Dual Orbital Degeneracy Lifting in a Strongly Correlated Electron System
R. J. Koch, R. Sinclair, M. T. McDonnell, R. Yu, M. Abeykoon, M. G. Tucker, A. M. Tsvelik, S. J. L. Billinge, H. D. Zhou, W.-G. Yin, E. S. Bozin
DDual Orbital Degeneracy Lifting in a Strongly Correlated Electron System
R. J. Koch, ∗ R. Sinclair, M. T. McDonnell, † R. Yu, ‡ M. Abeykoon, M. G. Tucker, A. M. Tsvelik, S. J. L. Billinge,
1, 5
H. D. Zhou, W.-G. Yin, and E. S. Bozin § Condensed Matter Physics and Materials Science Division,Brookhaven National Laboratory, Upton, NY 11973, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Photon Sciences Division, Brookhaven National Laboratory, Upton, NY 11973, USA Department of Applied Physics and Applied Mathematics,Columbia University, New York, NY 10027, USA (Dated: October 1, 2020)The local structure of NaTiSi O is examined across its Ti-dimerization orbital-assisted Peierlstransition at 210 K. An atomic pair distribution function approach evidences local symmetry break-ing preexisting far above the transition. The analysis unravels that on warming the dimers evolveinto a short range orbital degeneracy lifted (ODL) state of dual orbital character, persisting up toat least 490 K. The ODL state is correlated over the length scale spanning ∼ The rich physics associated with the emergence oftechnologically relevant quantum orders in materials [1]stems from complex interaction of electronic charge,spin, and orbitals, and their coupling to the host lat-tice [2, 3]. In transition metal systems with partialfilling of d -manifolds novel properties often engage theorbital sector [4]. Systems exhibiting orbital degener-acy and/or electronic frustration imposed by their lat-tice topology are of particular interest, as orbitals cou-ple both to the spin, via electronic interactions, andto the lattice, via Jahn-Teller type mechanisms [5, 6].The removal of this orbital degeneracy and the subse-quent relief of frustration then impact symmetry low-ering and material properties. The electronic complex-ity of the low temperature ordered symmetry-brokenstates has been thoroughly studied in systems display-ing diverse emergent behaviors such as frustrated mag-netism [7, 8], colossal magnetoresistivity [9], charge andorbital order [10, 11], metal-insulator transition [12–14],pseudogap [15, 16] and high temperature superconductiv-ity [17, 18]. Their understanding employs Fermi surfacenesting [19, 20], Peierls [21, 22], and band Jahn-Tellermechanisms [23, 24], among others.In systems where orbital degeneracies are anticipated,crystallographic symmetry lowering at the temperaturedriven structural phase transitions is often assumed toimply simultaneous orbital degeneracy lifting (ODL) byengaging some cooperative mechanism [5, 6, 25]. Con-sequently, seemingly mundane high temperature regimespossessing high crystallographic symmetry remain muchless explored. In contrast to this concept, recent uti-lization of probes sensitive to local symmetry have qual-ified the ODL as a local electronic effect existing attemperature well above [26, 27] the global symmetrybreaking transitions. Focusing on the CuIr S [26] andMgTi O [27] spinel systems in the weak electron cou-pling limit, the studies unmasked a highly localized ODL state at high temperature involving two transition metalions, which serves as a precursor to an orbitally drivenmetal-insulator transition [28]. Albeit discontinuouslyconnected to the ground state, the ODL in these spinelsproves to be a prerequisite state for charge & orbital orderand spin singlet dimerization observed at low tempera-ture [10, 29, 30], thus enabling the transition. The ODLstate further rationalizes the apparent order of magni-tude discrepancy between the energy scales of the ob-served phenomena ( ∼ ∼ ∼ O clinopyroxene [33], one ofthe rock-forming silicate minerals constituting the upperEarth’s mantle [34]. It is a paramagnetic strongly cor-related Mott insulator with a ∼ (Fig. 1(a), (b)),with Ti in d ( S = 1 /
2) nominally triply degenerate (cid:127)
2g orbital configuration. A nonmagnetic ground state ofNaTiSi O with a 53 meV spin gap [36] establishes oncooling through a 210 K [37] structural transition whereorbital ordering stabilizes intrachain Ti-Ti spin singletdimerization [38]. Once thought to host Haldane S = 1chains [35, 39, 40], NaTiSi O is considered a candidatefor quantum liquid with strong orbital fluctuations [41].By combining neutron and x-ray total scattering basedatomic pair distribution function (nPDF and xPDF) ap-proaches [42] we find compelling local structural evidence a r X i v : . [ c ond - m a t . s t r- e l ] S e p FIG. 1. Properties of NaTiSi O : (a) 2¸c structure; (b)Quasi-1D zigzag TiO chains; (c) Undistorted TiO plaque-ttes of the 2¸c phase featuring uniform Ti-Ti and O-O dis-tances; (d) Distorted TiO plaquettes of the dimerized P1phase with Ti-Ti and O-O distances bifurcated (S=short,L=long); (e) The Curie law subtracted DC magnetic suscep-tibility; (f) The c axis parameter from P1 model fits to thexPDF data; (g)-(i) Temperature evolution of a selected seg-ment of neutron total scattering data. Note: zigzag chainsrun along c axis in 2¸c, and along a axis in P1. for a fluctuating ODL state of dual orbital characterin NaTiSi O at high temperature. The spatial extentof associated short range structural correlations impliesPeierls-like instability at 1/6 filling and a relevance of allthree Ti (cid:127)
2g orbitals in that regime. The PDF observa-tions establish that the ODL phenomenology does extendto materials with strong electron correlations, reinforcingthe notion of its ubiquity. The results account for a num-ber of high temperature anomalies reported in previousstudies of this system [36, 43, 44], and provide new insightinto the understanding of the transition mechanism.Polycrystalline NaTiSi O used in powder diffractionmeasurements was obtained via a solid state route [36,37, 45] and shows a transition to a nonmagnetic state be-low T s = 210 K, Fig. 1(e). Total scattering data for PDFanalysis were collected over 100 K ≤ T ≤
300 K (neutrons),and over 10 K ≤ T ≤
300 K range and at 490 K (x-rays).The approach utilizes both Bragg and diffuse scattering,and provides information on the average structure and onthe local deviations from it [42]. Robust crystallographicsymmetry change at the transition is evident in the x-ray(Fig. 1(f)), and neutron data (Figs. 1(g)-(i)), where thereis an abrupt discontinuity in the number, intensity, andstrength of observed Bragg peaks in reciprocal space. De-tails of data collection and reduction, and PDF analysisprotocols used [42] are provided elsewhere [45].NaTiSi O crystallizes in a monoclinic 2¸c structure,Fig. 1(a), featuring characteristic zigzag chains of edge-sharing TiO octahedra, Fig. 1(b), giving the system aquasi-one-dimensional character [46]. The chains are em-bedded in a somewhat disordered SiO network encom-passing Na [45]. Within the chains, the shared-edge Opairs and Ti centers constitute TiO plaquettes, iden- FIG. 2. Comparison of simulated PDFs: Crystallographic P1(blue) and 2¸c (red) models with their differential (green) offsetfor clarity for neutron probe over a wide (a) and a narrow (b) r range. (c) Neutron Ti-Ti partial PDF for the two models.Corresponding PDFs for x-ray probe are shown in (d)-(f).Simulations use uniform 0.001 ˚A ADPs for all atoms, andare scaled to match the data shown in Fig. 3. tical in 2¸c, which alternate in orientation, as shown inFig. 1(c). Magnetically active Ti have +3 valence in 3 d configuration [47], confirmed by neutron Rietveld re-finement based bond valence sum calculations [45]. Thedominant octahedral crystal field splits the Ti 3 d orbitalsinto a partially filled t g triplet and an empty e g dou-blet [37]. Nominally triply degenerate t g orbitals [5] areoriented toward the TiO edges: xy and zx point towardthe common edges of the zigzag chains, while yz is per-pendicular to the general chain direction (for illustrationsee the top right corner inset in Fig. 5). Partial degener-acy alleviation is expected from slight trigonal distortionof TiO [43], placing the single electron into a two folddegenerate low lying t g -derived ( zx , xy ) doublet [41] andrendering the third ( yz ) orbital inert [36, 43, 48]. Theedge-sharing topology fosters direct ( xy, xy ) and ( zx, zx )overlaps of t g orbitals belonging to neighboring Ti alongthe chains. This promotes Ti-Ti dimerization [43] in theorbitally ordered regime [46] upon cooling below T s , lift-ing the t g degeneracy and lowering the average symme-try to tricilinic (P1) [47].The average structure change observed in diffractionacross the transition is associated with the splitting of Ti-Ti pair distances in the zigzag chains. The dimerizationtakes place within the TiO plaquettes of just one of thetwo available orientations (zig or zag, Fig. 5(b)). Conse-quentially, the Ti-Ti and O-O interatomic distances onthe plaquettes bifurcate, Fig. 1(d): Ti-Ti (3.18 ˚A) andO-O (2.74 ˚A) contacts on the plaquettes in 2¸c become(3.11 ˚A, 3.22 ˚A) and (2.69 ˚A, 2.81 ˚A) in P1, respec-tively, with the short Ti-Ti distance (long O-O distance)on a dimerized plaquette [47] (see Fig. 5(a)). Neighbor-ing TiO plaquettes become inequivalent, reflecting Tidimer and associated bond charge order formation, thusremoving the zx/xy degeneracy. The 2¸c and P1 modelsexplain our neutron Bragg data in the high and low tem-perature regimes, respectively [45]. All Ti sites partici-pate in dimerization in P1 but remain equivalent (+3 va- FIG. 3. Comparison of experimental PDFs: Data at temper-ature below (150 K) and above (230 K) the transition tem-perature, T s , for neutrons (nPDF) over broad (a) and narrow(e) r ranges. Matching X-ray data (xPDF) scaled to nPDFare shown in (c) and (g). Comparison of nPDFs within thesame crystallographic phase, 2¸c, at 230 K and 300 K, is shownin (b) and (f). The same for xPDF is shown in (d) and (h).Differential PDFs, ∆ G ( r ) are shown underneath each data,offset for clarity. The vertical dotted lines in panels (a)-(d)and (e)-(h) correspond to the fifth and the first Ti-Ti near-est neighbor distances along the zigzag chains, respectively,marked also by vertical double arrows as 6 Ti (2 Ti) intra-chain interatomic separations. lence) [45, 47]. The fingerprint of the average structuralchange across T s , simulated from crystallographic dataabove and below the transition [47] for nPDF (Fig. 2(a),(b)) and xPDF (Fig. 2(c), (d)), illustrates the expectedPDF response should the local structure follow the aver-age behavior. In Figs. 2(c) and 2(f) the crystallographi-cally observed Ti-Ti splitting [47] is shown by scattering-weighted partial PDFs, revealing considerably weakersignal in nPDF than in xPDF case. Significantly, thepair contributions to PDF of Ti-Ti when compared to O-O are order of magnitude stronger in xPDF case, whereasin nPDF they are 3 times weaker.While crystallography may seem to dictate that thelifting of Ti orbital degeneracy and associated dimer for-mation occur at T s , the complexity increases when thelocal structure information from PDF data is considered.If we compare the PDF signal from T = 150 K (well below T s ) to that from T = 230 K (just above T s ), the differencesignal ∆ G for interatomic distances r >
15 ˚A is large andsignificant, as is to be expected when passing through astructural transition (Fig. 3(a) and 3(c)). However, andin contrast to the average structure based expectationsshown in Fig. 2, ∆ G is substantially smaller over theshorter distances ( r <
15 ˚A) reflecting local structure, ashighlighted in Fig. 3(e) and 3(g), especially in the nPDFcase, which is less sensitive to Ti.
FIG. 4. The spin-singlet dimer disappearance: Comparisonof xPDF data at T s = 210 K with (a) 90 K and (b) 300 Kdata. Differentials ∆ G = G ( T ) − G ( T s ) are offset for clarity,revealing the spin-singlet signature (shaded signal) for 90 Kset. (c) Waterfall view of the temperature evolution of thexPDF differential using 300 K reference for 90 K ≤ T ≤
300 Krange (∆T= 2 K). Corresponding integrated intensity in therange between vertical black arrows is shown in (d). Dottedgray lines are guides to the eye. Dashed red line in (c) and(d) marks T s . The nearest neighbor Ti-Ti distances from P1-based model (see text) fits over (e) 15 ˚A ≤ r ≤
30 ˚A and (f)1 ˚A ≤ r ≤
15 ˚A ranges. Corresponding r (Ti-Ti) splittingsare shown in (g). (h) The spin-singlet dimer and ODL statessketched as (cid:127)
2g orbital manifold overlaps. The transparencyof turquoise color indicates the bond charge filling, as noted.
In fact, the local ∆ G observed across the transition iscomparable in magnitude to that observed in a 70 K dif-ference ∆ G which is fully above the transition (Fig. 3(e)-(h)), where only small changes due to thermal motionamplitude variations would be expected. While the struc-tural transition associated with the dimer formation isclearly apparent in the average structure, the same can-not be said regarding the local structure, revealing acurious local vs average disparity in NaTiSi O . Thismay suggest that spin singlet dimers do not disassem-ble locally on warming across T s , in contrast to magneticsusceptibility measurements according to which the spinsinglet dimers cannot be retained in the high tempera-ture regime. We argue below that the transition is notof a trivial order-disorder type, as one may deduce fromthe nPDF analysis alone [45], Figs. 3(a) and 3(b), butthat it has an ODL-type character [26] evident from thexPDF data analysis. In contrast to the differential nPDFsignal implying minute change across the transition overthe length scale corresponding to ∼ G signal in xPDF, Figs. 3(c) and 3(d), suggests that somelocal structural modification actually does occur at T s .This motivates a closer look at the temperature re-solved xPDF data. Temperature evolution of the PDFdifferential, ∆ G ( T ), underneath the Ti-Ti PDF peak at ∼ T
30 ˚A range adoptstwo unique Ti-Ti pair distances below T s , and these twodistances become degenerate above T s , Fig. 4(e), consis-tent with dimer elimination. When the average structureportion of the PDF is excluded and 1 ˚A < r <
15 ˚A rangeis used instead, the model structure again adopts twounique Ti-Ti pair distances below T s , but these two dis-tances remain distinct above T s , albeit with significantlyreduced splitting, Fig. 4(f). Thus, at T >T s NaTiSi O shows a regularization of the Ti chains over long struc-tural length scales, but this regularization is not presentlocally. Some degree of degeneracy lifting is apparentabove T s observed up to 300 K (Fig. 4(g)), with split-ting of 0.12(4) ˚A still present in our 490 K xPDF data.The behavior where short spin singlet dimer bondsgive way to longer local-symmetry-breaking transitionmetal contacts upon heating above the crystallographicsymmetry breaking transition is established as a hall-mark of the ODL phenomenology in several spinel dimersystems proximal to a localized-to-itinerant crossover.Initially observed in CuIr S [26], and recently also inMgTi O [27], the ODL state is evidenced in their hightemperature metallic cubic regimes. There, at the metal-insulator transition, the spin singlet dimers comprisedof pairs of strongly bonded holes (in CuIr S ) or elec-trons (in MgTi O ) dismount by means of bond chargetransfer upon warming, and are succeeded in the metallicphase by twice as many spatiotemporally fluctuating sin-gle charge Hund-Mulliken molecular-orbital-like states [5]that lift the (cid:127)
2g degeneracy [26, 27]. The dimer and ODLstates are shown in Fig. 5(f) using energy diagram repre-
FIG. 5. NaTiSi O orbital considerations. (a) TiO dimer-ization plaquettes. (b) The two choices: the zig and thezag. (c) Dimerization of the zx variety within the P1 struc-ture. (d) Uniform chain with degenerate (cid:127)
2g manifolds asportrayed by the 2¸c structure model. (e)
Local model ofthe chain for T ≥ T s featuring ODL states with a 6-Ti pe-riod. (f) Molecular-orbital (MO) view, counterclockwise, ofTi-Ti contacts with degenerate or non-bonding Ti-Ti con-tacts, degeneracy-lifted MO, and dimerized Ti-Ti contacts.In the legend, DEG/NONBOND (Ti ), ODL (1e − per Ti-Ti bond), and DIMER (2e − per Ti-Ti bond). Corner insets:orbital geometry of the (cid:127)
2g manifold (upper right), and yz defect discussed in text (lower right). sentation. The observed high crystallographic symmetryensues from three-dimensional spatiotemporal averaging.In addition to thermal evolution of the transition metalsublattices, the similarity of Ti pyroxenes and the spinelsextends to observed pressure effects. In the spinels, pres-sure increases (cid:127)
2g orbital overlaps and stabilizes the tran-sition [49, 50] and the ODL state [26]. In LiTiSi O ,which is isostructural and isoelectronic to NaTiSi O , ∼ T s also to higher tem-perature [37, 45], further corroborating the equivalenceof the underlying orbital behaviors. Following the spinelODL phenomenology, in NaTiSi O the Ti dimers exhibita “2e-0e”-type bond charge order along the zigzag chainsfor T
2g filling asin NaTiSi O and whose ODL states have a two-orbital(2O-ODL) character [27] sketched in Fig. 4(h), encoun-ters two challenges. First, in NaTiSi O not what is seen experimentally.The PDF observations impose that, similar to the dimerregime, both shorter (ODL) and longer (non-ODL) Ti-Ticontacts along the zigzag chains are present. Second, thethree-dimensional pyrochlore network of corner-sharedtransition metal tetrahedra in the spinels provides a ba-sis for reconciling the discrepancy between the averageand the local behavior. Quasi-one-dimensional topologyof Ti zigzag chains in NaTiSi O necessitates a modifiedscenario for reconciling the different length scales.Both challenges can be addressed by considering pres-ence of “orbital” defects along the chains. The Ti sitesexhibiting either mixed zx/xy orbital character, such asthese depicted in Fig. 5(e), or whose nonbonding yz or-bitals are engaged (bottom right inset to Fig. 5) representpossible defect types. Notably, local structural correla-tions extend to ∼ ∼ O which can be considered tohave dual ( xy,xy )/( zx,zx ) character indicated in Fig. 5(e)by solid blue lines connecting affected Ti sites. In ad-dition to providing entropy driven stabilization, due tolack of direct overlaps with other (cid:127) T >T s regime of NaTiSi O and attributedto various electronic instabilities. Anomalies include un-usual temperature dependence of magnetic susceptibil-ity [37], the lack of recovery in the muon asymmetry atlonger times and lack of sharp change in electronic relax-ation rate λ at T s in µ SR measurements [51], anomalousand unusually broad phonon modes in Raman [43, 48]and neutron scattering [36] and infrared reflectivity [52],glasslike temperature evolution of thermal conductiv-ity [44], as well as anomalous peak broadening in x-raydiffraction [36]. They were assigned to short-range cor-relations enhancing spin singlet dimer fluctuations [51],orbital disorder [43, 48, 52], rapidly fluctuating orbital oc-cupancy [44], and presence of bond disorder due to orbitalfluctuations [36], respectively. Observation of the ODLstate, which is presumably dynamic, provides a concreterationale for their understanding and invites reexamina-tion of the transition mechanism [36, 53, 54]. Such hightemperature anomalies could in fact be indicators of theODL state in a diverse class of transition metal systemswith active orbital sector [44, 55–57], reinforcing the ideaof ubiquitous ODL precursor states, extending the phe-nomenology to strongly correlated electron systems.Work at Brookhaven National Laboratory was sup-ported by U.S. Department of Energy, Office of Science,Office of Basic Energy Sciences (DOE-BES) under con-tract No. DE-SC0012704. R.S. and H.Z. thank the sup-port from the U.S. Department of Energy under awardDE-SC-0020254. Neutron total scattering data were col-lected at the NOMAD beamline (BL-1B) at the Spalla-tion Neutron Source, a U.S. Department of Energy Officeof Science User Facility operated by the Oak Ridge Na- tional Laboratory. X-ray PDF measurements were con-ducted on beamline 28-ID-1 of the National SynchrotronLight Source II, a U.S. Department of Energy (DOE)Office of Science User Facility operated for the DOE Of-fice of Science by Brookhaven National Laboratory underContract No. DE-SC0012704. ∗ [email protected] † Present address: Computer Science and Mathematics Di-vision, Oak Ridge National Laboratory, Oak Ridge, TN37831, USA ‡ Present address: Institute of Physics, Chinese Academyof Science, Beijing 100190, Peoples Republic of China § [email protected][1] Y. Tokura, M. Kawasaki, and N. Nagaosa, Nat. Phys. ,1056 (2017), URL .[2] Editorial, Nat. Phys. , 105 (2016), URL .[3] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida,and J. Zaanen, Nature , 179 (2015), URL .[4] Y. Tokura and N. 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