends more strongly on the energy and number of such states. Ac susceptibility measurements performed under transverse magnetic fields provide then a rather powerful, and general, method for quantifying the degree of S-mixing in SMMs.

We acknowledge the assistance of Tomoaki Yamasaki

  FIG. 1: A: Molecular structure of the [(C6H15N3)6Fe8O2(OH)12] molecular magnet, briefly referred to as Fe8. B: Two equivalent tunneling paths linking states with m = +S and m = -S. A magnetic field along the hard axis (denoted by x) shifts the relative phases of these trajectories, thus leading to constructive and destructive interferences. C: Dependence of the quantum tunnel splittings ∆m on Hx calculated with Eq. (1) for states ±m, with m = 10 (bottom curve) to m = 1 top curve).

FIG. 3 :

 3 FIG. 3: Left: χ ′ and χ ′′ susceptibility components of Fe8 measured at ω/2π = 333 Hz and for three different Hx values. Right: blocking temperature T b as a function of Hx (hard axis) and Hy (medium axis) for the same frequency. Dotted and solid lines are theoretical predictions following from, respectively, the giant spin model [Eq. (1)] and the two-spin model [Eq. (2)], which includes S-mixing effects, for ϕ = 4 deg. The inset compares blocking temperatures measured with ⃗ H along the hard (x) and medium (y) axes.

FIG. 4 :

 4 FIG.4:A: χ ′ of Fe8 vs Hx measured at T = 2.6 K and ω/2π = 333 Hz. B: transverse field dependence of the magnetic relaxation time τ determined at different temperatures. C: temperature dependence of the quantum oscillation periods estimated from Landau-Zener relaxation measurements (⋆)[11], χ ′ vs T = K (•), and χ ′ vs T at different Hx ( * ). Dotted and solid lines are theoretical predictions, for ϕ = 4 deg., that follow from the giant spin model [Eq. (1)] and the two-spin model [Eq. (2)], respectively.

2 FIG. 5 :

 25 FIG. 5: Scheme of exchange pathways connecting Fe 3+ ions in the Fe8 core. Approximate values of the exchange constants are [29] J1/kB = -36 K, J2/kB = -201 K, J3/kB = -26 K, and J4/kB = -59 K.

and Dr. Miki Ueda in the synthesis of the samples. The present work has been partly funded trough the Spanish MINECO (grants MAT2009-13977-C03 and the CON-SOLIDER project on Molecular Nanoscience) and the Gobierno de Aragón (project MOLCHIP).


Mathieu Dubois

Paola K Rozo

Alexander Gepperth

Fabio A González

David Filliat

A Comparison of Geometric and Energy-Based Point Cloud Semantic Segmentation Methods

The recent availability of inexpensive RGB-D cameras, such as the Microsoft Kinect, has raised interest in the robotics community for point cloud segmentation. We are interested in the semantic segmentation task in which the goal is to find some relevant classes for navigation, wall, ground, objects, etc. Several effective solutions have been proposed, mainly based on the recursive decomposition of the point cloud into planes. We compare such a solution to a non-associative MRF method inspired by some recent work in computer vision. The MRF yields interesting results that are however less good than those of a carefully tuned geometric method. Nevertheless, MRF still has some advantages and we suggest some improvements.

I. INTRODUCTION

The recent availability of inexpensive RGB-D cameras, such as the Microsoft Kinect, has made it possible to combine techniques developed for LIDAR and standard cameras for scene understanding (sometimes called image parsing). Therefore a lot of recent work in the computer vision community has been devoted to using RGB-D cameras for scene understanding while fewer works have studied the problem directly from a robotic point of view.

In the robotics community, recent years have seen the emergence of semantic mapping. The goal is to build robots able to autonomously map indoor environments and to recognize and localize significant elements such as objects, furnitures or rooms. RGB-D sensors are very interesting in this context because they give access to much more information than laser scans. However, object recognition, particularly at the category level, is a difficult task.

Therefore it is interesting to separate segmentation and recognition steps. In this article we present a system for the segmentation task: we try to recognize the structure of the environment (such as walls, ground), which is useful for navigation, and the presence of objects without identifying them. The precise identification of objects can be handled by a specialized module using a different set of techniques. We present a system developed for semantic segmentation of RGB-D images based on a probabilistic framework to incorporate color, 3D shape and priors. This approach is similar to other energy minimization approaches found in the literature. We will compare the performance of our algorithm against a baseline approach purely based on the geometric segmentation of the point cloud [START_REF] Filliat | RGBD object recognition and visual texture classification for indoor semantic mapping[END_REF]. 1 ENSTA ParisTech -INRIA Flowers Team 828 boulevard des Maréchaux 91762 Palaiseau CEDEX, France first.last@ensta.fr 2 Universidad Nacional de Colombia, Of. 114 Edif. 453, Bogotá, Colombia {pkrozob, fagonzalezo}@unal.edu.co Both methods have been developed in the context of the CAROTTE semantic mapping challenge 1 . In this challenge the robot explores an unknown environment and must provide a map containing as much information as possible. The environment covers approximately 120m 2 and contains several rooms typically 10 or more, with variable grounds and various difficulties (fitted carpet, tiling, grid, sand, stones). Several kinds of objects are present, either isolated or gathered, in multiple specimens, which must be detected, located, and identified by the robot. Figure 1 shows the robot and a typical environment.

The article is organized as follows. We review related works in section II and present our system in section III. Experiments are presented in section IV. Finally we conclude and present future works in section V.

II. STATE OF THE ART

There is an extensive body of research in the field of 2D scene understanding and semantic segmentation [START_REF] Li | Towards total scene understanding: Classification, annotation and segmentation in an automatic framework[END_REF], [START_REF] Murphy | Object detection and localization using local and global features[END_REF] and a comprehensive review is out of the scope of this paper. A lot of modern approaches [START_REF] Sudderth | Learning hierarchical models of scenes, objects, and parts[END_REF], [START_REF] Tighe | Superparsing -scalable nonparametric image parsing with superpixels[END_REF] use methods based on visual words [START_REF] Csurka | Visual categorization with bags of keypoints[END_REF]: the idea is to use an unsupervised vector quantization algorithm to find clusters (called visual words) in the set of image signatures. When a new image is presented the signatures are computed and matched to the closer cluster. This reduce the noise (because close signatures are mapped to the same word) and, because it acts as a form of discretization, allows to simplify learning.

There is also a lot of work on point cloud segmentation. Typical techniques are region growing methods [START_REF] Jiang | Robust and fast edge detection and description in range images[END_REF], 3D Hough Transform [START_REF] Rabbani | Efficient Hough transform for automatic detection of cylinders in point clouds[END_REF] or ad-hoc methods exploiting domain knowledge [START_REF] Rusu | Close-range scene segmentation and reconstruction of 3D point cloud maps for mobile manipulation in domestic environments[END_REF] often based on decomposing the environment into planes.

Markov Random Field (MRF) [START_REF] Geman | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[END_REF] segmentation techniques have previously been applied to the point cloud segmentation and classification problem. Early works [START_REF] Anguelov | Discriminative learning of Markov random fields for segmentation of 3D scan data[END_REF], [START_REF] Triebel | Instancebased AMN classification for improved object recognition in 2D and 3D laser range data[END_REF], [START_REF] Lim | 3D terrestrial LIDAR classifications with super-voxels and multi-scale conditional random fields[END_REF], [START_REF] Shapovalov | Non-associative Markov networks for 3D point cloud classification[END_REF] use Markov network that segments point clouds based on geometric features.

More recent works [START_REF] Micusik | Semantic segmentation of street scenes by superpixel co-occurrence and 3D geometry[END_REF], [START_REF] Collet | Structure discovery in multi-modal data: a region-based approach[END_REF], [START_REF] Silberman | Indoor segmentation and support inference from RGBD images[END_REF], [START_REF] Anand | Contextually guided semantic labeling and search for three-dimensional point clouds[END_REF] use similar models to incorporate color and 3D information and demonstrate that combining RGB and 3D information leads to superior results. Most of those works use first a bottom-up segmentation to simplify labeling. To model relationships between objects parts, it is necessary to use non-associative edge energies [START_REF] Shapovalov | Non-associative Markov networks for 3D point cloud classification[END_REF]. However, those studies are more oriented toward object recognition. Each paper use a different database and evaluation protocol so it's hard to compare MRF-based and ad-hoc methods for semantic segmentation ( [START_REF] Silberman | Indoor segmentation and support inference from RGBD images[END_REF] compare their approach to a geometric approach for the task of support inference).

In the next section, we present a model combining visual words and MRF applied to semantic scene segmentation.

III. USING VISUAL WORDS AND MRF FOR SEMANTIC

SEGMENTATION

Our algorithm is composed of two main steps: we first segment the point cloud into N compact, homogeneous regions thanks to a bottom-up algorithm and then use a MRF model to assign a label to each region. Later, contiguous regions with the same label can be merged to form the final segments. The low-level algorithm is described in section III-A. The MRF model is described in section III-B and following. As the Kinect provides organized point clouds, each pixel in the 2D image corresponds to one point in the point cloud. In the following, we will use region and superpixel on one hand and point and pixel on the other hand, interchangeably.

A. Over-segmentation

For segmentation, we have adapted the SLIC superpixel algorithm [START_REF] Achanta | SLIC superpixels compared to state-of-the-art superpixel methods[END_REF] to take into account depth information. The algorithm allows to set the desired size of the regions S (and then the number of regions). The algorithm is an adaptation of the well-known K-means clustering algorithm to enforce compact regions: starting from initial seeds, at each step, points are attributed to the closest cluster and then clusters positions are recomputed as the average position of the points assigned in each cluster.

Given 2 pixels i and j, the distance between them D(i, j) is:

D(i, j) = d c (i, j) 2 + m 2 S 2 d 2 s (i, j) (1)
where:

d c (i, j) = (l j -l i ) 2 + (a j -a i ) 2 + (b j -b i ) 2 (2) d s (i, j) = (x j -x i ) 2 + (y j -y i ) 2 + (z j -z i ) 2 (3)
Where the z components are given by the depth map and [l, a, b] T is the color of the point in the Lab color space.

The key difference with classical K-means is that the distance measure combines color and spatial proximity. As the expected size is controlled by the parameter S , the search region in the assignment step is limited to a neighborhood of size 2S × 2S . The parameter m allow to weight the relative importance between color similarity and spatial proximity: large values of m enforce compact superpixels while small values enforce adherence to image boundaries. Compared to other bottom-up segmentation algorithms it takes into account both the color and the 3D information.

Depth maps obtained by structured light methods may contain missing values (due to infrared-absorbing surfaces, such as glass, or objects "shadow"). To remove those holes we use the method developed in [START_REF] Janoch | A category-level 3-D object dataset: Putting the kinect to work[END_REF] which uses a leastsquare interpolation on missing data.

B. MRF model

Following [START_REF] Micusik | Semantic segmentation of street scenes by superpixel co-occurrence and 3D geometry[END_REF], we model the semantic segmentation problem in a probabilistic framework. We note Λ = {ℓ 1 , . . . , ℓ m } the set of classes (i.e. of possible label for each region). Given a point cloud segmented into N segments, we seek for the labeling

⋆ x = (x 1 , . . . , x N ) that satisfies: ⋆ x = arg max x P (x|A, S, G) (4)
where A = [a 1 , . . . , a N ] is the appearance features matrix, S = [s 1 , . . . , s N ] is the matrix of 3D shape descriptors and G = [g 1 , . . . , g N ] is the matrix of geometrical features. We use vector quantization for appearance and shape features discretization. The matrix A is then approximated by the vector of appearance word W A = w A 1 , . . . , w A N composed of the index in the visual vocabulary. Similarly S is approximated by the vector of shape words W S = w S 1 , . . . , w S N . Using Bayes' rule, Equation 4 can be written:

⋆ x = arg max x P W A , W S , G|x P (x) P (W A , W S , G) = arg max x P W A , W S , G|x P (x) (5)
The last line follows from the fact that P W A , W S , G doesn't depend on x. We assume that the geometrical features, the shape and the appearance features are independent conditionally to x. Therefore:

⋆ x = arg max x P W A |x P W S |x P (G|x) P (x) (6)
The appearance and shape features and the vocabulary construction are explained in section III-E.

The inference problem in Equation 6is formulated in a binary MRF framework. The image is encoded as an undirected graph G = (V , E ) where the nodes V are the regions and the edges E are determined as follows: two nodes are connected if they share at least one boundary point.

Classically, we assume conditional independence between sites in the formulation of P W A |x , P W S |x and P (G|x). Note that [START_REF] Micusik | Semantic segmentation of street scenes by superpixel co-occurrence and 3D geometry[END_REF] propose alternatives to this hypothesis.

In the MRF framework, the distribution P (x) can be factorized over the cliques of G. Most studies use only binary cliques potential in P (x). However, [START_REF] Celeux | EM-based image segmentation using Potts models with external field[END_REF] showed that unary terms (called external field) can be useful for segmentation especially when the class distribution is uneven which is true in our case. This term can be used to model the prior distribution of the classes.

The binary potential ensures spatial regularization but must be able to detect frontiers between classes. In the kind of environment we are interested in, object frontiers can be detected by an abrupt change in depth (e.g. boundary between an object and the wall behind it) or by a difference in normal orientation (e.g. between the ground and a wall).

Therefore we model P (x) as:

P (x) = exp - N ∑ i=1 g prior i (x i ) -∑ (i, j)∈E g depth ij (x i , x j ) -∑ (i, j)∈E g normals ij (x i , x j ) (7)
The (unary) prior potential g prior is:

g prior i (x i ) = -log P (x i ) (8)
Following [START_REF] Micusik | Semantic segmentation of street scenes by superpixel co-occurrence and 3D geometry[END_REF], the binary potential g depth is:

g depth ij (x i , x j ) = 1 -e if x i = x j (9a) δ depth + e else ( 9b
)
where e = exp -|∆z| 2 2σ 2 depth (∆z is the depth difference between the 2 superpixels i and j).

To take into account the difference of normal orientation, the angle φ i j ∈ [0, π] between the 2 normals is discretized into K = 6 bins ( φi j ∈ 0, K denotes the discretized angle). The binary potential g normals is then given by: g normals ij (x i , x j ) = -log P x i , x j | φi j [START_REF] Geman | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[END_REF] Putting it all together, the optimal label is found by minimizing the energy:

E = λ color N ∑ i=1 E A (i) + λ shape N ∑ i=1 E S (i) + λ geom N ∑ i=1 E G (i) + λ prior N ∑ i=1 E prior (i) + λ normals ∑ (i, j)∈E E normals (i, j) + λ depth ∑ (i, j)∈E E depth (i, j) (11)
where:

E A (i) = -log P w A i |x i (12)
E S (i) = -log P w S i |x i (13)
E G (i) = -log P (g i |x i ) (14)
E prior (i) = -log P (x i )

E normals (i, j) = g normals ij (x i , x j ) (15)
E depth (i, j) = g depth ij (x i , x j ) (16)
Note that the model is non-associative.

C. Learning

For a given class ℓ ∈ Λ, learning the distributions P w A |ℓ and P w S |ℓ is very easy thanks to the discretization of appearance and shape descriptors. Similarly the discrete probability distribution P ℓ, ℓ ′ | φ and the prior distribution P (ℓ) are estimated on the training set. We use the Laplace method to avoid null probabilities.

For geometric features, we consider the position in the 2D image g = [x, y] T . Given the fact that the robot navigates, the x position of an object may not be relevant. To take this into account, we use the distance to the center of the image: the coordinate x is mapped in the range [0,W /2] (where W is the image width). The distribution P (g|ℓ) is approximated by a two-dimensional Gaussian distribution:

P (g|ℓ) = 1 2π |Σ ℓ | exp - 1 2 (g -µ ℓ ) T Σ -1 ℓ (g -µ ℓ )
We therefore have to evaluate 5 parameters per class: the mean positions µ x ℓ and µ y ℓ , the variances σ x ℓ and σ y ℓ and the correlation between the coordinates ρ ℓ .

The parameters of g depth and the weights are manually set (see section IV).

D. Inference

To solve the inference problem, we use the approach of [START_REF] Werner | A linear programming approach to max-sum problem: A review[END_REF] which formulates a relaxed version of the problem as a linear program. A specific algorithm is designed to efficiently solve this problem.

Because it is a relaxed version of the problem, a crisp class is not attributed to each site. Instead, for a given node i ∈ V , every class ℓ receives a confidence value 0 ≤ ϕ i (ℓ) ≤ 1. Usually the class with the larger value ϕ i is attributed to the region. However, using the distribution of ϕ i can be useful to detect regions which are not well classified for instance because they correspond to a previously unseen object. As learning is incremental the robot can then focus on the unknown regions and update the probability distributions used in the model.

E. Features and visual words

For each region, the appearance features are computed by concatenating the 128-dimensional SIFT features [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] (computed at the morphological center of each region) and the average Lab values on the region to give a 131dimensional feature vector. For shape features, we use the 496-dimensional features of [START_REF] Shotton | Real-time human pose recognition in parts from single depth images[END_REF] (also computed at the morphological center).

The visual and shape vocabularies are trained using the optimized learning rate LVQ1 procedure [START_REF] Kohonen | New developments of Learning Vector Quantization and the Self-Organizing Map[END_REF] (see section IV).

IV. EXPERIMENTS

A. Database

We use a database of point clouds acquired during the CAROTTE challenge. The point clouds where acquired while the robot moves autonomously as described in [START_REF] Filliat | RGBD object recognition and visual texture classification for indoor semantic mapping[END_REF]. This avoid the potential bias in databases acquired by manually guided robots or cameras as in [START_REF] Anand | Contextually guided semantic labeling and search for three-dimensional point clouds[END_REF].

The database consists of 100 point cloud with high accuracy annotations: each pixel in the RGB image is assigned a class. Care was taken to perform an appearance-based labeling which means that parts of, e.g., a wall which are visually strongly different will be assigned different identities. This was done in order to permit appearance-based segmentation and is interesting in the context of the CAROTTE challenge. In this article, all types of ground are counted as a single "ground" class, and similarly for wall and object types which together form the "wall" and "object" classes2 . Figure 2 shows an example of image from the database. Due to the structured light method, the depth can't be measured on some border pixels of the RGB image (this is a different problem than the missing data in section III-A). The RGB image provided by the Kinect is cropped to 410 × 560 pixels (starting at position (50, 40)) to remove border pixels.

B. Baseline system

We compare our system to the one described in [START_REF] Filliat | RGBD object recognition and visual texture classification for indoor semantic mapping[END_REF] which is based on geometric segmentation using only distance information. This system is inspired by [START_REF] Rusu | Close-range scene segmentation and reconstruction of 3D point cloud maps for mobile manipulation in domestic environments[END_REF], doesn't require training but has several parameters which were manually tuned. It showed interesting performances during the CAROTTE competition.

The point cloud is filtered to remove distant outliers, and re-sampled using a voxel grid to have a constant spatial resolution (2 cm).

An off-line calibration phase, is used to detect the ground plane in an ideal setting using RANSAC [START_REF] Fischler | Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography[END_REF]. We assume that the coefficients in its defining equation stay roughly constant wherever the robot goes. During segmentation, this equation is used to detect the ground: surface normals are calculated for each point in the point cloud and all points sufficiently close to the theoretical ground plane, and having a normal pointing upwards, are recognized as part of the ground and removed from the cloud.

Next walls are iteratively detected using RANSAC. A wall is defined as a plane of sufficient size with normals parallel to the ground plane. Points on and close to detected walls are removed as well, and the procedure is repeated until no more walls are detected.

The remaining points are then projected onto the theoretical ground plane and subsequently grouped according to their (projected) distance. For this purpose, we use a regiongrowing type of algorithm to form groups of 3D points that may correspond to objects. These groups form the initial segmentation results. To segment objects placed on planar surfaces of other objects, such as desks, tables or shelves, we continue to analyze the point cloud that is left after floor and wall removal to detect horizontal planes, and we remove all points below this level. The remaining points are grouped using the same region-growing clustering method as before, leading to additional segmentation results. In the event that several horizontal planes are detected (for example if the image contains two shelves), this process is iterated from the highest to the lowest detected horizontal plane, always producing segmentation results from the points above the current horizontal plane, and repeating the whole step for all the points below it. Figure 3 shows an example of segmentation with the baseline algorithm.

Due to the fact that the system don't use color information, it is unable to detect object smaller than a few centimeters high on the ground which may block the robot. The geometric model was manually tuned and is used as a black box here.

C. Results

For superpixels, we set S = 15pixels and m = 10. The vocabularies are trained on a random subsample of the features (using 30 % of the features). We use 200 words for the color and depth vocabularies.

We have run different experiments in order to assess the influence of the non-learned parameters. In all experiments, the values of δ depth and σ depth (for computing g depth ) are fixed (respectively to 0.8 and 0.05). Similarly, the weights λ color , λ shape , λ geom and λ prior are set to 1.0.

Regularization weights λ normal and λ depth are always equal and can take 3 different values: 0 (i.e. no regularization), 0.2 (i.e. weak regularization) and 1.0 (i.e. strong regularization). For each value, we run a 5-fold cross validation procedure on the database to evaluate the performances. The evaluation metrics are the pixel-wise precision and recall.

Table I summarizes the results. The performances of the MRF on the "wall" class is comparable to the baseline (precision is better but recall a bit less good). For the "ground" class, results are less good but still rather high. For those 2 classes (which are well represented in the training set), the influence of the regularization weights is not very important.

However for the class "objects", the MRF model lags behind the baseline in particular in terms of recall. Several hypotheses can be made to explain this fact. First, the object class is far more diverse in shape and appearance and thus harder for a learning based method. Second, our regularization terms might not capture enough relation between classes. In particular, objects and ground tend to have similar angles than ground and walls. Therefore the MRF tend to label as objects as "wall" (because this class is far more frequent).

V. CONCLUSION AND FUTURE WORKS

A. Conclusion

The results shows that the geometric method gives superior results for the task of semantic segmentation in particular for the object class. This can be explained by the fact that it incorporates a lot of domain knowledge (namely that indoor environments are made of planes and that objects lie on top of them).

However, MRF segmentation gives interesting results and has several advantages. First most of it's components can be used for other purpose or in other, less constrained, environments where domain knowledge is not available. For instance we could try to recognize more precisely the objects. Second it requires less tuning since most parameters are learned from the database. Third, it uses the appearance information which could help to identify different types of ground or wall (this was one of the goal in the CAROTTE challenge). Last but not least, as it gives a probabilistic output, it allows the robot to draw hypothesis on the environment and adapt its behavior. Therefore we think it is interesting to investigate improvements.

B. Future works

For now our algorithm doesn't fully exploit the structure of the point clouds. For instance, the SLIC segmentation procedure could be modified to use the real distances in meters in the distance computation and search in the full 3D neighborhood. Similarly the neighborhood and the geometric positions of the classes could be computed with the realworld coordinates. As we have mentioned, it seems necessary to investigate more complex edge potentials. Finally more investigations could be done on learning the parameters.

Fig. 1 .

 1 Fig. 1. Illustration of the CAROTTE challenge: (a) the robot we used is based on a Pioneer 3DX from Mobile Robots Inc., equiped with a Kinect to acquire RGB-D data (b) a typical room of the environment.

Fig. 2 .

 2 Fig. 2. Examples of image annotations in the PACOM RGBD segmentation database. (a): Original image. (b): annotated image. Each pixel has an identity associated to it, where different pixel colors indicate different identities.

Fig. 3 .

 3 Fig. 3. Example of the segmentation with the baseline system. Left: original point cloud after re-sampling. Right (upper row): detected walls which are subsequently removed. Right (lower row): Segmented objects including objects placed on top of each other. The whole group is segmented as a single object at first. Detection of the horizontal place on the metal suitcase allows to separate the suitcase from the three objects placed on top of it.

TABLE I RESULTS

 I OF THE SEMANTIC SEGMENTATION TASK. "BASELINE" STANDS FOR THE GEOMETRIC MODEL FROM[START_REF] Filliat | RGBD object recognition and visual texture classification for indoor semantic mapping[END_REF]. MRF STANDS FOR THE ENERGY-BASED MODEL.

	Algorithm	Precision Walls Ground Objects Walls Ground Objects Recall	Overall
	Baseline	93.3	97.8	65.0	87.7	91.2	98.1	94.9
	MRF strong regul. 96.4	89.5	46.8	77.6	79.5	41.6	76.0
	MRF weak regul. 94.7	89.4	88.1	82.8	80.4	23.5	77.86
	MRF no regul	94.7	88.8	64.8	81.1	78.5	32.1	76.8

The CAROTTE competition is organized by the French research funding agency (ANR) and the French armament procurement agency (DGA). Website: http://www.defi-carotte.fr

The database can be freely downloaded at http://cogrob.ensta. fr/pacom


E Burzurí

F Luis

B Barbara

R Ballou

E Ressouche

O Montero

J Campo

S Maegawa

Magnetic dipolar ordering and quantum phase transition in Fe 8 molecular magnet

Keywords: numbers: 75.45.+j, 73.43.Nq, 75.50.Xx, 75.40.Gb

We show that a crystal of mesoscopic Fe8 single molecule magnets is an experimental realization of the quantum Ising model in a transverse field, with dipolar interactions. Quantum annealing has enabled us to explore the quantum and classical phase transitions between the paramagnetic and ferromagnetic phases, at thermodynamical equilibrium. The phase diagram and critical exponents we obtain agree with expectations for the mean-field universality class.

Quantum Phase Transitions (QPT) [1,[START_REF] Sachdev | Quantum Phase Transitions[END_REF] have been extensively studied in recent years. Physical realizations include the superconductor insulator transition in cuprates [START_REF] Sachdev | [END_REF][4][5], the onset of antiferromagnetism in heavy fermions [6], the pressure driven insulator-metal transition in V 2 O 3 [7], and the magnetic transitions driven by field (LiHoYF 4 [8]) or concentration (Cr x V 1-x alloys [9]). In addition to their intrinsic interest, a plethora of new properties arise at nonzero temperature.

In magnetism, the archetypal QPT is realized by a lattice of N coupled Ising spins in a transverse magnetic field H ⊥ [10], the Hamiltonian of which reads as follows

H = -2S 2 N ∑ i<j J ij σ z i σ z j -∆ S N ∑ i σ x i (1)
Here, σ ′ s are the Pauli spin operators, J ij the longitudinal couplings (here of dipolar origin) and ∆ S the ground-state tunnel splitting which depends on and vanishes with H ⊥ . The classical long-range order that exists for H ⊥ = ∆ S = 0 (ferromagnetic or antiferromagnetic) competes with the field-induced quantum fluctuations ([H, σ] ̸ = 0). It is completely destroyed at the critical point when ∆ S (H ⊥ ) ∆ Sc , where ∆ Sc = 2JS 2 ∼ k B T c , J = (1/N ) ∑ i<j J ij , and T c is the Curie temperature. The ground-state becomes then a superposition of "up" and "down" spin states.

Crystals of lanthanide-based insulators seem to be natural candidates to observe a QPT [8,11]. Because of the weak interactions, quantum fluctuations can become noticeable at moderate transverse fields. However, the strong hyperfine interactions seriously limit the observation of the intrinsic quantum criticality in these materials [12,13]. Single-molecule magnets (SMMs) based on 3d transition metals show some properties that make them better suited for such QPT studies: dipolar interactions are weak because of the very large inter-molecular distances [14] and hyperfine interactions are also generally weak enough not to block the QPT. Finally, SMMs show phenomena, like tunneling [15], interference [16] and superpositions of spin states [17], which reveal the existence of important quantum spin fluctuations. Yet, the search of a QPT faces some experimental challenges, mainly associated with structural disorder. In Mn 12 acetate, where indications of a possible QPT were inferred from magnetic neutron diffraction experiments [18], the local anisotropy axes are slightly tilted due to the presence of several isomers giving rise, in the presence of a perpendicular field, to random fields [19]. It has been argued [20] ) has a spin S = 10 and a strong uniaxial magnetic anisotropy. In the presence of a magnetic field -→ H , the magnetism of Fe 8 can be described by the following spin hamiltonian

H 0 = -DS 2 z + E ( S 2 x -S 2 y ) -gµ B -→ H -→ S (2)
where D/k B = 0.294 K, E/k B = 0.046 K, and g = 2 [21]. Equation (2) defines x, y and z as the hard, medium and easy magnetization axes. In the triclinic structure of Fe 8 , these axes are common to all molecules [24]. Susceptibility experiments were performed on a 1.6 mg single crystal [24] of approximate dimensions 1 × 2 × 1 mm 3 . The complex magnetic susceptibility χ = χ ′ (T, ω) -iχ ′′ (T, ω) was measured down to 90 mK and in the frequency range 3 Hz ≤ ω/2π ≤ 20 kHz using a homemade ac susceptometer thermally anchored to the mixing chamber of a 3 He-4 He dilution refrigerator. The magnetic field -→ H was applied with a 9 T×1 T×1 T superconducting vector magnet.

The magnetic anisotropy axes were first approximately located by X-ray diffraction, using the information reported in [24]. The magnetic easy axis z was oriented parallel to the ac excitation magnetic field of amplitude h ac = 0.01 Oe. The precise alignment of -→ H perpendicular (±0.05 • ) to z and close (±20 • ) to the medium y axis was done at low temperatures (T = 2.6 K), using the strong dependence of χ ′ (T, ω) on the magnetic field orientation [25]. The sample was completely covered by a non-magnetic epoxy to prevent it from moving under the action of -→ H . Neutron diffraction experiments were performed on the thermal neutron two-axis diffractometer D23 at the Institute Laue Langevin. A single crystal, thermally anchored to the mixing chamber of a 3 He-4 He dilution refrigerator (T ≥ 50 mK), was mounted so as to have z orthogonal to -→ H . In this experiment, -→ H was close to the hard axis x. Longitudinal field components arising from the nonperfect orientation of the crystal were compensated by means of a superconducting mini-coil inserted in the space between the dilution refrigerator and the superconducting magnet. The intensities of different Bragg peaks, in particular (010), (-100), and (-212), were measured as a function of T (under a constant H ⊥ ) and H ⊥ (for a constant T ). The nonmagnetic contributions were measured at high T (5 K) and H ⊥ = 0. The fit of the magnetic intensities enabled us to estimate the magnetization components along the external magnetic field M ⊥ and along the anisotropy axis M z .

The ac-susceptibility (Figure 1) deviates from equilibrium for low H ⊥ and low T , as shown by the vanishing of χ ′ and by the relatively large values of χ ′′ . The superparamagnetic blocking temperature T b (associated with the χ ′′ maximum) strongly depends on frequency (not shown). In these dipolar Ising-like systems, out of equilibrium conditions arise from the presence of energy barriers larger than the typical interaction energies (i.e. T b > T c ), a situation which can be inverted by quantum annealing via the application of H ⊥ [26,27]. When H ⊥ increases, the height of the energy barrier decreases and spins are able to tunnel via lower energy states, which leads to enhanced spin-lattice relaxation rates. This explains the decrease of T b , as shown in the inset of Fig. 1(a). Besides enhancing the spin dynamics, the magnetic field also lowers the paramagnetic susceptibility. This decrease can be associated with the reduction of the effective S z by quantum fluctuations as well as with the decrease in the paramagnetic Weiss temperature (see below and Ref. [25]). Both effects become more noticeable for H ⊥ 1 T, as seen in Fig. 1(a). At moderate H ⊥ , e.g. at µ 0 H ⊥ = 1.7 T, χ ′ is not fully out of equilibrium. The relaxation time τ = χ ′′ /χ ′ ω at the same field is already very short and, more importantly, it remains constant below approximately 0.25 K (inset of Fig. 1(b)). χ ′ becomes independent of frequency, thus it reaches full equilibrium, for µ 0 H ⊥ 2 T as seen in Fig. 2(a).

We next show the existence of a magnetic phase transition and discuss its critical behavior. As shown in Fig. 2(a), 1/χ ′ follows the Curie-Weiss law at sufficiently high T , becoming independent of T below 0.34 K (which we take as the critical temperature T c at this field µ 0 H ⊥ = 2.25 T). The saturation value 1/χ max = 9.5(5) cm 3 Oe/emu agrees well with the demagnetizing factor of our sample N = 10(1) cm 3 Oe/emu, as expected for an equilibrium ferromagnetic phase transition. A very similar behavior is observed as H ⊥ is varied at constant T (Fig. 2(b)). Again, 1/χ ′ depends linearly on H ⊥ until it saturates (to the same value ≃ N ) below µ 0 H c = 2.65(5) T, which we take as the critical magnetic field. The intrinsic susceptibility χ ′ i , corrected from demagnetizing effects, is plotted vs the reduced temperature (at 2.33 T) and field (at 0.11 K) in Fig. 2(c). Under these conditions, χ ′ i should follow, as it approximately does, the power laws

χ ′ i = ( T -T c T c ) -γ cl , χ ′ i = ( H ⊥ -H ⊥,c H ⊥,c ) -γqu (3)
The slopes give critical exponents γ cl = 1.1(1) and γ qu = 1.0(1), in good agreement with γ = 1 of the mean-field universality class. This result agrees with the prediction that the marginal dimensionality for mean-field behavior is d * = 3 in an Ising dipolar ferromagnet [29] and with the fact that the critical exponents for the field-induced transition at T → 0 become equivalent to those of the classical transition in (d + 1) dimensions [30].

The H c -T c phase diagram shown in Fig 3 was obtained by repeating the different procedures described in Fig. 2. The equilibrium reciprocal susceptibility, measured above T b , was fitted with a Curie-Weiss law. The extrapolations of these fits to 1/χ ′ → 1/χ max were used to determine either T c , at fixed H ⊥ (Fig. 2a), or H c , at fixed T (Fig. 2b). A third method consisted in using the scaling plots of Fig. 2c to determine T c independently.

The three methods give a very consistent H c (T c ) curve. As expected, T c decreases when quantum fluctuations in- 5) T (quantum). The ferromagnetic nature of the ordered phase agrees with theoretical predictions [22,23]. The solid line is obtained by solving numerically the mean-field Hamiltonian for the S = 10 spin

H = -DS 2 z + E ( S 2 x -S 2 y ) -gµ B H ⊥ (S x cos φ + S y sin φ) -2J⟨S z ⟩ T S z (4)
where φ is the angle between the magnetic field and the hard axis, 2J is the molecular field coefficient and ⟨S z ⟩ T is the thermal average of S z . We set φ = 68 • that, as shown by the inset of Fig. 3, accounts for µ 0 H c (T = 0) ≃ 2.65 T. A very good fit is obtained for J/k B = 2.85 × 10 -3 K, which gives T c = 0.6 K. This value agrees, within 10 %, with the Curie temperature T c = 0.54 K determined by classical MC simulations [22]. An extension [31] of these simulations to the case of nonzero H ⊥ gives the classical phase boundary (dashed line in Fig. 3), which is well approximated by

H c (T c ) = H c (0) [1 -T c /T c (H ⊥ = 0)] 1/2 . In this model H c (0) equals the anisotropy field H K = 2 [ D -E ( sin 2 φ -cos 2 φ )]
/gµ B S ≃ 3.8 T, which clearly overestimates the experimental critical field due to the absence of quantum fluctuations.

Additional evidence supporting the existence of a transition to a ferromagnetic phase is found in the results of neutron diffraction experiments. Figure 4 shows M 2 z de-termined from magnetic diffraction intensities measured at µ 0 H ⊥ = 1.5 T. The inset shows field-dependent data measured at T = 0.14 K. Despite the high noise level, typical of these experiments, the results show the onset of a net spontaneous M z in, respectively, the low-T (≤ 0.6(1) K) and low-H ⊥ (≤ 3(1) T) regions. Summarizing, we have shown that the SMM Fe 8 undergoes a dipolar ferromagnetic to paramagnetic phase transition of (i) classical nature (µ 0 H c = 0 T and T c = 0.6 K) and (ii) quantum nature (T c = 0.11 K and µ 0 H c = 2.65 T). Long range magnetic order was enabled by the use of quantum annealing allowing the spin-bath to attain equilibrium despite high energy barriers preventing spinreversal. The H c -T c phase diagram and critical exponents are in excellent agreement with results obtained from exact diagonalisation of the mean-field Hamiltonian Eq. ( 4). As expected, the critical behavior belongs to the mean-field universality class of the Ising model in a transverse magnetic field. The classical portion of the phase diagram (near zero field), agrees very well with classical MC calculations, giving in particular nearly the same T c . Deviations of the classical model become important above ∼ 2 T, when quantum fluctuations dominate largely over thermal fluctuations

We are grateful to Julio Fernández for useful suggestions and for sharing with us the results of his classical MC simulations. We acknowledge D10 and the CRG D23 lines of ILL for allocating measuring time for the neutron diffraction experiments. We acknowledge Eric Bourgeat-Lami, Xavier Tonon, Pascal Fouilloux, and Bruno Vettard for their help in designing, fabricating, and installing the superconducting mini-coil used in these ex-periments, and Tomoaki Yamasaki and Miki Ueda for the synthesis of the samples. The present work was partly funded by Spanish MICINN grants MAT2009-13977-C03 (MOLCHIP) and Consolider-Ingenio on Molecular Nanoscience.

FIG. 1 :

 1 FIG.1:(Color online) χ ′ (a) and χ ′′ (b) measured at ω/2π = 333 Hz and different H ⊥ . The inset in (a) shows the shift of χ ′′ maxima with increasing H ⊥ . The solid line shows theoretical predictions for quantum spin-phonon relaxation that follow from Pauli's master equation as described in[28]. The inset in (b) shows the magnetic relaxation time at µ0H ⊥ = 1.7 T.

FIG. 2 :

 2 FIG.2:(Color online). (a) and (b): 1/χ ′ measured at µ0H ⊥ = 2.25 T as a function of T and at T = 0.110 K as a function of H ⊥ , respectively. The crossovers between the "Curie-Weiss" law, observed at either high T or high H ⊥ (dotted blue lines), and the ferromagnetic limit 1/χ ′ = N (solid red lines) give Tc (= 0.34(1)K) and µ0Hc (= 2.65(5) T), respectively. (c) Log-log plot of demagnetization-corrected χ ′ i vs the reduced temperature (for µ0H ⊥ = 2.33 T with Tc = 0.31 K, •) and field (at T = 0.110 K with µ0Hc = 2.65 T, •). The linear fits give critical exponents γ cl ≃ 1.1(1) and γqu ≃ 1.0(1).

FIG. 3 :

 3 FIG. 3: (Color online) Hc-Tc phase diagram determined from the linear extrapolation of 1/χ ′ to 1/χmax, as shown in Figs. 2a and b. • and • correspond to Tc > T b and Tc < T b , respectively; ⋆, data determined from susceptibility scaling plots, as those shown in Fig. 2(c). Solid line, quantum mean-field calculation of the phase boundary using Eq. (4); dashed line, classical phase diagram, derived from MC simulations. Inset: zero-temperature critical field vs angle ϕ. The horizontal line represents the experimental Hc(Tc = 0).

FIG. 4 :

 4 FIG.4:(Color online) Squared longitudinal magnetization of a Fe8 crystal determined from neutron diffraction intensities measured at µ0H ⊥ = 1.5 T. The inset shows data measured at T = 0.14 K as a function of H ⊥ . The lines are mean-field calculations using Eq. (4) with ϕ = 0.

6 H 15 N 3 ) 6 Fe 8 O 2 (OH) 12 ]

  that such fields turn Mn 12 acetate into a realization of the classical random-field Ising model.

				The existence
	of a pure QPT in a crystal of molecular nanomagnets is,
	therefore, not yet established. Our aim here is to explore
	the experimental realization of the Ising QPT model in a
	crystal of Fe 8 SMMs [21] where (i) classical Monte Carlo
	(MC) simulations suggest a ferromagnetic ground state
	with T c = 0.54 K [22, 23] (ii) hyperfine frequencies, much
	smaller than Larmor frequencies, cannot perturb quan-
	tum dynamics of SMMs and (iii) disorder is weak enough
	to avoid sizable random fields.
	Each	Fe 8	molecule	(brief	for
	[(C

* Electronic address: fluis@unizar.es


M Brando

W J Duncan

D Moroni-Klementowicz

C Albrecht

D Grüner

Rafik Ballou

F M Grosche

Logarithmic Fermi-Liquid Breakdown in NbFe2

Keywords: numbers: 71.10.Hf, 72.80.Ga, 75.40.-s

  L'archive ouverte pluridisciplinaire

The standard model of metals, Fermi liquid theory, is expected to break down in clean systems at the low temperature border of magnetism, or magnetic quantum phase transition [START_REF] See | [END_REF]. A wide range of possibilities exists, however, regarding the precise nature of this breakdown and of the resulting non-Fermi liquid states. Among these, arguably the mildest form of Fermi liquid breakdown involves a logarithmic energy dependence of the real part of the quasiparticle self-energy, while the imaginary part becomes linear in energy. In this logarithmic breakdown [START_REF] Baym | Landau Fermi liquid theory[END_REF], the quasiparticle relaxation rate becomes comparable to the quasiparticle energy all the way down to the Fermi level, so that the quasiparticle resonance marginally cannot be resolved even in the low-energy limit. More drastic forms of Fermi liquid breakdown are suggested for those materials on which most of the available evidence for Fermi liquid breakdown near magnetic quantum phase transitions has been collected: in Ce-or Yb-based f-electron metals. In some of these, anomalous temperature dependences of the resistivity and of the heat capacity over extended ranges in temperature, as well as Hall coefficient discontinuities [START_REF] Mathur | [END_REF][4][5][6], have been taken to suggest non-Fermi liquid states in which local moments survive down to very low temperatures. These more extreme scenarios contrast with the "conventional" approach, in which the effects of quantum critical magnetic fluctuations are computed within a band picture of magnetism [START_REF] See | Electron[END_REF]. Clear-cut experimental evidence for the conventional breakdown scenario, paradoxically still scarce, may be found among the d-electron systems, because for many of these a band picture is much more clearly appropriate than for the inherently more complex f-electron compounds. In particular, the logarithmic Fermi liquid breakdown scenario has long been expected to be realised at the quantum critical point (qcp) of 3D band ferromagnets (fm). The experimental signature of logarithmic Fermi liquid breakdown in such systems are a leading-order logarithmic temperature depen-dence in C/T and a sub-leading order T 5/3 contribution to the electrical resistivity [START_REF] See | [END_REF][START_REF] See | Electron[END_REF]. More detailed calculations (e.g. [START_REF] Belitz | [END_REF]9]) suggest that the fm qcp may be inherently unstable, either towards first order transitions, or towards long-wavelength modulated structures. Even in these comparatively simple systems a range of phenomena may therefore be expected near the threshold of magnetism. The precise composition of these nearlystoichiometric samples has been determined from their lattice constants (from powder x-ray diffraction), which were compared against a scale drawn up by measuring the composition of a large number of samples using wavelength-dispersive xray spectroscopy, WDXS (inset).

A number of magnetic transition metal compounds have been investigated in the search for logarithmic Fermi liquid breakdown, including MnSi [10], ZrZn 2 [11,12], Ni 3 Al [13] and Fe [14]. High pressure has been used in these cases to approach the fm qcp. Non-Fermi liquid forms have indeed been observed in transport properties, such as a distinctive T 3/2 power-law resistivity in MnSi over three orders of magnitude in temperature and, very recently, a striking linear-in-T contribution to the thermal resistivity in ambient-pressure ZrZn 2 [15]. However, the technical constraints intrinsic to high pressure experiments have so far hindered multi-probe investigations. Alloying offers an alternative approach to the fm qcp [16], but the dopant concentration must be minimal. Otherwise, disorder-induced scattering may smear or completely remove phase transitions, or lead to diffusive transport, and site disorder or inhomogeneity may cause cluster or even spin-glass formation, all of which complicate the theoretical description and obscure the objective of the study.

Balanced on the threshold of magnetism at ambient pressure, the hexagonal C14 Laves phase compound NbFe 2 provides a new perspective to this long-standing problem. Stoichiometric NbFe 2 has been reported as a rare example of low-temperature spin-density-wave (SDW) order (T N ≃ 10 K) in a d-metal system [17,18]. The SDW state grows out of low-ordered-moment fm order, which can be induced by slight Fe-excess (Fig. 1). Slight Nb-excess, on the other hand, suppresses the SDW ordering temperature, and a qcp (T N → 0) is approached close to ≃ Nb 1.015 Fe 1.985 . Here, we report the first detailed study of a slightly niobium-rich single crystal of NbFe 2 , which is very close to a magnetic qcp at ambient pressure. We find that the resistivity follows a T 3/2 power-law temperature dependence between 0.1 K and 10 K and that the heat capacity varies logarithmically with temperature between 0.1 K and 4 K. These results represent the first demonstration of the key experimental signatures of logarithmic Fermi liquid breakdown in a pure transition metal compound.

We present data from an iron-rich polycrystal (sample A, Nb 0.985 Fe 2.015 ), a stoichiometric polycrystal (sample B, NbFe 2 ) and a niobium-rich single crystal (sample C, Nb 1.01 Fe 1.99 ). The polycrystals have been prepared by melting together the elements (99.95% Nb, 4N vacuum-remelted iron) in an argon atmosphere using radio-frequency induction heating on a water-cooled copper boat, immediately followed by a short (1-3 hour) anneal near 1000 • C . Similarly, the single crystal has been obtained using a Czochralski method from an inductionheated melt in an argon atmosphere. Measurements of the resistivity, heat capacity, magnetisation and magnetic susceptibility down to 1.8 K have been carried out in a 9 T Physical Properties Measurement System (Quantum Design). High-resolution measurements of the resistivity at temperatures down to 50 mK were obtained in an adiabatic demagnetisation refrigerator (Cambridge Magnetic Refrigeration) by a standard 4-terminal AC technique. The low temperature heat capacity was determined in the same refrigerator by a relaxation-time technique. Slightly Fe-rich NbFe 2 represents the point of departure of our investigation (Fig. 2). At 1.5% excess iron, we observe (i) Curie-Weiss behaviour in the inverse susceptibility down to T m ≃ 32 K, (ii) a kink in ρ(T ) at a slightly lower temperature T ρ ≃ 25 K and (iii) fm hysteresis loops in the lowest-temperature M -H isotherms. Whereas the low-T state clearly appears ferromagnetic, no ordered moment is observed between T m and T ρ . This suggests that the transition at T m may be into a non-fm ordered state and that the transition to fm order happens only at a lower temperature T c ≃ T ρ < T m .

Reducing the Fe-excess lowers the magnetic transition temperature and changes the nature of the ordered state. Our findings confirm the reported interpretation that stoichiometric NbFe 2 forms a magnetically ordered, but not ferromagnetic, low temperature state (Fig. 3). We note that this state is characterised by a very high magnetic susceptibility χ ∼ 0.02, which corresponds to a Stoner enhancement factor χ/χ 0 ≃ 180, where χ 0 ≃ 10 -4 is the bare band structure susceptibility estimated from the calculated density of states g(E F ) ≃ 40(Ry f.u. spin) -1 [19]. The enhancement of the specific heat capacity Sommerfeld coefficient γ = C/T ≃ 45 mJ/molK 2 , with respect to its band structure value γ 0 ≃ 15 mJ/molK 2 , by contrast, is about 3, leaving a Wilson ratio R W = χ/χ0 γ/γ0 ≃ 60, which suggests extreme proximity to ferromagnetism. We suspect that the so far unidentified magnetic order in NbFe 2 is a long-wavelength modulated state, possibly helical. The ordering wave-vector Q may be estimated as ∼ 0.05 Å-1 , by inserting the measured χ q=0 ≃ 0.02 at T m into the dispersion χ -1 q ≃ c|q -Q| 2 with a typical c ≃ 2.4 • 10 4 Å2 (value determined for ZrZn 2 by neutron scattering). The very small magnetisation M ≃ 0.02 µ B /atom at just above the critical field leads us to expect, then, a weakly modulated structure with a wave-length of about 130 Å.

If the iron content in NbFe 2 is reduced beyond stoichiometry, our own results as well as literature data [17,18] show that T m decreases, giving rise to a qcp at a composition close to Nb 1.015 Fe 1.985 . We have investigated the vicinity of this qcp in a slightly niobiumrich single crystal (Nb 1.01 Fe 1.99 ), which despite its offstoichiometric composition, has the highest residual resistance ratio (RRR ≃ 18) of all the samples presented.

The H -T map of the susceptibility of this single crystal (Fig. 4) suggests magnetic order below T m ≃ 2.8 K and a very small critical field µ 0 H c ≃ 0.2 T. Transition-metal compounds are governed by band energy scales which are orders of magnitude larger than those of narrow-band f-electron compounds. Rescaling temperature by the ratio of the Sommerfeld coefficients brings T m to a similar relative magnitude as, for example, in YbRh 2 Si 2 . Despite its residual, weak low temperature order the resistance of the single crystal follows a T 3/2 power law down to T < 0.1 K, and its heat capacity divided by temperature, C/T , increases logarithmically without any indication of a cross-over to conventional Fermi-liquid behaviour (Fig. 5). We have observed qualitatively similar behaviour in a number of single crystals FIG. 4: Temperature-field map of the magnetic susceptibility (H||c) of a nearly quantum critical single crystal (sample C), constructed from χ -H isotherms at 1.6, 2, 2.5, 3, 4, 5, 6, 8, 10 and 12 K. On cooling at zero field, χ(H, T ) builds up into a ridge, which bifurcates into two branches below 2.8 K.

of similar composition, but note that the precise value of the resistivity power-law exponent varies between 3/2 and 5/3. By contrast, in stoichiometric samples of NbFe 2 (T N ≃ 10 K) ρ(T ) recovers a T 2 form below 0.6 K, and C/T saturates for T < 1 K (inset of lower panel, Fig. 5) at values far below those observed in the single crystal. The precise form of ρ(T ) may depend on stoichiometry and sample orientation. The key result, however, anomalous forms for both the resistivity and the heat capacity, which are characteristic of logarithmic Fermi-liquid breakdown, is robust.

Our findings raise questions as to (i) why ρ and C/T appear unperturbed by the onset of magnetic order, (ii) the role of disorder, and (iii) the nature of magnetic order in Nb 1.01 Fe 1.99 . Firstly, the ordered moment below T m (< 0.02 µ B ) represents a tiny fraction of the fluctuating moment: magnetic order is then expected to have only a minor effect on the magnetic fluctuation spectrum and hence on the low-T properties (cf. small C anomaly at T N ≃ 10 K even in stoichiometric sample, inset of Fig. 5). Secondly, although a T 3/2 form for ρ(T ) has also been observed in spin-glasses (e.g. [START_REF] Mydosh | Spin Glasses[END_REF]), (i) in spin glasses this form for ρ(T ) is only seen below the freezing temperature, whereas here it also holds above T m , (ii) the dynamic range of the anomalous resistivity is far larger in the pure compounds measured here than in reported spin glass measurements and (iii) the measured C/T is completely inconsistent with that found in spin glasses. Finally, while the observed form for ρ(T ) cannot decide unambiguously between fm and SDW quantum criticality, the measured C/T is consistent with predictions for a fm qcp. Naively, one would expect only a sub-leading √ T correction to the Fermi-liquid form for the heat capacity close to a SDW qcp, but the situation is less clear, if the SDW ordering wavevector Q is very small. Our bulk measurements as well as theory work [START_REF] Belitz | [END_REF]9] suggest a small wavevector SDW state (Q ∼ 0.05 Å-1 ) for stochiometric NbFe 2 . Quantitative calculations will be needed to clarify whether a small-Q SDW qcp explains the observed experimental signatures, while renewed neutron studies are planned to identify the small-moment ordered state of NbFe 2 .

Among the pure transition metal compounds -a class of materials believed to be well understood and yet full of surprising twists and discoveries -these findings represent the first clear indication of a logarithmic break-down of the Fermi liquid state, as expected at a 3D fm qcp. NbFe 2 may be sufficiently close to ferromagnetism for its critical behaviour to fall into this category. It presents a unique opportunity for multi-probe studies of band-magnet quantum criticality at ambient pressure.

FIG. 1 :

 1 FIG.1: By adjusting the precise stoichiometry within a narrow homogeneity range, NbFe2 can be tuned from ferromagnetism (FM) (y > 0.01) via an intermediate spin density wave modulated state (SDW) to a quantum critical point (QCP, y ≃ -0.015). This paper discusses samples A (Ferich, y = 0.015), B (stoichiometric) and C (Nb-rich single crystal, y = -0.01). The precise composition of these nearlystoichiometric samples has been determined from their lattice constants (from powder x-ray diffraction), which were compared against a scale drawn up by measuring the composition of a large number of samples using wavelength-dispersive xray spectroscopy, WDXS (inset).

FIG. 2 :

 2 FIG.2: Magnetic properties of slightly iron-rich NbFe2 (sample A). The inverse susceptibility follows a Curie-Weiss behaviour from 100 K down to Tm ≃ 32 K, with fluctuating moment ≃ 1.0 µB/atom. Low temperature ferromagnetism is inferred from Arrot plot isotherms (upper left inset), which exhibit typical hysteretic behaviour with a remanent magnetisation ≃ 0.04µB /atom. The T -dependence of the resistivity ρ (lower right inset) exhibits a kink (arrow) just below Tm. Both above and below this kink ρ(T ) follows a T 5/3 power-law form.

FIG. 3 :

 3 FIG.3: Susceptibility map of the low-field, low-temperature sector of the NbFe2 phase diagram (sample B), compiled from isotherms at 2, 4, 6, 8, 10, 12, 15 and 20 K. A line of χmaxima separates the H = 0, T << Tm state from the high temperature state of NbFe2, suggesting magnetic order at low temperature.

2 FIG. 5 :

 25 FIG. 5: Quantum criticality in single-crystal Nb1.01Fe1.99 (sample C). Upper panel: ρ(T ) follows a power-law form with exponent significantly less than the Fermi-liquid value 2 over a wide range in T , down to 0.1 K. A detailed analysis involving piece-wise three-parameter fitting of the data (inset, horizontal bars indicate T -range, vertical bars indicate precision of exponent estimate) confirms that the power-law exponent locks into a stable value close to 3/2 over nearly two decades in temperature. Lower panel: The Sommerfeld coefficient C/T displays a distinct logarithmic T -dependence, which increases the already high electronic contribution to C/T at 4K by about 30% on cooling to 0.1 K.

We are indebted to A. Chubukov, B. Fåk, S. Grigera, G. Kreiner, G. G. Lonzarich, P. Niklowitz, C. Pfleiderer, J. Quintanilla, J. Saunders, A. Schofield, B. Simons and M. Sutherland for useful discussions, and to the Max-Planck Inter-Institutional Research Initiative "The Nature of Laves Phases" for supporting key characterisation activities. This work was funded by the Engineering and Physical Sciences Research Council of the UK.