Origin of the transitions inversion in rare-earth vanadates

Erik Koch; Eva Pavarini; Xue-Jing Zhang

REVIEW 3 major objections 4 minor 50 references

Origin of the transitions inversion in rare-earth vanadates

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Vanadate ordering inversion traced to super-exchange shift

desk verdict Novel tensor decomposition plus LDA+DMFT gives a credible microscopic origin for the RVO3 transition inversion, but the headline crossover definition (p=0.5) needs robustness testing. read the letter →

arxiv 2411.16351 v1 pith:EF2KFZOE submitted 2024-11-25 cond-mat.str-el cond-mat.mtrl-sci

Xue-Jing Zhang , Erik Koch , Eva Pavarini This is my paper · ORCID

classification cond-mat.str-elcond-mat.mtrl-sci

keywords rare-earthvanadatesorbitalorderingmagneticsuper-exchangedynamicalmean-fieldtheoryirreducibletensordecompositionKugel-Khomskiitransitiontemperatureinversion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to explain why, in the rare-earth vanadate series RVO3, the magnetic ordering temperature TN and the orbital-ordering temperature TOO cross as the rare-earth ion grows, a reversal seen in almost no other orbitally ordered material. It claims the cause is not simply lattice distortion but a shift in the hierarchy of super-exchange interactions: for large rare-earth ions, xz-xz quadrupolar super-exchange takes over orbital physics, while the dipolar spin-spin interaction with orbital rank r=0 dominates magnetism, and spin-orbital coupling changes from frustrating to reinforcing. Using LDA+DMFT with the order parameter decomposed into irreducible tensors, the paper reproduces TN < TKK < TOO for small rare-earth ions and TN > TKK ~ TOO for large ones, matching experiments. If right, it gives concrete criteria for finding or engineering the same inversion in other materials.

What carries the argument

The key machinery is an irreducible-tensor decomposition of both the Kugel-Khomskii super-exchange Hamiltonian and the orbital order parameter. The super-exchange is split into channels labeled by orbital rank r (monopole, dipole, quadrupole) and spin rank q (monopole, dipole), separating pure spin-spin, orbital-orbital, and spin-orbital entangled terms; the order parameter is similarly decomposed into components $\langle \hat{\tau}^{r\mu;q\nu}_i \rangle$. This decomposition lets the authors attribute the inversion to the growth of the (s,s) dipolar spin-spin term and to the competition between q=0 and q=1 channels, and it explains why TN>TKK requires both TOO~TKK and dominant r=0 dipolar spin order.

What would settle it

Measure the magnetic and orbital ordering temperatures of a single RVO3 compound under applied pressure or epitaxial strain that continuously weakens the GdFeO3 distortion; the model predicts that reducing the distortion should drive the system from TN<TOO toward TN>TOO. A direct numerical falsifier is to repeat the LDA+DMFT calculation with U and J varied by about 1 eV and see whether the predicted inversion and the stated energy balance survive.

Watch

Extended reading notes

Core claim

The central discovery is that the temperature inversion has a microscopic origin in the competition between distinct super-exchange channels. As the rare-earth radius increases, the GdFeO3-type distortion weakens, suppressing off-diagonal hoppings and shifting orbital physics from crystal-field control to xz-xz quadrupolar super-exchange. In the antiferromagnetic state, the orbital-monopole dipolar spin-spin interaction, the (s,s) channel, grows and becomes the dominant magnetic coupling; simultaneously the q=0 and q=1 spin-orbital contributions, which cancel for higher orbital ranks, cease to frustrate G-type order. The result is that G-type antiferromagnetism can set in above the orbital-ordering transition for large rare-earth ions, while C-type order appears around it, reproducing the observed inversion.

Load-bearing premise

The whole explanation depends on the assumption that the simplified electron model with fixed interaction strengths U=5 eV and J=0.68 eV, a t2g-only Hubbard Hamiltonian, and the super-exchange expansion truncated at quadrupolar orbital and dipolar spin rank correctly tracks how the ordering temperatures move across the series.

Editorial extensions

If this is right

For large rare-earth ions, G-type antiferromagnetic order is predicted to appear above the orbital-ordering transition, while C-type order sits near it.
Small-rare-earth systems keep the conventional hierarchy TN<TKK<TOO, so the inversion is not universal but tied to the distortion-suppressed hopping balance.
The two criteria, TOO~TKK and r=0 dipolar spin dominance, provide a screening rule for other materials that might show the inversion.
The decomposition method can be applied to other problems, including octupolar order in spin-orbit-coupled materials, as the paper notes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One could test the mechanism by tuning the GdFeO3 distortion via epitaxial strain or pressure in a single compound: the model implies that compressing the lattice should move the system toward the small-radius hierarchy and restore TN<TOO.
The criteria suggest a search strategy: among t2g perovskites with comparable crystal-field and super-exchange scales, those with dominant r=0 dipolar spin-spin coupling are the natural candidates for temperature inversion.
Recomputing the phase diagram with U and J varied independently, and with the super-exchange expansion extended to higher ranks, would establish how robust the inversion is to model truncation.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The short version: this is a solid, genuinely new explanation for the TN/TOO inversion in the RVO3 series, backed by serious LDA+DMFT calculations and a clever tensor decomposition. It deserves a careful referee. My main reservation is that the headline inversion rests on a crossover criterion (p=0.5) that the paper neither justifies nor tests for sensitivity.

What's new: the decomposition of the order parameter into irreducible tensor components is a real methodological contribution. The authors use it to show that for large rare-earth ions the xz-xz quadrupolar super-exchange takes over orbital physics, while the pure spin-spin (s,s) channel becomes dominant and the spin-orbital coupling shifts from frustrating to reinforcing. That is a concrete, falsifiable mechanism, and it explains why the inversion is rare. The calculations are state of the art: LDA+DMFT with continuous-time QMC, literature U and J, experimental structures for five rare earths. The schematic phase diagram matches experiments qualitatively.

Soft spots, in order of importance. First, TOO is defined as where p(T)=0.5, and p(T) is a smooth crossover curve. No susceptibility or Binder cumulant is computed, and no threshold sensitivity is shown. If the definition were changed to p=0.3 or p=0.7, the relative ordering across the series could shift; the stress-test note makes this point and it is fair. The authors should show robustness of the inversion to the crossover definition, or better, compute a genuine transition diagnostic. Second, there are no error bars on TN, TKK, or TOO, and no variation of U and J. The mechanism might be robust, but the crossing of TN and TOO is a quantitative claim. Third, the two criteria for inversion are extracted from the same calculations they are meant to explain; that's post-hoc, though not circular in a damaging way. The paper would be stronger if it predicted a material outside the fitted series. Fourth, the comparison to experiment is qualitative; the figure shows TS but not the experimental TN values.

None of these are fatal. The core logic is coherent and the decomposition genuinely illuminates why the inversion appears. For a reader working on orbital ordering or transition-metal oxides, this is worth a close read and likely a citation.

Recommendation: send it to a strong referee. It is not desk-reject material. The referee should push on the crossover definition and ask for a sensitivity analysis.

Best.

Referee Report

3 major / 4 minor

Summary. The paper addresses the inversion of magnetic (TN) and orbital (TOO) ordering temperatures in the rare-earth vanadate series RVO3 (R = Lu, Y, Tb, Pr, La) as the rare-earth radius increases. Using LDA+DMFT calculations for a t2g-only Hubbard model with literature values U = 5 eV and J = 0.68 eV, the authors compute orbital polarization p(T) and magnetization m(T) for real and idealized (zero crystal-field) structures. They define TOO as the temperature where p(T) = 0.5 and TKK as the corresponding temperature in the zero-crystal-field case. They report that the numerical calculations reproduce the experimental trend: TN < TKK < TOO for small rare-earth ions and TN > TKK ~ TOO for large ones. The mechanism is analyzed by decomposing the super-exchange Hamiltonian and the order parameter into irreducible tensor components. The authors conclude that the inversion arises when two conditions hold: (i) TOO ~ TKK, i.e., orbital ordering is controlled by super-exchange rather than lattice distortion, and (ii) the magnetic structure is governed by the orbital-monopole (r = 0) dipolar spin-spin interaction, with spin-orbital coupling changing from frustrating to reinforcing. They present these as criteria for finding similar inversions in other materials.

Significance. If the central result is robust, the paper provides a physically appealing explanation for a puzzling and material-specific phase diagram, showing that the inversion is not accidental but follows from a hierarchy of super-exchange channels. The use of an irreducible-tensor decomposition of both the super-exchange Hamiltonian and the order parameter is a systematic and potentially transferable analysis tool. A notable strength is that the inversion is not fitted: U and J are taken from the literature, the crystal structures are experimental, and the inversion emerges from the calculations. The stated criteria for inversion are falsifiable in principle, although the paper does not yet test them on a non-RVO3 material. The main weaknesses are the arbitrary p = 0.5 definition of TOO, the lack of uncertainty estimates for the transition temperatures, and the absence of a sensitivity analysis with respect to U and J. Because these issues directly affect the central claim, the paper needs revision before the conclusions can be considered established.

major comments (3)

[Fig. 2 and Sec. 'Orbital ordering in the paramagnetic phase'] The definition of TOO as the temperature at which p(T) = 0.5 is an arbitrary crossover threshold. Figure 2(a) shows a smooth, featureless p(T) curve, and the text itself states that for small rare-earth ions 'all ⟨τ⟩ rise smoothly: there is no phase transition in any channel.' Since the central inversion TN vs TOO and the criterion (i) TOO ~ TKK are built on this threshold, the authors must demonstrate that the ordering of the extracted temperatures across the RVO3 series is unchanged for other reasonable thresholds (e.g., p = 0.3 and p = 0.7), or better, identify a thermodynamic signature of the orbital ordering. Without such a test, the claimed inversion may be an artifact of the p = 0.5 convention.
[Model and Method, Eqs. (1)-(3); Fig. 2] The transition temperatures are read from LDA+DMFT order-parameter curves without any estimate of statistical or systematic error. The super-exchange tensor elements in Eq. (3) depend directly on U and J, so the hierarchy of channels in Fig. 3 and the resulting TN, TOO, and TKK values are expected to shift with these parameters. The authors should provide at least representative error bars on p(T) and m(T) and report the resulting uncertainties in TN, TOO, and TKK. In addition, a sensitivity analysis over a plausible range of U and J (e.g., U = 4-6 eV at fixed J/U, and J = 0.5-0.8 eV) is needed to establish that the inversion and the proposed criteria are robust.
[Conclusion] The two criteria for the inversion are extracted from the same RVO3 calculations and are not tested against any material in which the inversion does not occur. The abstract claims that the constraints are 'strict' and explain why the inversion is rare; to support this claim, the authors should apply the same decomposition to at least one conventional orbitally ordered material (e.g., LaMnO3 or KCuF3) and show that at least one condition fails there. Without such a test, the criteria remain a restatement of the RVO3 results rather than a falsifiable explanation of rarity.

minor comments (4)

[Conclusion] There is a typo in the conclusion: 'vandates' should be 'vanadates'.
[Sec. 'Antiferromagnetic phase'] The inequality 'TN > TKK < TOO' for large rare-earth ions appears to be a typographical error; the preceding sentence and the phase diagram indicate the intended relation is 'TN > TKK ~ TOO'.
[Fig. 1 caption] The definition of TOO as the temperature where p(T) = 0.5 appears only in the Fig. 2 caption; it should also be stated in the Fig. 1 caption, since Fig. 1 is the main phase diagram.
[Fig. 3 caption] The notation for the tensor components (e.g., 's,s', 's,x', 'z,z', 'xz,xz') is not defined in the caption. A brief explanation of the spin rank q, orbital rank r, and component indices would make the figure much easier to interpret.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the TN/TOO inversion emerges from LDA+DMFT with literature parameters; the tensor decomposition is an explanatory analysis, not a fitted input.

full rationale

The paper's central numerical result, the inversion of TN and TOO across the RVO3 series, is an emergent output of LDA+DMFT calculations using literature Coulomb parameters (U=5 eV, J=0.68 eV) and experimental crystal structures; the paper does not fit any parameter to the inversion. The irreducible-tensor decomposition of the super-exchange Hamiltonian (Eq. 3) and of the order parameter is the authors' own formalism (Refs. [30,35]), but it is a mathematical decomposition that does not encode the target result, and the DMFT transition temperatures are computed from the full Hubbard model before any decomposition is invoked. The two criteria stated in the Conclusion (TOO~TKK and r=0 dipolar dominance) are post-hoc explanations inferred from the same calculations, not inputs used to produce those calculations, so they do not make the derivation circular. Self-citations appear (Refs. [29,30,32,35,42]) but none is load-bearing in the sense of importing the inversion by construction. The arbitrary p=0.5 crossover definition of TOO is a robustness concern about whether the reported temperatures reflect true transitions, not a circular reduction: no equation or fitted parameter forces TN>TOO for La. No specific circular step can be exhibited.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central mechanism is derived from a Hubbard model with fixed screened Coulomb parameters and a truncated super-exchange decomposition; no parameters are fitted to the inversion itself, but the model inputs and the decomposition truncation are load-bearing assumptions. No new physical entities are introduced.

free parameters (2)

U = 5 eV
Screened on-site Coulomb repulsion in the Hubbard model (Eq. 1), taken from prior literature [29,41,42]. The central results depend on this value, but no sensitivity analysis is provided.
J = 0.68 eV
Hund's coupling in the Hubbard model, taken from prior literature. It affects the competition between orbital and magnetic super-exchange terms, and its value is not varied.

assumptions (4)

domain assumption The t2g manifold alone (Wannier functions of xy, xz, yz orbitals) is sufficient to describe the low-energy physics of RVO3.
The Hubbard model Eq. (1) is built on t2g Wannier states; eg orbitals are assumed irrelevant for the ordering transitions.
domain assumption LDA+DMFT with continuous-time quantum Monte Carlo accurately determines the relative trends of orbital and magnetic transition temperatures across the RVO3 series.
The method is applied to each compound with experimental structures and standard U, J, but no independent benchmark or error bars are provided for these specific transition temperatures.
domain assumption The Kugel-Khomskii super-exchange Hamiltonian, expressed through irreducible tensor components with orbital rank r <= 2 and spin rank q <= 1, is complete for the interactions that drive ordering in these systems.
The decomposition in Eq. (3) truncates the SE interaction at quadrupolar orbital and dipolar spin rank; higher-rank terms are assumed negligible without explicit justification.
domain assumption The screened Coulomb parameters U=5 eV and J=0.68 eV are transferable across the entire RVO3 series.
The paper cites Refs. 29, 41, 42 for these values but does not test whether the inversion mechanism is robust to variations in U and J across the series.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Origin of the transitions inversion in rare-earth vanadates." pith.science (2026). https://pith.science/paper/EF2KFZOE

@misc{pith2026241116351,
  author       = {Pith},
  title        = {Pith review of: Origin of the transitions inversion in rare-earth vanadates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EF2KFZOE}},
  note         = {Machine review of arXiv:2411.16351}
}

read the original abstract

The surprising inversion of the orbital- and magnetic-order transition temperatures in the RVO3 series with increasing the rare-earth radius makes the series unique among orbitally-ordered materials. Here, augmenting dynamical mean-field theory with a decomposition of the order parameter into irreducible tensors, we show that this anomalous behavior emerges from an unusual hierarchy of interactions. First, increasing the rare-earth radius, orbital physics comes to be controlled by xz-xz quadrupolar super-exchange rather than by lattice distortion. Next, for antiferromagnetic spin order, orbital super-exchange terms with different spin rank compete, so that the dipolar spin-spin interaction dominates. Eventually, G-type magnetic order (anti-ferro in all directions) can appear already above the orbital ordering transition, and C-type order (anti-ferro in the ab plane) right around it. The strict constraints we found explain why the inversion is rare, giving at the same time criteria to look for similar behavior in other materials.

Figures

Figures reproduced from arXiv: 2411.16351 by the authors.

**Figure 2.** FIG. 2. LDA+DMFT results. Orbital and magnetic transi [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. SE energy gain ∆ [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Irreducible components of the order parameter, [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 48 canonical work pages

[1]

Khomskii, Transition Metal Compounds, Cambridge University Press (2014)

D.I. Khomskii, Transition Metal Compounds, Cambridge University Press (2014)

work page 2014
[2]

Tokura and N

Y. Tokura and N. Nagaosa, Science 288, 462-468 (2000)

work page 2000
[3]

Dagotto, Science 309, 257-262 (2005)

E. Dagotto, Science 309, 257-262 (2005)

work page 2005
[4]

H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Nagaosa, and Y. Tokura, Nat. Mat. 11, 103-113 (2012)

work page 2012
[5]

J. Yan, A. Kumar, M. Chi, M. Brahlek, T. Z. Ward, and M.A. McGuire, Phys. Rev. Mat. 8, 024404 (2024)

work page 2024
[6]

A. S. Disa, J. Curtis, M. Fechner, A. Liu, A. von Hoegen, M. F¨ orst, T. F. Nova, P. Narang, A. Maljuk, A. V. Boris, B. Keimer, and A. Cavalleri, Nature 617, 73-78 (2023)

work page 2023
[7]

Horsch, A

P. Horsch, A. M. Ole´ s, L. F. Feiner and G. Khaliullin, Phys. Rev. Lett. 100, 167205 (2008)

work page 2008
[8]

M. J. Mart ´ ınez-Lope, J. A. Alonso, M. Retuerto, and M. T. Fern´ andez-D ´ ıaz, Inorg. Chem.47, 2634 (2008)

work page 2008

Show all 50 references

[9]

J.-S. Zhou, Y. Ren, J.-Q. Yan, J.F. Mitchell, and J. B. Goodenough, Phys. Rev. Lett. 100, 046401 (2008)

2008
[10]

Y. Ren, A. A. Nugroho, A. A. Menovsky, J. Strempfer, U. R¨ utt, F. Iga, T. Takabatake and C. W. Kimball, Phys. Rev. B 67, 014107 (2003)

2003
[11]

Yan, J.-S

J.-Q. Yan, J.-S. Zhou, and J. B. Goodenough, Phys. Rev. Lett. 93, 235901 (2004)

2004
[12]

M. H. Sage, G. R. Blake, G. J. Nieuwenhuys, and T. T. M. Palstra, Phys. Rev. Lett. 96, 036401 (2006)

2006
[13]

P. N. Shanbhag, F. Fauth, and A. Sundaresan, Phys. Rev. B 108 134115 (2023)

2023
[14]

Nasir, I

M. Nasir, I. Kim, Ki. Lee, S. Kim, K. Hyoung Lee, and H. Jung Park Phys. Chem. Chem. Phys. 25, 3942 (2023)

2023
[15]

Meley, M

H. Meley, M. Tran, J. Teyssier, J. A. Krieger, T. Prokscha, A. Suter, Z. Salman, M. Viret, D. van der Marel, and S. Gariglio, Phys. Rev. B 103, 125112 (2021)

2021
[16]

J.-Q. Yan, W. Tian, H. B. Cao, S. Chi, F. Ye, A. Llobet, A. Puretzky, Q. Chen, J. Ma, Y. Ren, J.-G. Cheng, J.-S. Zhou, M. A. McGuire, and R. J. McQueeney, Phys. Rev. B 100 184423 (2019)

2019
[17]

Benckiser, L

E. Benckiser, L. Fels, G. Ghiringhelli, M. Moretti Sala, T. Schmitt, J. Schlappa, V. N. Strocov, N. Mufti, G. R. Blake, A. A. Nugroho, T. T. M. Palstra, M. W. Haverkort, K. Wohlfeld, and M. Gr¨ uninger, Phys. Rev. B, 88 205115 (2013)

2013
[18]

Novelli, D

F. Novelli, D. Fausti, J. Reul, F. Cilento, P. H. M. van Loosdrecht, A. A. Nugroho, T. T. M. Palstra, M. Gr¨ uninger, and F. Parmigiani, Phys. Rev. B86, 165135 (2012)

2012
[19]

Ruotsalainen, M

K. Ruotsalainen, M. Gatti, J.M. Ablett, F. Yakhou- Harris, J.-P. Rueff, A. David, W. Prellier, and A. Nico- laou, Phys. Rev. B 103, 235158 (2021)

2021
[20]

M. H. Sage, G. R. Blake, C. Marquina, and T. T. M. Palstra, Phys. Rev. B 76, 195102 (2007)

2007
[21]

G. R. Blake, T. T. M. Palstra, Y. Ren, A. A. Nugroho, and A. A. Menovsky, Phys. Rev. Lett.87, 245501 (2001)

2001
[22]

Reehuis, C

M. Reehuis, C. Ulrich, K. Prokeˇ s, S. Mat´ aˇ s, J. Fujioka, S. Miyasaka, Y. Tokura, and B. Keimer, Phys. Rev. B 83, 064404 (2011)

2011
[23]

R. Saha, F. Fauth, V. Caignaert, and A. Sundaresan, Phys. Rev. B 95, 184107 (2017)

2017
[24]

Reehuis, C

M. Reehuis, C. Ulrich, P. Pattison, M. Miyasaka, Y. Tokura, and B. Keimer, Eur. Phys. J. B64, 27–34 (2008)

2008
[25]

Miyasaka, Y

S. Miyasaka, Y. Okimoto, M. Iwama, and Y. Tokura, Phys. Rev. B 68, 100406(R) (2003)

2003
[26]

J.-S. Zhou, J. B. Goodenough, J.-Q. Yan, and Y. Ren, Phys. Rev. Lett. 99, 156401 (2007)

2007
[27]

Munoz, J

A. Munoz, J. A. Alonso, M. T. Cas´ ais, M. J. Mart ´ ınez- Lope, J. L. Mart ´ ınez, and M. T. Fern´ andez-D ´ ıaz, Phys. Rev. B 68, 144429 (2003)

2003
[28]

J. Reul, A. A. Nugroho, T. T. M. Palstra, and M. Gr¨ uninger, Phys. Rev. B86, 125128 (2012)

2012
[29]

De Raychaudhury, E

M. De Raychaudhury, E. Pavarini, and O. K. Andersen, Phys. Rev. Lett. 99, 126402 (2007)

2007
[30]

Zhang, E

X.-J. Zhang, E. Koch, and E. Pavarini, Phys. Rev. B 105, 115104 (2022)

2022
[31]

Kugel and D.I

K.I. Kugel and D.I. Khomskii, Zh. Eksp. Teor. Fiz. 64, 1429 (1973) [Sov. Phys. JETP 37, 725 (1973)]

1973
[32]

Zhang, E

X.-J. Zhang, E. Koch, and E. Pavarini, Phys. Rev. B 106, 115110 (2022)

2022
[33]

Pavarini, E

E. Pavarini, E. Koch, A.I. Lichtenstein, Phys. Rev. Lett. 101, 266405 (2008)

2008
[34]

Pavarini and E

E. Pavarini and E. Koch, Phys. Rev. Lett. 104, 086402 (2010)

2010
[35]

Zhang, E

X.-J. Zhang, E. Koch, and E. Pavarini, Phys. Rev. B 102, 035113 (2020)

2020
[36]

Musshoff, G

J. Musshoff, G. Zhang, E. Koch, E. Pavarini, Phys. Rev. B 100, 045116 (2019)

2019
[37]

Autieri, E

C. Autieri, E. Koch, and E. Pavarini, Phys. Rev. B 89, 155109 (2014)

2014
[38]

Blaha, K

P. Blaha, K. Schwarz, P. Sorantin, and S. Trickey, Com- put. Phys. Commum. 59, 399 (1990)

1990
[39]

Marzari and D

N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997)

1997
[40]

A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Van- derbilt, and N. Marzari, Comput. Phys. Commun. 178, 685 (2008); J. Kune´ s, R. Arita, P. Wissgott, A. Toschi, H. Ikeda, and K. Held, ibid. 181, 1888 (2010)

2008
[41]

Mizokawa and A

T. Mizokawa and A. Fujimori, Phys. Rev. B 54, 5368 (1996)

1996
[42]

Pavarini, A

E. Pavarini, A. Yamasaki, J. Nuss, and O. K. Andersen, New J. Phys. 7, 188 (2005)

2005
[43]

E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, Rev. Mod. Phys. 83, 349 (2011)

2011
[44]

Flesch, E

A. Flesch, E. Gorelov, E. Koch, and E. Pavarini, Phys. Rev. B 87, 195141 (2013)

2013
[45]

Musshoff, G

J. Musshoff, G. R. Zhang, E. Koch, and E. Pavarini, Phys. Rev. B 100, 045116 (2019)

2019
[46]

Musshoff, A

J. Musshoff, A. Kiani, and E. Pavarini, Phys. Rev. B 103, 075136 (2021)

2021
[47]

In the monoclinic case, the relation |θ, ϕ⟩3∼| −θ−δθ, ϕ±δϕ⟩ maximizes the energy gain with δθ∼20◦ and δϕ∼110◦ in the PM phase

Defining V1 and V2 two neighboring sites in one layer and V3 and V4 the corresponding ones in the layer above, with V3 on top of V1, in the GdFeO 3-type structure, if |θ, ϕ⟩1 is the hole orbital at site V1, the hole orbitals at the other sites are |θ, ϕ⟩2=|θ, π 2 −ϕ⟩1, |θ, ϕ⟩3...
[48]

For additional information see Supplemental Material
[49]

This is because, for a specific hole orbital, ⟨ˆτ r,µ i ⟩α=⟨θ, ϕ|ˆτ r,µ;qν i |θ, ϕ⟩α, depend on two variables only, θ and ϕ [48]

For a given spin rank q, there are only two independent saturation ( T →0) values. This is because, for a specific hole orbital, ⟨ˆτ r,µ i ⟩α=⟨θ, ϕ|ˆτ r,µ;qν i |θ, ϕ⟩α, depend on two variables only, θ and ϕ [48]. These constraints do not hold at finite temperature, however
[50]

The CF affects TN more for C-type than G-type magnetic order, making the first more stable

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Khomskii, Transition Metal Compounds, Cambridge University Press (2014)

D.I. Khomskii, Transition Metal Compounds, Cambridge University Press (2014)

work page 2014

[2] [2]

Tokura and N

Y. Tokura and N. Nagaosa, Science 288, 462-468 (2000)

work page 2000

[3] [3]

Dagotto, Science 309, 257-262 (2005)

E. Dagotto, Science 309, 257-262 (2005)

work page 2005

[4] [4]

H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Nagaosa, and Y. Tokura, Nat. Mat. 11, 103-113 (2012)

work page 2012

[5] [5]

J. Yan, A. Kumar, M. Chi, M. Brahlek, T. Z. Ward, and M.A. McGuire, Phys. Rev. Mat. 8, 024404 (2024)

work page 2024

[6] [6]

A. S. Disa, J. Curtis, M. Fechner, A. Liu, A. von Hoegen, M. F¨ orst, T. F. Nova, P. Narang, A. Maljuk, A. V. Boris, B. Keimer, and A. Cavalleri, Nature 617, 73-78 (2023)

work page 2023

[7] [7]

Horsch, A

P. Horsch, A. M. Ole´ s, L. F. Feiner and G. Khaliullin, Phys. Rev. Lett. 100, 167205 (2008)

work page 2008

[8] [8]

M. J. Mart ´ ınez-Lope, J. A. Alonso, M. Retuerto, and M. T. Fern´ andez-D ´ ıaz, Inorg. Chem.47, 2634 (2008)

work page 2008