A M Nikitin
email: a.nikitin@uva.nl

Y Pan

X Mao

R Jehee

G K Araizi

Y K Huang

C Paulsen

S C Wu

B H Yan

A De Visser
email: a.devisser@uva.nl

Magnetic and superconducting phase diagram of the half-Heusler topological semimetal HoPdBi

Keywords: Superconductivity, Antiferromagnetism, Topological insulators PACS numbers: 74.10.+v, 74.25.-q, 74.20.Pq

We report a study of the magnetic and electronic properties of the non-centrosymmetric half-Heusler antiferromagnet HoPdBi (T N = 2.0 K). Magnetotransport measurements show HoPdBi has a semimetallic behaviour with a carrier concentration n = 3.7×10 18 cm -3 extracted from the Shubnikov-de Haas effect. The magnetic phase diagram in the field-temperature plane has been determined by transport, magnetization and thermal expansion measurements: magnetic order is suppressed at B M ∼ 3.6 T for T → 0. Superconductivity with T c ∼ 1.9 K is found in the antiferromagnetic phase. Ac-susceptibility measurements provide solid evidence for bulk superconductivity below T c = 0.75 K with a screening signal close to a volume fraction of 100 %. The upper critical field shows an unusual linear temperature variation with B c2 (T → 0) = 1.1 T. We also report electronic structure calculations that classify HoPdBi as a new topological semimetal, with a non-trivial band inversion of 0.25 eV.

Introduction

The compound HoPdBi is member of the large family of half-Heuslers that crystallize in the non-centrosymmetric space group F 43m [1]. Ternary Heuslers with composition 2:1:1 and half Heuslers with composition 1:1:1 attract ample attention because of their flexible electronic structure, which gives rise to applications as multifunctional materials in, for example, the fields of spintronics and thermoelectricity [2]. Recently, a new incentive to investigate half-Heusler compounds was provided by first principles calculations [3][4][5]. By employing the hybridization strength and the magnitude of the spin-orbit coupling is was established that "heavy-element" half-Heusler compounds may show a non-trivial band inversion, similar to the prototypical topological insulator HgTe [6][7][8]. Topological half-Heuslers are zero-gap semiconductors, where the topological surface states can be created by applying strain that opens the gap. The emergence of magnetic order and superconductivity at low temperatures in half-Heuslers provides a new research direction to study the interplay of topological states, superconductivity and magnetic order [9].

Recently, much research has been devoted to the band-inverted half-Heusler platinum and palladium bismuthides, notably because some of these superconduct at low temperatures [10]: LaPtBi (T c = 0.9 K [11]), YPtBi (T c = 0.77 K [12,13]), LuPtBi (T c = 1.0 K [14]) and LuPdBi (T c = 1.7 K [15,16]). The s-p inverted band order (Γ 8 > Γ 6 ) makes these compounds candidates for topological superconductivity [START_REF] Kitaev | AIP Conf. Proc[END_REF][START_REF] Schnyder | AIP Conf. Proc[END_REF]. Topological superconductors are predicted to be unconventional superconductors, with an admixture of even-and odd-parity Cooper pair states in the bulk, and protected Majorana states at the surface [7,8]. Since the crystal structure lacks inversion symmetry these superconductors might also be classified as non-centrosymmetric. Here theory predicts a mixed even-and odd-parity Cooper pair state [START_REF] Frigeri | [END_REF] as well.

Recently, we reported the observation of superconductivity (T c = 1.22 K) and antiferromagnetic order (T N = 1.06 K) in the half-Heusler ErPdBi. In addition, electronic structure calculations predicted a sizeable band inversion, which makes ErPdBi a first candidate for the investigation of the interplay of topological states, superconductivity and magnetic order. Another candidate proposed is CePdBi (T c = 1.3 K, T N = 2.0 K), however, here superconductivity develops in a small sample volume and is associated with a disordered phase [20]. The coexistence of superconductivity and magnetic order in ErPdBi provided a strong motivation to search for similar phenomena in other REPdBi compounds (RE = Rare Earth), such as HoPdBi [1,21]. HoPdBi was reported to be an antiferromagnet with a Néel temperature T N = 2.2 K [21]. Its magnetic susceptibility follows a Curie-Weiss law with an effective moment µ eff = 10.6 µ B , in good agreement with the theoretical value for free RE 3+ ions with Russell-Saunders coupling (J = 8). The weak antiferromagnetic coupling is furthermore inferred from the low value of the Curie-Weiss constant Θ P = -6.4 K. Here we report results of transport, magnetic and thermal measurements on single crystalline HoPdBi. Our flux-grown crystals show antiferromagnetic order at T N = 2.0 K and superconductivity at T c ∼ 1.9 K. However, the transition to bulk superconductivity, inferred from the full diamagnetic screening signal (100 % volume fraction), sets in near T bulk c ∼ 0.75 K. We report the magnetic and superconducting phase diagram in the field-temperature plane, as well as electronic structure calculations that reveal a non-trivial band inversion of 0.25 eV at the Γ-point. In the course of our investigations, Nakajima et al. [22] reported that superconductivity occurs in almost the entire antiferromagnetic REPdBi series, with T c ∼ 1 K for HoPdBi.

Methods

Several single crystalline batches of HoPdBi were prepared using Bi flux. The high purity elements Ho, Pd and Bi (purity 3N5, 4N and 5N, respectively) served as starting material. First an ingot of HoPdBi was prepared by arc-melting and placed in an alumina crucible with excess Bi flux. The crucible was sealed in a quartz tube under a high-purity argon atmosphere (0.3 bar). Next the tube with crucible was heated in a furnace to 1150 • C and kept at this temperature for 36 h. Then the tube was slowly cooled to 500 • C at a rate of 3 • C per hour to form the crystals. The single-crystalline nature of the crystals was checked by Laue backscattering. After cutting the crystals in suitable shapes for the various experiments, the surfaces were cleaned by polishing. Powder X-ray diffraction confirmed the F 43m space group and the extracted lattice parameter, a = 6.605 Å, is in good agreement with the literature [1]. Several tiny extra diffraction lines point to the presence of a small amount of an impurity phase (< 5 %). These lines could not be matched to potential inclusions or impurity phases, such as Bi, PdBi, α-Bi 2 Pd and β-Bi 2 Pd. The results presented here are taken on samples that all come from the same single-crystalline batch. The measured samples are labeled #2-#4, and show very similar magnetic and superconducting properties.

Magnetic and transport measurements were carried out in a Physical Property Measurements System (PPMS, Quantum Design) in the temperature range 1.8-300 K. The magnetization and susceptibility were measured utilizing a Vibrating Sample Magnetometer, while the resistance and Hall effect were measured using a standard four-point low-frequency ac-technique with an excitation current of 1 mA. The specific heat was measured down to 2.0 K on a crystal with a mass of 10 mg in the PPMS as well. Low temperature, T = 0.24 -10 K, resistance and ac-susceptibility measurements were made in a 3 He refrigerator (Heliox, Oxford Instruments) using a low-frequency lock-in technique with low excitation currents (I ≤ 200 µA). The coefficient of linear thermal expansion, α = L -1 (dL/dT), with L the sample length, was measured using a threeterminal parallel-plate capacitance method using a home-built sensitive dilatometer based on the design reported in Ref. [23]. Magnetoresistance data were taken in a dilution refrigerator (Kelvinox, Oxford Instruments) for T = 0.03 -1 K and high magnetic fields up to 17 T. Additional low-temperature dc-magnetization and acsusceptibility measurements were made using a SQUID magnetometer, equipped with a miniature dilution refrigerator, developed at the Néel Institute. The superconducting transition measured by resistivity in zero and applied magnetic fields is reported in detail in fig. 2. The superconducting transition at B = 0 is fairly broad, with an onset temperature of 2.3 K and zero resistance at 1.6 K. By tracing the maximum in the derivative dρ/dT we obtain T c = 1.86 K. The broad transition is most likely related to the simultaneous development of antiferromagnetic order with T N = 2.0 K (see below). A similar rounded ρ(T ) near the onset for superconductivity was observed for ErPdBi, where T c ≃ T N as well [9]. We have determined the upper critical field B c2 by taking T c (B) as the maximum in dρ/dT obtained in constant magnetic fields. See the inset of fig. 2 for the data at B = 1.0 T. Here the structure at 2.0 K signals the antiferromagnetic transition. The temperature variation B c2 (T ) is reported in the phase diagram, fig. 7. In the upper inset of fig. 2 we show the zero-field data for sample #3. The resistivity data for this sample (data in field are not shown) are in good agreement with the results obtained for sample #2.

In fig. 3 we show ac-susceptibility (χ ac ) data taken on sample #3 for different driving fields B ac ranging from 0.0025 to 0.25 mT. Upon cooling χ ac (T ) first increases and has a pronounced maximum at 2.0 K, which locates the antiferromagnetic phase transition temperature. Upon further cooling a large superconducting signal appears below 0.75 K. For the smallest driving fields B ac ≤ 0.05 mT the screening fraction associated with this diamagnetic signal is ∼ 90 %, which provides strong support for bulk superconductivity. Upon increasing the driving field to 0.25 mT the magnitude of the diamagnetic signal is rapidly depressed, which indicates the lower critical field B c1 is very small. However, a close inspection of the data shows that a weaker diamagnetic signal develops near 1.8 K, which is close to midpoint of the superconducting transition at 1.9 K of sample #3 (see upper inset fig. 2). The amplitude of the diamagnetic signal varies with B ac . This clearly indicates a smaller volume fraction (∼ 20 %) of the sample becomes superconducting at 1.8 K. Here we have neglected the demagnetization factor that we estimate to be N = 0.07 ± 0.02 for our needle-shaped sample #3. We remark that after correcting for demagnetization effects the total superconducting volume fraction is close to ∼ 100 %. These observations are corroborated by the imaginary part of χ ac (T ) shown in the inset of fig. 3. The transition to the bulk superconducting state reveals the presence of a remaining sample inhomogeneity, as indicated by the double peak structure in χ ′′ ac for small values B ac ≤ 0.005 mT.

Antiferromagnetic order

Magnetic susceptibility measurements carried out in a field of 0.1 T confirm local moment behaviour of the Ho moments. For T > 50 K Curie-Weiss behaviour is observed with an effective moment µ ef f = 10.3 µ B and a negative Curie-Weiss constant Θ P = -5.7 K indicating the presence of antiferromagnetic interactions. These values are close to the ones reported in Ref. [21]. In fig. 4 we show the dc-susceptibility around the antiferromagnetic transition measured in fields up to 3 T down to very low temperatures (0.1 K). The Néel temperature is progressively shifted with field and can no longer be discerned at the largest field. The variation T N (B) is traced in the phase diagram (fig. 7). In the right panel we show the magnetization measured at T = 0.075 K, deep in the antiferromagnetic phase, and at 2 K and 5 K, measured along the [100] axis. At the lowest temperature the magnetization increases rapidly until it reaches a value of ∼ 6 µ B at 3 T after which the increase is more gradual. In a field of 8 T the magnetization attains a value of 7.2 µ B which is still considerably below the saturation value m s = gJ = 10 µ B (here g = 5/4 is the Landé factor). An explanation for this is offered by the sizeable magnetic anisotropy associated with the Ho ions (see section 4). The presence of magnetocrystalline anisotropy can also be inferred from the very weak signature of a metamagnetic-like transition observed in M(B) below T N . In fig. 5 we show the coefficient of linear thermal expansion α(T ) around the antiferromagnetic transition. Upon cooling a large positive step ∆α is observed, where the midpoint nicely agrees with T N determined by the magnetic susceptibility. The superconducting transition is not observed in the thermal expansion because of the relatively small entropy involved (see the discussion section). In the right panel we show the specific heat measured in the temperature range 2-50 K. The increase at low temperatures upon approaching 2 K is due to the antiferromagnetic transition. When we compare c(T ) with the lattice specific heat, estimated by a Debye function with a Debye temperature θ D = 193 K, a broad magnetic contribution centered around 15 K becomes noticeable. This magnetic contribution with an entropy of ∼ 1.7 × R, where R = 8.31 J/molK is the gas constant, is most likely related to the partly lifting of the 17-fold degenerated crystalline electric field ground state in the cubic symmetry. We remark that the magnetic susceptibility starts to deviate from the Curie-Weiss behaviour below 50 K as well. By combining the thermal expansion and specific heat data we can make an estimate of the pressure variation of T N using the Ehrenfest relation dT N / dp = V m ∆β/(∆c/T N ). Here the coefficient of volume expansion β = 3α and V m = 4.3 × 10 -5 m 3 /mol is the molar volume. Using the experimental values ∆α = 1.2 × 10 -4 K -1 and ∆c/T N ∼ 5 J/molK 2 we obtain dT N / dp ∼ 0.3 K/kbar. In an applied field, the longitudinal thermal expansion measured along the direction of the magnetic field, α , develops a pronounced minimum. The negative α indicates a large magnetocrystalline anisotropy, with an expansion along the field and a contraction perpendicular to the field, upon cooling below T N . Notice, in magnetic field β = α +2α ⊥ . The variation T N (B) extracted from the thermal expansion is traced in the phase diagram fig. 7.

Quantum oscillations

The magnetoresistance was measured in magnetic fields up to 17 T at low temperatures in the dilution refrigerator. A typical trace taken at T = 0.05 K is shown in fig. 6.

After the suppression of superconductivity, the kink at B M = 3.6 T locates the antiferromagnetic phase boundary. In fields exceeding 8 T we observed Shubnikovde Haas oscillations. Since the oscillatory component is rather small (∆ρ/ρ ∼ 0.1%) and superimposed on a non-monotonic background, it is difficult to extract the precise magnitude as a function of 1/B. In the lower inset of fig. 6 we trace the resulting oscillations in ∆ρ as a function of 1/B. A fast Fourier transform (upper inset) shows the frequency is 75 ± 5 T. For a circular extremal cross section A k (E F ) = πk 2 F we calculate with help of the Onsager relation a Fermi wave vector k F = 4.8 × 10 8 m -1 . Assuming a spherical Fermi surface pocket, the corresponding 3D carrier density n = k 3 F /3π 2 = 3.7 × 10 18 m 3 which is a factor ∼ 3.5 smaller than the value deduced from the Hall effect measurements. Additional measurements at T = 0.3 and 0.6 K show the thermal damping is very weak, which points to an effective mass much smaller than the free electron mass, m ef f < 0.5m e , as reported for other topological semimetals [12,24].

Phase diagram

In fig. 7 we present the magnetic and superconducting phase diagram of HoPdBi. The upper critical field B c2 shows a smooth variation with temperature and extrapolates to the value of 1.1 T at T = 0. This tells us B c2 (T ) is associated with the superconducting transition observed at 1.9 K in zero field. When we compare the B c2 -data to the Werthamer-Helfand-Hohenberg (WHH) curve for a weak-coupling spin-singlet orbitallimited superconductor, some departure appears below ∼ 0.7 K. We remark this is close to the temperature below which the diamagnetic screening signal shows the sudden increase towards bulk superconductivity (see fig. 2). In the WHH model the clean-limit zero-temperature orbital critical field can be calculated from the relation B orb c2 (0) = 0.72 × T c |dB c2 /dT | Tc and attains a value of 0.9 T (in the dirty limit the coefficient 0.72 is replaced by 0.69). From the experimental value B c2 (0) = 1.1 T we calculate, utilizing the relation B c2 = Φ 0 /2πξ 2 , where Φ 0 is the flux quantum, a superconducting coherence length ξ = 17 nm. A departure from the WHH model has also been observed for ErPdBi [9] and notably for YPtBi [12,13], where is was taken as a strong indication of an odd-parity component in the superconducting order parameter. We remark, the superconducting phase is completely embedded in the antiferromagnetic phase.

The antiferromagnetic phase boundary has been determined by resistivity, magnetization and thermal expansion measurements, see fig. 7. After an initial (B < 1 T) steep increase with field, T N smoothly decreases, and the antiferromagnetic phase transition is suppressed at B M (0) = 3.6 T. Values for T N extracted from the magnetization and thermal expansion are somewhat lower than the values obtained by transport, but all three experiments track the same phase boundary. These differences might be partly attributed to different orientations of the crystals in the magnetic field in connection to the large magnetic anisotropy. For the magnetization experiments the magnetic field was applied along [100], while for the other experiments the field was applied along an arbitrary crystal direction. In the figure we compare the phase boundary with the phenomenological order parameter function B M (T ) = B M (0)(1 -(T /T N ) α ) β with T N = 2.0 K, B M (0) = 3.6 T, α = 2 and β = 0.34. The latter value is close to the theoretical value β = 0.38 expected for a 3D Heisenberg antiferromagnet [START_REF] Domb | The Critical Point[END_REF].

Electronic structure

To understand the electronic properties, we have performed ab initio band structure calculations [START_REF] Kresse | [END_REF] based on the density-functional theory. The core electrons are represented by the projector-augmented-wave potential. We keep only three valence electrons of Ho (equivalent to La) and freeze all other f electrons as core electrons, to simplify the understanding of the topology. The calculated bulk band structure of HoPdBi exhibits a zero-gap, and is quite similar to that of ErPdBi [9] and other half-Heusler topological semimetals [3,4]. The Γ 6 band (contributed by Pd-s states) is below the Γ 8 bands, showing the hall mark of band inversion. Therefore, we can clearly conclude that HoPdBi is also a topological semimetal. The effect of magnetization of Ho-f states is expected to possibly split the spin degeneracy slightly, but to preserve the inverted band order because the band inversion strength (the energy difference between Γ 8 and Γ 6 bands) is as large as 0.25 eV. The measured sample is slightly p-doped, which will cause E F to lie marginally below zero. .

W L Γ X -3 -2 -1 0 1 2 Energy (eV) Γ8 Γ 6 Γ 7
Figure 8. Calculated bulk band structure of the half-Heusler HoPdBi. The projection of the Pd-s states is highlighted by the size of the filled red circles. Bands Γ 6,7,8 are labeled according to the symmetry of their wave functions. The Fermi energy, E F , is shifted to zero (horizontal dashed line). The measured sample is slightly p-doped, which will cause E F to lie marginally below zero.

Discussion

Our systematic transport, magnetic and thermal properties study demonstrates that HoPdBi combines local moment magnetism and superconductivity and thus behaves very similar to the related half-Heusler compound ErPdBi [9]. Superconductivity occurs at T c = 1.9 K as determined by the resistive transition and the onset in χ ac , however, the transition to a full diamagnetic screening fraction sets in at a lower temperature, near 0.75 K. These results are in line with the recent work by Nakajima et al. [22], but nevertheless there are some noticeable differences: the midpoint of the resistive transition is at ∼ 1.0 K, and the onset of the diamagnetic signal is at ∼ 0.9 K [22]. Moreover, the transition measured by χ ac [22] is very broad and extends to the lowest temperature measured (∼ 0.1 K). A comparison of both studies suggests the superconducting phase transition found below ∼ 0.9 K by Nakajima et al. [22] is to be associated with the transition to the bulk superconducting state at 0.75 K as reported in fig. 2. The presence of a small volume fraction of our sample that superconducts at 1.9 K remains a puzzling aspect. A possible explanation is the presence of an impurity phase. Among the binary Bi-Pd alloys α-Bi 2 Pd is reported to superconduct at 1.7 K [27]. However, the few extra tiny lines in the diffraction pattern could not be matched to the structure of α-Bi 2 Pd. Moreover, B c2 (0) of α-Bi 2 Pd is small [22], which is at variance with the measured value B c2 (0) = 1.1 T.

A second, appealing explanation for superconductivity at 1.9 K is the occurrence of surface superconductivity associated with the topological surface states. Surface superconductivity at temperatures higher than the bulk T c was recently proposed by Kapitulnik et al. [START_REF] Kapitulnik | E-MRS Fall meeting[END_REF][START_REF] Kapitulnik | APS March Meeting[END_REF] for the topological half-Heuslers YPtBi and LuPtBi. Experimental evidence for this was provided by (surface sensitive) Scanning Tunneling Spectroscopy measurements. Surface superconductivity may also offer an explanation for the "unknown impurity phase" with T c = 1.7 K, i.e. above the bulk T c of 1.2 K, reported for ErPdBi [9]. Interestingly, Nakajima et al. [22] claim that surface superconductivity with T c = 1.6 K can be induced by heat treatments at 200 • C for all REPdBi. These results may open a new, fascinating research direction in the field of topological superconductivity. Future experiments have to be directed to disentangle surface and bulk superconductivity in order to put these ideas on a firmer footing.

Local-moment antiferromagnetic order has been detected for most members of the REPdBi series [9,21,22,[START_REF] Riedemann | Heat capacities, magnetic properties and resistivity of ternary RPdBi alloys where R = La[END_REF], with T N following the De Gennes scaling: T N ∝ (g J -1) 2 J(J + 1) with g J the Landé factor (see e.g. [START_REF] Jensen | Rare Earth Magnetism[END_REF]). The Neél temperature T N = 2.0 K of our crystals is in between the values 2.2 K and 1.9 K reported in Refs. [21] and [22], respectively. The phase diagram (fig. 7) shows that antiferromagnetic order entirely encloses the superconducting phase. The low-temperature χ ac and dc-magnetization measurements indicate that superconductivity and antiferromagnetic order coexist. In order to provide solid proof for this muon spin rotation/relaxation (µSR) and/or nuclear magnetic resonance experiments (NMR) would be very helpful. Alternatively, both ordering phenomena could possibly occupy different sample volumes. From recent neutron diffraction experiments [22] it was concluded that HoPdBi has a type-II antiferromagnetic structure: ferromagnetic planes couple antiferromagnetically with a propagation vector along the cube diagonal, Q = (0.5, 0.5, 0.5). The magnetization curves (fig. 4) and the thermal expansion in field (fig. 5) reveal the magnetocrystalline anisotropy is rather strong. This offers the following scenario for the magnetization curves. When the field is increased along the [100] axis the spins progressively reorient towards the field, while still pointing along the [111] direction. Near the antiferromagnetic phase boundary the component of the spin along the [100] direction is m s / √ 3 = 5.8 µ B which is close to the experimental value. For fields above 3 T the spins are being pulled against the anisotropy field, which explains the reduced value with respect to m s = 10 µ B . Interestingly, theoretical work based on symmetry classifications predicts the type II antiferromagnetic structure in these half-Heuslers produces a new ground state: an antiferromagnetic topological insulator [START_REF] Mong R S K, Essin | [END_REF]. Recently, it was proposed that a first realization of this novel electronic state is found in the half-Heusler antiferromagnet GdPtBi (T N = 9 K) [33].

The large contribution of antiferromagnetic order to the thermal properties (fig. 5) hampers the detection of superconductivity in the thermal expansion and specific heat. An estimate for the step-size ∆c in the specific heat at T c can be calculated from the standard BCS relation (∆c/T c )/γ = 1.43, where γ is the Sommerfeld coefficient. It appears that γ cannot reliably be deduced from the experimental data (see fig. 5) because of the presence of the large magnetic contribution centered around 15 K. However, an estimate for γ can be obtained from the Shubnikov -de Haas data using the relation γ = π 2 k 2 B g(E F )/3, where g(E F ) = m * k F / 2 π 2 is the density of states. With the parameters calculated in section 3.3 we obtain γ < 0.1 mJ/molK 2 and ∆c < 0.1 mJ/K at T c , which is very small indeed. An estimate for the step size in the thermal expansion coefficient, ∆α, at T c can be obtained from the Ehrenfest relation dT c /dp = V m 3α/(∆c/T c ). Assuming a typical value for dT c / dp = 0.044 K/GPa [13] we calculate ∆α at T c is smaller than 4.5 × 10 -11 K -1 , which is beyond the resolution of the dilatometer.

The occurrence of superconductivity in a Ho-based compound is uncommon. Rare examples are found among the Chevrel phases [34] and the heavy-rare earth cuprates [35] and pnictides [36]. Coexistence of superconductivity and antiferromagnetic order is found in the borocarbide HoNi 2 B 2 C [37], but here T c = 10.5 K and exceeds T N = 6.0 K. An important new aspect in the REPdBi series is the non-centrosymmetric crystal structure and the topological nature of the electronic structure. This provides a new opportunity to investigate unconventional superconductivity, due to mixed even and odd parity Cooper pair states, and its interplay with long-range magnetic order.

Summary

We have investigated the transport, magnetic and thermal properties of the half-Heusler antiferromagnet HoPdBi. Electrical resistivity and ac-susceptibility data taken on flux-grown single crystals show superconductivity occurs below 1.9 K. We demonstrate HoPdBi is a bulk superconductor. However, the transition to bulk superconductivity, as indicated by a diamagnetic screening fraction close to 100 %, sets in at a lower temperature of 0.75 K. The Néel temperature T N is 2.0 K as determined by thermal expansion and dc-magnetization measurements. The superconducting and magnetic phase diagram in the B -T plane has been determined: superconductivity is confined to the antiferromagnetic phase. Electronic structure calculations show HoPdBi is a topological semimetal with a band inversion of 0.25 eV at the Γ-point. This is in-line with the semimetallic behaviour observed in the electrical resistivity and the low carrier concentration n h = 3.7 × 10 18 cm -3 extracted from the Shubnikov-de Haas effect. We conclude, HoPdBi belongs to the half-Heusler REPdBi series with a topological band structure and presents a new laboratory tool to study the interplay of antiferromagnetic order, superconductivity and topological quantum states [9,22].

Figure 1 .

 1 Figure 1. Resistivity and carrier concentration (inset) as a function of temperature of HoPdBi sample #2.

Figure 2 .

 2 Figure 2. Resistivity as a function of temperature for HoPdBi sample #2 measured in fixed magnetic fields: from right to left 0 T to 1.1 T in steps of 0.1 T, 1.2 T to 2 T in steps of 0.2 T, and 2.25 T, 2.5 T and 3 T. Lower inset: Derivative dρ/dT versus T for B = 1.0 T; arrows indicate T c and T N . Upper inset: ρ(T ) versus T in zero field for sample #3.

3. Results and analysis 3 . 1 .

 31 SuperconductivityIn fig.1we show the resistivity ρ(T ) of a single crystal of HoPdBi (sample #2). The resistivity values are rather large and ρ(T ) has a broad maximum around 80 K. The charge carrier concentration n h extracted from the low field (B < 2 T) Hall effect measurements is traced in the inset of fig.1. The carriers are holes and at the lowest temperature (T = 2.0 K) n h = 1.3 × 10 19 cm -3 . These transport parameters reveal semimetallic-like behaviour. At low temperatures, the resistivity drops to zero, which signals the superconducting transition.

Figure 3 .

 3 Figure 3. Real part of the ac-susceptibility of HoPdBi sample #3 for driving fields of 0.0025, 0.005, 0.025, 0.05, 0.10 and 0.25 mT, from bottom to top. Arrows indicate T N , T c and T bulk c . The inset shows the corresponding imaginary part of χ ac .

Figure 4 .

 4 Figure 4. Left panel: Magnetic susceptibility of HoPdBi sample #3 as a function of temperature measured in magnetic fields up to 3 T as indicated. Right panel: Magnetization in fields up to 8 T at temperatures of 0.075, 2.0 and 5.0 K.

Figure 5 .

 5 Figure 5. Left panel: Coefficient of linear thermal expansion as a function of temperature of HoPdBi (sample #4) measured in zero and applied fields as indicated. Right panel: Specific heat (sample #2) in a plot of c versus T up to 50 K. The solid line indicates the lattice contribution given by the Debye function with θ D = 193 K.

Figure 6 .

 6 Figure 6. Resistance of HoPdBi sample #2 as a function of the magnetic field at T = 0.05 K. The arrow indicates the antiferromagnetic phase boundary at B M . Lower inset: ∆ρ as a function of 1/B showing the Shubnikov -de Haas effect at 0.05, 0.3 and 0.6 K. Upper inset: fast Fourier transform of the data with frequency F = 75 ± 5 T.

Figure 7 .

 7 Figure 7. Magnetic and superconducting phase diagram of HoPdBi. Data points for the superconducting (SC) transition are taken on sample #2 where T c (B) is defined by the temperature of the maximum in dρ/dT at fixed fields (blue circles) or the maximum in dρ/dB at fixed temperatures (blue triangles). The blue solid line is a comparison of the B c2 (T ) data with the WHH model (see text). The blue dashed line represents a linear fit to the data. Data points for the antiferromagnetic (AFM) transition are taken from the resistivity (sample #2: red circles and triangles from dρ/dT and dρ/dB, respectively), the magnetization (sample #3: open stars) and thermal expansion (sample #4: open circles). The red solid line represents a phenomenological order parameter fit with T N = 2.0 K and B M = 3.6 T at T = 0 (see text)

Acknowledgments

This work is part of the research programme on Topological Insulators of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO). B.Y. acknowledges financial support from a European Research Council (ERC) Advanced Grant (291472).


Zineb Abderrahmane

Gowrishankar Ganesh

André Crosnier

Andrea Cherubini

Visuo-Tactile Recognition of Daily-Life Objects Never Seen or Touched Before

 with vision, which enables a multimodal recognition system. In our test with the PHAC-2 dataset, the system was able to get a recognition accuracy of 72% among 6 objects that were never touched or seen during the training phase.

I. INTRODUCTION

Object recognition is an important ability, and a fundamental pre-requisite for many of the cognitive and social abilities of robots. Most state of the art recognition approaches are based on multi-class classification. They classify any encountered object as one of the previously experienced objects during the training phase, and never as a novel object, the robot has not been previously trained on. However, there are many objects that the robot can encounter in real life, and training the robot on all of them is infeasible. This is due to the effort and time required for collecting sensory training data from each one of them. Thus, the robot is usually trained on a limited set of objects and encountering novel objects is very often. This makes important to recognize these latter without collecting training data from any of them.

Information from multiple senses can be used for object recognition; the most prominent ones for this task are vision and touch. Artificial object recognition systems use either vision [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF] or touch [START_REF] Schmitz | Tactile object recognition using deep learning and dropout[END_REF], but less frequently both together [START_REF] Falco | Cross-modal visuo-tactile object recognition using robotic active exploration[END_REF]. In this work, we provide probably the first visuo-tactile recognition system that can handle novel objects, i.e. objects that have neither been seen or touched during the training phase.

We cope with novel objects by so called Zero-Shot Learning (ZSL). A Zero-Shot Learning system generalizes the models learned from objects seen and/or touched during the training phase, to recognize novel ones. This can be done by describing training and novel objects using attributes, which are semantic properties a human can use to describe each object (e.g. round, soft and bumpy). Then, attributes are used to recognize novel objects based on visuo-tactile data collected from training ones. While tactile data are relevant to perceive the object's material, texture and compliance properties, the addition of vision can improve the performance by perceiving Zineb Abderrahmane is supported by the Ministry of Higher Education and Scientific Research of Algeria through the Excellence Fellowship

All authors are with LIRMM, Universit de Montpellier, CNRS, Montpellier, France firstname.lastname@lirmm.fr

Gowrishankar Ganesh is also with CNRS-AIST JRL UMI3218/RL 1-1-1

Umezono, 305-8560 Tsukuba g.ganesh@aist.go.jp properties such as shape and color. In this paper, we suggest an attribute-based framework that enables visuo-tactile ZSL.

To the best of our knowledge, several studies have been carried out on visual ZSL [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF], only one on haptic (tactile and shape) ZSL [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF], and there is no studies on visuo-tactile ZSL. A recent study suggested a hybrid system by combining visual and tactile data, but they performed tactile ZSL with trained visual features [START_REF] Liu | Cross-modal zero-shot-learning for tactile object recognition[END_REF]. Here, we exploited the PHAC-2 dataset [START_REF] Chu | Robotic learning of haptic adjectives through physical interaction[END_REF], that provides both haptic and visual data for 60 daily-life objects (see examples in Fig. 1). We improved our haptic ZSL framework proposed in [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF] as follows: First, we improved attributes learning by replacing hand-crafted feature extractors with deep Convolutional Neural Networks (CNNs). CNNs were used to classify attributes based on both tactile and visual data. Second, we adapted the Direct Attributes Prediction (DAP) model used in [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF] to take into account both visual and tactile modalities. For recognizing a novel object, three scenarios were investigated and compared: (1) the use of vision only, by assuming that the object cannot be touched, e.g. it is far from the robot, (2), the use of touch only, e.g. in case the robot operates in the dark, and (3), the use of both vision and touch. Finally, we improved the visuo-tactile ZSL by extending the PHAC-2 attribute set by adding visual ones capturing more object properties that cannot be felt using touch.

This paper is organized as follows. Sect. II presents related work on visual and tactile ZSL. In Sect. III, we present the theoretical framework of attribute-based ZSL. Next, we propose our solution for integrating visual and tactile modalities in Sect. IV. We then present our experimental setup and experimental evaluation in sections V and VI respectively. Finally, in Sect. VII, conclusions are provided.

II. RELATED WORK

A. Visual Zero-Shot Learning

Although many broad image datasets are available for object recognition (such as ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF]), image labeling for all possible classes is still intractable. This justifies the great attention gained by ZSL in visual recognition. Lampert et al. [START_REF] Lampert | Learning to detect unseen object classes by between-class attribute transfer[END_REF] designed the first attribute-based ZSL system for recognizing novel animal classes. By describing animals using attributes (e.g. furry, small and tail), they used available images to train a classifier per attribute. Then, these classifiers were used to classify novel classes, solely given their attribute-based description. Further improvements of this framework have been proposed, by adapting it to large scale datasets [START_REF] Russakovsky | Attribute learning in large-scale datasets[END_REF], generalizing to real-valued attributes [START_REF] Parikh | Relative attributes[END_REF], [START_REF] Cheng | Random forest classifier for zero-shot learning based on relative attribute[END_REF], developing an online incremental approach [START_REF] Kankuekul | Online incremental attribute-based zero-shot learning[END_REF], reducing human effort by automatically designing attributes [START_REF] Yu | Designing category-level attributes for discriminative visual recognition[END_REF], handling attributes unreliability [START_REF] Jayaraman | Zero-shot recognition with unreliable attributes[END_REF], designing a hierarchical transfer model [START_REF] Al-Halah | How to transfer? zero-shot object recognition via hierarchical transfer of semantic attributes[END_REF], and recovering missing class-attribute associations [START_REF] Al-Halah | Recovering the missing link: Predicting class-attribute associations for unsupervised zero-shot learning[END_REF]. Another approach [START_REF] Elhoseiny | Write a classifier: Zeroshot learning using purely textual descriptions[END_REF], [START_REF] Ba | Predicting deep zero-shot convolutional neural networks using textual descriptions[END_REF] performs ZSL based on classes textual descriptions available on linguistic databases (e.g. Wikipedia). A third approach [START_REF] Rohrbach | What helps where-and why? semantic relatedness for knowledge transfer[END_REF], [START_REF] Mensink | COSTA: Co-occurrence statistics for zero-shot classification[END_REF], [START_REF] Fu | Zero-shot object recognition by semantic manifold distance[END_REF] classifies novel classes based on their direct or hierarchical relationships with training classes. Several approaches have been reviewed and compared in [START_REF] Xian | Zero-shot learning -the good, the bad and the ugly[END_REF].

B. Tactile Zero-Shot Learning

Tactile recognition systems suffer not only from the difficulty of labeling data, but also from the difficulty of collecting them. Tactile data collection requires robot-object interaction, which is time consuming, especially because some sensors need a stable contact with the object's surface to obtain good-quality measures, e.g. [START_REF] Kerr | Material classification based on thermal and surface texture properties evaluated against human performance[END_REF] maintained the contact with the object for 20 seconds. Nevertheless, tactile ZSL has gained much less research attention than visual ZSL, which motivated us to propose a Zero-Shot haptic (tactile and shape) recognition system in [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF].

In [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF], we used the state of the art PHAC-2 dataset [START_REF] Chu | Robotic learning of haptic adjectives through physical interaction[END_REF]. This dataset was used in multiple studies to improve robot haptic perception using the provided haptic attributes . Gao et al. [START_REF] Gao | Deep learning for tactile understanding from visual and haptic data[END_REF] used deep learning for recognizing PHAC-2 attributes from haptic and visual data. The authors of [START_REF] Liu | Structured outputassociated dictionary learning for haptic understanding[END_REF] improved haptic perception by making use of attributes correlations and learned attributes in a multi-label setting instead of separately as in [START_REF] Gao | Deep learning for tactile understanding from visual and haptic data[END_REF].

C. Visuo-Tactile Fusion

Many studies showed the efficiency of combining visual and tactile data for improving different robotic tasks. For instance, Gao et al [START_REF] Gao | Deep learning for tactile understanding from visual and haptic data[END_REF] incorporated both modalities for understanding objects' properties, Ghanbari et al. [START_REF] Ghanbari | Haptic microrobotic intracellular injection assistance using virtual fixtures[END_REF] for assisting humans in cell injection task and Yamashiro et al. [START_REF] Yamashiro | Active estimation of friction properties with haptic vision[END_REF] for estimating friction properties. In a recent study, authors of [START_REF] Liu | Cross-modal zero-shot-learning for tactile object recognition[END_REF] proposed a visuo-tactile dictionary learning for ZSL of the eight material categories that group PHAC-2 objects. They showed that incorporating both visual and tactile modalities is effective for performing ZSL. This encouraged us to extend our previous haptic ZSL framework [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF] by adding vision. Since we use PHAC-2 objects which have simple and similar shapes, we do not consider kinesthetic (shape) data and we only take tactile data combined with visual data.

Our contribution w.r.t the state of the art is the design of a visuo-tactile ZSL framework which improves the haptic ZSL framework that we proposed in [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF]. This work is different from [START_REF] Liu | Cross-modal zero-shot-learning for tactile object recognition[END_REF] that uses visuo-tactile data for ZSL. They perform a material-based ZSL by considering the eight material classes, the PHAC-2 objects belong to, whereas, here, we perform an instance-based object recognition. In addition, they use visual data available for novel classes to perform tactile ZSL, whereas, here, we recognize novel objects having neither visual nor tactile data.

III. ATTRIBUTE-BASED ZERO-SHOT LEARNING

Zero-Shot Learning (ZSL) is the problem of training and testing a recognition system on two disjoint sets. Let Y = {y 1 , . . . , y N } be the set of objects the robot has been trained on, giving training set D train ⊂ X ×Y , where X is the feature space where collected sensory raw data are represented. During the test phase, the robot collects x test ∈ X by exploring an unknown object, which should be classified as one of the novel objects Z = {z 1 , . . . , z L }. Since Y ∩ Z = / 0, the robot has no training data for Z objects and needs an auxiliary information about objects to classify x test as one of Z objects based on sensory training data collected from Y objects.

The solution applied in [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF], illustrated in Fig. 2, is largely used by many studies. It consists in defining a set of attributes A = {a 1 , . . . , a M } (e.g. round, plastic, concave, etc. in [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF]). Then, each object o ∈ Y ∪ Z is described using A. This associates o with a deterministic vector

a o = [a o 1 , . . . , a o M ]
, where for m = 1, . . . , M: a o m = 1 if attribute a m is a property present in object o (e.g. a m = concave for o = cup) and a o m = 0 otherwise. Authors of [START_REF] Lampert | Learning to detect unseen object classes by between-class attribute transfer[END_REF] proposed two models for using the attributes layer to make use of training data collected from Y to recognize Z objects. In [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF], we chose the Direct attributes Prediction (DAP) model which is more popular and showed better performance in [START_REF] Lampert | Learning to detect unseen object classes by between-class attribute transfer[END_REF].

The DAP model uses D train to learn a classifier per attribute. During the training phase, for each attribute a m ∈ A, a probabilistic binary classifier

f m : X -→ [0, 1] is trained on D m train = {(x i , a y n m ), s.t. (x i , y n ) ∈ D train }.
During the test phase, the test sample x test is input to each trained f m which returns the posterior f m (x test ) = p(a m | x test ). It predicts the presence of attribute a m in the object from which x test was collected. Then, the posterior of each novel object z l ∈ Z is computed given a z l and all attributes posteriors as follows:

p(z l | x test ) = p(z l ) p(a z l ) M ∏ m=1 p(a z l m | x test ), (1)
By replacing object and attribute priors with a uniform distribution, the test sample x test is classified as the object having the highest posterior:

z test = argmax z l ∈Z p(z l | x test ). (2)
IV. VISUO-TACTILE ZERO-SHOT LEARNING

In this section, we adapt the DAP model to perform tactile, visual or visuo-tactile ZSL. A robot equipped with visual sensors can provide visual images in X V about the object when it is in its field of view. In addition, by physically interacting with the object, the robot tactile sensors provide tactile data samples in X T . X V and X T are feature spaces in which visual and tactile data are represented respectively. Thus, our aim is to adapt the framework presented in Sect. III for data samples

x = [x V , x T ] in X = X V × X T .

A. Attributes Learning

The first step to learn from training data how to predict the presence of each attribute in an object, i.e. to compute {p(a

1 | x), . . . , p(a M | x)} given x = [x V , x T ].
We propose three solutions. The first solution learns a binary classifier per attribute that uses tactile data x T only. The second one learns a binary classifier per attribute using visual data x V only. The advantage of separating tactile and visual modalities for attributes prediction is that the system is operational even when only one sensor modality is available. The third solution learns one classifier per attribute using x = [x V , x T ]. It makes use of both visual and tactile data to learn the attribute.

First, to learn a tactile classifier per attribute, we replace the hand-crafted feature extractor and the SVM classifier used in [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF] by a CNN, which requires representing tactile signals in the form of a tactile image. By deriving a tactile image from x T , the CNN automatically extracts discriminative features and predicts the presence of the attribute in the object at the same time. Likewise, we train a binary CNN per attribute that predicts its presence from visual images x V .

The third solution classifies both tactile and visual data at the same time using one CNN. As illustrated in Fig. 3, we extract features from each modality separately using an independent convolutional part per modality. Then, we concatenate the tactile and visual CNN features to form one visuo-tactile feature vector that we classify using a fully connected neural network.

B. Visuo-Tactile DAP

The second step is to use outputs of attribute classifiers to compute posteriors of Z objects according to [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF]. This requires the prediction of all attributes posteriors p(a m | x test ). However, choosing CNNs for attributes classification provides us with a classification score s m (x test ) ∈ R. To transform this score into a posterior probability, we use a sigmoid function as follows:

p(a m | x) = 1/(1 + e -s m (x) ). (3)
Thus, for each attribute, we have three posteriors given the test sample x test = [x V , x T ]: p(a m | x T ) returned by the tactile CNN, p(a m | x V ) returned by the visual CNN and p(a m | x V , x T ) returned by the visuo-tactile CNN. Thus, inferring p(a m | x test ) used in (1) can be performed based on:

1) Tactile data only: by replacing p(a m | x test ) in ( 1) with p(a m | x T ). We refer to this method as Tactile-ZSL (denoted T-ZSL); 2) Visual data only: by replacing p(a m | x test ) in (1) with p(a m | x V ). We refer to this method as Visual-ZSL (denoted V-ZSL); 3) Both tactile and visual data: we proceed in two ways: a) by replacing p(a m | x test ) in ( 1) with p(a m | x V , x T ). We refer to this method as "Visuo-Tactile Features Concatenation ZSL" (denoted VT-FC-ZSL); b) by combining the two independent visual and tactile attributes posteriors to compute a visuotactile attribute posterior:

p(a m | x test ) = tact(a m ) p(a m | x T ) + vis(a m ) p(a m | x V ), (4)
where tact(a m ) and vis(a m ) are user-tuned scores given to the importance of tactile and visual modalities for classifying attribute a m respectively, s.t. tact(a m ) + vis(a m ) = 1. We refer to this method as "Visuo-Tactile Scores Merging ZSL" ( denoted VT-SM-ZSL).

V. DATABASE DESCRIPTION A. PHAC-2 Dataset

In this work, we use the state of the art PHAC-2 dataset [START_REF] Chu | Robotic learning of haptic adjectives through physical interaction[END_REF]. This dataset describes a set of 60 objects, having various texture, material and compliance properties, using 25 haptic attributes. We reduce them to 19 binary attributes (as in [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF]): A = {absorbent, bumpy, compressible, cool, f uzzy, hard, hairy, metallic, porous, rough, scratchy, slippery, smooth, so f t, solid, springy, squishy,textured,thick}.

Each object has been explored 10 times using the gripper of a PR2 robot equipped with 2 BioTac sensors. Each exploration trial consists of a sequence of 4 exploration steps: squeeze, hold, slow slide and fast slide. In addition, 8 images have been taken from different viewpoints for each of 53 objects.

B. Data Augmentation and Pre-processing

The available data are very few; only 10 tactile samples and 8 visual samples per object. This requires both tactile and visual data augmentation. As in [START_REF] Gao | Deep learning for tactile understanding from visual and haptic data[END_REF], we augmented tactile data by combining data from both BioTacs and by sub-sampling the signals measured by each BioTac using five different starting points, resulting in 100 samples per Therefore, the space in which tactile data are represented is X T = R 32×30 . On the other hand, 8 RGB images of resolution (224 × 224) yield a visual features space X V = R 3×224×224 . This space was augmented by rotating each image multiple times and zooming in object's surface, resulting in 80 images per object instead of 8.

C. Objects Splits

By definition, performing ZSL requires the splitting of the object set into two disjoint sets: Y and Z. Since we aim at developing a visuo-tactile recognition system and visual data are not available for all of the 60 objects, we keep only the 53 objects for which visual data are available. We randomly select 6 objects (≈ 10%) having different attributes vectors as Z objects, and the remaining objects as Y objects. We repeat the process 7 times in order to generate 7 random (Z,Y ) splits to ensure the independence of the results from the choice of objects.

VI. EVALUATION AND RESULTS

A. Implementation Choices

We have implemented our framework using Python based on [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF], [START_REF] Chu | Penn-hapticsbolt[END_REF], [START_REF] Gao | deep learning for tactile understanding from visual and haptic data[END_REF]. CNNs were implemented using caffe [START_REF] Jia | Caffe: Convolutional architecture for fast feature embedding[END_REF]. The architecture of tactile CNN is the same as in [START_REF] Gao | Deep learning for tactile understanding from visual and haptic data[END_REF] and is illustrated in Fig. 4. Fig. 5 illustrates the visual CNN architecture, we used a pre-trained model of GoogleNet [START_REF] Szegedy | Going deeper with convolutions[END_REF] as a feature extractor for the visual data. We compared the BVLC [START_REF] Caffe | [END_REF] and MINC [START_REF] Bell | Material recognition in the wild with the materials in context database (supplemental material)[END_REF] GoogleNet pre-trained models and we found relatively similar results. Thus we chose the MINC model to have results comparable with [START_REF] Gao | Deep learning for tactile understanding from visual and haptic data[END_REF]. Then, the extracted visual features are averaged and classified using a fully connected neural network that predicts the presence of the attribute. Finally, the convolutional parts of each of Fig. 4 and Fig. 5 are used to extract tactile and visual features for the visuo-tactile CNN illustrated in Fig. 3. Attribute binary classification accuracies for all object splits averaged over all attributes (method of [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF] in blue and tactile CNN in red).

B. Attributes Learning

First of all, we focus on attributes learning using tactile data only. We compare our previous hand-crafted features extractor and SVM method [START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF] with the current deep classification method. In Fig. 6, we illustrate for each of the seven object splits defined in Sect. V-C the average classification accuracy of all attributes. We note that for all splits, CNN classification performs better than SVM with an average improvement of 4.37%. This shows the efficiency of deep learning in automatically extracting features and classifying them at the same time, compared to hand-crafted features and separate classification.

In Sect IV-A, we proposed three methods for learning attributes: based on tactile data only, based on visual data only and based on both visual and tactile CNN features. In Fig. 7, we compare the three classifiers of each attribute trained and tested based on split 1. We note that the performance changes from an attribute to another. Some attributes such as bumpy, metallic and squishy are better classified using tactile data. Some attributes such as rough, springy and textured are better classified using visual data. Others such as absorbent, compressible and hard are better classified using both tactile and visual data. Overall, 8 attributes are better classified using tactile data, 3 using visual data and 8 using visuotactile data. The fact that few attributes are better classified using vision only is obvious, since the attributes were defined in [START_REF] Chu | Robotic learning of haptic adjectives through physical interaction[END_REF] to describe the haptic sensation of the objects. Besides, for 8 attributes, merging visual and tactile data improved learning compared to learning from each modality separately, which is promising for combining both modalities using VT-SM-ZSL.

C. Visuo-Tactile DAP

Here we present results of classifying a test sample x test = [x V , x T ] as one of the 6 objects in Z. Knowing that we have zero training data for each of the 6 objects, classifying them with traditional classifiers gives an average classification accuracy of 16.67% which is equal to chance.

In table I, we compare DAP classification accuracies for classifying Z objects based on x T only (T-ZSL), x V only (V-ZSL) and [x V , x T ] (VT-FC-ZSL). Results show that most of splits are better classified with visuo-tactile data, some of them with tactile data only, and none of splits with visual data only. This was expected from results of attributes learning. However, we note that even though visual data alone are not very efficient for classifying objects, they efficiently improve tactile recognition in 5 out of 7 cases. Motivated by the good results obtained with the concatenation of visual and tactile features using VT-FC-ZSL, we continue investigating another method for performing visuotactile DAP which is VT-SM-ZSL. According to (4), the importance of both modalities for classifying each attribute tact(a m ) and vis(a m ) should be estimated. For this, we compare in table II three methods of computing tact(a m ) and vis(a m ). The first binary method gives a binary importance tact(a m ) = 1 and vis(a m ) = 0 if the attribute classification accuracy using tactile data is better than using visual data, and tact(a m ) = 0, vis(a m ) = 1 otherwise. The second weighted method gives a real valued importance: tact(a m ) = acc t m /(acc t m + acc v m ) and vis(a m ) = acc v m /(acc t m + acc v m ) where acc t m , acc v m are respectively the classification accuracies of tactile CNN and of visual CNN trained on classifying attribute a m and tested on validation data. The third, and simple uniform method assumes the same importance for both vision and tactile, i.e tact(a m ) = vis(a m ) = 0.5. Results show that the average accuracy of the binary method is greater than the other two methods. The elimination of the least performing modality for each attribute helped to perform visuo-tactile DAP by taking the best of each modality.

D. Adding Visual Attributes

In the experiments above, we used the haptic attributes provided with the PHAC-2 dataset to perform the ZSL. We obtained an improvement of 7, 74% when adding visual features to tactile ones (see tables I and II). This motivated us to try adding more visual attributes to further improve the visuo-tactile ZSL. While haptic attributes describe the texture, compliance and material properties, visual attributes can describe better the shape and color properties. Given that PHAC-2 objects have simple shapes with flat parallel sides, we assumed that adding visual attributes describing objects shapes would not be very effective. We therefore extended the attributes set with a set of color attributes. By observing objects' images, we defined a set of 7 colors shared between all objects, which are A c = {white, blue, yellow, red, beige, silver, black}. Each object-color pair is associated with a binary value given by a human operator.

First, for each color attribute, we trained a visual CNN having the same architecture as for the haptic attributes (see Fig. 5). Fig. 8 illustrates the classification accuracies of color attributes CNNs, obtained for each split. We note that classification performance varies from an attribute to another and from one set to another. Overall, all colors have been classified with more than 60% accuracy.

Next, we improved the DAP classification results by extending the haptic attributes with color attributes. We first improved V-ZSL by classifying all the 26 attributes (haptic and color) using visual data only. Results (reported in the first column of table III) show the significant improvement of recognition accuracy for almost all object splits, compared to V-ZSL in table I. This highlights the effectiveness of adding visual attributes along with haptic ones. Only split 3 shows a degradation in terms of accuracy, but this is coherent with the fact that it has the lowest average attributes classification accuracy of 83.15% and the lowest accuracy of 60.83% for classifying attribute yellow. Furthermore, in table III, we added color attributes for VT-FC-ZSL and VT-SM-ZSL by giving vis(a c ) = 1 and tact(a c ) = 0 for all color attributes. Compared to tables I and II, this addition improved object classification for almost all splits, with an accuracy of 86% for split 6.

VII. CONCLUSION

In this paper, we proposed a visuo-tactile recognition framework, capable of recognizing novel daily-life objects based on their attribute-based description and without collecting any visuo-tactile data about them. We showed how replacing hand-crafted feature extraction with CNNs improved attributes learning (see Fig. 6). In addition, integrating visual data to tactile data (see tables I and II) significantly improved the Zero-Shot recognition accuracy. Finally, extending haptic attributes with visual ones improved the recognition performance (see table III). The obtained improvement consolidates previous studies results which highlighted the importance of visuo-tactile collaboration for improving robotic tasks.

Fig. 1 .

 1 Fig. 1. Example images taken for PHAC-2 objects [7].

Fig. 2 .

 2 Fig. 2. Attribute-based ZSL (solution overview for N=5, M=4 and L=3): First, both Y and Z objects are described using A attributes. Then, a classifier f m is learned for each attribute. Last, attributes classifiers are used by the DAP model to infer the object class.

Fig. 3 .

 3 Fig. 3. Visuo-tactile CNN classifying an attribute as absent (0) or present (1) based on tactile and visual data.

Fig. 4 .

 4 Fig. 4. Architecture of CNN for predicting attribute presence (1) or absence (0) based on tactile data.

Fig. 5 .

 5 Fig. 5. Architecture of CNN for predicting attribute presence (1) or absence (0) based on visual data.

Fig. 6 .

 6 Fig. 6.Attribute binary classification accuracies for all object splits averaged over all attributes (method of[START_REF] Abderrahmane | Haptic Zero-Shot Learning: Recognition of objects never touched before[END_REF] in blue and tactile CNN in red).

Fig. 7 .

 7 Fig. 7. Attributes classification accuracies for split 1: purple with tactile data alone, yellow with visual data alone, and green with both visual and tactile data.

Fig. 8 .

 8 Fig. 8. Color attributes classification using visual images (bars are colored by the colors they represent).

TABLE I COMPARISON

 I OF TACTILE, VISUAL AND VISUO-TACTILE ZSL

		RECOGNITION ACCURACIES (%).
	set	T-ZSL V-ZSL VT-FC-ZSL
	1	60.5	31.46	71
	2	40.5	46.25	53.33
	3	43.67	54.17	61.95
	4	62.83	37.71	56.28
	5	41.83	28.13	42.97
	6	64.33	33.96	57.73
	7	33.33	35.21	54.88
	average 49,57	38,13	57,31

TABLE II VT

 II -SM-ZSL RECOGNITION ACCURACIES (%).

	set	binary weighted uniform
	1	55.27	52.87	53.65
	2	41.10	51.39	51.79
	3	48.25	59.74	61.13
	4	57.08	49.63	48.67
	5	33.86	42.23	42.34
	6	49.98	44.47	43.08
	7	51.53	34.84	33.84
	average 48.15	47.88	47,79

TABLE III RECOGNITION

 III ACCURACIES (%) WHEN ADDING COLOR ATTRIBUTES TO VISUAL AND VISUO-TACTILE ZSL.

	set	V-ZSL+C VT-FC-ZSL+C VT-SM-ZSL+C
	1	47.29	77.48	62.88
	2	54.58	66.67	54.29
	3	48.13	59.58	57.21
	4	66.25	75.1	73.19
	5	49.38	77.82	80.38
	6	62.5	77.82	69.52
	7	46.46	68.4	86.27
	average 53,51	71.74	66.53