I Yahniuk

A Kazakov

B Jouault

S S Krishtopenko

S Kret

G Grabecki

G Cywiński

N N Mikhailov

S A Dvoretskii

J Przybytek

V I Gavrilenko

F Teppe

T Dietl

W Knap

HgTe quantum wells for QHE metrology under soft cryomagnetic conditions: permanent magnets and liquid 4 He temperatures

HgTe quantum wells with a thickness of ~7 nm may have a graphene-like band structure and have been recently proposed to be potential candidates for quantum Hall effect (QHE) resistance standards under condition of operation in the fields above certain critical field Bc, above which the topological phase (with parasitic edge conduction) disappears . We present experimental studies of the magnetoresistance of different of HgTe quantum wells as a function temperature and magnetic field, determining the critical magnetic field Bc. We demonstrate that for QWs of specific width Bc becomes low enough to grant observation of remarkably wide QHE plateaus at the filling factor = -1 (holes) in relaxed cryomagnetic conditions: while using commercial 0.8T Neodymium permanent magnets and temperature of a few Kelvin provided by 4 He liquid system only. Realistic band structure calculations evidence that the observed phenomena can be qualitatively understood by considering two types of the charge carriesthe light and heavy holes. The formers are attributed to the Γ point of the Brillouin zone, while the latters are caused by the side maxima of the valence band. Our work clearly shows that the peculiar band structure of HgTe QWs make them a favorable platform for developing metrological devices under soft cryomagnetic conditions.

I. INTRODUCTION

The quantum Hall effect (QHE) is arguably the first topological phenomenon that has important practical applications. In a two-dimensional electron gas (2DEG) submitted to a high perpendicular magnetic field, the Hall resistance develops steps separated by plateaus whose value is quantized and given by 𝜌 𝑥𝑦 plateau = ℎ (𝑖𝑒 2 ) ⁄ , where RK = h/e 2 is the von Klitzing resistance, 𝑖 is an integer, h and e are the Planck constant and the electron charge, respectively [START_REF] Klitzing | New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance[END_REF][START_REF] Thouless | Quantized Hall conductance in a two-dimensional periodic potential[END_REF]. Thus, the QHE relates an electrical resistance to fundamental physical constants. At the same time, QHE measurements can be so precise that the resistances standards, currently maintained in various national metrology institutes, are based on the QHE and have a relative precision of a few parts per billion [START_REF] Jeckelmann | The quantum Hall effect as an electrical resistance standard[END_REF]. This precision was one of the triggers behind introducing the quantum International System of Units (SI), in which h and e, and therefore RK have fixed values [START_REF] Rigosi | The quantum Hall effect in the era of the new SI[END_REF].

For these metrological applications, the choice of the material is crucial. To reach ideal quantum limit conditions, one needs the highest possible Landau level (LL) splitting (low carrier mass and high magnetic fields), the lowest possible broadening of the LLs and of the Fermi-Dirac distribution function (high mobility, low temperatures). Up to recently, only Silicon metal-oxide-semiconductor field-effect transistors (MOSFETs) and GaAs/AlGaAs 2DEG heterostructures could be used for accurate measurements of the quantum Hall resistance. Usually, GaAs devices are preferred because of their superior properties [START_REF] Rigosi | The quantum Hall effect in the era of the new SI[END_REF][START_REF] Hartland | Direct comparison of the quantized Hall resistance in gallium arsenide and silicon[END_REF][START_REF] Jeckelmann | High-precision measurements of the quantized Hall resistance:Experimental conditions for universality[END_REF][START_REF] Schopfer | Quantum resistance standard accuracy close to the zero-dissipation state[END_REF]. However, both types of devices are accurate only with stringent experimental conditions: they operate at extremely low temperatures, 𝑇 ≃ 1.5 K or below, and under high magnetic fields, 𝐵 ≃ 10 T that can be provided only by superconducting magnets.

It has recently been demonstrated that graphene can operate as a quantum Hall resistance standard (QHRS) with quantization accuracy of a few parts per billion [START_REF] Tzalenchuk | Towards a quantum resistance standard based on epitaxial graphene[END_REF][START_REF] Ribeiro-Palau | Quantum Hall resistance standard in graphene devices under relaxed experimental conditions[END_REF][START_REF] Lafont | Quantum Hall resistance standards from graphene grown by chemical vapour deposition on silicon carbide[END_REF]. Furthermore, graphene can even work under less demanding experimental conditions than GaAs or Si semiconductorbased standards [START_REF] Ribeiro-Palau | Quantum Hall resistance standard in graphene devices under relaxed experimental conditions[END_REF][START_REF] Lafont | Quantum Hall resistance standards from graphene grown by chemical vapour deposition on silicon carbide[END_REF]. This results mainly from the significant LL energy separation which can be reached in graphene, thanks to the peculiarities of its energy band structure, which mimics the energy-momentum relation of massless Dirac fermions and has a high Fermi velocity [START_REF] Novoselov | Unconventional quantum Hall effect and Berry's phase of 2π in bilayer graphene[END_REF]. However, graphene is not the only material in which large LL energy gaps can be obtained at low fields. This distinct behavior is an inherent property of all systems hosting massless two-dimensional (2D) Dirac fermions. HgTe/HgCdTe quantum wells (QWs) are structures in which the 2D confinement can be tailored to obtain a Dirac-like band dispersion as in graphene [START_REF] Gerchikov | Interface States in Subband Structure of Semiconductor Quantum Wells[END_REF][START_REF] Büttner | Single valley Dirac fermions in zero-gap HgTe quantum wells[END_REF][START_REF] Konig | Quantum Spin Hall Insulator State in HgTe Quantum Wells[END_REF]. Indeed, with the decrease of the QW width, one may change the inverted band structure of a HgTe QW to a normal one. For rectangular HgTe/Cd0.7Hg0.3Te QWs grown on CdTe buffer, the band dispersion of massless Dirac fermions is realized at the critical thickness dc ~ 6.3 nm [START_REF] Büttner | Single valley Dirac fermions in zero-gap HgTe quantum wells[END_REF][START_REF] Konig | Quantum Spin Hall Insulator State in HgTe Quantum Wells[END_REF][START_REF] Olbrich | Giant photocurrents in a Dirac fermion system at cyclotron resonance[END_REF][START_REF] Kozlov | Quantum Hall effect in a system of gapless Dirac fermions in HgTe quantum wells[END_REF][START_REF] Kozlov | Quantum Hall effect in HgTe quantum wells at nitrogen temperatures[END_REF][START_REF] Khouri | High-temperature quantum Hall effect in finite gapped HgTe quantum wells[END_REF]. For example, the computed band structure for two rectangular HgTe quantum well structures of thickness d = 7.0 nm, and dc = 6.3 nm are shown in Figs. 1(a,b). The peculiar behavior of two Landau levels in HgTe QWs with the inverted band sequence, referred to as "zero-mode" Landau levels, leads to their crossing at a certain critical magnetic field Bc [see Fig. 1(c)]. According to Figs. 1(a,c), below (above) Bc, the QW hosts a topological (normal) bandgap. In the case of the Dirac-like energy dispersion, the critical field is zero, Bc = 0, as shown in Figs. 1(b,d). Details of the calculations and parameters are given in Ref. [START_REF] Krishtopenko | Pressure-and temperature-driven phase transitions in HgTe quantum wells[END_REF]. (b,d) dc = 6.3 nm thickness, respectively. The red curve (H1) corresponds to a heavy hole-like subband, and the blue curve (E1) corresponds to an electron-like subband. Symbol Bc is a critical magnetic field in which zero-mode LLs cross. The magnitude of Δ determines the energy gap between the two uppermost LLs originating from the valence band.

Thanks to the recent progress of molecular beam epitaxy (MBE) growth, HgCdTe-based QW structures with high carrier mobility µ > 10 5 cm 2 /Vs could be obtained. These structures allowed for the observation of the QHE with the filling factor = 1 in relatively low magnetic fields (B < 1 T) [START_REF] Konig | Quantum Spin Hall Insulator State in HgTe Quantum Wells[END_REF][START_REF] Olbrich | Giant photocurrents in a Dirac fermion system at cyclotron resonance[END_REF][START_REF] Kozlov | Quantum Hall effect in a system of gapless Dirac fermions in HgTe quantum wells[END_REF]. In Refs. 17 and 18, the QHE in HgTe QWs was observed even up to liquid nitrogen temperatures. These observations allow considering HgTe/HgCdTe QWs as promising candidates for QHRSs. It has been predicted in Ref. 20 that the QHE standards might be realized in relaxed cryomagnetic conditions: without any superconducting magnets and without any sub-Kelvin refrigerators required hitherto to observe the quantized Hall resistance in weak magnetic fields [START_REF] Parmentier | Quantum Hall effect in epitaxial graphene with permanent magnets[END_REF][START_REF] Götz | Precision measurement of the quantized anomalous Hall resistance at zero magnetic field[END_REF][START_REF] Fox | Part-per-million quantization and current-induced breakdown of the quantum anomalous Hall effect[END_REF].

In this work, we report on experimental studies in slightly inverted HgTe QW structures in view of their possible application for QHRSs. We demonstrate that for holes, one can reach the QHE states at = -1 in relatively favorable conditions: magnetic fields below 1 T and temperatures up to 10 K. We also show essential differences in the QHE plateaus widths between holes and electrons, and the temperatureinduced transition from the inverted to normal band ordering. Finally, we present the first demonstrator of the QHE resistance standard using Nd permanent magnets providing magnetic field B = 0.82 T, placed in 4 He cryogenic system that can be very convenient for educational purposes as well as for some less demanding metrological applications.

We also discuss the necessary steps to reach with HgTe QWs the ultimate/high metrological precision keeping relaxed cryomagnetic conditions. Our work clearly shows that the peculiar band structure properties of HgTe QWs with massless Dirac fermions make them an ideal platform for developing metrological devices.

II. EXPERIMENTAL

Our experiments were carried out in HgTe QW devices with top gate electrodes. The investigated (HgTe/Hg0.38Cd0.62Te) structures were grown on GaAs (013) substrate by MBE and had QW thicknesses 7.5 nm (device D1) and 7.1 nm (device D2) [START_REF] Dvoretsky | Growth of HgTe quantum wells for IR to THz detectors[END_REF]. First, Hall bars with dimensions LW = 53 µm 2 (device D1) and LW = 4020 µm 2 (device D2) were patterned by electron beam lithography. Then, a Si(ON) dielectric layer of thickness 140 nm was deposited onto the sample surface. Finally, a metallic gate completed the structure. The processing protocol was previously described in detail [START_REF] Majewicz | Low temperature processing of nanostructures based on II-VI semiconductors quantum wells[END_REF].

Magnetotransport experiments have been performed in a custom-made 4 He cryogenic system in the temperature range of 1.8 -20 K and magnetic fields up to 5 T provided by a superconducting coil. We have used SR830 lock-in amplifiers with low modulation frequency (~13 Hz) to determine the longitudinal and the Hall components of the resistance tensor, Rxx and Rxy, respectively. The current flowing through the devices has been limited to the range 10 nA -1.0 𝜇A.

In the experiments with permanent magnets, the sample was mounted in a Teflon spacer of thickness 2.5 mm inserted between two pairs of permanent Nd magnets (two circular cylinders with diameter D ≈ 22 mm and height h ≈ 10 mm) that providing an axial magnetic field B = 0.82 T at their center. Using finite element modeling, we estimated the field homogeneity to be around 2% in a radius of 5 mm.

III. RESULTS

Four probe measurements of Rxx and Rxy have been carried out for various gate voltages Vg at B = 0.2 T to determine carrier densities and mobilities. The Hall concentration was experimentally determined as nH = 1/eRH, where RH is the Hall coefficient. As in the case of field-effect transistors, 2D carrier density is also given by n = (Vg -V0), where 𝑉 𝑜 is a gate offset, and  depends on both geometric and quantum capacitances per surface unit. As the valence band side maxima trap holes,  is higher on the electron side (𝛼 𝑛 ) than on the hole side (𝛼 𝑝 ). For device D1, αn ≈ 1.4 × 10 11 cm -2 V -1 (and αp ≈ 1.2 × 10 10 cm -2 V -1 ), whereas for device D2 αn (αp) is 1.67 × 10 11 cm -2 V -1 (5.7 × 10 9 cm -2 V -1 ).

For both devices at T = 1.8 K, the hole mobility exceeds the electron mobility (see Fig. 1 in the Supplemental Material).

Figure 2(a) shows the Hall conductivity yx(Vg) for Device 1 in various magnetic fields ranging from 1.5 -4.0 T. According to the QHE theory yx = e 2 /h, where the filling factor ν determines the number of the filled LLs. Thus, transitions between quantum plateaus correspond to crossings of the Fermi level with the LL centers. We present experimental studies of magnetoresistances in such HgTe quantum wells versus temperature and magnetic field, which allow determining the critical magnetic field Bc above which the topological gap (and parasitic edge conduction channels) disappears. We show that for QWs for which Bc becomes very low or even vanishes, one observes remarkably wide QHE plateaus at filling factor  = -1 (holes) using only commercial Neodymium permanent magnets providing B = 0.82 T, and liquid 4 He system operating at a few Kelvins. 2(c)]. This means that the quantum spin Hall effect is eliminated for magnetic fields higher than Bc ≃ 1.2 T and replaced by the ordinary integer QHE [START_REF] Konig | Quantum Spin Hall Insulator State in HgTe Quantum Wells[END_REF][START_REF] Yahniuk | Magneto-transport in inverted HgTe quantum wells[END_REF].

For ordinary semiconductor systems, like GaAs/AlGaAs heterostructures, either the fractional plateaus or vanishing Hall conductivity (σxy = 0), are expected in high fields when the filling factor diminishes well below 1. For HgTe QWs, this QHE regime is observed in magnetic fields exceeding a critical value B > Bc, and for gate voltages in the vicinity of the charge neutrality point. At B = Bc, the QHE behavior is analogous to the case of massless Dirac fermions, with a direct transition between two states with yx = e 2 /h.

We present now the magnetotransport properties of device D2. In Figs. 3(a,b) we depict the Hall resistivity xy(B) as a function of electron and hole densities controlled by the gate voltage. As seen, the QHE plateaus ν = ±1 appear at a lower magnetic field for holes than for electrons. Overall, there are two prerequisites for finding a precise quantization: µB >> 1 and  > 100 kBT, i.e., large values of both carrier mobility µ and gap between neighboring Landau levels . Compared to other materials [START_REF] Rigosi | The quantum Hall effect in the era of the new SI[END_REF], HgTe/HgCdTe QWs exhibit sizable carrier mobility (~10 5 cm²/Vs) and large gap (Δ ≃ 10 meV). Accordingly, in device D2 quantized Hall conductance of holes appears already in B = 0.5 T at T = 2.0 K. Strikingly, in both devices, the quantized Hall plateaus are much wider for holes than for electrons. This may be explained by the existence of the valence band side maxima serving as charge reservoir specific to HgTe quantum wells.

The features of the QHE in HgTe QWs are strongly correlated to the behavior of the zero-mode LLs. Therefore, another objective of the present work is to investigate the properties of these zero-mode LLs versus temperature.

The color maps shown in Fig. 4 evidence the transition from the inverted band ordering to the normal band sequence with increasing temperature. This is confirmed by theoretical computations that show that with increasing temperature, the topological energy gap between the electron-like subband E1 and the heavy hole-like subband H1 decreases so that above critical temperature Tc a normal band ordering appears. Experimentally, according to results displayed in Fig. 4(b), Tc = 10 K for device D2. A further increase of temperature results in a gap opening, see Fig. 4(c). It is worth noting that analogous changes in band structure can be caused by hydrostatic pressure [START_REF] Krishtopenko | Pressure-and temperature-driven phase transitions in HgTe quantum wells[END_REF]. Finally, in agreement with Ref. 26, our results indicate that the carrier density does not change with temperature (see Fig. 2 in the Supplemental Material). As can be seen on Fig. 3b, the plateau v = -1 starts from fields about 0.5 T. Such fields are available using permanent Nd magnets. In order to construct a prototype device for resistance standard we have prepared a specific sample holder equipped with permanent magnets as shown in Figs. 5(a,b).

The results shown in Fig. 5(c) clearly demonstrate the appearance of QHE in device D2 in the magnetic field produced by permanent magnets. The Hall resistance as a function of gate voltage is measured in the temperature range 2.0 -5.0 K. Quantum Hall plateau  = -1 is observed up to 15 K, while the Fermi level remains pined in the valence band. Under our experimental conditions, we can only roughly estimate the accuracy of xy quantization as 0.01h/e 2 at T = 1.6 K and a minimum value of xx is below 0.001h/e 2 at T = 1.6 K and for I = 100 nA.

The robustness of the QHE states in the hole transport regime is presumably related to the Fermi level pinning by the side maxima in the valence band, playing the role of a charge reservoir [START_REF] Yahniuk | Magneto-transport in inverted HgTe quantum wells[END_REF][START_REF] Mantion | Quantum Hall states in inverted HgTe quantum wells probed by transconductance fluctuations[END_REF][START_REF] Saquib | Emergent quantum Hall effects below 50 mT in a two-dimensional topological insulator[END_REF].

IV. SUMMARY AND FUTURE DEVELOPEMENTS

To summarize, gated HgTe quantum wells of width around 7 nm exhibit graphene like band structure with low effective mass ("masless Fermions") and high hole mobilities. These properties grant well defined LLs with large energy gaps between them even at relatively low magnetic fields. These peculiar properties have allowed us to observe a wide  = -1 QHE plateaus using Nd permanent magnets. We also performed transport measurements of HgTe QWs at various temperatures, revealing a phase transition at which the twodimensional Dirac massless fermions emerge at 𝑇 = 10 K. Our results demonstrate that these HgTe quantum structures are potential candidates for quantum Hall resistance standards operating in relaxed cryomagnetic conditions There are a few challenges in further development of HgTe QW resistance standards. One of them is the reduction of the contact resistance, which induces local heating and precludes the use of high currents needed to reach a better metrological precision [START_REF] Weis | Metrology and microscopic picture of the integer quantum Hall effect[END_REF][START_REF] Haug | Edge-state transport and its experimental consequences in high magnetic fields[END_REF].

We suggest increasing carrier density under the contacts by using an additional gate in the contact proximity.

Also, one can consider a conductive substrate serving as a HgTe QWs bottom gate [START_REF] Baenninger | Fabrication of samples for scanning probe experiments on quantum spin Hall effect in HgTe quantum wells[END_REF]. Alternatively, one could also consider double HgTe QWs, where one of the QWs plays the role of the gate controlling the carrier concentration. Moreover, metals like Sn, In, or Au/Mo are required to form low-resistive ohmic contacts [START_REF] Leech | The specific contact resistance of Ohmic contacts to HgTe/Hg1-xCdxTe heterostructures[END_REF][START_REF] Liu | Ohmic contact of Au/Mo on Hg1-xCdxTe[END_REF], precluding Schottky barrier formation.

The band structure properties of HgTe QWs can be modified to obtain metrological QHRSs. It has been demonstrated theoretically that the positions of extrema in the valence band dispersion and the magnitude of LL gap can be controlled by axial strain [START_REF] Yahniuk | Magneto-transport in inverted HgTe quantum wells[END_REF][START_REF] Parijat | Design principles for HgTe based topological insulator devices[END_REF][START_REF] Leubner | Strain Engineering of the Band Gap of HgTe Quantum Wells Using Superlattice Virtual Substrates[END_REF]. Additionally, the band structure properties can be manipulated by adding Mn and Cd into the QWs barriers [START_REF] Saquib | Emergent quantum Hall effects below 50 mT in a two-dimensional topological insulator[END_REF][START_REF] Parijat | Design principles for HgTe based topological insulator devices[END_REF]. These additional degrees of freedom may increase the width and the robustness of the QHE plateaus, allowing further improvement of HgTe QWs based metrological devices operating with relaxed cryomagnetic conditions.

Fig. 1 .

 1 Fig. 1. Calculated band structure and Landau levels for rectangular HgTe quantum wells of (a,c) d = 7.0 nm and (b,d) dc = 6.3 nm thickness, respectively. The red curve (H1) corresponds to a heavy hole-like subband, and the blue curve (E1) corresponds to an electron-like subband. Symbol Bc is a critical magnetic field in which zero-mode LLs cross. The magnitude of Δ determines the energy gap between the two uppermost LLs originating from the valence band.

Figure 2 (

 2 b) presents the yx mapping as a function of the gate voltage Vg and the magnetic field B at T = 1.8 K. The QHE plateaus at  = 1 and  = -1 are marked by yellow and red areas and correspond to electron and hole transport, respectively. One can see that the QHE plateaus for holes ( = -1) are much wider than those for electrons ( = 1).

Fig. 2 .

 2 Fig. 2. (a) Device D1. Dependence of the Hall conductivity on the gate voltage in various magnetic fields at T = 1.8 K. (b) Map of the Hall conductivity (with contours and color scale) as a function of the magnetic field and gate voltage. The dashed lines are regularly spaced lines of the Hall conductivity. QHE plateaus at 𝜈 = ±1 correspond to yellow and red areas for electrons and holes, respectively. (c) Experimentally obtained Landau levels as a function of the gate voltage Vg. Color map shows the absolute values of the

Fig. 3 .

 3 Fig. 3. (a) Device D2. Hall resistivity for electrons and holes at various gate voltages Vg -V0. (b) Map of the Hall resistivity as a function of the magnetic field for different gate voltages with respect to the charge neutrality point. QHE plateaus at 𝜈 = ±1 correspond to blue and red areas for electrons and holes, respectively. Vertical dashed-dot line corresponds B =0.82 T.

Fig. 4 .

 4 Fig. 4. Device D2. Experimentally obtained Landau level fan charts as a function of the gate voltage Vg at various temperatures: (a) T = 1.8 K; (b) T = 10 K and (c) T = 20 K. Color maps depict the absolute values of the derivative dσxy/dVg. Symbols correspond to the onset of the regions with σyx = e 2 /h. The dashed-dot lines show the zero-mode LLs.

Fig. 5 .

 5 Fig. 5. (a) Schematic illustration and (b) photography of the sample holder with four permanent Nd magnets. (c) Hall resistance as a function of the gate voltage in the magnetic field of 0.82 T generated by the permanent magnets for different temperatures: 2.0 K (navy blue), 2.5 K (blue), 3.0 K (green), 3.5 K (yellow), 4.0 K (orange) 4.5 K (red) and 5.0 K (brown).

Fig. 1 .

 1 Fig. 1. The longitudinal resistance in device D1 (a) and device D2 (b) as a function of the gate voltage at T=1.8 K. (c) Hall carrier concentration and (d) Hall mobility in D1 and D2 for electrons and holes versus gate voltage at T=1.8 K.

Fig. 2 (

 2 Fig. 2 (a) Hall carrier concentration and (b) Hall mobility for electrons and holes as a function of gate voltage at temperature range from 2 K to 20 K. (c) Dependence of the Hall conductivity on gate voltage presented for different magnetic fields: (dark cyan) 1.5 T; (cyan) 2.0 T;(dark yellow) 2.5 T; (red) 3:0 T ; (blue) 3.5 T and (dark gray) 4.0 T. (d) Derivative of the transverse conductivity for (dark goldenrod curve) 2.5 T and (gray curve) 4.0 T from the gate voltage.

ACKNOWLEDGEMENTS

Research in CENTERA Laboratories and in the International Research Centre MagTop was supported in the framework of the International Research Agendas Program was supported of the Foundation for Polish Sciences cofinanced by the European Union under the European Regional Development Fund (MAB/2018/9 and MAB/2017/1), by the TEAM project POIR.04.04.00-00-3D76/16 (TEAM/2016-3/25), National Science Centre Poland (grant No. UMO-2017/25/N/ST3/00408) and by CNRS through IRP "TeraMIR".

Supplemental Materials

HgTe quantum wells for QHE metrology with relaxed cryomagnetic conditions: permanent magnets and liquid 4 He temperatures


S Mishra

D Gorbunov

D J Campbell

David Leboeuf

J Hornung

J Klotz

S Zherlitsyn

H Harima

J Wosnitza

D Aoki

A Mccollam

I Sheikin

Origin of the 30 T transition in CeRhIn 5 in tilted magnetic fields

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Unusual magnetic-field-induced transitions have recently become a matter of considerable interest in strongly correlated electron systems. In some heavyfermion compounds, intermetallic materials based on rare earths or actinides, such transitions are rather exotic in nature. Among them are field-induced quantum critical points [START_REF] Gegenwart | Magnetic-Field Induced Quantum Critical Point in YbRh2Si2[END_REF][START_REF] Harrison | Fermi Surface of CeIn3 above the Néel Critical Field[END_REF], field-induced or reinforced unconventional superconductivity [START_REF] Lévy | Magnetic Field-Induced Superconductivity in the Ferromagnet URhGe[END_REF][START_REF] Aoki | Extremely large and anisotropic upper critical field and the ferromagnetic instability in ucoge[END_REF][START_REF] Knebel | Field-Reentrant Superconductivity Close to a Metamagnetic Transition in the Heavy-Fermion Superconductor UTe2[END_REF], and Lifshitz transitions [START_REF] Daou | Continuous Evolution of the Fermi Surface of CeRu2Si2 across the Metamagnetic Transition[END_REF][START_REF] Rourke | Magnetic-Field Dependence of the YbRh2Si2 Fermi Surface[END_REF][START_REF] Aoki | Field-Induced Lifshitz Transition without Metamagnetism in CeIrIn5[END_REF]. Understanding the mechanisms and the origins of such transitions is of primary importance in modern condensed matter.

CeRhIn 5 , although discovered barely two decades ago [START_REF] Hegger | Pressure-Induced Superconductivity in Quasi-2D CeRhIn5[END_REF], is one of the best studied heavy-fermion materials. This compound crystallizes into a tetragonal HoCoGa 5 type crystal structure (space group 123, P 4/mmm) shown in Fig. 1(a). CeRhIn 5 undergoes an antiferromagnetic (AFM) transition at T N = 3.8 K. The zero-field magnetic structure, AFM1, shown in Fig. 1(b), is incommensurate (IC) with propagation vector Q IC = (1/2, 1/2, 0.297) [START_REF] Bao | Incommensurate magnetic structure of CeRhIn5[END_REF][START_REF] Fobes | Low temperature magnetic structure of CeRhIn5 by neutron diffraction on absorption-optimized samples[END_REF]. The Ce magnetic moments are antiferromagnetically aligned within the basal plane, and spiral transversally along the c axis.

At low temperatures, when a magnetic field B is applied in the basal plane, a first-order transition occurs at B m 2 T. The transition corresponds to a change of magnetic structure from IC AFM1 to commensurate (C) AFM3 with the propagation vector Q C = * ilya.sheikin@lncmi.cnrs.fr (1/2, 1/2, 1/4) [START_REF] Raymond | Magnetic structure of CeRhIn5 under magnetic field[END_REF][START_REF] Fobes | Tunable emergent heterostructures in a prototypical correlated metal[END_REF]. It is a collinear square configuration with "up-up-down-down" alignment. The magnetic moments are antiferromagnetically aligned in the basal plane along the direction perpendicular to the magnetic field, as shown in Fig. 1(c). When the magnetic field is tilted away from the basal plane, B m shifts to higher fields and follows the 1/ cos α dependence (α is the field angle from the basal plane) up to about 75 • -80 • . At higher angles, however, B m deviates from a 1/ cos α dependence towards lower fields. The transition was traced up to θ = 2 • from the c axis (α = 88 • ), where it occurs at B m ≈ 20 T [START_REF] Mishra | Fermi-Surface Morphology of CeRhIn5 across the Putative Field-Induced Quantum Critical Point[END_REF].

Recent results obtained in high magnetic fields applied along or close to the c axis suggest a remarkably novel behavior in CeRhIn 5 . The AFM order is suppressed at B c 50 T giving rise to a field-induced quantum critical point [START_REF] Jiao | Fermi surface reconstruction and multiple quantum phase transitions in the antiferromagnet CeRhIn5[END_REF][START_REF] Jiao | Enhancement of the effective mass at high magnetic fields in CeRhIn5[END_REF]. Furthermore, Moll et al. [START_REF] Moll | Field-induced density wave in the heavy-fermion compound CeRhIn5[END_REF] reported an observation of a hysteretic jump in the in-plane resistivity of CeRhIn 5 microstructures fabricated by focused ion beam (FIB) at B * ≈ 30 T applied along or close to the c axis. Later on, these experiments were advanced, allowing for simultaneous measurements of the in-plane resistivity along the [100] and [010], as well as along the [110] and [110] directions [START_REF] Ronning | Electronic in-plane symmetry breaking at field-tuned quantum criticality in CeRhIn5[END_REF]. These measurements revealed a strong in-plane resistivity anisotropy within each of the two inequivalent symmetry channels B 1g ([100],

[010]) and B 2g ([110], [110]), which emerges above B * ≈ 30 T tilted from the c axis towards the [100] and [110] directions, respectively. This electronic anisotropy was interpreted in terms of an electronic-nematic transition. To the best of our knowledge, these results, however, have not been reproduced on bulk samples so far.

For a long time, all the attempts to detect an anomaly at B * in various measurements on bulk single crystals remained ineffective. In particular, no noticeable anomaly was observed at B * in longitudinal magnetization [START_REF] Takeuchi | Magnetic and Thermal Properties of CeIrIn5 and CeRhIn5[END_REF] or reported in magnetic-torque data [START_REF] Moll | Field-induced density wave in the heavy-fermion compound CeRhIn5[END_REF][START_REF] Ronning | Electronic in-plane symmetry breaking at field-tuned quantum criticality in CeRhIn5[END_REF]. It is only very recently that a small magnetostriction anomaly was observed at B * in a magnetic field applied at ∼20 • from the c axis [START_REF] Rosa | Enhanced Hybridization Sets the Stage for Electronic Nematicity in CeRhIn5[END_REF]. The anomaly is similar in shape and size to the one at B m , which occurs at about 7.5 T for this field orientation. Furthermore, recent high-field nuclear magnetic resonance (NMR) measurements revealed a pronounced decrease in the 115 In Knight shift at B * for B c [START_REF] Lesseux | Orbitally defined fieldinduced electronic state in a Kondo lattice[END_REF]. These NMR measurements also suggest an IC magnetic structure of CeRhIn 5 above B * . Both the magnetostriction anomaly and the change of the NMR Knight shift at B * were interpreted as a signature of enhanced hybridization between the 4f and conduction electrons. This interpretation is in line with the conclusion of an earlier work, which suggests that the Ce f electrons, localized at low fields, become itinerant above B * [START_REF] Jiao | Fermi surface reconstruction and multiple quantum phase transitions in the antiferromagnet CeRhIn5[END_REF][START_REF] Jiao | Magnetic field-induced Fermi surface reconstruction and quantum criticality in CeRhIn5[END_REF]. This scenario, however, was later ruled out by high-field angular-dependent de Haas-van Alphen (dHvA) effect measurements, which demonstrated that the f electrons remained localized not only above B * but also above B c [START_REF] Mishra | Fermi-Surface Morphology of CeRhIn5 across the Putative Field-Induced Quantum Critical Point[END_REF]. Finally, a specific-heat anomaly has recently been observed at B * for B c, suggesting that a true thermodynamic phase transition, probably weakly first order, takes place at B * [START_REF] Mishra | Specific heat of CeRhIn5 in high magnetic fields: Magnetic phase diagram revisited[END_REF].

The exact origin of the transition at B * , especially in bulk crystals, remains, therefore, an important open question. One way to address this issue is via ultrasoundvelocity measurements, which are a powerful tool to detect phase transitions since ultrasound waves readily couple with electronic, magnetic, and structural degrees of freedom. Moreover, different ultrasound modes, characterized by different propagation directions and polarization, induce different symmetry-breaking strains in the crystal, as schematically shown in Fig. 1(d) for a tetragonal crystal structure. This, in turn, allows for the determination of the order-parameter symmetry in a phase as well as identification of broken symmetries at a phase transition.

Previous ultrasound velocity measurements performed in pulsed magnetic fields slightly tilted from the c axis revealed clear anomalies at both B m ≈ 20 T and B * ≈ 30 T [START_REF] Kurihara | High-field ultrasonic study of quadrupole ordering and crystal symmetry breaking in CeRhIn5[END_REF]. The latter was observed in the C T = (C 11 -C 12 )/2 mode only, which induces the symmetry strain ε xx -ε yy associated with the irreducible representation B 1g . This strain breaks the C 4 symmetry [Fig. 1(d)]. Another strain, which breaks the C 4 symmetry, is ε xy . This strain, induced in the C 66 mode, is associated with the B 2g symmetry channel [Fig. 1(d)]. If the anomaly at B * is attributed to the previously suggested electronic-nematic transition [START_REF] Ronning | Electronic in-plane symmetry breaking at field-tuned quantum criticality in CeRhIn5[END_REF], it is surprising that no anomaly was observed in the C 66 and C 11 modes. These inconsistencies motivated us to re-examine the elastic response of CeRhIn 5 in high magnetic fields.

In this paper, we report high-resolution ultrasoundvelocity measurements in CeRhIn 5 in high pulsed and static magnetic fields tilted away from the c axis. We observed clear field-induced anomalies at both B m and B * in all symmetry-breaking modes. This suggests that the transition at B * is bulk in origin contrary to what was previously suggested. Both anomalies are of similar shape, but of the opposite sign. Both of them exist within the AFM state only. Furthermore, when the magnetic field is tilted further away from the c axis, the anomaly at B * becomes broader and smaller in magnitude. It was traced up to an angle tilted by 30 • away from the c axis.

II. EXPERIMENTAL DETAILS

High-quality single crystals of CeRhIn 5 used in this study were grown by the In-self-flux technique, details of which can be found elsewhere [START_REF] Shishido | Fermi Surface, Magnetic and Superconducting Properties of LaRhIn5 and CeTIn5 (T: Co, Rh and Ir)[END_REF]. Ultrasound velocity measurements were performed in both pulsed (up to 60 T) and static (up to 36 T) magnetic fields. In pulsed fields, we used a 4 He flow cryostat, while static-field measurements were performed in a 3 He cryostat equipped with an insitu rotator. In both cases, the measurements were performed using a pulse-echo technique in transmis-sion mode. A pair of piezoelectric transducers was glued to opposite well-polished surfaces of the sample in order to generate and detect acoustic waves. The ultrasound frequencies and the absolute values of sound velocities at 4.2 K for different acoustic modes are given in Table I. For all modes except C 33 , the orientation of the magnetic field with respect to the c axis was determined from the position of the anomaly at B m , whose angular dependence was established in the previous torque measurements [START_REF] Mishra | Fermi-Surface Morphology of CeRhIn5 across the Putative Field-Induced Quantum Critical Point[END_REF]. For the C 33 mode, the orientation was determined from the frequencies of the magnetoacoustic quantum oscillations, which were compared to the previously established angular dependence of the de Haas-van Alphen (dHvA) frequencies [START_REF] Mishra | Fermi-Surface Morphology of CeRhIn5 across the Putative Field-Induced Quantum Critical Point[END_REF]. The oscillatory frequencies also allowed us to double check the field angle in other modes, in which the magnetoacoustic oscillations were strong enough. The non-symmetry-breaking (C 11 + C 12 )/2 mode related to the ε xx + ε yy strain belonging to the A 1g representation can not be measured directly using the pulseecho technique. The relative ultrasound velocity variation for this mode was, therefore, obtained from the C 11 and C T data. First, the absolute values of the zerofield elastic constants were calculated as C 0 = ρv 2 using the ultrasonic velocities v given in Table I and the density ρ = 8.316 g/cm 3 . Then, the field-dependent elastic constant variations were calculated as ∆C(B) = 2 ∆v v (B)C 0 using the measured field dependence of ∆v/v for both C 11 and C T modes, shown in Figs. 2(a determination of the ultrasound velocities for the C 11 and C T modes. One can see that, similar to the C 33 mode, the (C 11 + C 12 )/2 mode does not show any apparent anomalies within the experimental error.

III. RESULTS AND DISCUSSION

As mentioned above, both anomalies in the symmetrybreaking modes have a similar shape, but the corresponding changes of ∆v/v are of opposite sign. In the C 11 and C T modes, the anomalies at B m and B * appear as a sharp decrease and increase of the ultrasound velocity, respectively. These anomalies are remarkably similar to those previously observed in other compounds, in which field-induced spin-reorientation transitions occur [START_REF] Lüthi | Physical Acoustics in the Solid State[END_REF][START_REF] Dan'shin | Dynamic properties of YbFeO3 in the vicinity of an orientational phase transition[END_REF][START_REF] Fil | Acoustiv properties of Nd2CuO4 at low temperatures[END_REF][START_REF] Zherlitsyn | Magnetoelastic effects in Nd2-xCexCuO4 single crystals at low temperatures[END_REF][START_REF] Zherlitsyn | High field phase diagram of Nd2-xCexCuO4 single crystals[END_REF]. In the C 44 and C 66 modes, both characteristic fields result in similar, but somewhat reduced features. Another anomaly, a minimum in ∆v/v versus B, is clearly visible at B p ≈ 44 T in the C 11 mode. This anomaly was also observed in the previous ultrasound study, where it was interpreted as a transition (or crossover) into the polarized paramagnetic state [START_REF] Kurihara | High-field ultrasonic study of quadrupole ordering and crystal symmetry breaking in CeRhIn5[END_REF].

The anomaly at B m corresponds to the well-known first-order metamagnetic transition, where the magnetic structure changes from the IC AFM1 to the C AFM3 phase, as schematically shown in Fig. 1. Since the transition is of the first order, it is not surprising that it manifests itself in most of the ultrasound modes.

Regarding the transition at B * , its observation in all symmetry-breaking modes suggests that the transition is probably also of the first order, in agreement with recent specific heat measurements [START_REF] Mishra | Specific heat of CeRhIn5 in high magnetic fields: Magnetic phase diagram revisited[END_REF]. As mentioned, the anomalies at B m and B * have similar shapes but opposite signs. It is, therefore, natural to conclude that both transitions are of the same origin, i.e., that the transition at B * corresponds to another field-induced change of the magnetic structure, this time from the C AFM3 to an IC AFM4 phase. This conclusion is in line with previous NMR results, which unambiguously suggest an IC phase above B * [START_REF] Lesseux | Orbitally defined fieldinduced electronic state in a Kondo lattice[END_REF].

The above hypothesis regarding the origin of the transition at B * is further supported by the measurements performed at different temperatures, as shown in Fig. 4 for the C 11 (a), C T = (C 11 -C 12 )/2 (b), C 44 (c), and C 66 (d) acoustic modes. In all cases, the magnetic field was applied at θ 2 • . The data for the C T mode [Fig. 4(b)] were obtained in static fields. All the other curves were measured in pulsed magnetic fields.

The anomalies at both B m and B * remain sharp up to the highest temperature, at which they were observed. The anomaly at B m is observed up to higher temperatures than its counterpart at B * . Both anomalies are almost temperature independent at low temperatures. At temperatures approaching the antiferromagnetic phase boundary (∼2.5 K for B m and ∼2.4 K for B * ), B m and B * shift to slightly higher and lower field, respectively.

Both anomalies from all the measurements are traced in the resulting phase diagram together with the AFM boundary from the previous specific-heat results [START_REF] Mishra | Specific heat of CeRhIn5 in high magnetic fields: Magnetic phase diagram revisited[END_REF] (see Fig. 5). In agreement with previous reports [START_REF] Ronning | Electronic in-plane symmetry breaking at field-tuned quantum criticality in CeRhIn5[END_REF][START_REF] Rosa | Enhanced Hybridization Sets the Stage for Electronic Nematicity in CeRhIn5[END_REF][START_REF] Kurihara | High-field ultrasonic study of quadrupole ordering and crystal symmetry breaking in CeRhIn5[END_REF], is shown in Fig. 7(a). At 2 • , both anomalies manifest themselves as sharp steps [Fig. 7(b)]. With increasing angle, both features progressively become smaller. While the anomaly at B m remains sharp, the one at B * broadens with increasing angle, as evidenced by a progressive small increase of the FWHM of the first derivative shown in the inset of Fig. 7(d). This probably implies that the transition at B * at small angles changes to a crossover at larger angles. This assumption is supported by recent specific-heat [START_REF] Mishra | Specific heat of CeRhIn5 in high magnetic fields: Magnetic phase diagram revisited[END_REF] and magnetostriction [START_REF] Rosa | Enhanced Hybridization Sets the Stage for Electronic Nematicity in CeRhIn5[END_REF] measurements. While the specific heat, performed in magnetic fields applied parallel or very close to the c axis, suggests a phase transition, the magnetostriction, performed in fields tilted by ∼20 • , suggests a crossover at B * . The anomaly at B m is easily traceable all the way up to 40 • , while its counterpart at B * is still present at 30 • , but is no longer visible at 40 • , where the curve is completely dominated by strong low-frequency quantum oscillations [Fig. 7(c)]. The angular dependence of the anomaly at B * is strikingly different from that observed in resistivity measurements on FIB-fabricated microdevices [START_REF] Moll | Field-induced density wave in the heavy-fermion compound CeRhIn5[END_REF][START_REF] Ronning | Electronic in-plane symmetry breaking at field-tuned quantum criticality in CeRhIn5[END_REF], where the resistivity jump at B * remained sharp all the way up to 60 • , the highest angle to which it was traced. The size of the jump showed a nonmonotonic behavior with a maximum at about 20 • from the c axis. This difference between bulk samples and microfabricated devices is probably due to uniaxial stresses or strains inevitably present in FIB-fabricated devices [START_REF] Bachmann | Spatial control of heavyfermion superconductivity in CeIrIn5[END_REF]. CeRhIn 5 seems to be very sensitive to uniaxial strains, especially to orthorhombic strains, as evidenced by a particularly large size of the anomalies in the C T mode. Furthermore, previously reported NQR measurements on bulk and powder samples of CeRhIn 5 suggest that even small strains (or stresses) change the zero-field magnetic structure from IC to C [START_REF] Yashima | Incommensurate Antiferromag-netic Order under Pressure in CeRhIn5 Studied by 115 In-NQR[END_REF]. Additional measurements under uniaxial stress (or strain) at high magnetic fields on bulk samples are required to elucidate the role of the uniaxial stress in previous transport measurements on CeRhIn 5 microstructures [START_REF] Moll | Field-induced density wave in the heavy-fermion compound CeRhIn5[END_REF][START_REF] Ronning | Electronic in-plane symmetry breaking at field-tuned quantum criticality in CeRhIn5[END_REF].

The resulting angular dependence of both B m and B * is shown in Fig. 7(d). In agreement with previous results [START_REF] Mishra | Fermi-Surface Morphology of CeRhIn5 across the Putative Field-Induced Quantum Critical Point[END_REF], B m is strongly angle dependent. It is well fit by 1/ cos(90 • -θ) down to about 10 • from the c axis but deviates at lower angles. B * , on the other hand, is only weakly angle dependent. It increases approximately linearly all the way from 2 • to about 30 • , above which it either disappears or shifts to beyond 36 T. The remarkable difference between the two angular dependencies is probably due to different magnetic moment arrangements in the vicinity of the two transitions. While the moments are aligned in the basal plane in the vicinity of B m , they are probably tilted towards the c axis close to B * .

Finally, we comment on the magnetoacoustic quantum oscillations. In agreement with the results of the previous high-field ultrasound study of CeRhIn 5 [START_REF] Kurihara | High-field ultrasonic study of quadrupole ordering and crystal symmetry breaking in CeRhIn5[END_REF], we observed strong low-frequency magnetoacoustic quantum oscillations in the C T mode between B m and B * , as shown in the inset of Fig. 8. In this mode, these oscillations with a frequency denoted A (F A 0.6 kT) disappear above B * , as shown in Fig. 8, also in agreement with the previous study. This experimental result was interpreted as evidence of a possible Fermi-surface reconstruction at B * in the previous work [START_REF] Kurihara | High-field ultrasonic study of quadrupole ordering and crystal symmetry breaking in CeRhIn5[END_REF]. Similarly, additional oscillatory frequencies emerging above B * were observed in the dHvA effect [START_REF] Jiao | Fermi surface reconstruction and multiple quantum phase transitions in the antiferromagnet CeRhIn5[END_REF][START_REF] Jiao | Magnetic field-induced Fermi surface reconstruction and quantum criticality in CeRhIn5[END_REF] and magnetostriction [START_REF] Rosa | Enhanced Hybridization Sets the Stage for Electronic Nematicity in CeRhIn5[END_REF]. The appearance of these additional frequencies was discussed in terms of the Fermi-surface reconstruction at B * due to f -electrons delocalization [START_REF] Jiao | Fermi surface reconstruction and multiple quantum phase transitions in the antiferromagnet CeRhIn5[END_REF][START_REF] Jiao | Magnetic field-induced Fermi surface reconstruction and quantum criticality in CeRhIn5[END_REF].

In Ref. [START_REF] Kurihara | High-field ultrasonic study of quadrupole ordering and crystal symmetry breaking in CeRhIn5[END_REF], quantum oscillations were observed in the C T mode only, and only between B m and B * . In our experiments, on the contrary, we observed magnetoacoustic quantum oscillations in all modes both in static and pulsed-field measurements. Furthermore, the oscillations are the strongest in the C 33 mode.

Figure 9(a) shows the field dependence of the ultrasound-velocity variation in the C 33 mode measured in pulsed magnetic fields. Since the anomalies at B m and Our static-field data in C 33 mode obtained lower temperatures, an of which is shown in the inset of Fig. 10, reveal many more quantum-oscillation frequencies. Some of the frequencies are clearly visible at as low field as 5 T. After subtracting a nonoscillating background from the data shown in the inset of Fig. 10, we performed FFTs over small equidistant 1/B ranges to keep the same frequency resolution. The resulting field dependence of the FFT spectra is shown in Fig. 10, where the field corresponding to the transition at B * is highlighted. It is clear from this FFT analysis that no frequencies emerge or disappear above B * . This is in agrement with the results of our recent angle-dependent dHvA study [START_REF] Mishra | Fermi-Surface Morphology of CeRhIn5 across the Putative Field-Induced Quantum Critical Point[END_REF]. However, some of the dHvA frequencies observed at high fields in our previous study [START_REF] Mishra | Fermi-Surface Morphology of CeRhIn5 across the Putative Field-Induced Quantum Critical Point[END_REF] are not observed in our ultrasound-velocity data.

IV. CONCLUSIONS

In summary, we performed high-field ultrasoundvelocity measurements on bulk single crystals of CeRhIn 5 . For a magnetic field slightly tilted away from the c axis, we observed distinct ultrasound-velocity anomalies at both B m 20 T and B * 30 T at low temperatures in all symmetry-breaking modes, i.e., C 11 , C 44 , C 66 , and C T . In all these modes, the anomalies are of similar shape but of opposite sign at B m and B * . Both anomalies are absent in the symmetry preserving C 33 mode. Furthermore, our temperature-dependent measurements reveal that both anomalies exist within the AFM state only. Given that the transition at B m corresponds to a change of magnetic structure from IC below B m to C above B m , we argue that the transition at B * is of the same origin, i.e., from the C phase below B * to a new IC phase above it. This makes CeRhIn 5 one of the rare compounds, in which the application of a high magnetic field induces a C to IC transition. High-field neutron diffraction measurements would be of interest to definitely confirm our hypothesis.

FIG. 1 .

 1 FIG. 1. (a) Crystal structure of CeRhIn5. Magnetic structure of CeRhIn5 (b) in zero magnetic field (IC) and (c) in a magnetic field higher than 2 T applied along the [100] direction (C). Arrows indicate the orientation of the magnetic moments. Only Ce atoms are shown in (b) and (c) for clarity. (d) Schematic illustration of the symmetry strains, εij, induced by different ultrasound modes for a tetragonal crystal structure. For each mode, the propagation (k) and polarization (u) directions are shown by arrows. The associated irreducible representations are shown in brackets.

Figure 2

 2 Figure 2 shows low-temperature relative ultrasound velocity, ∆v/v, as a function of B applied close to the c axis for several ultrasound modes. Two distinct anomalies are observed at B m ≈ 20 T and B * ≈ 30 T in all symmetry-breaking modes, i.e., C 11 , C 44 , C 66 , and C T . Both anomalies are absent in the C 33 mode, which does not break any tetragonal symmetry.The non-symmetry-breaking (C 11 + C 12 )/2 mode related to the ε xx + ε yy strain belonging to the A 1g representation can not be measured directly using the pulseecho technique. The relative ultrasound velocity variation for this mode was, therefore, obtained from the C 11 and C T data. First, the absolute values of the zerofield elastic constants were calculated as C 0 = ρv 2 using the ultrasonic velocities v given in TableIand the density ρ = 8.316 g/cm3 . Then, the field-dependent elastic constant variations were calculated as ∆C(B) = 2 ∆v v (B)C 0 using the measured field dependence of ∆v/v for both C 11 and C T modes, shown in Figs.2(a) and 2(b), respectively. Since (C 11 + C 12 )/2 = C 11 -C T and ∆(C 11 + C 12 )/2 = ∆C 11 -∆C T , the field dependence of ∆C/C for the (C 11 +C 12 )/2 mode was obtained. Finally, the relative ultrasound velocity variation was calculated as ∆v/v = (∆C/C)/2.The resulting velocity variation for the (C 11 + C 12 )/2 mode is shown in Fig.3. Here, the line thickness represents the resulting error due to the uncertainties in the

FIG. 2 .

 2 Figure 2 shows low-temperature relative ultrasound velocity, ∆v/v, as a function of B applied close to the c axis for several ultrasound modes. Two distinct anomalies are observed at B m ≈ 20 T and B * ≈ 30 T in all symmetry-breaking modes, i.e., C 11 , C 44 , C 66 , and C T . Both anomalies are absent in the C 33 mode, which does not break any tetragonal symmetry.The non-symmetry-breaking (C 11 + C 12 )/2 mode related to the ε xx + ε yy strain belonging to the A 1g representation can not be measured directly using the pulseecho technique. The relative ultrasound velocity variation for this mode was, therefore, obtained from the C 11 and C T data. First, the absolute values of the zerofield elastic constants were calculated as C 0 = ρv 2 using the ultrasonic velocities v given in TableIand the density ρ = 8.316 g/cm3 . Then, the field-dependent elastic constant variations were calculated as ∆C(B) = 2 ∆v v (B)C 0 using the measured field dependence of ∆v/v for both C 11 and C T modes, shown in Figs.2(a) and 2(b), respectively. Since (C 11 + C 12 )/2 = C 11 -C T and ∆(C 11 + C 12 )/2 = ∆C 11 -∆C T , the field dependence of ∆C/C for the (C 11 +C 12 )/2 mode was obtained. Finally, the relative ultrasound velocity variation was calculated as ∆v/v = (∆C/C)/2.The resulting velocity variation for the (C 11 + C 12 )/2 mode is shown in Fig.3. Here, the line thickness represents the resulting error due to the uncertainties in the

FIG. 3 .

 3 FIG. 3. Field dependence of the relative ultrasound-velocity variation, ∆v/v, for the (C11 + C12)/2 mode at T 1.4 K, calculated from the C11 and CT data shown in Figs. 2(a) and 2(b), respectively, as explained in the text. The field was applied at 2 • from the c axis. Dashed lines indicate Bm and B * observed in the symmetry-breaking modes.

FIG. 4 .FIG. 5 .

 45 FIG. 4. Relative variation of the ultrasound velocity as a function of magnetic field at different temperatures for the (a) C11, (b) CT = (C11 -C12)/2, (c) C44, and (c) C66 acoustic modes. The magnetic field was applied at θ 2 • . Curves are vertically shifted in (a), (c), and (d) for clarity.

FIG. 6 .FIG. 7 .

 67 FIG. 6. Magnetic torque divided by field, τ /B, vs B applied at θ = 2 • from [001] towards the [110] direction at T = 30 mK.

FIG. 8 .

 8 FIG.8. Fast Fourier transform (FFT) spectra of the magnetoacoustic quantum oscillations in the CT mode (shown in the inset) both below and above B * . A nonoscillating background was subtracted prior to performing the FFTs. The FFTs were performed over equidistant 1/B ranges indicated by rectangles in the inset.

FIG. 9 .

 9 FIG. 9. (a) Field dependence of the relative ultrasoundvelocity variation for the modes C33 and CT with the field applied at θ = 4 • and 2 • , respectively. (b) FFT spectra of the magnetoacoustic quantum oscillations in the C33 mode from (a), both below and above B * . A nonoscillating background was subtracted prior to performing the FFTs. The FFTs were performed over equidistant 1/B ranges indicated by rectangles in (a).

FIG. 10 .

 10 FIG.10. FFT spectra of the static-field magnetoacoustic quantum oscillations, obtained by subtracting a nonoscillating background from the ∆v/v vs B curve in the C33 mode shown in the inset, with B tilted by 4.5 • from the c axis. The FFT spectra are obtained over the same 1/B range. For the bottom curve, the range is from Bmin = 10 T to Bmax = 10.71 T (Bavg = 10.34 T). For each successive curve, Bmin is increased by 1 T up to 20 T, and by 0.5 T from there on. Curves are vertically shifted for clarity.

TABLE I .

 I Ultrasound frequencies and velocities at T = 4.2 K for all measured acoustic modes. The error in ultrasound velocities is estimated to be 80 m/s.

		C11	CT	C33	C44	C66
	f (MHz)	115.6	134	104	173	105
	v (m/s)	3514	2127	4484	2083	2016

ACKNOWLEDGMENTS

We thank M.-B. Lepetit, C. Simon, R. Settai, and Y. Nemoto for fruitful discussions. We acknowledge the support of the LNCMI-CNRS, the HLD-HZDR, and the HFML-RU, members of the European Magnetic Field Laboratory (EMFL), the ANR-DFG grant "Fermi-NESt", the DFG through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter -ct.qmat (EXC 2147, Project ID 39085490), and JSPS KAKENHI Grant Nos. JP15H05882, JP15H05884, JP15H05886, JP15K21732 (J-Physics).