Reentrant Phase Diagram of Y b 2 T i 2 O 7 in ⟨111⟩ Magnetic Field
A. Scheie, J. Kindervater, S. Säubert, C. Duvinage, C. Pfleiderer, H. J. Changlani, S. Zhang, L. Harriger, K. Arpino, S.M. Koohpayeh, O. Tchernyshyov, C. Broholm
RReentrant Phase Diagram of Yb Ti O in (cid:104) (cid:105) Magnetic Field
A. Scheie,
1, 2
J. Kindervater,
1, 2
S. Säubert,
3, 4
C. Duvinage, C. Pfleiderer, H. J. Changlani,
1, 2
S.Zhang,
1, 2
L. Harriger, K. Arpino,
6, 2
S.M. Koohpayeh,
1, 2
O. Tchernyshyov,
1, 2 and C. Broholm
1, 2, 5, 7 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218 Institute for Quantum Matter, Johns Hopkins University, Baltimore, MD 21218 Physik-Department, Technische Universität München, D-85748 Garching, Germany Heinz Maier-Leibnitz Zentrum, Technische Universität München, D-85748 Garching, Germany NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899 Department of Chemistry, Johns Hopkins University, Baltimore, MD 21218 Department of Materials Science and Engineering,Johns Hopkins University, Baltimore, MD 21218 (Dated: September 24, 2018)We present a magnetic phase diagram of rare-earth pyrochlore Yb Ti O in a (cid:104) (cid:105) magneticfield. Using heat capacity, magnetization, and neutron scattering data, we show an unusual field-dependence of a first-order phase boundary, wherein a small applied field increases the orderingtemperature. The zero-field ground state has ferromagnetic domains, while the spins polarize along (cid:104) (cid:105) above 0.65T. A classical Monte Carlo analysis of published Hamiltonians does account forthe critical field in the low T limit. However, this analysis fails to account for the large bulge inthe reentrant phase diagram, suggesting that either long-range interactions or quantum fluctuationsgovern low field properties. Yb Ti O may be one of the most famous materi-als in frustrated magnetism, and yet its ground statehas not been fully established. Yb ions, each form-ing a Kramers doublet, occupy the vertices of a (py-rochlore) lattice of corner-sharing tetrahedra which frus-trates the development of conventional long range order[1–3]. Much of the recent attention to Yb Ti O has beendriven by the suggestion that this material forms a quan-tum spin-ice at low temperatures [4–8], wherein the spinsare constrained to point into or out of tetrahedra with atwo-in-two-out "ice rule". This exotic state of matter ispredicted to have a spin-liquid ground state with its owneffective field theory [9, 10]. The quantum spin ice (QSI)hypothesis is supported by evidence of monopoles in theparamagnetic phase [6–8], and diffuse zero-field inelasticneutron scattering [11, 12]. Challenging the QSI hypoth-esis, however, is evidence that stoichiometric Yb Ti O ferromagnetically orders around 270mK (though the spe-cific ordered structure is contested) [11, 13, 14] with mag-netic order enhanced under pressure [15]. It is unclearhow to reconcile the ground state order of Yb Ti O withits more unusual behavior, especially since the groundstate is not fully understood. What is more, there islimited experimental information about collective prop-erties of Yb Ti O due to the lack of stoichiometricallypure crystals.Here we report the phase diagram of stoichiometric Yb Ti O in a (cid:104) (cid:105) magnetic field. The (cid:104) (cid:105) field inpyrochlore compounds like Yb Ti O harbors the possi-bility of a quantum kagome ice phase [16]; but our datadoes not reveal such a phase in Yb Ti O . Instead, wefind a reentrant phase diagram where magnetic order isenhanced under small magnetic fields–a behavior that ex-tant models of Yb Ti O fail to explain when quantumfluctuations are neglected.An unfortunate obstacle to studying Yb Ti O is that most single crystals are plagued by site disordered "stuff-ing", which causes large variations in the critical temper-ature [17–20]. This extreme sensitivity to disorder makesit difficult to compare experimental results to each otheror to theory. Recently, however, high-quality stoichio-metric single crystals were successfully grown with thetraveling solvent floating zone method [21]. We reportthe first field-dependent measurements on stoichiometricsingle crystals of Yb Ti O , and we use them to build aphase diagram of Yb Ti O in a (cid:104) (cid:105) magnetic field. Inour analysis, we used three experimental methods: heatcapacity, magnetization, and neutron scattering.The heat capacity of Yb Ti O at various magneticfields is shown in Fig. 1. We collected heat capacity dataon a 1.04 mg sample of Yb Ti O in a (cid:104) (cid:105) orientedmagnetic field using a dilution unit insert of a QuantumDesign PPMS [22]. The heat capacity data were collectedmostly with a long-pulse method, in which we applied along heat pulse, tracked sample temperature as the sam-ple cooled, and computed heat capacity from the timederivative of sample temperature (see ref. [23] and sup-plemental materials for more details). The advantage ofthe long-pulse method is sensitivity to first order transi-tions, which Yb Ti O is reported to have [24–26]. Someadiabatic short-pulse data were taken as well, and Fig. 1shows the overall agreement between these two methods.The magnetic fields in Fig. 1(b) have been corrected forthe internal demagnetizing field. The demagnetizationcorrection (see inset) is H int = H ext − D M ( H int ) , where D is the demagnetization factor (determined by sam-ple geometry), and M ( H int ) is magnetization (measuredseparately–see below). This correction enables quanti-tative comparison between measurements on differentlyshaped samples.The magnetization of Yb Ti O (Fig. 2) was mea-sured by means of a bespoke vibrating coil magnetometer a r X i v : . [ c ond - m a t . s t r- e l ] J un () Short ( )Long ( ) (a) ( T ) (b) () (T) () (T) ( T ) Figure 1. Heat capacity data for Yb Ti O with magneticfield along (cid:104) (cid:105) . (a) C vs. T at various fields. Solid tracesindicate long-pulse data, while discrete symbols indicate adi-abatic short-pulse data. The fields in the legend are externalfields. (a, inset) Isothermal field scans of heat capacity usingthe short-pulse method. (b) Color map of long-pulse heat ca-pacity data vs. temperature and internal magnetic field. (b,inset) Relationship between µ H int and µ H ext . (VCM) as combined with a TL400 Oxford Instrumentstop-loading dilution refrigerator [22, 27–29]. We mea-sured the temperature dependence of the magnetizationwhile cooling and while heating, with field-heating mea-surements performed on both a zero-field-cooled and afield-cooled state. Similarly, we measured field depen-dence magnetization with field sweeps from →
1T per-formed on a zero-field-cooled sample, followed by fieldsweeps from +1T → -1T and -1T → +1T. Further detailsare provided in the supplementary material. All mag-netization measurements were carried out on a 4.7 mmdiameter, 0.40 g sphere of Yb Ti O , which was groundfrom a larger stoichiometric single crystal and polishedinto a spherical shape. The spherical geometry ensures auniform demagnetization factor of D = 1 / .Finally, we collected neutron diffraction data at theSPINS cold neutron triple axis spectrometer at theNCNR. Our sample for these experiments was a 4.7mm Yb Ti O sphere (ground from the same crystal as themagnetization sample) in a dilution refrigerator with the (cid:104) (cid:105) direction perpendicular to the scattering plane andalong a vertical magnetic field, with E i = E f = 5 meV neutrons and a full width at half maximum incoherentelastic energy resolution of .
23 meV . To explore thephase boundaries seen in the heat capacity and magne-tization measurements, we focused our attention on the (2¯20) peak, which was reported to be magnetic [11, 14].We first allowed the sample to settle into the ground stateat zero field by cooling from 300 K over 17 hours and al-lowing the sample to sit for an additional seven hours at65 mK. Following this, we scanned the applied magneticfield at 100 mK from 0 T to 1 T, and then performedslow temperature scans at various fields. The results areshown in Fig. 3. The neutron scattering measurementswere taken with the detector sitting at the (2¯20)
Braggpeak’s maximum intensity, with periodic rocking scansto ensure alignment after cryogenic operations.All three methods–heat capacity, magnetization, andneutron diffraction–point to a reentrant phase diagramof Yb Ti O in a (cid:104) (cid:105) magnetic field. The heat capac-ity plot in Fig. 1 presents the clearest manifestation. Atzero field, C ( T ) has a sharp peak at 270 mK, as reportedfor stoichiometric powders [18, 21], indicating a phasetransition. A magnetic field initially causes this phaseboundary to shift up in temperature, reaching a maxi-mum temperature of 0.42K at an internal field of 0.24 T.At higher fields, the phase boundary sweeps back to 0Kat 0.65 T. The data in Fig. 1(a) shows two heat capacitypeaks at fields between 0.02 and 0.1 T. This is consistentwith the result of field inhomogeneity from nonuniformdemagnetizing fields in the plate-like specific heat sam-ple. In a weak external field (below 0.1 T), the centerof the sample still has no net internal field, giving riseto a residual sharp peak with the same T c as in zero ex-ternal field. The residual peak disappears as soon as theentire sample has a non-zero net field. This field inhomo-geneity would also broaden the peak in finite fields (seesupplemental materials for more details).The magnetization data in Fig. 2 contain several im-portant features. First, Fig. 2(a-e) confirm the reentrantphase diagram: the kinks and changes of slope in magne-tization follow the same curved shape as the anomalies inheat capacity. The derivative d M/ d B shown in Fig. 2(e)underscores this observation. Second, the temperature-dependent magnetization data in Fig. 2(a-c) and (d, in-set) clearly show the ferromagnetic (FM) nature of thelow-temperature phase: at base temperature there is aspontaneous moment that vanishes above T C . Ferromag-netism is also indicated by the characteristic field sweepsin panels (j-k). Note, however, the difference betweenthe field-cooled and zero-field cooled magnetization inFig. 2(c) at 0.02T below the transition temperature,indicating some difference in field-cooled vs. zero-fieldcooled magnetic order for low fields. For higher fields,(panels a and b), there is no visible difference betweenfc-fh and zfc-fh data. Third, the considerable hystere-sis in temperature sweeps in Fig. 2(a-c) confirms previ-ous reports of a first-order phase transition in Yb Ti O [24, 26], which occurs discontinuously via nucleation anddomain growth, causing significant hysteresis in the or- Figure 2. Magnetization of Yb Ti O in applied magnetic fields along (cid:104) (cid:105) . (a-c) Temperature dependence of the magne-tization where we distinguish between data recorded according to procedure (i) zero-field-cooled / field-heated (zfc-fh), (ii)field-cooled (fc) and (iii) field-cooled / field-heated (fc-fh). (d) Magnetization and (e) numerical derivative of the experimentaldata of Yb Ti O as function of internal magnetic field after correction for demagnetization fields. The inset in (d) showsthe spontaneous magnetization as a function of temperature obtained from magnetization field sweeps (protocol (A1); seesupplemental material). (f-k) Magnetization in field cycles of sweep types protocol (A2) and (A3) (see supplemental material). (a) +B- B (b) = +1.2mK/min-1.2mK/min (c) = +0.6mK/min-0.6mK/min (d) = -0.6mK/min+0.6mK/min0.2 0.4 0.6 0.8 1.0 (T) () I n t e n s i t y ( c t s p e r s e c ) Figure 3. Field and temperature dependence of the (2¯20)
Bragg peak intensity. (a) Magnetic field scan going up (+B)and down (-B) in field. Note the lack of hysteresis. (b-d) Tem-perature scans at external magnetic fields of 550mT, 300mT,and 0mT (internal fields of 473mT, 239mT, and 0mT). Red in-dicates increasing temperature, blue indicates decreasing tem-perature. Error bars indicate one standard deviation aboveand one standard deviation below the measured value. der parameter vs. temperature. The first order natureis also confirmed by the spontaneous moment (Fig. 2(d,inset), computed from field-dependent magnetization (asdescribed in supplementary material) having no temper-ature dependence below T C . Fourthly and finally, thefield sweeps in Fig. 2(f-k) show asymmetric minor hys-teresis loops for temperatures between 0.3 K and 0.4 K(where the field scan passes through the phase boundarytwice). This hysteresis is an additional indication of thediscontinuous first-order phase boundary.The neutron diffraction measurements in Fig. 3 clearlyshow the onset of the magnetic order seen in the magne-tization, and corroborate the reentrant phase diagramof Yb Ti O : the temperature scans in Fig. 3(b-d)show transition temperatures (defined as the tempera-ture where the Bragg intensity begins to increase) fol-lowing the same field-dependence as heat capacity andmagnetization. Additionally, the data in Fig. 3(b-d) con-firm the first-order nature of the phase transition, withmassive hysteresis in the temperature scans, even thoughthe scans were extremely slow (the scans in panels (c) and(d) had sweep rates of 0.6mK/min). Note, however, thatno hysteresis is apparent in the 100mK field sweep of the (2¯20) peak (Fig. 2(a)). This suggests either a secondorder phase transition, or a weakly first order transition.Closer examination of the (2¯20) neutron diffractionprovides more clues about the magnetic order. In partic-ular, the field-dependent scattering in Fig. 3(a) is incon-sistent with that of a kagome ice phase. We comparedthe Yb Ti O data to the (2¯20) scattering for Ho Ti O entering the kagome ice phase [30], which has step-likeincreases in (2¯20) intensity signaling entry and exit fromthe kagome-ice state. For Yb Ti O , the steady increasein scattering suggests that spins continuously cant froma ferromagnetic ordered state as field increases, until at0.57 T they undergo a transition to a state polarized ( T ) (a) (+B) ( T) (+B) (T) ( ) (B) ( ) (T) ( ) (b) Clapeyron
Figure 4. (a) Phase diagram of Yb Ti O in a (cid:104) (cid:105) orientedfield, built from heat capacity, magnetization, and neutronscattering. Heat capacity points denote peak location (seeFig. 1), magnetization points denote inflection points (seeFig. 2), and neutron scattering points denote where intensitybegins increasing (see Fig. 3). Error bars indicate the dif-ference in transition temperature upon heating vs. cooling.Theoretically predicted phase boundaries are shown with thesmall data points which denote the location of simulated heatcapacity peaks. The colored lines are guides to the eye. (b)Change in entropy ( ∆ S ) extracted from heat capacity com-pared to ∆ S computed from the Clapeyron relation. Thegreen line is a guide to the eye. along (cid:104) (cid:105) , causing a drop in (2¯20) intensity.To determine the low T ordered spin state we collecteddifference data at (2¯20) , (4¯40) , and (311) . We performeda refinement to the observed Bragg intensities using thestructures reported by Gaudet et. al. (two canted in, twocanted out) [11] and Yaouanc et. al. (all canted in allcanted out) [14], allowing the canting angle and momentsize to vary. More details are provided in the supple-mentary information. The results are shown in Table I.Although our refinement contained only three peaks anddid not account for extinction, some basic conclusions canbe drawn. First, we found that fitting peak intensities toeither structure requires the existence of ferromagneticdomains. Evidence for ferromagnetic domains was previ-ously observed [26], and the presence of domains is con-sistent with the vanishing zero field magnetization in Fig.2(f-k). Second, our refined moment and angle are consis-tent with the Gaudet et. al. structure, but not with theYaouanc et. al. structure. Given the limited data in ourrefinement, this should not be taken as conclusive, but ascorroborating evidence for the two-in-two-out structure. Structure µ ( µ B ) θ χ χ domain µ fit ( µ B ) θ fit Gaudet[11] . ◦ . ◦ Yaouanc[14] . . ◦ . . ◦ Table I. Refinement to neutrons scattering intensities, allow-ing canting angle and ordered moment size to vary.
We can amalgamate the anomalies in heat capacity,magnetization, and neutron scattering to build a phasediagram of Yb Ti O in a (cid:104) (cid:105) oriented field, shownin Fig. 4(a). All measurements concur on the phaseboundary’s location. We double-checked for consistencybetween the various data sets by computing ∆ S usingthe Clapeyron equation for a first order phase boundary ∆ S ∆ M = − µ ∂H∂T , and then compared the result to ∆ S computed from heat capacity, shown in Fig. 4(b). (Seesupplemental materials for more details.) The agreementcorroborates the first-order nature of the phase boundary.Three model spin Hamiltonians have been determinedfor Yb Ti O by Ross et. al. [4], Robert et. al. [31], andThompson et. al. [12] through neutron scattering mea-surements, and we used these as the basis for classicalMonte Carlo simulations. The specific heat and averagemagnetization along (cid:104) (cid:105) were evaluated by measuringthermal averages employing up to × samples perspin. The simulations were carried out on a pyrochlorelattice with N = 4 L spins and periodic boundary condi-tions. Here, L is the number of unit cells along each di-rection, which varied from 6 to 30 in our simulations. Theresults shown are for L = 10 ; other simulations confirmedthat finite size effects were small away from phase bound-aries. More details of the Monte Carlo calculations andresults are provided in the supplementary material. Theoverall field and temperature scale of the computed phaseboundaries to FM order are in accord with the data, withthe Robert et. al. parameters coming the closest. Thesimulations also predicts a first order phase boundarythroughout. However, the marked lobe-like shape of thephase diagram is not reproduced, except for a small bulgepredicted by the Hamiltonian parameters of Robert et.al. that is five times too small in temperature.There are two obvious potential sources of the discrep-ancy. Firstly, long-range magnetic dipolar interactions—not included in our simulation—may cause the spinsto align more easily under a field. Alternatively, theenhancement of magnetic order in a small field maybe interpreted as a suppression of magnetic order inzero field relative to the classical MC result. In otherwords, quantum fluctuations may suppress the zero-fieldordering temperature. Various studies have predictedground state quantum fluctuations from competition be-tween ferromagnetic and antiferromagnetic phases [31–33]; the fact that the simulations using the Robert et. al.Hamiltonian—which is near the FM-AFM boundary—comes the closest to the observed phase diagram maylend credence to this theory. Given the evidence formonopoles in the paramagnetic phase [7, 8], it is alsoworth noting that the non-collinear spin structure in the Yb Ti O ordered phase (FM canted 2-in-2-out) does notpreclude collective ground state quantum fluctuations:even though the order is ferromagnetic, the ice-rule re-quired for the QSI effective field theory is approximatelypreserved in the lattice. In that case, the pocket of phasespace that opens up between the MC phase boundary andthe observed phase boundary could be a finite tempera-ture manifestation of a U(1) quantum spin liquid. Suchquantum fluctuations would lower the transition temper-ature and might persist in the zero–field ground state.Indeed zero field spin fluctuations in Yb Ti O have beenfound to be extremely broad in energy [11]. This is incon-sistent with conventional spin waves of the ordered stateand points instead to remnant fractionalized excitationsof a spin liquid regime.In summary, we have used stoichiometric single crys-tals of Yb Ti O to reveal a peculiar reentrant phasediagram in a (cid:104) (cid:105) oriented field, which current modelHamiltonians cannot explain within a classical shortrange Monte Carlo simulation. The zero-field orderedstate is ferromagnetic with domains, the spins seem topolarize along (cid:104) (cid:105) above an internal field of 0.65 T,and magnetization hysteresis hints at a correlated do-main structure. The peculiar decrease in ordering tem- perature for (cid:104) (cid:105) fields below 0.2 T may be a first tangi-ble indication of the proximity of Yb Ti O to a quantumspin liquid phase.This work was supported through the Institute forQuantum Matter at Johns Hopkins University, by theU.S. Department of Energy, Division of Basic EnergySciences, Grant DE-FG02-08ER46544. 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This supplemental material explains the details of theexperimental and computational methods used to ana-lyze Yb Ti O and simulate its magnetic behavior. I. EXPERIMENTAL METHODSA. Crystal Synthesis
All measurements were performed on two Yb Ti O single crystals grown under identical conditions with thetraveling-solvent floating zone method described in detailby Arpino et. al. [S1]. The heat capacity sample wascut from one crystal, and the neutron diffraction andmagnetization spheres came from adjacent parts of theother crystal. Both crystals had no noticeable variationsin lattice constant or purity between them or along theeach crystal. B. Heat Capacity
As noted in the main text, our heat capacity measure-ments were performed mostly using a long-pulse methodwherein we applied a long heat pulse and tracked sampletemperature as the sample cooled. The data processingwas performed with the LongHCPulse software package[S2]. In this software, heat capacity is computed usingthe equation
C dT s dt = κ ( T s − T b ) m s M s , (S.1)where T s is the sample temperature, T b is the temper-ature of the dilution refrigerator, κ is the thermal con-duction between the sample and refrigerator, m s is thesample mass, and M s is the sample molar mass. Theheat pulses we applied varied between 20 min. and 1hr., with temperature rises of up to 300%. We show onlysingle-slope analysis on cooling curves because the dual-slope analysis of a first-order hysteretic curves producesunphysical double-peak features. In addition, we foundthat the heating power the PPMS applied to the samplefluctuated somewhat, yielding substantial experimentalerror for heat capacity computed from heating pulses.The short-pulse data was taken using the standard PPMSheat capacity routine. Our sample for these heat capacitymeasurements was a near-rectangular 1.04 mg Yb Ti O plate with dimensions 0.20 mm × × ? ].We corrected for demagnetizing fields by first computingsample-independent magnetization M ( H int ) from spheremagnetization, and solving for B int = µ H int numeri-cally with the equation H int = H ext − D M ( H int ) . Forall heat capacity data, the heat capacity of the emptysample holder was measured and subtracted.2While the average demagnetizing factor for the rectan-gular heat capacity sample was 0.68, we calculate that theactual demagnetizing factor varies between 0.23 (at thesmallest edge) to 0.81 (at the center of the sample). Thisinhomogeneity is likely responsible for the two peaks inheat capacity in low fields. In a weak external field (below0.1 T), the center of the sample still has domain coexis-tence and no net field, while the edges are already fullymagnetized and experience a nonzero net field. Hence asplit transition: the central region has the same T c as inthe absence of the external field, while the edges have the T c appropriate for the local value of the net field H int .The estimated field inhomogeneity of approximately0.05 T can also be translated into a broadening ofthe transition temperature with the aid of the phase-boundary slope dH/dT c , which is of the order of 1 T/K(see Fig. 4). At low fields we thus expect the broadeningof the transition on the order of 0.05 K, in agreementwith the observations (Fig. 1). At µ H int = 0 . T, dH/dT c = 0 and we expect peak broadening due to fieldinhomogeneity of only 0.004 K, less than what is ob-served. Therefore, at higher fields an additional broad-ening effect seems to be present. C. Magnetization
The magnetization of Yb Ti O was measured bymeans of a bespoke vibrating coil magnetometer (VCM)as combined with a TL400 Oxford Instruments top-loading dilution refrigerator [S4–S7]. As its main advan-tage our VCM offers excellent thermal coupling withoutrisk of mechanical vibrations with respect to the appliedmagnetic field, which is highly homogeneous. This con-trasts Faraday magnetometers, in which the sample isexposed to a field gradient, or extraction magnetometers,where the sample is moved with respect to the field, ei-ther of which may generate parasitic signal contributionsdue to uncontrolled field and temperature histories.Data were recorded at temperatures down to ∼ . Kunder magnetic fields up to 5 T at a low excitation fre-quency of Hz and a small excitation amplitude of ∼ . mm. The sample temperature was measured with a RuO sensor mounted next to the sample and addition-ally monitored with a calibrated Lakeshore RuO sensorattached to the mixing chamber in the zero-field region.As noted in the text, the sample for magnetization wasa 4.7 mm sphere. To suspend the sample in the VCM itwas glued with GE varnish into an oxygen-free Cu sampleholder composed of two matching sections fitting accu-rately the size of the sphere (see Fig. ?? ). The sampleholder with the sample mounted was firmly bolted into aCu tail attached to the mixing chamber of the dilution re-frigerator. This provided excellent thermal anchoring ofthe sample across the entire surface of the sphere, whilekeeping its position rigidly fixed without exerting strain[S4]. The sample holder is shifted with respect to theaxis of the cold finger in a way that the sample is centred (a)(b) (c)4.7mm sampleholder coldfingersample mm Figure S1. (a) TSFZ technique (solvent = 30wt% rutileTi02 and 70wt% Yb Ti O ) produces a large single crystalof Yb Ti O that is clear and colourless (image taken fromArpino et al. [S1]). (b) Spherical sample ground from thestoichiometric single crystal and the oxygen-free Cu sampleholder composed of two matching sections fitting accuratelythe size of the sphere. (c) Sample holder mounted on the coldfinger which is then bolted into the Cu tail attached to themixing chamber of the dilution refrigerator. on the cold finger axis and therefore also on the axis ofthe VCM coil.To determine the signal contribution of the sample,the empty sample holder was remeasured and subtracted.The signal of the empty sample holder was found to besmall with a highly reproducible field dependence and anessentially negligible temperature dependence. The sig-nal of the sample was calibrated quantitatively at Kand K against the magnetization measured in a Quan-tum Design physical properties measurement system de-termined also at K and K, as well as a Ni standardmeasured separately in the VCM [S8].To measure temperature dependence, three procedureswere used: (i) After cooling at zero magnetic field from ∼ µ H ext = → +1 T, denoted (A1). (v) A field sweep+1T → -1T, denoted (A2). (vi) A related field sweep3from -1T → +1T, denoted (A3). For temperatures above0.05 K all data were recorded while sweeping the fieldcontinuously at 15 mT/min, whereas measurements at0.028 K, the lowest temperature accessible, were carriedout at a continuous sweep rate of 1.5 mT/min to mini-mize eddy current heating of the Cu mount.The spontaneous magnetization M shown in the insetin Fig. 2(d) was obtained by extrapolating the low fieldbehaviour of the zero-field-cooled magnetization data lin-early (Fig. 2(d)) to zero field. The black and redlines are the transition temperatures T C determined byspecific heat and magnetization measurements, respec-tively. The red shaded area indicates the difference in T C upon heating vs cooling in the temperature depen-dence of the magnetization. M remains almost constantat ∼ . µ B Yb − before vanishing at T C . The lack ofa temperature dependence of M below T C further sup-ports the first-order nature of the phase transition.The remnant magnetization of the superconductingmagnet in zero-field can be estimated from the hysteresisloops of Fig. 2 [S5]. When we look at the hysteresis looptaken at 900mK, where the hysteresis is closed, we findat M = 0 a difference in up- and down-sweep of <10mT.This is an instrumental offset of the VCM. Because ofthis, we can be confident that the zero-field cooled datahad a remnant field of <10mT, well within the rangewhere the internal magnetic field is zero. Relating Heat Capacity to Magnetization
As a consistency check, we can relate the heat capac-ity data to the magnetization data using the expression T ∂ M∂T = ∂C∂H for temperature ranges where our system isin equilibrium. The results for T = 0 . and T = 0 . (away from the first-order transition) are shown in Fig.S2. The large error bars are from taking a numericalsecond derivative of slightly noisy data (uncertainty wasestimated by computing the numerical derivative withcombinations of data points between n − and n + 3 andtaking the standard error of the mean). Even thoughthese measurements were taken on two different sampleswith different geometries in different instruments, the re-lation between them holds to within uncertainty (withthe exception of 0.52 T at 0.12 K, which is near a phaseboundary and not in equilibrium), and C relates to M as expected. D. Neutron Diffraction
We collected neutron diffraction data using the SPINStriple axis spectrometer at the NCNR with E i = E f =5 meV neutrons. We used a beryllium filter before theanalyzer, 40’ collimators before and after the sample, anda flat analyzer. Using the ResLib software package [S9],we calculate that this configuration gives a full width () = . K (a) () = . K (b) Figure S2. Calculation relating C to M with T ∂ M∂T = ∂C∂H at(a) 0.55 K and (b) 0.12 K. Error bars indicate one standarddeviation. at half maximum incoherent elastic energy resolution of .
23 meV . II. COMPUTATIONAL METHODSA. Magnetic Structure Refinement
The magnetic refinement to the neutron diffractionpeaks was carried out on three peaks available in ourinstrumental configuration. In zero-field temperaturescans, the (2¯20) , (4¯40) , and (311) peaks change in inten-sity by (2 . ± . , (0 . ± . , and (4 . ± . ,respectively. (Uncertainties indicate one standard devi-ation.) We computed the magnetic Bragg intensity bymultiplying the computed nuclear structure factor by theobserved change in intensity. This was done in order toeliminate the effects of different wavelengths used on thedifferent peaks ( (2¯20) and (311) data were collected with5 meV neutrons, (4¯40) data was collected with 20 meVneutrons).We first refined the structure assuming a net FM mo-ment along the (100) direction with no domains, but thefit was not very good. χ was reduced by 85% by assum-ing randomly distributed ferromagnetic domains along (100) , (¯100) , (010) , (0¯10) , (001) , or (00¯1) , as shown inTable 1.4 () (a) () (g) (b) (c) (d) (e) (f) exp (h) (i) () (j) (k) (l) exp (fc-fh) exp (fc) Figure S3. Classical Monte-Carlo simulations of the specific heat (for 864 spins) and the magnetization measured along the [11 1] direction (for 4000 spins) per spin compared to the corresponding experimental data. (a-f) Heat capacity vs. temperature.(g-l) Magnetization vs. temperature. Fields displayed are internal magnetic fields.
B. Classical Monte Carlo Simulation