Signature of spin-triplet exciton condensations in LaCoO3 at ultrahigh magnetic fields up to 600 T

Bose-Einstein condensation of electron-hole pairs, exciton condensation, has been effortfully investigated since predicted 60 years ago. Irrefutable evidence has still been lacking due to experimental difficulties in verifying the condensation of the charge neutral and non-magnetic spin-singlet excitons. Whilst, condensation of spin-triplet excitons is a promising frontier because spin supercurrent and spin-Seebeck effects will be observable. A canonical cobaltite LaCoO3 under very high magnetic fields is a propitious candidate, yet to be verified. Here, we unveil the exotic phase diagram of LaCoO3 up to 600 T generated using the electromagnetic flux compression method and the state-of-the-art magnetostriction gauge. We found the continuous magnetostriction curves and a bending structure, which suggest the emergence of two distinct spin-triplet exciton condensates. By constructing a phenomenological model, we showed that quantum fluctuations of excitons are crucial for the field-induced successive transitions. The spin-triplet exciton condensation in a cobaltite, which is three-dimensional and thermally equilibrated, opens up a novel venue for spintronics technologies with spin-supercurrent such as a spin Josephson junction.


Referee report to
Authors: A. Ikeda, Y.H. Matsuda, K. Sato, Y. Ishii, H. Sawabe, D. Nakamura, S. Takeyama, and J. Nasu Title: "Spin triplet condensations in LaCoO 3 at ultrahigh magnetic fields up to 600 T" Spin state crossover processes induced by changes in the electronic configuration of transition metal ions dramatically change physical properties of solids.
Thus, these processes are also considered as new spin-state degree of freedom and have been studied in detail for more than 50 years. Among other compounds, LaCoO 3 in particular is the most analysed substance in which spin-crossover occurs.
In addition to resulting effects such as conductivity changes, magnetization jumps, etc., magnetostriction measurements have been carried out because the electronic states strongly depend on and are influenced by the Co-O and Co-Co bond lengths.
More than 100 publications prove the great interest in the spin-crossover problem, which Ikeda et al. also address in the present draft.
In the meantime, there is a consensus that the pure existence of LS-IS-HS states is not sufficient for the description and that a more complex scenario exists. From a large number of experiments and theoretical simulations, LS-HS coexistence with temperature-and magnetic field-dependent distribution is derived in contrast to a pure IS story. The present work, like some modern papers, links the problem to exciton condensation or the formation of exciton superlattices. In any case, the performance of magnetostriction measurements up to 600 T using the FBG method and electromagnetic flux compression is new and shows a great experimental experience. These measurements are currently only possible worldwide at the ISSP.
But, it should be noted here that measurements (magnetization) at LaCoO 3 up to 500 Specifically, the following question arises: 1) Ikeda et al. show in their results a change in the slope of the magnetostriction curves at about 400 T and deduce from this the existence of a new gapless phase () in the phase diagram between 50 K and about 100 K. According to general statements (compare also Fig. 1), however, the gapless phase used for the description is characterised by a positive slope of the expansion curve. Furthermore: The upper limit of the  phase at approx. 100 K is not proven, since measurements at higher temperatures are not available. Measurements at higher temperatures would be helpful to confirm the B c3 transition. And: The reduction of the sharpness of B c3 at 108 K is rather due to the higher temperatur 2) The measurement results are combined with a mean-field calculation (in comparison to previous model calculations, e.g. Sotnikov et al. Sci. Rep. 6 (2016) 30510 extended). In addition to the well-known phases SSO and EC (spin-state order or exciton condensation), new phases are postulated (ES becomes ES1 and ES2, possibly connected to  and ) and shown by spin numbers in Fig. 4. The energy curves in Fig. 4, however, show only the states |0.0>, |0.2> and |2.2> as states of the lowest energy. But, this field dependent scenario would be already known.
3) Why the transition from EC1 to EC2 should be of second order, as later discussed in the text, is also not clearly justified.
4) The assignment of the calculated phases LS, EC1 and EC2 to ,  and  and the real magnetic field to the "normalized field" seems arbitrary ("may correspond", "we speculate"). Some more concrete (quantitative) results of the simulation would make the connection between Figs. 3 and 4 more clear. 5) Some formal errors are: Fig. 3(d) does not exist. "Isotoropic" on page 4.
Overall, I have to come to the conclusion that the results presented are very valuable in the context of the long-lasting discussion due to the new suggestions and ideas and, especially the experimental results, should definitely be published. However, it must also be stated that the proposed conclusions bring only little clarification of the basic problem and do not resolve the existing controversies (even the possible LS-IS-HS hybridizations already discussed are not clarified any more than elsewhere).
Reviewer #2 (Remarks to the Author): The manuscript is clearly written and linguistically correct, the illustrations instructive and of very good quality. However, the connection between experiment and theory does not yet seem to be fully harmonized. Because of this fact and because I cannot see enough significant new aspects for clarification on the basic problem of the spincrossover in LaCoO3, I do not want to recommend the publication in nat.comm at the present time.

Comments from reviewer #1 -101
The present article is devoted to a long-standing problem of LaCoO3

Our reply
We appreciate the positive comment. Fig.3 (I mean the B(t) dependences at different temperatures). Are they occasional or they reflect different operational modes of the facility?     The variation is largely due to the variation in the number of capacitor banks used. As you see in the bottom row in the Table, the larger the total energy, the earlier the Bmax is reached.

Our Reply
We note that the variation of the total energy arises from the accidental failure of the synchronization of the energy discharge of the capacitor banks. In addition, we note that Bmax is practically determined by when the B dot sensor is broken, which is not always the same as the real Bmax.  The key idea is temperature dependence. If the oscillating feature in 108 K data is actually a vibration, it should have appeared in the 78 K data with similar amplitude and frequency. We see only a small oscillating feature in the 78 K data above Bc2. The sharp slope features in 78 K are more apparent. 108 K is only 30 K above 78 K. The mechanical property is similar in this temperature change as reported in an ultrasound study [T. Sin Naing, et al, J. Phys. Soc. Jpn. 75 , 084601 (2006)]. Similarly, at the data 185 K, no negative slope is observed. This also indicates that the negative slopes of the data at 78 K and 108 K are intrinsic features.
Quantitatively, If the feature in 108 K is vibration, the shock propagation speed corresponds to 1.5 km/s with 250 kHz vibration and a sample size of 3 mm. In the same way, we obtain 3 km/s with 600 kHz vibration and a sample size of 3 mm. According to [T. Sin Naing, et al, J. Phys. Soc. Jpn. 75, 084601 (2006)], the sound velocity change from 5 K to 108 K is 15%. Thus the oscillating feature in 108 K data is too slow for a shock propagation vibration.
We described the above argument in Section I of the supplemental material and put the reference in the main manuscript as follows.

Can an axial magnetic field gradient in the experimental facility produce a sizable axial force on the sample (or strain in the sample)?
Our Reply As the reviewer indicated the spatial variation of the B field becomes severe especially when the Bmax is obtained. We used samples of 3 mm in axial length. It is ±1.5 mm from the B field center. The axial variation of B in EMFC was reported in [D. Nakamura et al., Rev. Sci. Instrum. 85, 036102 (2014).] as shown in the figure below. The axial field variation is small (<20 T) up to 500 T. It gets significant (>50 T) above 600 T. The study was performed on our previous EMFC system, whose record Bmax is about 700 T [S. Takeyama et al., J. Phys. D 44, 425003 (2010)].
In the present study, we used a new EMFC system whose record Bmax is 1200 T [D.
Nakamura (2018)] at the compression energy of 3.2 MJ. In the present measurements,we used smaller compression energy of 1.9 -2.5 MJ to reduce the Bmax down to 600 T. Using the smaller compression energy, one obtains a lower Bmax with better B field variation. Though we have not directly measured the axial distribution of B-field in the current EMFC system, we estimate that the B-field variation is similar to the reported one at t = 39.7 μs in the bottom figure below because we have sufficiently reduced the Bmax, where the 20 mm plateau of the B-field should be realized due to the relaxed Bmax. Thus, in the present study, the sample size is negligibly small compared to the field variation in the axial direction. We note that we plan to measure the B-field distribution in the future for the machine study.

Comments from reviewer #1 -105
There is also a question on the theoretical section of the work:

Our Reply
It is true at zero temperature within the mean-field theory. It may not hold, however, at finite temperatures and when strong quantum fluctuation exists.  (2015)]. Thus, experimental verification is achieved only by collecting collateral evidence. Observation of microscopic and macroscopic properties like magnetic excitations, lattice changes, thermal properties, and magnetic supercurrent will be a great help. The present study focuses on the condensation of spin-triplet excitons. Thus, magnetic properties can also be a great help. The present magnetostriction study is the first attempt at such a study.
There are misprints in the titles of the left and right panels in Fig.4 ("Exiciton condensation 1 (2)").

Our Reply
They are now corrected. Thank you.
Corrected Fig 4 d- Fig. 1), however, the gapless phase used for the description is characterized by a positive slope of the expansion curve.

Our Reply
The part of revised Fig. 2 c is shown above As the reviewer stated, we have simply indicated in Fig. 2 c that a positive slope infers the excitonic condensations. Now, we have revised Fig. 2 c so that a negative slope can also infer excitonic condensation. We argue that a "Slope" is fundamentally important because of its continuous change of lattice, regardless of its sign. The slope indicates the finite compressibility, which is one characteristic of exciton condensation.  Figs.4d, 4e, and 4i. The EMFC experiment has been carried out with the simultaneous ΔL measurement of two samples that are located at the field center and 7 mm away from the center. As you can see in Fig. 4e the results from the 2 samples are identical up to 300 T, where the off-centered sample disappears. Above 300 T, data for the centered sample is shown. In Fig. 4e, one can see the absence of the negative slope which is observed in Fig. 4f and 4g. This shows that the γ phase is absent at 185 K. On the other hand, we note that the transition corresponding to Bc2 is observed even at 185 K, which is indicated by a star-shaped symbol in Fig. 4i. Thus, the upper boundary of the β phase is shown to be beyond 185 K.
We described above argument in the main text as follows.
the revised part of the main text that appears in page 3 2) The measurement results are combined with a mean-field calculation (in comparison to previous model calculations, e.g. Sotnikov et al. Sci. Rep. 6 (2016) Fig. 4. The energy curves in Fig. 4, however, show only the states |0.0>, |0.2> and |2.2> as states of the lowest energy. But, this field dependent scenario would be already known.

Our Reply
As shown below, Figs 4a-4c have been updated. The calculated mean field energy E_MF is shown with the orange-dashed line, which is lower than any of the known classical state |0.0>, |0.2>, and |2.2>. Previously, the figure was so compressed that the difference is not apparent. We have now vertically enlarged the figure and changed the line style and thickness so that our present calculated result E_MF is apparent.

Comments from reviewer #2 -204
3) Why the transition from EC1 to EC2 should be of second order, as later discussed in the text, is also not clearly justified.

Our Reply
We omit this argument from the text.
The finding of the second-order transition is the result of the calculation, not the experiment.
EC1's order is characterized by the nonzero order parameter of <τ+> ≠ 0 and <ρ+> ≠ 0, a spatially uniform exciton condensation. In the EC2 phase, <τ-> and <ρ-> are additionally nonzero, which causes translational symmetry breakings of exciton number per site. Thus, EC1 and EC2 are allowed to be connected via second-order phase transition.
Experimentally, we observed that continuous changes seem to occur between EC1 and EC2, that is, there is no sudden jump of ΔL. However, we admit that this is not conclusive because the B field sweep is very fast.

Our Reply
We revised Figs 4a-4c so that the horizontal axis has a connection to the real magnetic field. The horizontal axis of Figs 4 a-f has been magnetic fields normalized to E1. The fact is not apparent in the previous version. Now, we have clearly indicated the label of the horizontal axis " h / E 1 " in the revised figure as shown below. Besides, we now use (g_1, g_2) = (2, 4) instead of the previously used (1, 2) because of the physical correspondence to the IS and HS states.

The bottom part of revised Figures 4 b
In the mean-field calculation, E 1 is taken to be unity, which is the energy gap between vacuum (LS) and isolated exciton (IS) states at B = 0. When we use the value of E1 to be 200 meV which is estimated for the CoO6 cluster in Ref. [M. W. Haverkort et al., Phys Rev Lett 97, 176405 (2006).], the reading of horizontal axis of h / E 1 = 0.1 corresponds to 20 meV. Then, with g = 2, Sz = 1, and μB = 5.79e-2(meV/T), we obtain the value of B = 172.7 T for h / E 1 = 0.1 based on the relation of Zeeman energy. In Fig. 4 b , the transition fields to EC1 and EC2 are 117, 263 T, respectively. In Fig. 4 c , the transition fields to EC1 and EC2 are 73, 307 T, respectively. As an order estimation, the calculation in Figs

Our Reply
Corrected.

Comments from reviewer #2 -207
Overall, I have to come to the conclusion that the results presented are very valuable in the context of the long-lasting discussion due to the new suggestions and ideas and, especially the experimental results, should definitely be published. However, it must also be stated that the proposed conclusions bring only little clarification of the basic problem and do not resolve the existing controversies (even the possible LS-IS-HS hybridizations already discussed are not clarified any more than elsewhere). The manuscript is clearly written and linguistically correct, the illustrations instructive and of very good quality. However, the connection between experiment and theory does not yet seem to be fully harmonized. Because of this fact and because I cannot see enough significant new aspects for clarification on the basic problem of the spin-crossover in LaCoO3 , I do not want to recommend the publication in nat.comm at the present time.

Our Reply
The first point " the connection between experiment and theory " is accomplished in the reply to #3-304 . Here, we try to reply to the second remark. " enough significant new aspects for clarification on the basic problem of the spin-crossover in LaCoO3 ".
We have re-considered what is new and what we did to the long-standing problem of LaCoO 3 . The summary is described as follows.
The very recent situation of the LaCoO 3 study: • The HS-IS duality is claimed by Tomiyasu et al. in 2018[K. Tomiyasu et al., arXiv:1808. It is an important new idea that explains why LS-IS-HS hybridization occurs very simply from a microscopic perspective. Also, it explains why people sometimes observe HS and sometimes observe IS depending on measurements. Tomiyasu claimed the HS-IS duality to explain the result of inelastic neutron scattering that indicates the existence of ferromagnetic 7-site clusters. This idea is further verified with a study by Hariki using model calculation and resonant inelastic x-ray scattering [A. Hariki PRB (2020)]. Further experimental verification has been difficult because no ordered phase appears in LaCoO3, except for those that appear under high magnetic fields. With the observation of only short-range order, we can not be sure whether the state mixing is due to the thermal effect or the quantum effect.
• In the meantime, spin-triplet exciton condensation is claimed for LaCoO3 at high magnetic fields to account for the field-induced phases. However, for the sake of • The present experiment indicates multiple field-induced abnormal phases (β and γ phases), which are possible exciton condensations. They can not be explained with conventional field-induced EC theories with only LS-IS considered because they show only one kind of exciton condensation with increasing magnetic fields. It motivated us to include the HS state in the model calculation, which has never been done before.
• In the present theoretical calculation with the HS-IS duality term, we have succeeded in finding two kinds of exciton condensation with increasing magnetic fields, which is a new finding. The calculated result qualitatively reproduced the experimental observation of two possible exciton condensations under magnetic fields thanks to the inclusion of the HS-IS duality term.
Thus, this study has new indications: • It is crucial to consider the HS-IS duality in LaCoO3. It is supported for the first time based on the stabilization of the ordered phase, not in the disordered phase. We observed two candidates for exciton condensation (β and γ) which are not explained without the HS-IS duality. As a general message, our study suggested that the HS-IS duality is a strong candidate for the origin of the controversy of LaCoO3, where people see both features of IS and HS in this material.
The above argument is now described in the main text as follows.
The revised part of the main text in page 6 (page 21 )

Comments from reviewer #3 -301
The manuscript experimentally investigated the magnetostriction of LaCoO 3 under magnetic fields as high as 600 T, with the aim of finding evidence for spin-triplet exciton condensation. The authors also proposed a theoretical model to further support their findings. However, the data presented do not convince me that they have achieved this goal. Therefore, I cannot recommend its publication in Nature Communications, at least not in its current version. My concerns are as follows:

Our Reply
We thank you for this criticism.

(page 22 )
Comments from reviewer #3 -302 1)Where is Fig. 3d? Without it, I cannot understand the results shown in the paragraph spanning pages 3 to 4 at all.
In particular, what does "gapless magnetostriction slopes" mean?

Our Reply
It is "a magnetostriction slope". "Gapless" is omitted.
How did the authors arrive at "These gapless behaviors evoke the Bose-Einstein condensation of spin-triplet excitons"?

Our Reply
In Bose systems, the expectation values of a creation operator b^+ and the compressibility should be nonzero when BEC occurs. The latter corresponds to the magnetic susceptibility in the present system with spin-triplet excitons.
In LaCoO3, the continuous increase of exciton number is an indication of exciton BEC, while a plateau of exciton number indicates an exciton solid. The exciton number is directly correlated with the lattice volume. Thus, our measurement of ΔL indirectly senses the exciton number changes as a function of the external magnetic field. The continuous increase of ΔL indicates the Bose-Einstein condensation of spin-triplet excitons". This is explained in more detail in the next Reply.

Comments from reviewer #3 -303
2) Figure 2c shows the theoretical correspondence of the experimental work. The authors expected to verify the exciton condensate from observing such "Slope" and "Plateau" behaviors. Given the importance of this theoretical basis and the distinctly different backgrounds of the readers of Nature Communications, it would be better for the authors to elaborate on its relationship with the spin-triplet exciton condensation in more detail, rather than simply provide three references [14,15,26] .
In my opinion, this hinders the readability and continuity of the manuscript. What makes me more confused is that there seems to be no work on this aspect in the references [14,15,26]. The word magnetostriction did not appear in any of the three publications. I wonder if the authors submitted the wrong version of the manuscript?

Our Reply
Thank you for this criticism.
In the present study, we have tried to measure the number of excitons as a function of external magnetic fields, instead of experimentally measuring the order parameter of exciton condensation, which is known as impractical. We have been inspired by the theoretical papers, where the evolution of the number of excitons as functions of external and internal parameters are reported. In Refs. [14,15,26], they report the change of the number of excitons as a function of external magnetic fields. As the reviewer pointed out, it is not a magnetostriction study. But, we regard that the number of excitons is firmly correlated with magnetostriction as follows.
In the current system of LaCoO 3 , the number of excitons is directly connected to the lattice volume. It is because the higher spin-states possess the larger ionic volume, thanks to the electrons occupying more extended orbitals of e g . For these reasons, we resorted to the magnetostriction measurement up to 600 T using our state-of-the-art magnetostriction monitor, in order to follow the evolution of exciton number as a function of magnetic fields.
respectively. We use J11 > J02 to stabilize the IS-IS pairs which are based on recent theoretical and neutron papers.
t and V are the quantum delocalization of IS to LS site, and HS resolving into IS-IS pair.
Previously we show only one parameter. Now, we have shown in Figs. 4 the variation of the parameter V, which is the origin of the unique excitonic condensation in the present study, EC2.
As one sees in the revised Figs. 4 that, the EC1 and EC2 phase appears with V = 0.17, which presumably account for the β and γ phase. While, with V = 0.14, they almost disappear and an excitonic solid (ES) appears in a wide range of magnetic fields h. ES is one possible origin that is induced at low temperatures with changing parameters of V. On the other hand, V = 0.2 show a re-entrant to EC1 after EC2 is destroyed. Because we have not reached the magnetic saturation, we are not sure whether V = 0.17 or 0.2 is more appropriate to account for our study.
We note that, for the appearance of EC2, a sufficient amount of V is mandatory as seen in the figure below. Also, J_02 needs to be sufficient. It is because EC2 is a hybridization between the HS-LS state and IS-IS state. For the appearance of EC1, a sufficient amount of t in addition to V is important because EC1 state is a hybridization of the LS, IS and HS states. In the present calculation, V is the most important parameter, which needs to be larger than twice the value of t. A systematic study of the parameter space is a topic of our future study.
As reviewer #3 pointed out we did not succeed in reproducing α phase in the previous model calculation, despite the fact α phase is the low-temperature phase and that the model is within the zero temperature limit. As you see in the revised Fig. 4a-4c, we have shown the variation of the parameter V which describes the HS-IS duality. It results in the appearance of the non-trivial phases of ES, EC1, and EC2, which are candidates for α, β, and γ phases. By the variation of the HS-IS duality parameter V, we have succeeded in reproducing ES phase as an origin of the α phase in the model.
ES appeared in the different parameter of V (= 0.14) from those (V = 0.17 and 0.2) for the appearance of EC1, and EC2. It is due to the situation that the α phase and β, and γ phases have distinct lattice parameters, resulting in distinct interaction parameters. We suspect that, in the α phase, the electron hopping is reduced in t_2g orbitals, resulting in the enhancement of the repulsive interactions for excitons so that more localized phases like super-lattices are formed in the α phase. This should be the origin of the plateau behavior of the magnetostriction in the α phase.
The change of interaction parameters in α and β, γ phases should have happened with the strong lattice expansion in α → β transition. Such lattice shrinkage is actually reported in the paper [A. Ikeda et al., Phys. Rev. Lett. (2020)] with increasing temperature with constant high magnetic fields ~100 T, which is shown below.
Further analysis of the changing interaction parameters with regard to the lattice changes is an interesting topic of research. One needs to take into account the bond angle, length, and symmetry change. Although it is beyond the scope of the present paper, we are now collecting x-ray diffraction data in the α phase, based on the state-of-the-art technique utilizing a portable ultrahigh magnetic field generator [A. Ikeda et al., Appl. Phys. Lett. (2022)].
We note this point in the main text as follows.
Overall view of the edition made to the latter half of the manuscript. The blue-colored part in the present version summarizes the theoretical part. originate from quantum effects. These results rule out the possibility of spin-state crystals and suggest that the magnetostriction slopes observed here are ascribed to the quantum mixing of distinct exciton states. Therefore, exciton condensation is the most plausible to understand the experiment for the β and γ phases. Hence, we state that the experimental observation of the nonthermal slopes (continuous changes) of the magnetostriction is the findings in the present study which we believe are the signatures of the exciton condensations. To our knowledge, there is no other simple explanation to account for the slope of magnetostriction.
The revised part on page 3 The revised part on page 4 Please note that our experiment is not an exception to the well-known notion that no experiment can directly confirm the order parameter of exciton condensation, which is the alignment of the phase factor of the wavefunctions of every exciton. Any resultant macroscopic or microscopic phenomena can not be a direct probe. For instance, the resultant change of the charge gap is observable. However, it can also occur as a result of other phenomena like lattice distortions which are not directly related to the quantum coherence of the exciton condensation.
We agree with the reviewer's opinion that our experiment is not compelling evidence of the spin-triple exciton condensation. So we decide to soften our tone for the conclusion. We state that we observed the signature of exciton condensations, which is the nonthermal slopes of the magnetostriction.
We agree with the reviewer on the suggestion that the previous manuscript sounds too conclusive, while our experimental result and calculation are only suggestive. In the revised manuscript, we have weakened our tone as to how the discussion is conclusive. Representatively, we have added the word "Signature of" on top of the title of our manuscript. We have colored the revised text in magenta color regarding these changes.
Last but not least, we emphasize that our work established a new technique to observe electronic states in extreme environments under strong magnetic fields. We are confident that the present study significantly advances the science of transition metal oxides and high magnetic fields. Despite the softened expressions, our work remains quite a significant scientific advance in the scientific fields of transition metal oxides and high magnetic fields.