Two spin-state crystallizations in LaCoO$_{3}$

We report a magnetostriction study of a perovskite $\rm{LaCoO}_{3}$ with our state-of-the-art strain gauge up to 200 T to investigate the interplay between electron correlations and spin crossover. There has been a controversy whether two novel high field phases in $\rm{LaCoO}_{3}$ are originated in crystallizations or Bose-Einstein condensation in spin crossover, which are the manifestations of localization and delocalization in spin states, respectively. We show that both phases are crystallizations rather than a condensation, and that the two crystallizations are different ones, based on our observations that both of the two phases exhibit magnetostriction plateaux, with distinct heights of the plateaux. The crystallizations of spin states have emerged as a manifestation of the localizations and the interactions in spin crossover. The spin crossover reshaped by electron correlations has brought about nontrivial orderings with cooperative lattice modifications.

In coordinated transition metal ions, the balance of Hund's couplings and crystal field splittings gives rise to the spin-state degree of freedom (SSDF). Spin-crossover (SCO) among them occurs with the variations of temperature, pressure, magnetic fields and also with optical excitations [Figs. 1(a)-(c)]. SCO in transition metal oxides is markedly enriched by electron correlations. Nontrivial orders such as spin-state crystallizations (SSC) and excitonic condensation (EC) are anticipated to emerge as manifestations of localizations and delocalizations of the interacting spin-states [1].
Mechanism of the successive SCO in LaCoO 3 has been an open issue for over 6 decades [2,3]. A trivial low-spin (LS: t 6 2g e 0 g , S = 0) band insulator appears below 30 K with a welldefined charge and spin gap [4]. The spin gap collapses with a crossover to a Curie paramagnet in the middle temperature (MT) range of 30-500 K, bewilderingly with a well sustained charge gap. The charge gap successively collapses above 500 K with a further increase of effective magnetic moments [5]. Any conventional models does not coherently describe the successive phase transformations.
Origin of the local moment in the MT range has been controversial between high-spin state (HS: t 4 2g e 2 g , S = 2) and intermediate-spin state (IS: t 5 2g e 1 g , S = 1) [ Fig. 1(b)]. IS is stabilized due to d-p hybridization [6], and is supported by the probes of the orbital degree of freedom [7][8][9][10]. Later, studies oppositely supporting HS outgrown in number [11][12][13][14][15][16][17], re- jecting IS. Meanwhile, a strong repulsive interaction between HS states (Effective HS-LS attraction) has been indicated in the analysis of thermodynamical measurements [18,19]. A microscopic probe of spin-state occupation using a resonant inelastic x-ray scattering (RIXS) also supports the measurement [20]. Such Ising like interactions leads to a long-range order in classical [21] and quantum [22][23][24] calculations of SCO. Though, any HS-LS order or spin-state disproportionation is not experimentally observed in the x-ray diffraction study [25] or by the recent nuclear magnetic resonance in any temperature range [26], respectively. As a solution to the HS-IS controversy, the duality of IS and HS is recently claimed [27,28]. A HS is dual with a pair of IS in a tight binding picture, which may be a root to the indistinguishability between HS and IS in LaCoO 3 . The nearest neighbor ferromagnetic correlation is observed in an inelastic neutron scattering experiment [27]. This supports the existence of such a neighboring pair of IS because a pair of IS tends to couple ferromagnetically in contrast to a pair of HS which tends to couple antiferromagnetically [28]. In actual pseudo-cubic LaCoO 3 , two IS is delocalized in a unit of sevensites on the background of LS which is formed with a center site and tetragonally surrounding 6 site [27]. A calculated result of the effective 28 orbital Hubbard model employed with a density functional theory (DFT) calculation supports the stability of such a unit [27].
Furthermore, it is controversially predicted that the Bose-Einstein condensation (BEC) of excitons may emerge in LaCoO 3 at high magnetic fields [30,31], called EC, as a significant manifestation of the itineracy of IS [32][33][34][35][36][37]. IS is regarded as a triplet exciton created from a LS vacuum. The dispersion of IS excitiation is actually observed in an angle resolved RIXS study, which is also consistent with a simulation based on dynamical mean field theory of hard core boson [28]. EC is characterized by a coherent hybridization of IS and LS, whose wave function is expressed as a |IS + b |LS on a site. In LaCoO 3 , it is speculated that triplet ECs and SSCs are induced in the SCO by a high magnetic field of ∼ 100 T in the analyses of two orbital Hubbard model with the dynamical mean field theory [31] and the mean field approximation of an  effective model in the strong coupling limite [30]. High field experiments can verify these possibilities [ Fig. 1(c)]. Previously, magnetic field induced SCO in LaCoO 3 has been observed at ∼ 65 T below 30 K [38,39] with a small magnetization jump less than 1/4 of the saturation, which we here term α phase. Another phase above 30 K beyond 100 T is present in a B-T phase diagram up to 135 T [40], which we here term β phase. As possible origins of α and β phases, EC and SSC are claimed based on the calculations [30,31]. The experimental verification requires the observation of IS or HS fraction ( f IS,HS ) as a function of magnetic fields. A slope and a plateau of f IS,HS in high fields should indicate a potential EC and a SSC, respectively [30,41], in analogy with a magnetization slope and a plateau indicating a magnon BEC and a magnon crystal in quantum magnets, respectively [42]. Quantitative differences of α and β phases have been elusive in the previous magnetization study [40], because magnetization is not a direct probe of f IS,HS , due to a possible disturbance by antiferromagnetic interactions. Instead, magnetovolume effect is one of the most direct probe of f IS,HS [39,43], considering the strong coupling between the spin-state and its ionic volume. In HS and IS, more extended e g orbitals are occupied leading to a larger volume than LS [See Fig. 1(c)]. We recently devised a state of the art high-speed strain gauge for the use with destructive pulse magnets beyond 100 T with a few µs duration [29], with which we should be able to verify SSCs and ECs in LaCoO 3 .
In this paper, we report magnetostriction measurements of LaCoO 3 up to 200 T by means of a new high-speed magne-tostriction monitor [29]. In both α and β phases, ∆L/L show plateaux, where ∆L/L is ∼ 1.4 times larger in β phase than that in α phase. This indicates that a SSC is preferred over an EC in each of α and β phases. Based on the observations, origins of α and β phases are discussed, considering also the duality of IS and HS, their magnetism and a recent calculation of the spin-state crystals [27].
High magnetic fields up to 200 T are generated with a single turn coil in ISSP Univ. of Tokyo, Japan [44], which is a destructive pulse magnet with a duration of 7 µs. Magnetostriction measurements are performed by means of a high-speed strain gauge of 100 MHz, where an optical fiber with a fiber Bragg grating (FBG) is directly glued onto a sample [29]. Polycrystalline and single crystalline samples of LaCoO 3 [38] have been measured. All the magnetostriction data in the present paper are measured in the longitudinal direction, ∆L B.
Figs. 2(a)-2(c) shows all the results of the magnetostriction measurements for polycrystalline samples of LaCoO 3 . The data for single crystalline samples is summarized in the supplemental material (SM) [45] and partly shown in Fig. 5(a) . The spin-state transitions are indicated by the arrows in Figs. 2(a) and 2(b), which are summarized on the B-T plane in Fig. 3 which is in good agreement with the previous data of the magnetization study [40]. In Fig. 3, with increasing temperature, the transition fields in the up-sweep of pulsed B suddenly increases above 30 K, approaching and exceeding 100 T. This indicates that β and α phases are distinct phases, as is discussed in Ref. [40]. At above 100 K, a sharp increase of the transition fields is found with a slight increment of tem- perature, indicating a flattening of the upper boundary of the β phase at ∼ 120 K. An advantage of ∆L/L for sensing SCO is demonstrated in the temperature dependence of ∆L/L plotted in Fig. 4(a).
∆L/L at 50 T show significant response from 25 K to 100 K with a peak at ∼ 70 K, being consistent with the previous data at 22 T [46]. At above 100 K, ∆L/L at 50 T is vanishing with increasing temperature. In contrast, a significant response of M at 45 T is observed at room temperature (RT) [46]. This indicates that, at RT, the spin moments of the HS or IS is reoriented with the magnetic fields, while f IS,HS does not vary [47]. Below 100 K, on the other hand, ∆L/L increases along with M as shown in Fig. 4(a), indicating that the increase of M accompanies the increase of f IS,HS . These behavior is reproducible with a single ion SCO model [46,48]. As a crude approximation, hereafter in this paper, we regard that ∆L/L ∼ f IS,HS based on above observations. We identify three temperature ranges as (T1) T < 30 K, (T2) 30 K < T < 130 K, (T3) T > 130 K, colored in red, blue, and green, respectively in Fig. 2(c). (T1) and (T2) correspond to temperatures that hosts α and β phases, respectively. A gap like behavior is seen in (T1), whereas a gapless behavior is observed in (T2) and (T3). (T1) and (T2) exhibit plateaux of ∆L/L above 100 T, while it is absent in (T3) even at 150 T. The heights of the plateaux in (T1) and (T2) are plotted in Fig. 4(b), where it is clearly shown that the height is sharply increased by a factor of ∼ 1.4 in (T2) as compared to that in (T1). This supports that α and β phases are distinct phases separated by a first order transition line [The horizontal dashed line in Fig. 3]. In Fig. 4(b), the height of the plateaux in β phase decreases at elevated temperatures, which should originate in the increasing preoccupation of IS or HS at zero fields. Note that single crystalline samples are measured up to 200 T as shown in the inset of Fig. 3 and in the SM [45]. Single crystalline samples show qualitatively similar results and with sharper transitions [45], whose transition points are also plotted in Fig. 3. Note also that the oscillatory features that overlap the plateaux in (T1) and (T2) as shown in Fig.  2(a) should be originated in a shockwave propagating inside the sample as discussed in the SM [45].
First, we discuss that a SSC is favored over a EC in each of α and β phases based on the present observation that each of α and β phase shows a ∆L/L plateau rather than a slope as shown in Figs. 2(a) and 2(c). With EC, spin-states can evolve in a gapless manner with magnetic fields with a |IS + b |LS where a/b can smoothly evolve with B [30,31]. On the other hand, in SSC, f IS,HS is gapfull, resulting in a spin-state plateau [30,31]. This indicates that, macroscopically, the localization dominates over the coherent delocalization of spin-states both in α and β phases.
Secondly, we discuss that α and β phases are distinct SSCs. Based on the presently observed jump in the temperature de- pendence of the heights of the ∆L/L plateaux [ Fig. 4(b)] and the jump of the entrance fields [ Fig. 3] at ∼ 30 K, we conclude that α and β phases are separated with a first order transition line as indicated by a dashed horizontal line in Fig. 3. Thirdly, we discuss the distinct natures of the SSCs in α and β phases. For this, we point out the distinct behavior of ∆L/L and M in α and β phases. In contrast to ∆L/L showing a jump at the α-β phase boundary in Fig. 4(b), magnetization is more smoothly connected as shown in Fig. 4(c), which is indicative of distinct spin-spin interactions in α and β phases. As representatively shown in Figs. 5(a) and 5(b), in β phase, a linear magnetization [40] and a constant ∆L/L is observed with increasing magnetic fields. The behavior infers that, in the SSC of β phase, spin moments are forced to align gradually to the field direction owing to the antiferromagnetic coupling of spins. This idea of antiferromagnetism is also consistent with the increasing M with the increasing temperature as shown in Fig. 4(c), where, with a paramagnetism or a ferromagnetism, an opposite behavior would be expected. On the other hand, in α phase, a constant magnetization [38][39][40] with constant ∆L/L are observed with increasing magnetic fields [39,43] as shown in Figs. 5(a) and 5(b), and also with increasing temperature as shown in Fig. 4(c). In the SSC of α phase, it is inferred that M is saturated with a paramagnetic or a ferromagnetic coupling of spins.
Lastly, based on above arguments, we tentatively propose the structures of the SSCs for α and β phases, which are schematically drawn in Figs. 6(a)-(c). In β phase, a HS-LS SSC with a 2 × 2 × 2 superlattice in a body-centered cubic (bcc) lattice (SSC-β) is anticipated as shown in Fig. 6(a). In α phase, a SSC of a seven-site IS cluster with the same superlattice in a cubic lattice (SSC-α) is anticipated as shown in Fig. 6(b). The two SSCs are connected by the HS-IS duality, where the seven-site IS cluster and a HS are dual, whose process is schematically depicted in Fig. 6(c) in a tight binding view [28]. Actually, in a recent calculation [27], SSCs corresponding to SSC-β and SSC-α are shown to become most stable when the lattice is expanded from the LS phase by 2% and 0.5%, respectively [49]. This is qualitatively in good agreement with the present observation that ∆L/L is 1.4 times larger in β phase than that in α phase. Besides, the SSCs in the calculation [27] corresponding to SSC-β and SSC-α are shown to have antiferromagnetic and ferromagnetic spin couplings, respectively. This is also in good agreement with our anticipation for the magnetic interactions in α and β phase [50]. The observed value of the magnetization in α phase ∼ 0.5 µ B /Co is in good agreement with the saturation of the model SSC-α, 0.5 µ B /Co with g = 2.0. The larger magnetization observed in β phase is also consistent with the model SSC-β. The appearance of SSCs in expanded lattices under high magnetic fields may resemble the case of thin films where various long-range orders emerges in expanded lattices of epitaxial LaCoO 3 [51][52][53].
Note that the feature of the first order transitions does not exclude EC. Actually, a first order transition to EC is demonstrated in a calculation [41]. We comment that both longitudinal and transverse strain should be measured to fully comprehend the magnetovolume effect, such a study is underway. Also, further theoretical evaluation considering further neighbor interactions is needed to verify above arguments.
In conclusion, we have reported magnetostriction measurements of LaCoO 3 up to 200 T by means of a new magnetostriction monitor [29]. α and β phases beyond 100 T are found to exhibit ∆L/L plateaux, which is larger in β phase by a factor of 1.4 than that of the low temperature α phase, indicating that α and β originates in SSCs rather than an EC. Model of SSCs are tentatively proposed based on the observed SCO, considering its magnetism and the duality of HS and IS states. The field induced SCO in LaCoO 3 is reshaped by electron-correlations leading to the crystallizations of the localized and interacting spin-states in the expanded lattice, thereby suppressing the itineracy of IS and the appearance of EC. The emergence of further exotic orders in expanded lattices at higher fields and epitaxial thin films are expected with the interplay of SSDF, electron correlations and the lattice modifications.
We thank T.

II. THE OSCILLATORY FEATURE IN THE MAGNETOSTRICTION DATA
The oscillatory features overlap the plateaux in (T1) and (T2) as shown in Figs. S1(b) (this text) and 2(a) (main text), whose origin is considered to be a shockwave propagating inside the sample. The oscillatory feature after the transition in the down sweep is dumped in (T1) while it is sustained in (T2). The velocity of the shockwave calculated from the sample size of 2.5 mm and the frequency ∼ 0.8 MHz is ∼ 5 km/s, which is roughly the typical sound velocity of TMO [1].
Deduced v has an error bar of 10 % which is too large to extract any information on the elastic properties of α and β phases. Here, we discuss the microscopic mechanism of the inter spin-state interactions and magnetic interactions, which are relevant to the spin-state crystals (SSCs) proposed in the main text. The HS-LS duality proposed in [2,3] is schematically depicted in Fig. S3(a). The a pair of the electron hoppings in the e g and t 2g orbitals transforms a pair of neighboring HS and LS into a pair of neighboring two ISs, and vice versa. The duality stabilizes the energy of the paired ISs and also the pair of HS-LS because of the itineracy of IS as compared to the cases of the completely localized state [3].
In Fig. S3(b), the microscopic mechanism of the ferromagnetic interaction between neighboring ISs in the SSC-α is depicted. The double exchange like interactions between the t 2g orbitals of neighboring ISs ferromagnetically align the neighboring spins via the Hund's coupling.
In Fig. S3(c), the microscopic mechanism of the antiferromagnetic interaction between the next-nearest-neighboring HSs. The 2 step electron hopping from a HS to the next-nearest-neighboring HS via a LS site and 2 more step of returning to the original site act as the superexchange like interactions between the antiferromagnetically aligned next-nearest-neighboring HSs because if the spins are aligned ferromagnetically the 4 step process is prohibited. The kinetic energy gain in the propagation of the electron stabilizes the antiferromagnetism. This may be relevant to the SSC-β proposed in the main text.
In Fig. S3(d), the microscopic origin of the the itineracy of IS is depicted. The IS adjacent to a LS will be delocalized between the sites by hopping of the IS between the sites. The hopping of a IS to the adjacent LS site occurs via a pair of a electron hopping n e g orbital and in t 2g orbital. One IS acts as an itinerant boson on the sea of the LS vacuum in the field theoretical view. The boson in this case is a spin-full exciton, which may condense or crystalized depending on the hopping parameters in the e g and in t 2g orbitals [4].