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LaVO$_3$: a true Kugel-Khomskii system

LaVO$_3$: a true Kugel-Khomskii system LaVO : a true Kugel-Khomskii system 1 1, 2 1, 2 Xue-Jing Zhang, Erik Koch, and Eva Pavarini Institute for Advanced Simulation, Forschungszentrum Julich,  52425 Julich,  Germany JARA High-Performance Computing, 52062, Aachen, Germany. (Dated: September 13, 2022) We show that the t perovskite LaVO , in its orthorhombic phase, is a rare case of a system 2g hosting an orbital-ordering Kugel-Khomskii phase transition, rather than being controlled by the Coulomb-enhanced crystal- eld splitting. We nd that, as a consequence of this, the magnetic transition is close to (and even above) the super-exchange driven orbital-ordering transition, whereas typically magnetism arises at much lower temperatures than orbital ordering. Our results support the experimental scenario of orbital-ordering and G-type spin correlations just above the monoclinic- to-orthorhombic structural change. To explore the e ects of crystal- eld splitting and lling, we compare to YVO and t titanates. In all these materials the crystal- eld is suciently large to 2g suppress the Kugel-Khomskii phase transition. I. INTRODUCTION Almost 50 years ago, Kugel and Khomskii showed in a classic paper that, in strongly-correlated materials, or- bital ordering (OO) can arise from pure super-exchange (SE) interactions [1]. It can, however, also result from the crystal- eld (CF) splitting via a lattice distortion, i.e., from electron-lattice coupling [2]. In typical cases both mechanisms lead to similar types of ordering, so that identifying which one actually drives the transition is a \chicken-and-egg problem" [3]. Despite the intensive search, it has therefore been hard to nd an undisputed realization of a Kugel-Khomskii (KK) system. Initially it was believed that the e perovskites KCuF g 3 and LaMnO could be KK materials [3{6]. In recent years, however, it was proven that neither in these nor in other e systems, super-exchange interactions are strong g FIG. 1. Left: LDA, highest energy crystal- eld (CF) state enough to drive the OO transition alone [7{12]. In fact, j; i . Center and right: DMFT hole orbital at 90 K. CF j; i : experimental structure. j; i : idealized case in order to explain the presence of OO at high tempera- OO KK with no CF splitting. Top line: YVO . Bottom line: LaVO . 3 3 ture, lattice distortions, arising from the Jahn-Teller ef- fect, the Born-Meyer potential or both [13], have to be present. The CF splitting generated by the distortions is then e ectively enhanced by the Coulomb repulsion, type structure) phase. LaVO has drawn a lot of at- which suppresses orbital uctuations [14{16], leading to tention due to its peculiar properties and phase dia- a very robust OO state. The actual form of the ordered gram [18{24]. In the low-temperature monoclinic phase state is then essentially determined by the crystal eld. (T < T  140 K), it exhibits a C-type magnetic struc- str Recently, the case of t perovskites was explored as pos- ture. Due to the small distortions of the VO octahedra, 2g sible alternative [17]. In fact, the CF is typically weaker it was suggested that C-type spin order could arise from in -bond than in -bond materials, while, at the same strong xz=yz orbital uctuations [25]. Calculations ac- time, the orbital degeneracy is larger. This can poten- counting for the GdFeO -type distortions have however tially turn the balance of interactions in favor of SE. shown that orbital uctuations are suppressed below T , str It was indeed shown that T , the critical temperature KK and classical OO is sucient to explain C-type spin order of super-exchange driven orbital ordering, is remarkably [16]. The debate remains open for what concerns the in- large in LaTiO and YTiO , about 300 K [17]. At the triguing phase just above the structural phase transition, 3 3 same time it became, however, also clear that even in 0 T < T < T 145 K. Here thermodynamic anomalies str N these systems OO is dominated by correlation-enhanced and weak magnetic peaks suggest a change in spin-orbital CF splitting rather than the SE interaction. order, either long or short ranged [22{24], the origin of which is, however, unclear. Our results show that the In this paper we identify the rst compound in KK nature of LaVO provides a natural explanation. which KK multi-orbital super-exchange yields an orbital- ordering phase transition at observable temperatures: To understand what makes orthorhombic LaVO spe- the t perovskite LaVO in its orthorhombic (GdFeO - cial, we compare it with the similar but more distorted 3 3 2g arXiv:2209.04912v1 [cond-mat.str-el] 11 Sep 2022 2 YVO and the two iso-structural t titanates. We start 3 YVO YTiO 2g 3 3 OO from the paramagnetic (PM) phase. We show that, for 0.8 the vanadates, T is actually smaller than for titanates, KK OO 0.6 KK CF T  190 K in LaVO and T  150 K in YVO , KK 3 KK 3 KK but at the same time, the crystal eld is weaker, for 0.4 LaVO more so than for YVO . The surprising result 3 3 0.2 of the energy-scale balance is that, while in YVO the occupied state is still, to a large extent, determined by 0 KK the CF splitting, for LaVO it substantially di ers from 3 1 LaVO LaTiO 3 3 CF CF-based predictions (Fig. 1). Staying with LaVO , we ∆φ/∆φ 0.8 OO nd that the actual orbital-ordering temperature, T , OO 0 5 10 is close to T . Even more remarkable is the outcome T/T KK 0.6 KK KK of magnetic calculations. They yield a G-type antifer- OO 0.4 romagnetic (AF) phase with T > T  T > T . N OO KK str This is opposite to what typically happens in orbitally- 0.2 KK ordered materials, in which T is smaller than T , often N OO 0 500 1000 0 500 1000 sizably so. Our results provide a microscopic explanation T (K) T (K) of the spin-orbital correlations found right above T in str experiments [22{24]. FIG. 2. Lines: Orbital polarization p(T ), de ned as p(T )=(1) ((n +n )=2n ), as a function of temperature 1 2 3 T ; here n is the occupation of natural orbitals, with n <n i i+1 i for n=2 and n >n for n=1. Bloch spheres: angles i+1 i II. ORBITAL-ORDERING TRANSITION de ning the least (n=2) or most (n=1) occupied orbital, j; i= sin  cos jxzi+ cos jxyi+ sin  sin jyzi. Empty cir- In order to determine the onset of the super-exchange cles (OO): experimental structure. Filled circles (KK): Ideal- driven orbital-ordering transition, T , we adopt the ized case with no CF splitting. Triangles (CF): crystal- eld KK orbital. Left: vanadates. Right: titanates. Equatorial line: approach pioneered in Refs. [7, 8]. It consist in pro- =90 . Thick-line meridian: =0 . The equivalent solution gressively reducing the static CF splitting to single at j; +i is on the far side of the sphere. For LaVO , an out the e ects of pure super-exchange from those of inset shows explicitly the sudden variation of  across T ; KK Coulomb-enhanced structural distortions. Calculations =  is the di erence with respect to the CF value CF are performed using the local-density approximation plus of . dynamical-mean- eld theory method (LDA+DMFT) [26] for the materials-speci c t Hubbard model 2g XX X (empty circles). For the titanates, the orbital polariza- i;i y 0 0 H = t c c + U n ^ n ^ 0 i m  im" im# tion p (T ) is already very large at 1000 K; instead, in im mm OO 0 0 im ii  mm the vanadates, and LaVO in particular, p (T ) ! 1 3 OO X X at much lower temperatures [16]. The OO ground state, 0 0 0 + (U2JJ )n ^ n ^ (1) ; im im for the t case, is characterized, a given site, by the least 2g 0 0 i m6=m occupied (or hole) natural orbital y y y y 0 0 0 J (c c c c +c c c c ): im " im # im# 0 im " im" im# im" im # j; i = sin  cos jxzi + cos jxyi + sin  sin jyzi; (2) im6=m represented via open circles on the Bloch spheres in the i;i Here t (i 6= i ) are the LDA hopping integrals [27] mm gure. The hole orbitals at the neighboring sites, yield- 0 0 from orbital m on site i to orbital m on site i . They are ing the spatial arrangement of orbitals, can be obtained obtained in a localized Wannier function basis via the lin- from (2) using symmetries [34]. For t systems (2) rep- 2g earized augmented plane wave method [28{30]. The op- resents the most occupied orbital. As the gure shows, erator c (c ) creates (annihilates) an electron with im im in the t case, at any temperature j; i is very close OO 2g spin  in Wannier state m at site i, and n = c c . to j; i , the lowest energy crystal- eld orbital (open im im CF im For the screened Coulomb parameters we use values es- triangle). Switching to t systems, for YVO , the hole 2g tablished in previous works [14{16, 31]: U =5 eV with orbital is j; i  j72 ;1 i, quite close to the crystal- OO J =0:68 eV for YVO and LaVO , and J =0:64 eV for eld state with the highest energy, j; i  j71 ; 9 i. 3 3 CF YTiO and LaTiO . We then solve this model using dy- Up to here, the results conform to the established pic- 3 3 namical mean- eld theory (DMFT). The quantum im- ture: the CF splitting, enhanced by Coulomb repulsion, purity solver is the generalized hybridization-expansion determines the shape of the ordered state [7{12]. continuous-time quantum Monte Carlo method [32], in The conclusion changes dramatically as soon as we the implementation of Refs. [10, 12, 33]. turn to LaVO . Here, p (T ) has a sharp turn upwards 3 OO The main results are summarized in Fig. 2. We start at T T 190 K. Furthermore, lowering the temper- KK OO from the experimental structure, with full CF splitting ature, the hole orbital j; i turns markedly away from OO p (T) p (T) 3 the CF high-energy state, j; i  j142 ; 25 i, showing 150 150 150 150 150 150 150 150 150 150 CF 2 2 2 2 2 2 2 2 2 2 Y d Y d Y d Y d Y d Y d Y d Y d Y d Y d PM PM PM PM PM PM PM PM PM PM AF AF AF AF AF AF AF AF AF AF F F F F F F F F F F that the ordering mechanism works against the crystal- 100 100 100 100 100 100 100 100 100 100 eld splitting. At the lowest temperature we could reach 50 50 50 50 50 50 50 50 50 50 numerically, j; i  j130 ;8 i. This can be seen OO 0 0 0 0 0 0 0 0 0 0 on the Bloch sphere, where the empty circles move away -50 -50 -50 -50 -50 -50 -50 -50 -50 -50 from the triangle as well as in the inset showing the sud- θθθθθθθθθθ =60 =60 =60 =60 =60 =60 =60 =60 =60 =60°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =90 =90 =90 =90 =90 =90 =90 =90 =90 =90°°°°°°°°°° θθθθθθθθθθ =129 =129 =129 =129 =129 =129 =129 =129 =129 =129°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =84 =84 =84 =84 =84 =84 =84 =84 =84 =84°°°°°°°°°° θθθθθθθθθθ =57 =57 =57 =57 =57 =57 =57 =57 =57 =57°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =85 =85 =85 =85 =85 =85 =85 =85 =85 =85°°°°°°°°°° M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 den variation of the most occupied orbital across T . KK Next we analyze the results in the zero CF splitting 150 150 150 150 150 150 150 150 150 150 2 2 2 2 2 2 2 2 2 2 La d La d La d La d La d La d La d La d La d La d θθθθθθθθθθ =135 =135 =135 =135 =135 =135 =135 =135 =135 =135°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =99 =99 =99 =99 =99 =99 =99 =99 =99 =99°°°°°°°°°° limit, which yields the pure Kugel-Khomskii transition. M M M M M M M M M M M M M M M M M M M M 100 100 100 100 100 100 100 100 100 100 The results are shown as lled circles in Fig. 2, and 50 50 50 50 50 50 50 50 50 50 the orbital polarization curve is p (T ). For the ti- KK 0 0 0 0 0 0 0 0 0 0 tanates, p (T ) exhibits a small tail at high tempera- KK ture and then sharply rises at T 300 K; at this tran- -50 -50 -50 -50 -50 -50 -50 -50 -50 -50 KK θθθθθθθθθθ =54 =54 =54 =54 =54 =54 =54 =54 =54 =54°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =112 =112 =112 =112 =112 =112 =112 =112 =112 =112°°°°°°°°°° θθθθθθθθθθ =54 =54 =54 =54 =54 =54 =54 =54 =54 =54°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =104 =104 =104 =104 =104 =104 =104 =104 =104 =104°°°°°°°°°° sition p (T ) has, however, long saturated. For the M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M OO -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 vanadates the rise in p (T ) is much sharper, and at a KK 150 150 150 150 150 150 150 150 150 150 2 2 2 2 2 2 2 2 2 2 markedly lower temperature, T 150 K in YVO and KK 3 cubic d cubic d cubic d cubic d cubic d cubic d cubic d cubic d cubic d cubic d 100 100 100 100 100 100 100 100 100 100 T 190 K in LaVO , while the high-temperature tail KK 3 is virtually absent. Furthermore, for LaVO , the gure 50 50 50 50 50 50 50 50 50 50 shows that p (T )p (T ) for temperatures suciently OO KK 0 0 0 0 0 0 0 0 0 0 below T T . At the same time, decreasing the tem- KK OO -50 -50 -50 -50 -50 -50 -50 -50 -50 -50 perature, the OO hole orbital for the experimental struc- θθθθθθθθθθ =48 =48 =48 =48 =48 =48 =48 =48 =48 =48°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =135 =135 =135 =135 =135 =135 =135 =135 =135 =135°°°°°°°°°° θθθθθθθθθθ =48 =48 =48 =48 =48 =48 =48 =48 =48 =48°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =135 =135 =135 =135 =135 =135 =135 =135 =135 =135°°°°°°°°°° M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 ture (empty circles), rapidly moves towards the KK hole ( lled circles). These results together identify LaVO 150 150 150 150 150 150 150 150 150 150 3 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 120 120 120 120 120 La d La d La d La d La d La d La d La d La d La d 135 135 135 135 30 30 30 30 30 30 30 30 30 as the best known representation of a Kugel-Khomskii 100 100 100 100 100 100 100 100 100 100 150 150 150 45 45 45 45 45 45 45 45 180 180 60 60 60 60 60 60 60 system, i.e., a system hosting an orbital-ordering KK- 90 90 90 90 90 90 50 50 50 50 50 50 50 50 50 50 driven phase transition. The KK super-exchange inter- 0 0 0 0 0 0 0 0 0 0 action both determines the value of T T and pulls KK OO -50 -50 -50 -50 -50 -50 -50 -50 -50 -50 the hole away from the crystal- eld orbital towards the θθθθθθθθθθ =54 =54 =54 =54 =54 =54 =54 =54 =54 =54°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =97 =97 =97 =97 =97 =97 =97 =97 =97 =97°°°°°°°°°° θθθθθθθθθθ =52 =52 =52 =52 =52 =52 =52 =52 =52 =52°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =72 =72 =72 =72 =72 =72 =72 =72 =72 =72°°°°°°°°°° θθθθθθθθθθ =54 =54 =54 =54 =54 =54 =54 =54 =54 =54°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =104 =104 =104 =104 =104 =104 =104 =104 =104 =104°°°°°°°°°° M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 KK orbital. YVO is on the borderline, but still on the 0 0 0 0 0 0 0 0 0 0 ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 2 2 2 2 2 2 2 2 2 2ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 ππππππππππ 0 0 0 0 0 0 0 0 0 0 ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 2 2 2 2 2 2 2 2 2 2ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 ππππππππππ 0 0 0 0 0 0 0 0 0 0 ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 2 2 2 2 2 2 2 2 2 2ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 ππππππππππ side where the Coulomb-enhanced CF interaction dom- θθθθθθθθθθ θθθθθθθθθθ θθθθθθθθθθ inates; that means, the hole stays close to the crystal eld orbital even below T . Furthermore, for YVO , OO 3 FIG. 3. Total super-exchange energy gain for YVO (top the critical temperature T  150 K itself is smaller KK panel) and LaVO (second panel from the top) in the para- that the T  200 K, the temperature at which the str (PM), antiferro- (AF) and ferro-magnetic (F) case, GdFeO - orthorhombic-to-monoclinic phase transition occurs. For type structure. Lines: di erent  values in the (0; ) interval, LaVO the opposite is true (T T >T ). 3 KK OO str see caption. Orange curve:  , yielding the minimum. Third panel from the top: cubic case. Bottom panel: energy gain for the hopping integrals of LaVO , but in a hypothetical t 3 2g con guration. III. SUPER-EXCHANGE HAMILTONIAN AND ENERGY SURFACES Further support for these conclusions comes from the tensor operator only counts the number of electrons on analysis of super-exchange energy-gain surfaces, Fig. 3. the neighboring site, here n=2. The r6=0; r 6=0 quadratic We obtain them adopting the approach recently intro- terms are those that can actually lead to a phase tran- duced in Refs. [17, 35], representing the multi-orbital KK sition. Terms with q=0 are purely paramagnetic, those super-exchange Hamiltonian via its irreducible-tensor de- with r=0 purely paraorbital. In Fig. 3 we show results composition for the ab-initio hopping integrals and, for reference, an idealized cubic case. Comparing the results for the PM XXX 0 0 r;;q ij;q r ; ;q phase with those in Fig. 2, one may see that the angles H =  ^ D  ^   : (3) 0 0 SE q;0 ;0 i r;r  j ;  minimizing E(; ) yield j; i , the state ob- 0 0 M M KK i;j  r;r tained in LDA+DMFT calculations in the zero CF limit. The maximum quadratic SE energy gain, jE( ;  )j, Here r = 0; 1; 2 is the orbital rank and  the associ- M M increases from  35 meV for YVO to  41 meV for ated components; the spin rank is q with components LaVO , to  46 meV for LaTiO and YTiO , explaining . The analytic expressions of the tensor elements can 3 3 3 the progressive increase in T obtained in the DMFT KK be found in Refs. [17, 35]. The terms with q=q =0 and calculations. r=0; r 6=0 (or viceversa) describe the (linear) orbital Zee- man interaction [12]. These contributions behave as a Fig. 4 shows the various contributions to E( ;  ). M M site-dependent crystal- eld splitting, since the r=q=0 The most important are the quadratic SE terms, those Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) YVO LaVO 111111111 Total Cubic 3 3 RRRRRRR::::::: PPPPPPPMMMMMMM 0 AF AF AF AF AF AF mmmmm(((((TTTTT))))) -10 III ::: PPPMMM m(T) -20 -30 000000000.........555555555 IIIIIIIII Quadratic RRRRRRRRR RRRRRRRRR -10 -20 000000000 500 500 500 500 500 500 500 500 500 1000 1000 1000 1000 1000 1000 1000 1000 1000 TTTTTTTTT TTTTTTTTT -30 KKKKKKKKKKKKKKKKKK NNNNNNNNN TTTTTTTTT (((((((((KKKKKKKKK))))))))) FIG. 4. Decomposition of E(; ) at the angles ( ;  ) M M FIG. 5. LaVO , orbital and magnetic transition. Circles: that yield the absolute minimum in the paramagnetic case. orbital polarization. Pentagons: magnetization. Empty sym- Top: result from the total E(; ). Bottom: only quadratic bols: DMFT for experimental structure (R). Full symbols: terms are included. The components are speci ed on the hor- DMFT for an ideal structure without CF splitting (I ). Trian- izontal axis; missing non-diagonal terms can be obtained via 0 0 gle: LDA. Rhombi: angles yielding the maximum SE energy- the transformation r $ r  . gain (AF case), obtained from quadratic terms (dark grey) and linear terms (light grey) only. which can lead to a transition. For YVO several chan- nels have comparable weight { similarly to the titanates 2;xz 2;xz [34]. In contrast, for LaVO , a single term, the  ^  ^ 3 IV. ORBITAL ORDERING AND THE G-TYPE i j interaction, dominates, as in the cubic limit; the total ANTI-FERROMAGNETIC PHASE energy gain from quadratic SE terms is, however, signi - cantly larger than in the cubic limit, since small positive and negative contributions cancel out. To obtain the nal con rmation of the dominant role of super-exchange in LaVO we perform calculations The linear orbital Zeeman terms [34], while not giv- allowing for G-type antiferromagnetism. For conven- ing rise to a phase transition, can generate a nite po- tional orbitally ordered materials, where OO is driven larization tail already for T > T [12], which can ei- KK by the Coulomb-enhanced crystal eld, T  T . In N OO ther assist or hinder the transition. For the titanates LaVO , instead, we nd that T and T are compa- 3 N OO there is a sizable tail in the p (T ) curves in Fig. 2; fur- KK rable. In fact, as illustrated in Fig. 5, the magnetic or- thermore, the linear terms cooperate with the quadratic dering happens already above the orbital ordering tran- terms, favoring the occupation of the same orbital. For sition, T > T  T . To explain this remarkable N OO KK the vanadates, instead, the tail is negligible. The sup- result, we return to the SE energy surfaces in Fig. 3, and pression of the orbital Zeeman e ect turns out to be due compare the PM case, left column, to the AF case, cen- to lling, rather than to the band structure: Assuming ter column. The gure shows that the AF curves are identical hopping integrals in the two families of com- ij 00 ij 00 shifted uniformly downwards. This is due to the paraor- pounds, D (n=1)=D (n=2)=W =V , as de ned 1 1 00;r 00;r bital (r=0) term with spin rank q=1; comparing to the in [35], Table III; the right hand side depends only on bottom row of the gure, one may see that the latter J=U . The prefactor W for the t con guration can be 2g 2 1 is much larger in the t than in the t con guration. 2g 2g sizably larger than the corresponding V for the t case 2g Furthermore, the quadratic energy gain for orbital or- { up to a factor of 10 for realistic J=U values [35]. This dering alone (obtained subtracting the paraorbital term) 2 1 can be seen comparing the La d and d panels in Fig. 2. decreases going from the PM to the AF case; at the same Summarizing, in the PM phase, quadratic SE interac- time, the orbital Zeeman linear terms increase in impor- tions are weaker in the vanadates than in the titanates, tance, but favor    +180 , hence competing with M M and the orbital Zeeman terms are negligible. LaVO is, the quadratic terms. Thus, at a given nite tempera- however, characterized by a very small CF splitting; the ture, with respect to the PM case, the OO hole orbital KK Hamiltonian form is close to the cubic limit, but (orange open circle on the sphere) remains closer to the the distortions actually increase the energy gained from highest-energy CF orbital (triangle). Importantly, we ob- ordering the orbitals. This makes LaVO unique, and re- tain such a behavior only for LaVO ; in the case of YVO , 3 3 3 sults in low-temperature orbital physics being controlled for the experimental structure we do not nd any mag- by super-exchange interactions. netic phase above T or T , in line with experiments. str KK ΔE (meV) ΔE (meV) 0s 1z 0s 1z 0s 1x 0s 1x 2 2 2 2 0s 2(3z - r ) 0s 2(3z - r ) 0s 2xz 0s 2xz 2 2 2 2 0s 2(x - y ) 0s 2(x - y ) 1z 1z 1z 1z 1z 1x 1z 1x 2 2 2 2 1z 2(3z - r ) 1z 2(3z - r ) 1z 2xz 1z 2xz 2 2 2 2 1z 2(x - y ) 1z 2(x - y ) 1x 1x 1x 1x 2 2 2 2 1x 2(3z - r ) 1x 2(3z - r ) 1x 2xz 1x 2xz 2 2 2 2 1x 2(x - y ) 1x 2(x - y ) 2 2 2 2 2 2 2 2 2(3z - r ) 2(3z - r ) 2(3z - r ) 2(3z - r ) 2 2 2 2 2(3z - r ) 2xz 2(3z - r ) 2xz 2 2 2 2 2 2 2 2 2(3z - r ) 2(x - y ) 2(3z - r ) 2(x - y ) 2xz 2xz 2xz 2xz 2 2 2 2 2xz 2(x - y ) 2xz 2(x - y ) 2 2 2 2 2 2 2 2 2(x - y ) 2(x -y ) 2(x - y ) 2(x -y ) ppppppppp (((((((((DDDDDDDDDMMMMMMMMMFFFFFFFFFTTTTTTTTT))))))))) 5 V. CONCLUSION Taking into account that DMFT, as all mean- eld theo- ries, somewhat overestimates ordering temperatures, our results provide a natural explanation for the proposed In conclusion, we have identi ed orthorhombic LaVO picture of orbital order and weak G-type magnetism (or as a rare case of a system hosting a Kugel-Khomskii short range spin-orbital order) right above the structural orbital-ordering transition. The relevance of SE in t 2g orthorhombic-to-monoclinic transition [22{24]. vanadates has been previously suggested based on ide- alized model calculations [25], however, within a strong xz=yz uctuations picture. In fact, we have shown that orbital uctuations, large at room temperature, are sup- ACKNOWLEDGMENTS pressed when approaching T . Furthermore, we have str shown that SE is key only for LaVO . In all other cases considered, the conventional picture of the Coulomb- We would like to acknowledge computational time on enhanced CF splitting applies. This is further con rmed JURECA and the RWTH-Aachen cluster via JARA, by the fact that T  T only in orthorhombic LaVO . which was used for the actual computations, as well as N OO 3 More strikingly, we nd that T > T  T , op- computational time on JUWELS, which was used in par- N KK OO posite to what happens in conventional OO materials. ticular for code development. [1] K. I. Kugel' and D. I. Khomskii, Zh. Eksp. Teor. Fiz. 64, [19] S. Miyasaka, Y. Okimoto, M. Iwama, and Y. Tokura, 1429 (1973) [Sov. Phys. JETP 37, 725 (1973)]. Phys. Rev. B 68, 100406(R) (2003). [2] J. Kanamori, J. Appl. Phys. 31, S14 (1960). [20] J.-Q. Yan, J.S. Zhou, and J.B. Goodenough, Phys. Rev. [3] D. I. Khomskii, Transition metal compounds (Cambridge Lett. 93, 235901 (2004). University Press 2014), p. 204. [21] J.-Q. Yan, J.-S. Zhou, J.G. Cheng, J.B. Goodenough, Y. [4] A.I. Liechtenstein, V.I. Anisimov, and J. Zaanen, Phys. Ren, A. Llobet, and R.J. McQueeney, Phys. Rev. B 84, Rev. 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LaVO$_3$: a true Kugel-Khomskii system

Zhang, Xue-Jing; Koch, Erik; Pavarini, Eva
Condensed Matter , Volume 2022 (2209) – Sep 11, 2022
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LaVO : a true Kugel-Khomskii system 1 1, 2 1, 2 Xue-Jing Zhang, Erik Koch, and Eva Pavarini Institute for Advanced Simulation, Forschungszentrum Julich,  52425 Julich,  Germany JARA High-Performance Computing, 52062, Aachen, Germany. (Dated: September 13, 2022) We show that the t perovskite LaVO , in its orthorhombic phase, is a rare case of a system 2g hosting an orbital-ordering Kugel-Khomskii phase transition, rather than being controlled by the Coulomb-enhanced crystal- eld splitting. We nd that, as a consequence of this, the magnetic transition is close to (and even above) the super-exchange driven orbital-ordering transition, whereas typically magnetism arises at much lower temperatures than orbital ordering. Our results support the experimental scenario of orbital-ordering and G-type spin correlations just above the monoclinic- to-orthorhombic structural change. To explore the e ects of crystal- eld splitting and lling, we compare to YVO and t titanates. In all these materials the crystal- eld is suciently large to 2g suppress the Kugel-Khomskii phase transition. I. INTRODUCTION Almost 50 years ago, Kugel and Khomskii showed in a classic paper that, in strongly-correlated materials, or- bital ordering (OO) can arise from pure super-exchange (SE) interactions [1]. It can, however, also result from the crystal- eld (CF) splitting via a lattice distortion, i.e., from electron-lattice coupling [2]. In typical cases both mechanisms lead to similar types of ordering, so that identifying which one actually drives the transition is a \chicken-and-egg problem" [3]. Despite the intensive search, it has therefore been hard to nd an undisputed realization of a Kugel-Khomskii (KK) system. Initially it was believed that the e perovskites KCuF g 3 and LaMnO could be KK materials [3{6]. In recent years, however, it was proven that neither in these nor in other e systems, super-exchange interactions are strong g FIG. 1. Left: LDA, highest energy crystal- eld (CF) state enough to drive the OO transition alone [7{12]. In fact, j; i . Center and right: DMFT hole orbital at 90 K. CF j; i : experimental structure. j; i : idealized case in order to explain the presence of OO at high tempera- OO KK with no CF splitting. Top line: YVO . Bottom line: LaVO . 3 3 ture, lattice distortions, arising from the Jahn-Teller ef- fect, the Born-Meyer potential or both [13], have to be present. The CF splitting generated by the distortions is then e ectively enhanced by the Coulomb repulsion, type structure) phase. LaVO has drawn a lot of at- which suppresses orbital uctuations [14{16], leading to tention due to its peculiar properties and phase dia- a very robust OO state. The actual form of the ordered gram [18{24]. In the low-temperature monoclinic phase state is then essentially determined by the crystal eld. (T < T  140 K), it exhibits a C-type magnetic struc- str Recently, the case of t perovskites was explored as pos- ture. Due to the small distortions of the VO octahedra, 2g sible alternative [17]. In fact, the CF is typically weaker it was suggested that C-type spin order could arise from in -bond than in -bond materials, while, at the same strong xz=yz orbital uctuations [25]. Calculations ac- time, the orbital degeneracy is larger. This can poten- counting for the GdFeO -type distortions have however tially turn the balance of interactions in favor of SE. shown that orbital uctuations are suppressed below T , str It was indeed shown that T , the critical temperature KK and classical OO is sucient to explain C-type spin order of super-exchange driven orbital ordering, is remarkably [16]. The debate remains open for what concerns the in- large in LaTiO and YTiO , about 300 K [17]. At the triguing phase just above the structural phase transition, 3 3 same time it became, however, also clear that even in 0 T < T < T 145 K. Here thermodynamic anomalies str N these systems OO is dominated by correlation-enhanced and weak magnetic peaks suggest a change in spin-orbital CF splitting rather than the SE interaction. order, either long or short ranged [22{24], the origin of which is, however, unclear. Our results show that the In this paper we identify the rst compound in KK nature of LaVO provides a natural explanation. which KK multi-orbital super-exchange yields an orbital- ordering phase transition at observable temperatures: To understand what makes orthorhombic LaVO spe- the t perovskite LaVO in its orthorhombic (GdFeO - cial, we compare it with the similar but more distorted 3 3 2g arXiv:2209.04912v1 [cond-mat.str-el] 11 Sep 2022 2 YVO and the two iso-structural t titanates. We start 3 YVO YTiO 2g 3 3 OO from the paramagnetic (PM) phase. We show that, for 0.8 the vanadates, T is actually smaller than for titanates, KK OO 0.6 KK CF T  190 K in LaVO and T  150 K in YVO , KK 3 KK 3 KK but at the same time, the crystal eld is weaker, for 0.4 LaVO more so than for YVO . The surprising result 3 3 0.2 of the energy-scale balance is that, while in YVO the occupied state is still, to a large extent, determined by 0 KK the CF splitting, for LaVO it substantially di ers from 3 1 LaVO LaTiO 3 3 CF CF-based predictions (Fig. 1). Staying with LaVO , we ∆φ/∆φ 0.8 OO nd that the actual orbital-ordering temperature, T , OO 0 5 10 is close to T . Even more remarkable is the outcome T/T KK 0.6 KK KK of magnetic calculations. They yield a G-type antifer- OO 0.4 romagnetic (AF) phase with T > T  T > T . N OO KK str This is opposite to what typically happens in orbitally- 0.2 KK ordered materials, in which T is smaller than T , often N OO 0 500 1000 0 500 1000 sizably so. Our results provide a microscopic explanation T (K) T (K) of the spin-orbital correlations found right above T in str experiments [22{24]. FIG. 2. Lines: Orbital polarization p(T ), de ned as p(T )=(1) ((n +n )=2n ), as a function of temperature 1 2 3 T ; here n is the occupation of natural orbitals, with n <n i i+1 i for n=2 and n >n for n=1. Bloch spheres: angles i+1 i II. ORBITAL-ORDERING TRANSITION de ning the least (n=2) or most (n=1) occupied orbital, j; i= sin  cos jxzi+ cos jxyi+ sin  sin jyzi. Empty cir- In order to determine the onset of the super-exchange cles (OO): experimental structure. Filled circles (KK): Ideal- driven orbital-ordering transition, T , we adopt the ized case with no CF splitting. Triangles (CF): crystal- eld KK orbital. Left: vanadates. Right: titanates. Equatorial line: approach pioneered in Refs. [7, 8]. It consist in pro- =90 . Thick-line meridian: =0 . The equivalent solution gressively reducing the static CF splitting to single at j; +i is on the far side of the sphere. For LaVO , an out the e ects of pure super-exchange from those of inset shows explicitly the sudden variation of  across T ; KK Coulomb-enhanced structural distortions. Calculations =  is the di erence with respect to the CF value CF are performed using the local-density approximation plus of . dynamical-mean- eld theory method (LDA+DMFT) [26] for the materials-speci c t Hubbard model 2g XX X (empty circles). For the titanates, the orbital polariza- i;i y 0 0 H = t c c + U n ^ n ^ 0 i m  im" im# tion p (T ) is already very large at 1000 K; instead, in im mm OO 0 0 im ii  mm the vanadates, and LaVO in particular, p (T ) ! 1 3 OO X X at much lower temperatures [16]. The OO ground state, 0 0 0 + (U2JJ )n ^ n ^ (1) ; im im for the t case, is characterized, a given site, by the least 2g 0 0 i m6=m occupied (or hole) natural orbital y y y y 0 0 0 J (c c c c +c c c c ): im " im # im# 0 im " im" im# im" im # j; i = sin  cos jxzi + cos jxyi + sin  sin jyzi; (2) im6=m represented via open circles on the Bloch spheres in the i;i Here t (i 6= i ) are the LDA hopping integrals [27] mm gure. The hole orbitals at the neighboring sites, yield- 0 0 from orbital m on site i to orbital m on site i . They are ing the spatial arrangement of orbitals, can be obtained obtained in a localized Wannier function basis via the lin- from (2) using symmetries [34]. For t systems (2) rep- 2g earized augmented plane wave method [28{30]. The op- resents the most occupied orbital. As the gure shows, erator c (c ) creates (annihilates) an electron with im im in the t case, at any temperature j; i is very close OO 2g spin  in Wannier state m at site i, and n = c c . to j; i , the lowest energy crystal- eld orbital (open im im CF im For the screened Coulomb parameters we use values es- triangle). Switching to t systems, for YVO , the hole 2g tablished in previous works [14{16, 31]: U =5 eV with orbital is j; i  j72 ;1 i, quite close to the crystal- OO J =0:68 eV for YVO and LaVO , and J =0:64 eV for eld state with the highest energy, j; i  j71 ; 9 i. 3 3 CF YTiO and LaTiO . We then solve this model using dy- Up to here, the results conform to the established pic- 3 3 namical mean- eld theory (DMFT). The quantum im- ture: the CF splitting, enhanced by Coulomb repulsion, purity solver is the generalized hybridization-expansion determines the shape of the ordered state [7{12]. continuous-time quantum Monte Carlo method [32], in The conclusion changes dramatically as soon as we the implementation of Refs. [10, 12, 33]. turn to LaVO . Here, p (T ) has a sharp turn upwards 3 OO The main results are summarized in Fig. 2. We start at T T 190 K. Furthermore, lowering the temper- KK OO from the experimental structure, with full CF splitting ature, the hole orbital j; i turns markedly away from OO p (T) p (T) 3 the CF high-energy state, j; i  j142 ; 25 i, showing 150 150 150 150 150 150 150 150 150 150 CF 2 2 2 2 2 2 2 2 2 2 Y d Y d Y d Y d Y d Y d Y d Y d Y d Y d PM PM PM PM PM PM PM PM PM PM AF AF AF AF AF AF AF AF AF AF F F F F F F F F F F that the ordering mechanism works against the crystal- 100 100 100 100 100 100 100 100 100 100 eld splitting. At the lowest temperature we could reach 50 50 50 50 50 50 50 50 50 50 numerically, j; i  j130 ;8 i. This can be seen OO 0 0 0 0 0 0 0 0 0 0 on the Bloch sphere, where the empty circles move away -50 -50 -50 -50 -50 -50 -50 -50 -50 -50 from the triangle as well as in the inset showing the sud- θθθθθθθθθθ =60 =60 =60 =60 =60 =60 =60 =60 =60 =60°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =90 =90 =90 =90 =90 =90 =90 =90 =90 =90°°°°°°°°°° θθθθθθθθθθ =129 =129 =129 =129 =129 =129 =129 =129 =129 =129°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =84 =84 =84 =84 =84 =84 =84 =84 =84 =84°°°°°°°°°° θθθθθθθθθθ =57 =57 =57 =57 =57 =57 =57 =57 =57 =57°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =85 =85 =85 =85 =85 =85 =85 =85 =85 =85°°°°°°°°°° M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 den variation of the most occupied orbital across T . KK Next we analyze the results in the zero CF splitting 150 150 150 150 150 150 150 150 150 150 2 2 2 2 2 2 2 2 2 2 La d La d La d La d La d La d La d La d La d La d θθθθθθθθθθ =135 =135 =135 =135 =135 =135 =135 =135 =135 =135°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =99 =99 =99 =99 =99 =99 =99 =99 =99 =99°°°°°°°°°° limit, which yields the pure Kugel-Khomskii transition. M M M M M M M M M M M M M M M M M M M M 100 100 100 100 100 100 100 100 100 100 The results are shown as lled circles in Fig. 2, and 50 50 50 50 50 50 50 50 50 50 the orbital polarization curve is p (T ). For the ti- KK 0 0 0 0 0 0 0 0 0 0 tanates, p (T ) exhibits a small tail at high tempera- KK ture and then sharply rises at T 300 K; at this tran- -50 -50 -50 -50 -50 -50 -50 -50 -50 -50 KK θθθθθθθθθθ =54 =54 =54 =54 =54 =54 =54 =54 =54 =54°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =112 =112 =112 =112 =112 =112 =112 =112 =112 =112°°°°°°°°°° θθθθθθθθθθ =54 =54 =54 =54 =54 =54 =54 =54 =54 =54°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =104 =104 =104 =104 =104 =104 =104 =104 =104 =104°°°°°°°°°° sition p (T ) has, however, long saturated. For the M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M OO -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 vanadates the rise in p (T ) is much sharper, and at a KK 150 150 150 150 150 150 150 150 150 150 2 2 2 2 2 2 2 2 2 2 markedly lower temperature, T 150 K in YVO and KK 3 cubic d cubic d cubic d cubic d cubic d cubic d cubic d cubic d cubic d cubic d 100 100 100 100 100 100 100 100 100 100 T 190 K in LaVO , while the high-temperature tail KK 3 is virtually absent. Furthermore, for LaVO , the gure 50 50 50 50 50 50 50 50 50 50 shows that p (T )p (T ) for temperatures suciently OO KK 0 0 0 0 0 0 0 0 0 0 below T T . At the same time, decreasing the tem- KK OO -50 -50 -50 -50 -50 -50 -50 -50 -50 -50 perature, the OO hole orbital for the experimental struc- θθθθθθθθθθ =48 =48 =48 =48 =48 =48 =48 =48 =48 =48°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =135 =135 =135 =135 =135 =135 =135 =135 =135 =135°°°°°°°°°° θθθθθθθθθθ =48 =48 =48 =48 =48 =48 =48 =48 =48 =48°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =135 =135 =135 =135 =135 =135 =135 =135 =135 =135°°°°°°°°°° M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 ture (empty circles), rapidly moves towards the KK hole ( lled circles). These results together identify LaVO 150 150 150 150 150 150 150 150 150 150 3 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 120 120 120 120 120 La d La d La d La d La d La d La d La d La d La d 135 135 135 135 30 30 30 30 30 30 30 30 30 as the best known representation of a Kugel-Khomskii 100 100 100 100 100 100 100 100 100 100 150 150 150 45 45 45 45 45 45 45 45 180 180 60 60 60 60 60 60 60 system, i.e., a system hosting an orbital-ordering KK- 90 90 90 90 90 90 50 50 50 50 50 50 50 50 50 50 driven phase transition. The KK super-exchange inter- 0 0 0 0 0 0 0 0 0 0 action both determines the value of T T and pulls KK OO -50 -50 -50 -50 -50 -50 -50 -50 -50 -50 the hole away from the crystal- eld orbital towards the θθθθθθθθθθ =54 =54 =54 =54 =54 =54 =54 =54 =54 =54°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =97 =97 =97 =97 =97 =97 =97 =97 =97 =97°°°°°°°°°° θθθθθθθθθθ =52 =52 =52 =52 =52 =52 =52 =52 =52 =52°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =72 =72 =72 =72 =72 =72 =72 =72 =72 =72°°°°°°°°°° θθθθθθθθθθ =54 =54 =54 =54 =54 =54 =54 =54 =54 =54°°°°°°°°°°, , , , , , , , , , Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ =104 =104 =104 =104 =104 =104 =104 =104 =104 =104°°°°°°°°°° M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M -100 -100 -100 -100 -100 -100 -100 -100 -100 -100 KK orbital. YVO is on the borderline, but still on the 0 0 0 0 0 0 0 0 0 0 ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 2 2 2 2 2 2 2 2 2 2ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 ππππππππππ 0 0 0 0 0 0 0 0 0 0 ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 2 2 2 2 2 2 2 2 2 2ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 ππππππππππ 0 0 0 0 0 0 0 0 0 0 ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 2 2 2 2 2 2 2 2 2 2ππππππππππ/3 /3 /3 /3 /3 /3 /3 /3 /3 /3 ππππππππππ side where the Coulomb-enhanced CF interaction dom- θθθθθθθθθθ θθθθθθθθθθ θθθθθθθθθθ inates; that means, the hole stays close to the crystal eld orbital even below T . Furthermore, for YVO , OO 3 FIG. 3. Total super-exchange energy gain for YVO (top the critical temperature T  150 K itself is smaller KK panel) and LaVO (second panel from the top) in the para- that the T  200 K, the temperature at which the str (PM), antiferro- (AF) and ferro-magnetic (F) case, GdFeO - orthorhombic-to-monoclinic phase transition occurs. For type structure. Lines: di erent  values in the (0; ) interval, LaVO the opposite is true (T T >T ). 3 KK OO str see caption. Orange curve:  , yielding the minimum. Third panel from the top: cubic case. Bottom panel: energy gain for the hopping integrals of LaVO , but in a hypothetical t 3 2g con guration. III. SUPER-EXCHANGE HAMILTONIAN AND ENERGY SURFACES Further support for these conclusions comes from the tensor operator only counts the number of electrons on analysis of super-exchange energy-gain surfaces, Fig. 3. the neighboring site, here n=2. The r6=0; r 6=0 quadratic We obtain them adopting the approach recently intro- terms are those that can actually lead to a phase tran- duced in Refs. [17, 35], representing the multi-orbital KK sition. Terms with q=0 are purely paramagnetic, those super-exchange Hamiltonian via its irreducible-tensor de- with r=0 purely paraorbital. In Fig. 3 we show results composition for the ab-initio hopping integrals and, for reference, an idealized cubic case. Comparing the results for the PM XXX 0 0 r;;q ij;q r ; ;q phase with those in Fig. 2, one may see that the angles H =  ^ D  ^   : (3) 0 0 SE q;0 ;0 i r;r  j ;  minimizing E(; ) yield j; i , the state ob- 0 0 M M KK i;j  r;r tained in LDA+DMFT calculations in the zero CF limit. The maximum quadratic SE energy gain, jE( ;  )j, Here r = 0; 1; 2 is the orbital rank and  the associ- M M increases from  35 meV for YVO to  41 meV for ated components; the spin rank is q with components LaVO , to  46 meV for LaTiO and YTiO , explaining . The analytic expressions of the tensor elements can 3 3 3 the progressive increase in T obtained in the DMFT KK be found in Refs. [17, 35]. The terms with q=q =0 and calculations. r=0; r 6=0 (or viceversa) describe the (linear) orbital Zee- man interaction [12]. These contributions behave as a Fig. 4 shows the various contributions to E( ;  ). M M site-dependent crystal- eld splitting, since the r=q=0 The most important are the quadratic SE terms, those Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) E (meV) YVO LaVO 111111111 Total Cubic 3 3 RRRRRRR::::::: PPPPPPPMMMMMMM 0 AF AF AF AF AF AF mmmmm(((((TTTTT))))) -10 III ::: PPPMMM m(T) -20 -30 000000000.........555555555 IIIIIIIII Quadratic RRRRRRRRR RRRRRRRRR -10 -20 000000000 500 500 500 500 500 500 500 500 500 1000 1000 1000 1000 1000 1000 1000 1000 1000 TTTTTTTTT TTTTTTTTT -30 KKKKKKKKKKKKKKKKKK NNNNNNNNN TTTTTTTTT (((((((((KKKKKKKKK))))))))) FIG. 4. Decomposition of E(; ) at the angles ( ;  ) M M FIG. 5. LaVO , orbital and magnetic transition. Circles: that yield the absolute minimum in the paramagnetic case. orbital polarization. Pentagons: magnetization. Empty sym- Top: result from the total E(; ). Bottom: only quadratic bols: DMFT for experimental structure (R). Full symbols: terms are included. The components are speci ed on the hor- DMFT for an ideal structure without CF splitting (I ). Trian- izontal axis; missing non-diagonal terms can be obtained via 0 0 gle: LDA. Rhombi: angles yielding the maximum SE energy- the transformation r $ r  . gain (AF case), obtained from quadratic terms (dark grey) and linear terms (light grey) only. which can lead to a transition. For YVO several chan- nels have comparable weight { similarly to the titanates 2;xz 2;xz [34]. In contrast, for LaVO , a single term, the  ^  ^ 3 IV. ORBITAL ORDERING AND THE G-TYPE i j interaction, dominates, as in the cubic limit; the total ANTI-FERROMAGNETIC PHASE energy gain from quadratic SE terms is, however, signi - cantly larger than in the cubic limit, since small positive and negative contributions cancel out. To obtain the nal con rmation of the dominant role of super-exchange in LaVO we perform calculations The linear orbital Zeeman terms [34], while not giv- allowing for G-type antiferromagnetism. For conven- ing rise to a phase transition, can generate a nite po- tional orbitally ordered materials, where OO is driven larization tail already for T > T [12], which can ei- KK by the Coulomb-enhanced crystal eld, T  T . In N OO ther assist or hinder the transition. For the titanates LaVO , instead, we nd that T and T are compa- 3 N OO there is a sizable tail in the p (T ) curves in Fig. 2; fur- KK rable. In fact, as illustrated in Fig. 5, the magnetic or- thermore, the linear terms cooperate with the quadratic dering happens already above the orbital ordering tran- terms, favoring the occupation of the same orbital. For sition, T > T  T . To explain this remarkable N OO KK the vanadates, instead, the tail is negligible. The sup- result, we return to the SE energy surfaces in Fig. 3, and pression of the orbital Zeeman e ect turns out to be due compare the PM case, left column, to the AF case, cen- to lling, rather than to the band structure: Assuming ter column. The gure shows that the AF curves are identical hopping integrals in the two families of com- ij 00 ij 00 shifted uniformly downwards. This is due to the paraor- pounds, D (n=1)=D (n=2)=W =V , as de ned 1 1 00;r 00;r bital (r=0) term with spin rank q=1; comparing to the in [35], Table III; the right hand side depends only on bottom row of the gure, one may see that the latter J=U . The prefactor W for the t con guration can be 2g 2 1 is much larger in the t than in the t con guration. 2g 2g sizably larger than the corresponding V for the t case 2g Furthermore, the quadratic energy gain for orbital or- { up to a factor of 10 for realistic J=U values [35]. This dering alone (obtained subtracting the paraorbital term) 2 1 can be seen comparing the La d and d panels in Fig. 2. decreases going from the PM to the AF case; at the same Summarizing, in the PM phase, quadratic SE interac- time, the orbital Zeeman linear terms increase in impor- tions are weaker in the vanadates than in the titanates, tance, but favor    +180 , hence competing with M M and the orbital Zeeman terms are negligible. LaVO is, the quadratic terms. Thus, at a given nite tempera- however, characterized by a very small CF splitting; the ture, with respect to the PM case, the OO hole orbital KK Hamiltonian form is close to the cubic limit, but (orange open circle on the sphere) remains closer to the the distortions actually increase the energy gained from highest-energy CF orbital (triangle). Importantly, we ob- ordering the orbitals. This makes LaVO unique, and re- tain such a behavior only for LaVO ; in the case of YVO , 3 3 3 sults in low-temperature orbital physics being controlled for the experimental structure we do not nd any mag- by super-exchange interactions. netic phase above T or T , in line with experiments. str KK ΔE (meV) ΔE (meV) 0s 1z 0s 1z 0s 1x 0s 1x 2 2 2 2 0s 2(3z - r ) 0s 2(3z - r ) 0s 2xz 0s 2xz 2 2 2 2 0s 2(x - y ) 0s 2(x - y ) 1z 1z 1z 1z 1z 1x 1z 1x 2 2 2 2 1z 2(3z - r ) 1z 2(3z - r ) 1z 2xz 1z 2xz 2 2 2 2 1z 2(x - y ) 1z 2(x - y ) 1x 1x 1x 1x 2 2 2 2 1x 2(3z - r ) 1x 2(3z - r ) 1x 2xz 1x 2xz 2 2 2 2 1x 2(x - y ) 1x 2(x - y ) 2 2 2 2 2 2 2 2 2(3z - r ) 2(3z - r ) 2(3z - r ) 2(3z - r ) 2 2 2 2 2(3z - r ) 2xz 2(3z - r ) 2xz 2 2 2 2 2 2 2 2 2(3z - r ) 2(x - y ) 2(3z - r ) 2(x - y ) 2xz 2xz 2xz 2xz 2 2 2 2 2xz 2(x - y ) 2xz 2(x - y ) 2 2 2 2 2 2 2 2 2(x - y ) 2(x -y ) 2(x - y ) 2(x -y ) ppppppppp (((((((((DDDDDDDDDMMMMMMMMMFFFFFFFFFTTTTTTTTT))))))))) 5 V. CONCLUSION Taking into account that DMFT, as all mean- eld theo- ries, somewhat overestimates ordering temperatures, our results provide a natural explanation for the proposed In conclusion, we have identi ed orthorhombic LaVO picture of orbital order and weak G-type magnetism (or as a rare case of a system hosting a Kugel-Khomskii short range spin-orbital order) right above the structural orbital-ordering transition. The relevance of SE in t 2g orthorhombic-to-monoclinic transition [22{24]. vanadates has been previously suggested based on ide- alized model calculations [25], however, within a strong xz=yz uctuations picture. In fact, we have shown that orbital uctuations, large at room temperature, are sup- ACKNOWLEDGMENTS pressed when approaching T . Furthermore, we have str shown that SE is key only for LaVO . In all other cases considered, the conventional picture of the Coulomb- We would like to acknowledge computational time on enhanced CF splitting applies. 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Published: Sep 11, 2022

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