Pith. sign in

REVIEW 4 major objections 5 minor 27 references

Sixteen-State Energy Mapping for First-Principles Four-Spin Ring Exchange: Validation on $La_2CuO_4$ and $SrFeO_2$

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read A sixteen-state collinear energy sum isolates the four-spin ring exchange from DFT, with a catch when longer loops contribute.

desk verdict A real methodological step forward on ring exchange from collinear DFT, with a strong algebraic core and an honest account of where the direct extraction stops being direct; worth a careful referee. read the letter →

arxiv 2607.18986 v2 pith:EF523N6S submitted 2026-07-21 cond-mat.mtrl-sci cond-mat.str-elquant-ph

Xavier Rocquefelte , Peter Blaha This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.str-elquant-ph PACS 71.15.-m75.10.-b
keywords four-spinringexchangecyclicenergymappingsixteen-statemethodLa2CuO4SrFeO2GGA+Umulti-spininteractions
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

This paper introduces a sixteen-state energy-mapping scheme that extracts the four-spin ring exchange J_ring directly from collinear GGA+U total energies, much as the standard four-state method extracts bilinear exchange J. The key move is flipping the four spins of a single nearest-neighbor plaquette through all 2^4 arrangements and summing the energies weighted by the product of the four spin signs; every spin-independent constant, bath molecular field, and pair coupling cancels exactly, leaving J_ring alone. As a corollary, the conventional four-state nearest-neighbor coupling is shown to be ring-renormalized by ±2 J_ring S^2 depending on the reference bath, a prediction confirmed in La2CuO4 to 0.2% across three independent routes. The method also reveals its own limitation: when six- or eight-spin loop terms are present, a single reference bath gives an effective, reference-dependent value, which additional baths separate into a bare J_ring plus a converging tower of loop couplings. SrFeO2 provides a negative control where the plaquette exists but J_ring/J is only 0.006, showing that a square plaquette is necessary but far from sufficient for ring exchange.

What carries the argument

The central object is the collinear reduction of the four-spin ring operator: in a collinear state, the bracket in Eq. (1) reduces to J_ring S^4 times the product of the four spin signs. This makes it possible to construct a signed sum over the sixteen spin patterns of a plaquette that cancels all lower-order terms, exactly analogous to the four-state method. The cancellation relies on the fact that two distinct nearest-neighbor plaquettes share at most an edge, so no pair or single-site term can mimic the plaquette product, and the weighted sum over the sixteen states annihilates every lower-degree contribution. The derivation also exposes the ring-renormalized four-state coupling, Eq. (5),

What would settle it

Perform the sixteen-state extraction on a well-isolated plaquette in La2CuO4 using five different reference baths; if the measured values do not lie on the plane predicted by Eq. (13) with a single bare J_ring and small loop amplitudes, the model fails. Separately, in any other material with significant ring exchange, compute the four-state nearest-neighbor coupling on both Néel and ferromagnetic references: if their difference is not exactly 4 J_ring S^2 (with J_ring obtained from an independent route such as energy mapping), the central renormalization claim is wrong.

Watch

Extended reading notes

Core claim

The central claim is that on any collinear spin configuration the cyclic ring operator collapses to the product of the four spin signs, which allows an exact decomposition of the total energy and a signed sum over the sixteen spin arrangements of a plaquette—Eq. (4)—that cancels all terms below fourth order, leaving J_ring = (1/(16 S^4)) Σ σ_i σ_j σ_k σ_l E_n. The derivation also yields Eq. (5): a four-state extraction of a nearest-neighbor coupling actually returns the ring-renormalized value J_NN − 2 J_ring S^2 on a Néel bath and J_NN + 2 J_ring S^2 on a ferromagnetic bath, so any four-state J quoted without naming its reference is ill-defined in the presence of ring exchange. The paper va

Load-bearing premise

The extraction formula assumes the spin Hamiltonian contains only two-spin exchanges plus the four-spin ring term on the probed plaquette, but the paper's own La2CuO4 test shows that when six- or eight-spin loop terms are present, a single-reference sixteen-state sum returns a reference-dependent effective value rather than the bare J_ring.

Editorial extensions

If this is right

  • J_ring can now be extracted from standard collinear GGA+U calculations at a cost comparable to the four-state method, since symmetry reduces the sixteen configurations to six or eight inequivalent energies.
  • Previously reported four-state nearest-neighbor couplings in ring-exchange materials are reference-dependent; in La2CuO4 a four-state J_1 quoted without naming its reference is wrong by 12%.
  • The difference between ferromagnetic- and Néel-reference four-state couplings provides a cheap diagnostic: it measures, at fourth order, how far a material departs from the pair-plus-ring Hamiltonian.
  • Using several reference baths resolves the effective J_ring into a bare ring coupling plus contributions from six- and eight-spin loop exchange, giving a hierarchical description of multi-spin interactions.
  • A square plaquette is a necessary but not sufficient condition for significant ring exchange: SrFeO2 has the same plaquette yet J_ring/J = 0.006, because orbital dilution and spin normalization suppress the cyclic process.

Reading between the lines

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

  • A practical consequence the authors leave implicit: for strongly correlated square-lattice magnets, a single-reference sixteen-state J_ring should be treated as an upper or lower bound bracketing the bare value, and at least two or more baths should be used to certify any local quadrilinear probe.
  • The loop-tower decomposition in Eq. (13) suggests a direct numerical bridge between DFT energy mapping and the t/U expansion of the Hubbard model, where the extracted J_ring and loop amplitudes could be compared with analytical fourth- and sixth-order coefficients.
  • The orthogonality of the four-spin column to all bilinear columns in the energy-mapping fit explains why J_ring is insensitive to supercell aliasing of pair shells; this robustness could motivate using four-spin ring terms as anchors when fitting complicated magnetic Hamiltonians.
  • The method could be extended to other plaquette geometries, such as triangular or kagome lattices, where cyclic exchange terms may stabilize non-collinear orders; the same weighted-sum construction, with appropriate collinear spin products, should isolate the corresponding multi-spin couplings.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper introduces a sixteen-state energy-mapping scheme that extracts the four-spin ring-exchange coupling J_ring from collinear spin configurations. The key identity, Eq. (4), is an exact algebraic consequence of the pair-plus-ring Hamiltonian of Eq. (1): a signed sum over the 2^4 collinear arrangements of a plaquette cancels the energy constant, the bath molecular fields, and all pair couplings. The derivation also predicts that the conventional four-state nearest-neighbor coupling is ring-renormalized by ±2J_ring S^2 depending on the reference bath, and that the next-nearest-neighbor coupling should be reference-independent. The method is applied to T-La2CuO4 and SrFeO2. For La2CuO4, three energy routes agree on J_ring ≈ 32.6–32.7 meV, while the direct sixteen-state extraction is reference-dependent (27.54 meV on the Néel bath, 34.52 meV on the ferromagnetic bath); the authors attribute this to six- and eight-spin loop terms and fit a bare J_ring = 29.57 meV. For SrFeO2, J_ring/J2 = 0.006, providing a negative control. The workflow is implemented in the open-source Mag4 package.

Significance. The algebraic machinery is clean and useful. Eq. (4) is a genuine exact identity within the stated model, and the ring-renormalization prediction of Eq. (5) is confirmed to high precision (0.2% agreement and a 6 µeV non-shift for the diagonal). The paper also supplies two valuable practical contributions: explicit gap and moment diagnostics for validating local-spin mappings, and a negative control (SrFeO2) showing that a plaquette is necessary but not sufficient for ring exchange. The open-source Mag4 package and the reproducible workflow strengthen the paper. However, the headline claim that Eq. (4) leaves J_ring alone 'for any reference bath' is not what the paper's own data show for the strongly coupled material that validates the method; the reference dependence is real and is resolved only by introducing a fitted tower of loop couplings. This limits the direct method precisely in the regime where ring exchange matters most, and the presentation should be reframed accordingly.

major comments (4)
  1. [Sixteen-state extraction: a genuine reference dependence (Table 3)] Eq. (4) is exact for the Hamiltonian of Eq. (1), but the paper's own validation shows that the direct sixteen-state extraction is not reference-independent: VASP gives J_ring = 27.54 meV on a Néel bath and 34.52 meV on a ferromagnetic bath, with WIEN2k reproducing the spread. The paper acknowledges this and resolves it by introducing the loop model of Eq. (13), whose four fitted parameters produce the quoted bare J_ring = 29.57 meV. That bare value is therefore not the output of Eq. (4) alone; it is the solution of a model-dependent deconvolution. Since La2CuO4 is the central validation material, the abstract and introduction overstate the method when they say that the sixteen-state sum cancels the bath and leaves J_ring alone. The authors should qualify this claim as holding only within the pair-plus-ring model and state prominently that in strongly coupled magnets the direct extraction
  2. [Bath decomposition, Eq. (13)] The decomposition that recovers the bare J_ring = 29.57 meV is based on a four-parameter fit (J_ring, J_e^(6), J_zz^(6), J^(8)) to five reference-bath values. The paper does not report the residuals of the fit, the parameter uncertainties, or the condition number/collinearity of the correlator matrix. The 1.3 µeV agreement for the stripe2 prediction is a strong consistency check, but it verifies only that no loop family reaches the out-of-plane Cu sites; it does not verify that the three loop families are sufficient, or that further t'-assisted or ten-spin loops are negligible. To support the claim of a converging tower, the authors should provide errors on all four fitted amplitudes and a sensitivity analysis to the choice of included loop families.
  3. [Quantitative confirmation of ring renormalization, Eqs. (9)–(10)] The text describes 'three independent routes' to J_ring, but Eq. (9) uses the energy-mapping J1 as an input; only Eq. (10) is a pure difference of two four-state calculations. The three routes therefore share data and are not fully independent determinations. The agreement to 0.2% is still a strong internal-consistency check, but it should be described as such rather than as three independent measurements. This matters because the independence of the routes is part of the paper's central validation argument.
  4. [SrFeO2 sixteen-state result, Table 5 and footnote d] The negative control also shows a discrepancy between the direct sixteen-state value and the mapping value: J_ring = 0.0526 meV from the sixteen-state method versus 0.0404 meV from the energy mapping, and a different supercell gives 0.074 meV with two metallic configurations. This is a 30–80% spread relative to the small mapping value. The paper presents SrFeO2 as a validation of the methodology, but this unexplained scatter in the direct extraction is not discussed. The authors should either explain this reference/cell dependence or explicitly list it as another instance where the direct extraction is not a black-box probe, separate from the La2CuO4 loop decomposition.
minor comments (5)
  1. [Eq. (3) and Supporting Information] When deriving the effective edge coupling ~J_ij, the paper should state explicitly which ring plaquettes contribute to the degree-two term with factor one (the neighboring plaquette) and which contribute to the degree-four term (the chosen plaquette). The connection to the factor 2 in Eq. (5) is then clearer.
  2. [Figure 4(b)] The data points for the five reference baths are shown without error bars. Given that the fitted loop amplitudes are small residuals of near-cancelling energies, error bars or at least a statement of the numerical uncertainty should be included.
  3. [Table 6 and WIEN2k comparison] The text notes that the WIEN2k 1×1×2 FM and Néel values differ by 4.7 meV because of different k-meshes, and that a matched-mesh pair 'is being computed.' For a published comparison this is unsatisfactory; either present the matched-mesh result or remove the statement.
  4. [Notation] The manuscript uses J_ring/J1 and J_ring/J2 interchangeably with J_ring/J in places. A consistent notation, plus explicit statement of the shell labeling for the two materials, would avoid confusion.
  5. [Introduction] The historical and personal introduction is engaging, but much of it (Whangbo, Hoffmann, 'du bout des doigts') is not needed for the technical content. It could be shortened without loss.

Circularity Check

1 steps flagged · score 2.0 of 10

Eq. (4) is a genuine algebraic identity and the La2CuO4/SrFeO2 tests are externally anchored; only the Eq. (9) 'independent/parameter-free' route reuses the fitted J1, so the 'three independent routes' claim is mildly overstated.

  1. fitted input called prediction [Results and discussion, La2CuO4, 'Quantitative confirmation of the ring renormalization of J', Eq. (9) and Table 2]
    "Inverting Eq. (5), that deficit is an independent determination of the ring coupling that uses only bilinear energy differences, Jring = (J1 - J1^{4-state}(Neel)) / 2S^2 = 16.31/0.5 = 32.62 meV, in agreement with the 32.69 meV of the energy mapping to 0.2%. This is a stringent, parameter-free check of the entire framework."

    The 'independent determination' takes J1 = 130.51 meV from the same energy-mapping fit that simultaneously returns Jring = 32.69 meV, the value being compared. Eq. (9) is simply Eq. (5) solved for Jring with that fitted J1 as input, so the agreement is a consistency check inside the fitted pair-plus-ring model, not parameter-free and not a third independent route. The two-bath route, Eq. (10), is the genuinely independent check; because it agrees, this overstatement does not invalidate the central result.

full rationale

The core derivation is not circular. Eq. (4) is an exact signed-sum identity under the Hamiltonian of Eq. (1): every lower-degree term cancels by the summing identities, leaving only the plaquette's degree-four term. The reference-dependence found for La2CuO4 (27.54 vs 34.52 meV, Table 3) is reported honestly and used to diagnose terms beyond Eq. (1); the Eq. (13) bath decomposition fits Jring = 29.57 meV together with loop amplitudes and makes a parameter-free stripe2 prediction confirmed to 1.3 micro-eV, so that fit is not a disguised renaming of the input. SrFeO2 is a negative control with an independent mapping value. The only circularity-adjacent step is the labeling of Eq. (9) as an independent, parameter-free route: it reuses the fitted J1 from the mapping whose Jring is the comparison value, so the 'three independent routes agree to 0.2%' claim rests on two genuinely independent determinations plus one internal rearrangement. This is a minor overstatement, not a construction-level circularity; the central extraction remains externally validated by the two-bath route and by agreement with Moreira et al. and experimental estimates.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. Its central method depends on the chosen Hubbard U, on the assumed spin-Hamiltonian form, on the grey-group symmetry reduction, and especially on the truncation of the loop expansion in Eq. (13). The fitted loop amplitudes and the bath-decomposed bare J_ring are the main free parameters beyond the standard DFT+U setup.

free parameters (5)
  • Ueff (Hubbard U) = 8 eV (La2CuO4), 4 eV (SrFeO2)
    DFT+U on-site correction chosen by hand; all reported couplings, including J_ring, depend on this choice.
  • J_e^(6) (six-spin edge-loop amplitude) = 3.49 meV (VASP); 6.45 meV (WIEN2k)
    Fitted to the reference-bath dependence in Eq. (13); essential to resolving the bare J_ring.
  • J_zz^(6) (six-spin across-plaquette loop amplitude) = 0.90 meV
    Fitted in Eq. (13); third-order correction to the quadrilinear channel.
  • J^(8) (eight-spin loop amplitude) = 0.57 meV
    Fitted in Eq. (13); smallest loop correction included.
  • J_ring (bare, bath-decomposed) = 29.57 meV
    Simultaneously fitted with the three loop amplitudes in Eq. (13) to four reference-bath values; not obtained by the direct Eq. (4) alone.
assumptions (5)
  • domain assumption Collinear DFT total energies are described by the classical Heisenberg Hamiltonian of Eq. (1) (pair + ring terms) in the localized-spin regime.
    Used to derive the exact decomposition Eq. (3) and extraction Eq. (4); the paper's own gap/moment diagnostics are designed to test this.
  • standard math The ring exchange operator reduces on any collinear state to the product of the four spin signs, Eq. (2).
    Direct consequence of the chosen operator; underlies every cancellation in the 16-state sum.
  • domain assumption The symmetry reduction of the sixteen configurations can be performed in the magnetic grey group with time reversal, and the resulting inequivalent classes are sufficient.
    Used to reduce computational cost; requires C4 or C4×T symmetry of the spin Hamiltonian and cell.
  • ad hoc to paper The bath decomposition (Eqs. 11-13) includes only six- and eight-spin loop terms in three families, with higher-order or t'-assisted partners suppressed.
    Four unknowns are fixed by four reference baths; the truncation of the loop tower is assumed, though the converging amplitudes and stripe2 prediction support it.
  • domain assumption DFT+U with PAW/VASP and WIEN2k at the chosen U reproduces the relevant localized-spin energetics.
    All numbers are computed with these codes; code comparison is used as a mutual check, but no experimental data are fitted.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Sixteen-State Energy Mapping for First-Principles Four-Spin Ring Exchange: Validation on $La_2CuO_4$ and $SrFeO_2$." pith.science (2026). https://pith.science/paper/EF523N6S

@misc{pith2026260718986,
  author       = {Pith},
  title        = {Pith review of: Sixteen-State Energy Mapping for First-Principles Four-Spin Ring Exchange: Validation on $La_2CuO_4$ and $SrFeO_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EF523N6S}},
  note         = {Machine review of arXiv:2607.18986}
}
abstract

Four-spin ring (cyclic) exchange $J_{ring}$ is an essential ingredient of the Heisenberg spin Hamiltonian of cuprates and other square-lattice magnets, yet it has lacked the kind of direct, local first-principles extraction that the four-state method provides for bilinear exchange, $J$. We supply it by generalizing that method to a sixteen-state ($2^4$) scheme. Symmetry reduces the sixteen configurations to six or eight inequivalent energies, so the cost is modest. The derivation also shows that the conventional four-state magnetic coupling, $J$, is itself ring-renormalized, by $\pm 2 J_{ring} S^2$ with the sign set by the reference state. T-La$_2$CuO$_4$ confirms this quantitatively: three independent routes agree on $J_{ring}$ to $0.2\%$, giving $J_{ring}/J_1 = 0.25$, and a four-state $J_1$ quoted without naming its reference is wrong by $12\%$ in this material. The direct sixteen-state extraction itself proves reference-dependent, the N\'eel and ferromagnetic baths bracketing the mapping value: a fourth-order fingerprint of interactions beyond the pair-plus-ring model, which additional reference baths resolve into a bare $J_{ring}$ and a converging tower of six- and eight-spin loop couplings. SrFeO$_2$ ($S = 2$), with the same plaquette yet $J_{ring}/J = 0.006$, provides the negative control: a plaquette is necessary for ring exchange, far from sufficient. The complete workflow, including the band-gap and local-moment diagnostics that certify any such extraction, is implemented in the openly available Mag4 package, so that $J_{ring}$ costs no more effort to obtain than $J$.

Figures

Figures reproduced from arXiv: 2607.18986 by the authors.

Figure 1
Figure 1. The two test materials. (a) T-phase La2CuO4 (K2NiF4 type, I4/mmm; Cu blue, O red, La green). Each Cu sits at the centre of a Jahn–Teller elongated CuO6 octahedron, the apical oxygen lying along c, and the CuO2 layers are stacked body-centred. (b) A CuO2 layer viewed along c: the Cu ions form a square lattice bridged by oxygen, so that the nearest-neighbor coupling proceeds through a 180◦ Cu–O–Cu superexchange path. … view at source ↗
Figure 2
Figure 2. Origin and structure of the four-spin ring exchange on a square plaquette. (a) [PITH_FULL_IMAGE:figures/full_fig_p034_2.png] view at source ↗
Figure 3
Figure 3. Energy mapping for T-La2CuO4 in the 2 × 2 × 1 supercell. (a) Model energies against DFT energies for the 13 inequivalent collinear configurations, both measured from the lowest-energy state. The line is y = x, not a fit to the points. The residuals, plotted below on a scale four orders of magnitude finer, lie within ±0.02 meV and give an RMS of 0.009 meV over an energy range of more than 1 eV, so the spin model of E… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Bath decomposition of the sixteen-state extraction for T-La [PITH_FULL_IMAGE:figures/full_fig_p036_4.png]
Figure 5
Figure 5. Figure 5: Energy mapping for SrFeO2: 2×2×2 supercell, 10×10×10 k-mesh, 22 inequivalent collinear configurations, tetrahedron integration. (a) Model energies against DFT energies, both referred to the lowest configuration, with the residuals below. The six configurations whose ga…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references

  1. [1]

    Ab initio methodology for magnetic exchange parameters: Generic four-state energy mapping onto a. Phys. Rev. B , volume =. 2020 , doi =

    2020

  2. [2]

    Xiang, H. J. and Wei, Su-Huai and Whangbo, M.-H. , title =. Phys. Rev. Lett. , volume =. 2008 , doi =

    2008

  3. [3]

    Xiang, H. J. and Kan, E. J. and Wei, Su-Huai and Whangbo, M.-H. and Gong, X. G. , title =. Phys. Rev. B , volume =. 2011 , doi =

    2011

  4. [4]

    and Lee, C

    Xiang, H. and Lee, C. and Koo, H.-J. and Gong, X. and Whangbo, M.-H. , title =. Dalton Trans. , volume =. 2013 , doi =

    2013

  5. [5]

    First-Principles Periodic Calculation of Four-Body Spin Terms in High-

    Moreira, Ib. First-Principles Periodic Calculation of Four-Body Spin Terms in High-. Phys. Rev. Lett. , volume =. 2006 , doi =

    2006

  6. [6]

    Toader, A. M. and Goff, J. P. and Roger, M. and Shannon, N. and Stewart, J. R. and Enderle, M. , title =. Phys. Rev. Lett. , volume =. 2005 , doi =

    2005

  7. [7]

    and Hayden, S

    Coldea, R. and Hayden, S. M. and Aeppli, G. and Perring, T. G. and Frost, C. D. and Mason, T. E. and Cheong, S.-W. and Fisk, Z. , title =. Phys. Rev. Lett. , volume =. 2001 , doi =

    2001

  8. [8]

    and Malrieu, Jean-Paul , title =

    Calzado, Carmen J. and Malrieu, Jean-Paul , title =. Phys. Rev. B , volume =. 2004 , doi =

    2004

Show all 27 references
  1. [9]

    and Tohyama, T

    Mizuno, Y. and Tohyama, T. and Maekawa, S. , title =. Phys. Rev. B , volume =. 1998 , doi =

    1998

  2. [10]

    and Tassel, C

    Tsujimoto, Y. and Tassel, C. and Hayashi, N. and Watanabe, T. and Kageyama, H. and Yoshimura, K. and Takano, M. and Ceretti, M. and Ritter, C. and Paulus, W. , title =. Nature , volume =. 2007 , doi =

    2007

  3. [11]

    Rahman, Mavlanjan and Nie, Yao-zhuang and Guo, Guang-hua , title =. Inorg. Chem. , volume =. 2013 , doi =

    2013

  4. [12]

    Takahashi, Minoru , title =. J. Phys. C: Solid State Phys. , volume =. 1977 , doi =

    1977

  5. [13]

    MacDonald, A. H. and Girvin, S. M. and Yoshioka, D. , title =. Phys. Rev. B , volume =. 1988 , doi =

    1988

  6. [14]

    and Hetherington, J

    Roger, M. and Hetherington, J. H. and Delrieu, J. M. , title =. Rev. Mod. Phys. , volume =. 1983 , doi =

    1983

  7. [15]

    Anisotropic Superexchange Interaction and Weak Ferromagnetism , journal =

    Moriya, T. Anisotropic Superexchange Interaction and Weak Ferromagnetism , journal =. 1960 , doi =

    1960

  8. [16]

    and Furthm

    Kresse, G. and Furthm. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set , journal =. 1996 , doi =

    1996

  9. [17]

    Projector augmented-wave method , journal =

    Bl. Projector augmented-wave method , journal =. 1994 , doi =

    1994

  10. [18]

    and Burke, Kieron and Ernzerhof, Matthias , title =

    Perdew, John P. and Burke, Kieron and Ernzerhof, Matthias , title =. Phys. Rev. Lett. , volume =. 1996 , doi =

    1996

  11. [19]

    Dudarev, S. L. and Botton, G. A. and Savrasov, S. Y. and Humphreys, C. J. and Sutton, A. P. , title =. Phys. Rev. B , volume =. 1998 , doi =

    1998

  12. [20]

    Blaha, Peter and Schwarz, Karlheinz and Tran, Fabien and Laskowski, Robert and Madsen, Georg K. H. and Marks, Laurence D. , title =. J. Chem. Phys. , volume =. 2020 , doi =

    2020

  13. [21]

    Whangbo, Myung-Hwan and Koo, Hyun-Joo and Dai, Dadi , title =. J. Solid State Chem. , volume =. 2003 , doi =

    2003

  14. [22]

    Whangbo, Myung-Hwan and Hoffmann, Roald , title =. J. Am. Chem. Soc. , volume =. 1978 , doi =

    1978

  15. [23]

    and Burdett, Jeremy K

    Albright, Thomas A. and Burdett, Jeremy K. and Whangbo, Myung-Hwan , title =

  16. [24]

    Dai, Dadi and Whangbo, Myung-Hwan , title =. J. Chem. Phys. , volume =. 2001 , doi =

    2001

  17. [25]

    and Bortis, Amad

    Fedorova, Natalya S. and Bortis, Amad. Four-spin ring interaction as a source of unconventional magnetic orders in orthorhombic perovskite manganites , journal =. 2018 , doi =

    2018

  18. [26]

    and Ederer, Claude and Spaldin, Nicola A

    Fedorova, Natalya S. and Ederer, Claude and Spaldin, Nicola A. and Scaramucci, Andrea , title =. Phys. Rev. B , volume =. 2015 , doi =

    2015

  19. [27]

    2026 , note =

    Rocquefelte, Xavier , title =. 2026 , note =

    2026

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.