Classical Spin Nematic Transition in LiGa_{0.95}In_{0.05}Cr_4O_8
R. Wawrzyńczak, Y. Tanaka, M. Yoshida, Y. Okamoto, P. Manuel, N. Casati, Z. Hiroi, M. Takigawa, G. J. Nilsen
CClassical Spin Nematic Transition in LiGa . In . Cr O R. Wawrzy´nczak, ∗ Y. Tanaka, † M. Yoshida, Y. Okamoto, P. Manuel, N. Casati, Z. Hiroi, M. Takigawa, and G. J. Nilsen ‡ Institut Laue-Langevin, 6 rue Jules Horowitz, 38042 Grenoble, France Institute for Solid State Physics, University of Tokyo,5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan Department of Applied Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan ISIS Neutron and Muon Source, Science and Technology Facilities Council, Didcot, OX11 0QX, United Kingdom Swiss Light Source, Paul Scherrer Institute, 5232 Viligen PSI, Switzerland (Dated: October 20, 2018)We present the results of a combined Li NMR and diffraction study on LiGa . In . Cr O , amember of LiGa − x In x Cr O “breathing” pyrochlore family. Via specific heat and NMR measure-ments, we find that the complex sequence of first-order transitions observed for x = 0 is replaced bya single, apparently second-order transition at T f = 11 K. Neutron and X-ray diffraction rule outboth structural symmetry lowering and magnetic long-range order as the origin of this transition.Instead, reverse Monte Carlo fitting of the magnetic diffuse scattering indicates that the low temper-ature phase may be described as a collinear spin nematic state, characterized by a quadrupolar orderparameter. This state also shows signs of short range order between collinear spin arrangementson tetrahedra, revealed by mapping the reverse Monte Carlo spin configurations onto a three-statecolor model describing the manifold of nematic states. Spinel materials, AB X , host a variety of interest-ing magnetic phenomena, including spin-orbital liquidstates (FeSc S ) [1], skyrmion lattices (GaV S ) [2], andmagneto-structural transitions [3–6]. Many of these orig-inate from the B site, which forms a frustrated py-rochlore lattice of corner-sharing tetrahedra. When thelarge spin degeneracy caused by the frustration is com-bined with strong magneto-elastic coupling, a typicalfeature of spinels, several possible magneto-structurallyordered and disordered states arise. For example, inthe chromate spinel oxides, A Cr O , collinear, copla-nar, and helical magnetic structures (and their accom-panying structural distortions) may all be realized byvarying the cation on the A site [3, 4, 6, 7]. Althoughthe low temperature behaviour of the chromates is com-plex, it is surprisingly well captured by the bilinear-biquadratic model [8, 9], the Hamiltonian of which is H = J (cid:80) i,j (cid:126)S i · (cid:126)S j + b (cid:80) i,j ( (cid:126)S i · (cid:126)S j ) + P , where (cid:126)S i,j areclassical Heisenberg spins and J is the nearest neighbourexchange. The second and third terms represent themagneto-elastic coupling, which assumes a biquadraticform if only local distortions are considered, and pertur-bative terms such as further neighbor or anisotropic cou-plings. These act on the degenerate manifold of states (cid:80) i ∈ tet (cid:126)S i = 0 (where the sum is over tetrahedra) gen-erated by the Heisenberg term, in turn selecting eithercollinear ( b <
0) or coplanar ( b >
0) configurations [8],then breaking the remaining degeneracy and establishingmagnetic order (
P (cid:54) = 0) [10, 11].Two recent additions to the chromate spinel family arethe so-called “breathing” pyrochlore systems, AA (cid:48) Cr O ,where A =Li + and A (cid:48) =Ga ,In [12]. Here, the alter-nation of the A and A (cid:48) cations on the A site leads toan alternation in tetrahedron sizes and, hence, magnetic exchange constants, J and J (cid:48) . This alternation is quan-tified by the “breathing” factor B f = J (cid:48) /J , which is ∼ . A (cid:48) =Ga and ∼ . A (cid:48) =In . The small B f notwithstanding, the phenomenology of the “breathing”pyrochlores at low temperature is similar to their undis-torted cousins. In both A (cid:48) =Ga ( x = 0) and A (cid:48) =In ( x = 1), a sequence of two transitions, as observed inlightly doped MgCr O [13], lead to structural and mag-netic phase separation (as in ZnCr O [14]), while the ex-citation spectra show gapped “molecular” modes in theordered states [6, 15]. For x = 0, the upper first-ordermagneto-structural transition at T u ∼
20 K results inphase-separation into cubic paramagnetic and tetragonalcollinear phases [16]. The cubic phase then undergoes an-other first-order transition into a second tetragonal phaseat T l = 13 . Li NMR 1 /T , implying proximityto a tricritical point [17].Studies of the solid solutions LiGa − x In x Cr O in-dicate that T l is rapidly suppressed when x is in-creased/decreased from x = 0 /
1. Starting from the x = 0composition, sharp peaks in the magnetic susceptibilityand specific heat persist until x ∼ .
1, beyond which theyare replaced by features characteristic of a spin glass [18].This side of the phase diagram resembles those of boththe undistorted chromate oxides [19] and Monte Carlo(MC) simulations of the bilinear-biquadratic model for b < P = J nnn (cid:80) i,j (cid:126)S i · (cid:126)S j [20].In this letter we will show that the low temperaturebehaviour of the x = 0 .
05 composition, apparently wellinside the magnetically ordered regime of the phase di-agram, differs drastically from the x = 0 composition. a r X i v : . [ c ond - m a t . s t r- e l ] J un Instead of two first-order transitions and phase separa-tion, we observe a single second-order transition using Li NMR and specific heat. Remarkably, this transi-tion neither corresponds to magnetic long range ordernor a structural transition. Rather, the magnetic diffuseneutron scattering implies that it shares features withboth the nematic transition predicted for the bilinear bi-quadratic model on the pyrochlore lattice, as well as thepartial ordering transition expected for the pyrochloreantiferromagnet with further neighbor interactions [11].Furthermore, the transition is shown to coincide withspin freezing, drawing parallels to other frustrated mate-rials, like Y Mo O [21, 22].The powder samples of LiGa . In . Cr O were pre-pared via the solid-state reaction method in Ref. [12],using Li enriched starting materials to reduce neutronabsorption. The specific heat was measured by the relax-ation method in a Physical Property Measurement Sys-tem (Quantum Design). The Li-NMR measurementswere carried out in a magnetic field of 2 T, and NMRspectra were obtained by Fourier transforming the spin-echo signal. At low temperature, the spectra were con-structed by summing the Fourier transformed spin-echosignals measured at equally spaced frequencies. The nu-clear spin-lattice relaxation rate 1/ T was determined bythe inversion-recovery method at the spectral peak [23].The samples were further characterised by powder syn-chrotron X-ray diffraction (SXRD) on the MS-X04SAbeamline at the Swiss Light Source, and powder neu-tron time-of-flight diffraction (ND) on the WISH instru-ment at the ISIS facility. For the former, several pat-terns were measured in the temperature range 6 −
20 Kusing a photon energy of 22 keV ( λ = 0 .
564 ˚A). TheND was measured at several temperatures between 1 . Q = 0 . − [23] . The validity of the subtrac-tion was verified by comparing with polarized neutrondiffraction data on the x = 1 composition at the sametemperature. The reverse Monte Carlo (RMC) analysisof the magnetic diffuse scattering was performed usingthe SPINVERT package [25].The temperature dependence of the Li-NMR spec-trum is shown in Fig. 1(a). In the paramagnetic state,it consists of a sharp single line without quadrupolestructure, similar to the A (cid:48) =Ga compound. Below11 K, however, the spectrum shows a marked broaden-ing, which indicates the development of a static inter-nal field at the Li site as a result of spin freezing. Thespectra in the low-temperature phase consist of two com-ponents: a relatively narrow line whose width saturatesbelow 9 K, and a broader one which broadens further / T ( s - ) T (K) C p / T ( m J / K - m o l C r - ) T (K) T f x = 0 para AF x = 0.05 T f (b) I n t e n s it y ( a r b . un it s ) -4 -2 0 2 4 (cid:39) f (MHz) L i n e w i d t h ( M H z ) T (K) T f FIG. 1. (a) Li-NMR spectra, normalized by the peak in-tensity. ∆ f = 0 corresponds to the center of gravity of thespectrum. The inset shows temperature dependence of the av-eraged linewidth of whole spectra (red crosses), the FWHM ofthe narrow component (closed circles), and the FWHM of thebroad ones (open circles) (b) The temperature dependence of1 /T for the x = 0 .
05 composition (solid circles), paramag-netic (open triangles) and AF (solid triangles) components forthe pure composition, respectively. The inset shows temper-ature dependence of the specific heat divided by temperature C p /T of the x = 0 .
05 composition. with decreasing temperature. The temperature depen-dence of each line-width extracted by a double Gaussianfit are shown in the inset of Fig. 1(a). The intensityratio of the sharp and broad components of spectra isestimated to be 1 : 4 . . /T for both the x = 0 .
05 (red circles) and x = 0 (blue trian-gles) compositions [26]. In the case of the former, 1 /T exhibits a sharp peak at T f =11.08(5) K, indicating crit-ical slowing down associated with a bulk second-ordermagnetic transition. This transition is also evidencedby a sharp anomaly at 11.29(3) K in specific heat, alsoindicative of a second-order transition [Fig. 1(b), inset].These behaviors are in contrast with the x = 0 com-pound, which shows two first-order magnetic transitions,and phase separation [Fig. 1(b)].Despite the clear hallmarks of a second-order transi-tion in both specific heat and Li NMR, both SXRDand ND surprisingly indicate an absence of structuralsymmetry breaking below T f , unlike all other chromatespinels with well-defined phase transitions [Fig. 2(a)].Should the transition corresponds to a purely magneticlong range ordering, the only antiferromagnetic structurecompatible with a cubic structural symmetry is the so-called all-in all-out structure, where the spins lie alongthe local (cid:104) (cid:105) axes. On the other hand, this structurewould imply zero internal field at the Li site, in contra-diction with the large width of the NMR line. Neitherdo any sharp features consistent with all-in all-out orderappear in ND.While no peak splittings are observed on crossing T f ,some broadening of Bragg peaks with indices ( h
00) and( hk
0) is observed on cooling towards T f . This implies Q (Å ) (a) B i s o C r ( Å ) x C r T (K) (b) (c) S S −0.015−0.01−0.0050 T (K) (b)
FIG. 2. (a) Synchrotron X-ray pattern at low and high T withits Rietveld refinement( R p =12.2 R wp =15.6 R e =4.05), (inset)T-dependence of (008) peak. Peaks visible in the measuredpattern and not signed with green markers come from a ∼ O impurity. Pattern refined with resulting fit parameters:Temperature dependence of (b) strain parameters S and S , extracted from SXRD, and (c) B iso and Cr fractionalposition parameter x obtained from ND. that the local symmetry is tetragonal, as expected frommagneto-elastic coupling within the bilinear-biquadraticmodel [8]. To identify the changes in the structure oncrossing T f , we plot the temperature dependence of the S and S strain broadening parameters [27] corre-sponding to these families of peaks, as well as the frac-tional Cr position parameter and isotropic displace-ment parameter in Fig. 2(b,c). All parameters are foundto evolve continuously before freezing at T f , with the lat-ter doing so in a step-like fashion. This emphasizes thestrong magneto-elastic coupling in the system. Further-more, the apparent freezing of all parameters below T f is consistent with the spin freezing observed in NMR.Turning to the magnetic diffuse scattering, the dataat 30 K, well above T f , show a broad, diffuse featurewith maximum centred around Q = 1 .
55 ˚A − [Fig. 3(a)].This is compatible with expectations for the undistortedpyrochlore lattice, and implies the presence of Coulomb-phase-like power law spin-spin correlations [28]. Coolingto 15 K, a weak, but sharp peak is observed at the (110)position, as for x = 0. Because this appears above T f and appears to be temperature independent on furthercooling, it is ascribed to the presence of a small amountof x = 0 phase in the sample. The only intrinsic changesin the magnetic scattering on crossing the transition aretherefore a slight redistribution of the diffuse scatteringtowards the positions Q = 0.8 ˚A − [near (100)], 1 . − (110), 1.73 ˚A − (210), and 1.87 ˚A − (112) [Fig. 3(a)]. To understand the apparent paradox of the presenceof a phase transition, on the one hand, and the ab-sence of any peak splittings or (intrinsic) magnetic peaks,on the other, we investigate the changes in the realspace spin-spin correlations by performing RMC fits ofthe magnetic diffuse scattering [29]. At 30 K, the ex-tracted normalized real-space spin-spin correlation func-tion (cid:104) S · S i (cid:105) /S ( S +1) indicates antiferromagnetic nearestneighbour correlations, with (cid:104) S · S (cid:105) /S ( S + 1) = − . (cid:104) S · S (cid:105) /S ( S + 1), which corresponds to thenext-nearest-neighbour distance along (cid:104) (cid:105) , is negative.Reconstructing the single crystal scattering from theRMC spin configurations in the ( hhl ) [Fig. 3(c)] and( hk
0) planes [25] provides some clues as to why this is: in-tensity is observed at positions consistent with the prop-agation vector k = (001), which manifests in our exper-iments as scattering around the Bragg positions listedpreviously. The scattering maps also allow us to iden-tify “bow-tie” features characteristic of the underlyingCoulomb liquid state [30] [Fig. 3(c-d)]. The overall pic-ture of k = (001) short range order superimposed on“bow-tie” features remains unchanged to 1.5 K, althoughthe former grow slightly in intensity below T f . This isalso true of (cid:104) S · S i (cid:105) /S ( S + 1), which is nearly indistin-guishable between the high and low temperature datasets[Fig. 3(b)], emphasising that the order parameter of thetransition cannot be dipolar.Classical MC simulations for the bilinear-biquadraticmodel with a Gaussian bond disorder ∆ (here related tothe substitution x ) indicate a quadrupolar nematic tran-sition to a collinear state with persistent Coulomb-likecorrelations for b < P = 0, and small ∆ [20]. As ∆ ( x )is increased, this nematic transition becomes concurrentwith spin freezing. Because of (i) the lack of magneticBragg peaks at T < T f , despite a clear phase transition;(ii) the tendency towards collinear spin arrangements in x = 0 ,
1, implying b < x ; (iii) the local sym-metry lowering, consistent with such spin arrangements;and (iv) the spin freezing observed in NMR, we specula-tively assign the phase transition to concurrent nematicorder and freezing. To test this assignment, we performfurther RMC simulations on the 1 . i.e. collinear spins [31]. Several directions are modelled dueto the cubic symmetry of the system; while all should re-produce the experimental scattering, the correspondingspin configurations generally differ. With collinear spins,the data is modelled nearly as well as the Heisenberg case( χ /χ ∼ .
05) for all spin directions.To verify that this assumption is consistent with ourNMR results, we simulated the NMR spectrum corre-sponding to the collinear RMC spin configurations [23].The main component of the simulated spectrum, ascribedto the broad part of the magnetic diffuse scattering (andhence the “bow-tie” features), is triangular, and repro-duces the shape of the broad component in the exper-imental NMR spectrum at 4 . k = (001) short-range order, is underestimated by ourRMC simulations. This discrepancy is likely due to thedifficulty of fitting k = (001) magnetic diffuse scatter-ing near nuclear positions in the experimental data, thedifferent temperatures of the NMR (4 . . site is estimatedto be gS = 1 . µ B at 4.2 K. Although this quantityis model-dependent, the reduced moment could indicatethe present of fast quantum or thermal fluctuations inthe ground state.The RMC spin configurations may be further analyzedby examining the collinear spin arrangements on indi-vidual tetrahedra: for all directions of the anisotropic N o r m a li z ed c oun t s ( a . u . ) Q (Å ) (a) r ( Å )(b) FIG. 3. (a) Diffuse scattering at 1.5 K and 30 K with RMC fitsusing free (Heisenberg) and constrained (Ising) spins parallelto (001) for the former. (b) Real space spin-spin correlationfunctions (cid:104) S · S i (cid:105) /S ( S + 1) versus r for both temperatures.The dashed lines indicate the expected envelope for dipolarcorrelations ∝ /r , and the green square the expectationfor (cid:104) S · S (cid:105) /S ( S + 1) as T →
0. (d) Reconstructed sin-gle crystal scattering from the 30 K Heisenberg fit comparedwith (c) “bow-tie” scattering (highlighted by the white dashedpolygon) calculated for the “breathing” pyrochlore antifer-romagnet within the self-consistent Gaussian approximation(SCGA), for B f = 0 . T /J = 0 . axis, configurations with two spins up and two down( uudd ) are favored over other configurations. The localconstraint (cid:80) i ∈ tet (cid:126)S i = 0 obeyed by these configurationsis responsible for the “bow-tie” scattering in the recon-structed single crystal patterns. The ratio of uudd tothree-up one-down ( uuud , or vice versa, uddd ) configu-rations is typically in excess of 8 for the (001) axis, versus2 − uudd tetra-hedron may be given a color according to the arrange-ment of the two ferro- and four antiferromagnetic bondson each tetrahedron, and correspondingly, the directionof the local tetragonal distortion. Ferromagnetic (long)bonds along (cid:104) (cid:105) correspond to blue (B), (cid:104) (cid:105) green(G), and (cid:104) (cid:105) red (R). The refined spin structures ex-hibit a majority of B tetrahedra. In the case of the x = 0and x = 1 compounds, the magnetic orders respectivelycorrespond to RG and BG color arrangements on the di-amond lattice formed by the tetrahedra, with the spinsdirected along the c axis of the tetragonal cell. The ob-served growth of the k = (001) scattering may thus beassociated with domains containing combinations of BRand BG. Interestingly, this means that the ground stateof the present material lies closer to the x = 1 orderedstructure, despite its chemical proximity to the x = 0compound. The BR/BG short-range order is furthermoreconsistent with the negative (cid:104) S · S (cid:105) /S ( S + 1) found inthe Heisenberg fits, as well as the broadening (but lackof splitting) of the structural diffraction peaks.To quantify the spatial extent of the opposite colorcorrelations, we compute the color correlation function (cid:104) C C i (cid:105) , defined such that (cid:104) C C i (cid:105) = 1 for the same and (cid:104) C C i (cid:105) = − (cid:104) C C (cid:105) = − .
14 and opposite-color shortrange order, with an exponential radial decay character-ized by the correlation length ξ c =2 ˚A. While this isshort, two-color (BG and BR) domains as large as 20 ˚Acan be identified by visual inspection of the color configu-rations [Fig 4(b)]. The scattering around (001) (0.8 ˚A − )can be understood as resulting from disorder between theminority colors (R and G) in the two-color pattern – inthe case of perfect color order, it would vanish. In thissense, the low temperature state shows some commonal-ities with the partially ordered phase predicted from MCsimulations of the pyrochlore lattice with further neigh-bor couplings in [11] (this state is also found in the three-state Potts model [32]). The partially ordered phase hasa single color on all up-pointing tetrahedra, and a distri-bution of other colors on down-pointing tetrahedra.There are, however, some uncertainties concerning theabove interpretation. Most importantly, the existing MCsimulations only consider the undistorted pyrochlore lat- r (Å) -0.1-0.0500.050.1 C C i 〉〉 (a) (b) FIG. 4. (a) Real space plot of a color RMC configuration at1.5 K (2 × × uuud tetrahedra (or vice versa), and the highlighted area shows aregion with predominantly blue-green correlations. (b) Ra-dial dependence of color-color correlations (cid:104) C C i (cid:105) at 1 . ξ c = 2 ˚A. tice, and it is not known how the phase diagram changeswhen the “breathing” distortion is introduced. Indeed,in the undistorted case, the transition is expected to beweakly first-order for small x , only becoming second-order deep inside the spin glass regime, where the ne-matic transition coincides with spin freezing. Both ofthese statements contradict our observation of a second-order transition and spin freezing at small x , as wellas the appearance of a conventional spin glass phase atlarger x [18]. On the other hand, the theoretically pre-dicted spin glass state at large x is expected to be excep-tionally robust towards magnetic field, like the presentlow-temperature phase [23] – similar behaviour is ob-served below T f in pure Y Mo O , where no clear ne-matic transition is observed. It is finally not evident whythe nematic transition occurs at all, given the presenceof the further neighbor couplings which presumably playa role in the ordering of the x = 0 and x = 1 compounds.These points will hopefully be clarified by future MCsimulations and experimental work.To conclude, we have shown that LiGa . In . Cr O undergoes a single, apparently second-order, transition at T f = 11 K. This transition corresponds neither to mag-netic long range order nor a structural symmetry break-ing, but is rather ascribed to nematic (collinear) spinfreezing. Upon cooling, correlations corresponding to thepropagation vector k = (001) are enhanced. Assumingthat the spins lie along (001), these correspond to short-range order between collinear spin configurations on thetetrahedra. The transition thus shares features with boththe nematic and partial ordering transitions anticipatedfor the pyrochlore Heisenberg model with perturbations.We gratefully acknowledge T. Fennell for his carefulreading of and useful comments on the manuscript. Wealso thank N. Shannon and O. Benton for sharing theirSCGA data, and S. Hayashida and J. A. M. Paddisonfor useful discussions. This work was supported by JSPSKAKENHI (Grant Nos. 25287083 and 16J01077). Y. T. was supported by the JSPS through the Program forLeading Graduate Schools (MERIT). ∗ Email address: [email protected] † Email address: [email protected] ‡ Email address: [email protected][1] V. Fritsch, J. Hemberger, N. B¨uttgen, E.-W. Scheidt, H.-A. K. von Nidda, A. Loidl, and V. Tsurkan, Phys. Rev.Lett. , 116401 (2004).[2] I. K´ezsm´arki, S. Bord´acs, P. Milde, E. Neuber, L. M.Eng, J. S. 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Hiroi, PSI experimental report: SLS proposal no.20150562 , available at: http://duo.psi.ch.[34] R. Saha, F. Fauth, M. Avdeev, P. Kayser, B. J.Kennedy, and A. Sundaresan, Phys. Rev. B ,064420 (2016), URL http://link.aps.org/doi/10.1103/PhysRevB.94.064420 .[35] P. W. Stephens, J. Appl. Cryst. , 281 (1999).[36] Y. Yamada and A. Sakata, J. Phys. Soc. Jpn. , 1751(1986). SUPPLEMENTAL MATERIALNuclear spin-lattice relaxation rate 1/ T The nuclear spin-lattice relaxation rate 1/ T was de-termined by fitting the recovery curves of the spin-echointensity at the spectral peak after an inversion pulse as afunction of time t to the stretched-exponential function: I ( t ) = I eq − I exp (cid:104) − ( t/T ) β (cid:105) , (1)where I eq is the intensity at the thermal equilibrium and β is a stretch exponent that provides a measure of in-homogeneous distribution of 1/ T . The case of homoge-neous relaxation corresponds to β = 1. Figure S1 showsthe temperature dependence of β , which decreases from1 on cooling below 30 K and reaches a minimum around T f . A stretched exponential relaxation with β < /T arises due to inho-mogeneity. S t r e t c h e xpon e n t β T (K) T f FIG. S1. Temperature dependence of stretch exponent β for Li nuclei in a field of 2 T.
NMR spectra and spin-echo decay time T In order to track the temperature-dependence of thesignal intensity and spin-echo decay time T , we per-formed NMR spectrum measurements with differentpulse separation times τ in the spin-echo pulse sequence π/ τ - π . Figure S2(a) shows the τ -dependence of NMRspectra at the base temperature. Although both sharpand broad components of the spectra have fast T , andconsequently show a reduction of the intensity for longer τ , the spectral shapes are almost identical within therange of the τ probed. The spectra shown in Fig.1(a) inthe main paper were measured with τ = 25 µ s.The integrated intensities of the spectra I (multipliedby T to cancel out Curie law for the nuclear moments)are plotted against 2 τ in Fig. S2(b); the I correspondingto the sharp and broad components at base temperature were extracted by a double Gaussian fit, and are plot-ted separately. The fitting function for each dataset is I ( t ) T = I T exp( − t/T ), where T is the spin-echo de-cay time. The T thus extracted are 425(1) (30 K > T f ),54(3) (11 K ∼ T f ), 45(1) (4 . µ s(4 . T drops sharply at T f =11.1 K and doesnot recover even at base temperature, as is the case forother frustrated systems with slow low-temperature dy-namics. In order to quantify the loss of NMR intensitybelow T f , all data were normalized by the extrapolated I (0) T at 30 K [Fig. R2(b)]. The resulting I (0) T are0.66(2), 0.16(0), and 0.75(4) at 11.1 K, and for the sharpand broad component at 4.2 K, respectively. This indi-cates that 91 % of the intensity is conserved at the basetemperature. Finally, we note that the ratio of the inten-sities corresponding to the two components is estimatedto be 1 : 4 . Neutron and X-ray diffraction
The sample used in the synchrotron X-ray powderdiffraction experiment was enclosed in a silica capillaryof 3 mm diameter, and loaded into a He flow cryostatwith a base temperature of 5 K. Neutron powder diffrac-tion patterns were measured on 5.8 g of powder loadedin an 8 mm diameter vanadium can.The measured diffraction patterns were analysed bymeans of Rietveld refinement. Part of one refined neutrondiffraction pattern is presented in Fig. S3, and parame-ters showing the goodness of fit for all detector banks andtemperatures are presented in Tab. I. Due to the fact thatthe peak profile of WISH is difficult to describe with thecommonly used back-to-back exponential function andhigh- Q parts of the patterns suffer from significant peakoverlap, we present the weighted R parameters for the Ri-etveld refinement and LeBail profile matching (represent-ing the best possible fit for the given set of profile param-eters) along with the expected R value based on the dataset’s statistics. The strain ( S hkl ) parameters, whose tem-perature dependence is presented in main body of thiswork, were extracted from the SXRPD data, due to thesuperior resolution of these measurements. On the otherhand, B iso and the fractional x coordinate of chromiumwere derived from refinements of neutron powder diffrac-tion data. The refined structural parameters are shown inTable II, where position values are retrieved from SXRDpatterns and B iso parameters come from NPD data. Magnetic diffuse scattering and reverse Monte Carlorefinement
To extract the diffuse magnetic scattering from theneutron powder diffraction data, a flat background C ( C (cid:39) .
3) was subtracted from the data such that the I n t e n s it y ( a r b . un it s ) -4 -2 0 2 4 ∆ f (MHz) T = 4.2 K τ = 7 µ s τ = 16 µ s τ = 25 µ s I • T ( a r b . un it s ) τ ( µ s)4.2 Kbroad 30 K T f = 11.1 K4.2 Ksharp (a) (b) FIG. S2. (a) τ dependence of spectra at 4.2 K. (b) τ dependence of the integrated intensity of NMR spectra. TOF (µs) −5000050001000015000 I n t e n s i t y ( a . u . ) LiGa In Cr O T=1.5 K. Y obs Y calc Y obs -Y calc FIG. S3. Measured time-of-flight neutron diffraction pattern (black points) at 1 . R Rwp and R Lwp are weighted R paraneters for Rietveld and LeBail refinements respectively and R exp is R expected. T (K) Banks 5 & 6 Banks 4 & 7 Banks 3 & 81.5 R Rwp =7.11 R Lwp =6.32 R exp =0.64 R Rwp =6.86 R Lwp =6.13 R exp =0.65 R Rwp =6.32 R Lwp =5.81 R exp =0.676 R Rwp =7.15 R Lwp =6.37 R exp =0.66 R Rwp =6.88 R Lwp =6.15 R exp =0.68 R Rwp =6.34 R Lwp =5.83 R exp =0.6911 R Rwp =7.98 R Lwp =7.12 R exp =0.57 R Rwp =7.53 R Lwp =7.16 R exp =0.58 R Rwp =6.85 R Lwp =6.80 R exp =0.6015 R Rwp =7.87 R Lwp =7.09 R exp =0.56 R Rwp =7.75 R Lwp =7.24 R exp =0.40 R Rwp =7.37 R Lwp =6.98 R exp =0.4130 R Rwp =7.96 R Lwp =7.19 R exp =0.39 R Rwp =7.88 R Lwp =7.39 R exp =0.40 R Rwp =7.51 R Lwp =7.10 R exp =0.41TABLE II. Structural parameters obtained by refinement of neutron and synchrotron X-ray powder diffraction data at 6 K.Atom Wyckoff position x y z Occupancy B iso (˚A )Li1 4 a d a d d e e e mean intensity at Q < . − was zero. This choicewas justified by the fact that the high temperature dif-fuse scattering in the first Brillouin zone is zero (indeed,polarized neutron scattering on the x = 1 compound,where the high temperature scattering is similar, alsosuggests this is the case [6]), and that the WISH instru-mental background is flat when using a V can and radialoscillating collimator. The nuclear Bragg peaks were re-moved from the ranges where the build-up of diffuse scat-tering was observed. RMC refinements were performedon boxes of spins containing 6 × × × × W = 0 . T Ising-type collinear spin models and W = 1 forthe high- T Heisenberg-type spin models.The analysis of the RMC-refined spin configurationsconsisted of several steps. Firstly, for each simulationbox, the normalized real space spin-spin correlation func-tion was evaluated: (cid:104) S · S i (cid:105) /S ( S + 1) = (cid:104) S (0) · S ( r ) (cid:105) /S ( S + 1) = 1 n ( r ) S ( S + 1) N (cid:88) j Z jk ( r ) (cid:88) k S j (0) · S k ( r ) , (2)where S i is the vector of the i -th spin in the simula-tion box, N is the total number of sites inside the box,and Z ij ( r ) is the number of spins in the coordinationshell at distance r . This was done using the SPINCOR-REL application in the SPINVERT suite [29]. Despitethe noticeable differences between the form of the dif-fuse scattering at high and low temperatures, it is diffi-cult to distinguish the differences between the real spacespin-spin correlations corresponding to the fits of thosedatasets. This is most likely due to the dominant con-tribution of the broad Coulomb liquid-like component toboth patterns. Following evaluation of (cid:104) S · S i (cid:105) , the sin-gle crystal patterns were reconstructed in the ( hk
0) and( hhl ) planes using SPINDIFF, also part of the SPIN-VERT suite. This allowed for a more diagnostic compar-ison with theoretical calculations, and permitted identi-fication of the Coulomb-like and short-range ( k = (001))ordered components of the scattering.The third step, the calculation of color populationsand correlations, applied only to the low-temperaturecollinear spin configurations. In the case of a single tetra-hedron, the bilinear-biquadratic model yields six possiblemagneto-structural ground state configurations, three ofwhich are independent with respect to a global rotationof spins. These may be given a color c = { R, G, B } ac-cording to the direction of the ferromagnetic bond (whichcorresponds to the long bond in the tetragonally dis-torted tetrahedron), as explained in the main text andRef. [8]. In the pure bilinear-biquadratic model on thepyrochlore lattice, the colors in the low-temperature ne-matic state are uncorrelated, whereas in the bilinear-biquadratic model with long-range couplings, they arefully ordered. By calculating the correlations between thecolors, we may thus effectively separate the correlationsdue to the nematic and short-range ordered componentsidentified in the second step. Since the diamond latticeformed by the tetrahedra is bipartite, a color correlation function (order parameter) which distinguishes betweensame ( e.g. all R) and different color ( e.g. RG) orders isconsidered sufficient: (cid:104) C C i (cid:105) = (cid:104) C (0) C ( r ) (cid:105) = N bs ( r ) − ¯ p sd N bd ( r ) Z btot (3)where N bs ( r ) and N bd ( r ) are the number of bonds of thesame and different color at distance r , respectively, Z btot is the total number of bonds for that distance, and¯ p sd = (cid:80) c = c (cid:48) ( N tc ) /N tot (cid:80) c (cid:54) = c (cid:48) N tc N tc (cid:48) /N tot (4)is the mean ratio of the probabilities of finding the samecolor tetrahedron to a different color tetrahedron on aparticular bond. The final term is required as the popula-tions of the colors in the simulation box are generally notequal. This correlation function generates (cid:104) C C i (cid:105) = 1 forall shells for a same color order, and an alternation be-tween ± Spectrum simulation from RMC spin map
The Li NMR spectrum of the low temperature phasewas simulated from the spin configurations obtained byRMC. The shape of the powder averaged NMR spectraof magnetic substances is mainly determined by magni-tude of the internal field at the nuclear positions. Toobtain an internal field distribution at each Li site froma given spin map, we summed up the classical dipolefield and transfer hyperfine field from Cr spins withina 100 ˚A radius and from nearest-neighbor Cr sites, re-spectively. The transfer hyperfine coupling constant is0.10 T/ µ B from the 12 nearest-neighbor Cr spins, asestimated from a K - χ plot above 100 K. Figure S4 shows0a histogram of magnitude of the internal field B int calcu-lated from the spin map with ordered moments of 1 µ B per Cr ion. !"! ’ ( ) *+ , - . / ) ( * ) · · ) -12 !3%"!3%!!3&"!3&!!3!"! (*4 )567 FIG. S4. Histogram of magnitude of the internal field at Lisites. The size of ordered moment in the RMC spin map istaken as 1 µ B per Cr ion. I n t e n s it y ( a r b . un it s ) -0.4 -0.2 0 0.2 0.4 B (T) Experimental B = 2 T T = 4.2 K Simulated m = 1.3 m B FIG. S5. Experimental NMR spectrum of the ordered phaseat 4.2 K with pulse separation τ = 7 µ s(black curve) andsimulated spectrum from RMC spin map (red curve). Theunit of the horizontal axis of the experimental one is changedto magnetic field by dividing frequency by the gyromagneticratio of Li. The origin of the horizontal axis corresponds tothe center of gravity of both spectra.
When the external field B ext is sufficiently larger than B int , a rigid AF spin arrangement producing a sin-gle value of B int yields a rectangular NMR spectrumbounded by | B int − B ext | ≤ B ≤ B int + B ext for powdersamples [36] . We obtained our simulated NMR spectrumby piling up rectangle distributions whose half widthswere B int , the horizontal axis of Fig. S4. The height ofeach rectangle was normalized so that its area was pro-portional to the vertical value of each point in Fig. S4.The experimental NMR spectrum at 4.2 K and a sim-ulation from an RMC spin configuration at 1.5 K areshown in Fig S5; the vertical and horizontal scales ofthe simulated spectrum are both adjusted to match thebroad component of the experimental spectrum. Whilethe simulated spectrum appears triangular, the experi-mental one clearly consists of two Gaussian componentswith different linewidth. This discrepancy could be dueto a variety of factors, including the difficulty of fittingthe k = (001) scattering near nuclear positions, the dif-ferent temperatures of the NMR (4 . . spins in the wide com-ponent may be estimated to be 1.3 µ B at 4.2 K. The mo-ment size in the sharper component is more difficult toestimate. However, because the perfect two-color ordersproduce either no field distribution or a relatively narrowrectangular spectrum (the FWHM is 0.1 T at most for afull moment gS = 3 µ B ) depending on the color configu-rations, this contribution can readily be associated withtwo-color order domains. Magnetic susceptibility
The zero field cooled (ZFC) and field cooled (FC) mag-netic susceptibility curves for x = 0 .
05 (Fig. S6) show aclear splitting, indicating spin freezing below T f ; hence,the nematic spin freezing scenario appears likelier thanthe pure nematic transition. Unlike a regular spin glass,the splitting persists up to 5 T. This robustness to-wards applied magnetic field is also observed in materi-als like SrCr p Ga − p O and Y Mo O [21, 22], wherethe ground states are characterised by flat energy land-scapes with shallow minima. Such energy landscapes,and as a result, behaviors of the ZFC and FC suscepti-bility, have also been predicted for the pyrochlore latticewith bilinear-biquadratic interactions and bond disorder[20].1 - T (K)16