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arxiv: 2603.28355 · v2 · submitted 2026-03-30 · ❄️ cond-mat.mtrl-sci · cond-mat.str-el

Incipient magnetic instability in RuO₂ with random phase approximation

Diana Csontosov\'a , Kyo-Hoon Ahn , Jan Kune\v{s} This is my paper

Pith reviewed 2026-05-14 21:44 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.str-el
keywords RuO2altermagnetismmagnetic instabilityrandom phase approximationHubbard modelspin susceptibilityHartree-Fockcommensurate order
0
0 comments X

The pith

In stoichiometric RuO2 without spin-orbit coupling, commensurate altermagnetic order is the leading magnetic instability at low temperatures.

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

The paper applies the Hartree-Fock approximation and random phase approximation to a three-orbital Hubbard model of RuO2 without spin-orbit coupling. It finds that the spin susceptibility is the dominant response, leading to commensurate altermagnetic order in the stoichiometric compound at sufficiently low temperatures. At higher temperatures or with hole doping, incommensurate wave vectors become favored instead. This matters for understanding the magnetic properties of RuO2, a material of interest for altermagnetism, and for distinguishing altermagnetic order from conventional antiferromagnetism through band structure effects.

Core claim

Using Hartree-Fock followed by random phase approximation on a three-orbital Hubbard model without spin-orbit coupling, the analysis of static susceptibility eigenvalues shows spin susceptibility dominates. In the stoichiometric system, commensurate altermagnetic order is the leading instability at low temperatures, while incommensurate vectors emerge at higher temperatures or with hole doping. The staggered Weiss field splits bands differently in altermagnets than in antiferromagnets.

What carries the argument

Eigenvalues and eigenvectors of the static susceptibility matrix from the random phase approximation, identifying the dominant magnetic instability channel and wave vector.

If this is right

  • Commensurate altermagnetism is the ground state instability in undoped RuO2 at low T.
  • Incommensurate magnetic orders are preferred at elevated temperatures or upon hole doping.
  • The spin channel is the primary response over charge and other channels.
  • Qualitative differences in band splitting distinguish altermagnetic from antiferromagnetic order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Experiments on pure RuO2 at very low temperatures could confirm or refute the predicted commensurate order.
  • Similar modeling approaches may apply to other proposed altermagnets to predict their ordering temperatures.
  • Tuning via doping or temperature could be used to switch between magnetic phases in RuO2 devices.
  • Including spin-orbit coupling, omitted here, might stabilize different instabilities or suppress the altermagnetic one.

Load-bearing premise

The three-orbital Hubbard model without spin-orbit coupling combined with Hartree-Fock and RPA accurately describes the magnetic response of real RuO2.

What would settle it

If neutron scattering or other probes find no magnetic order or an incommensurate order in stoichiometric RuO2 at low temperatures, the prediction of commensurate altermagnetism as the leading instability would be falsified.

Figures

Figures reproduced from arXiv: 2603.28355 by Diana Csontosov\'a, Jan Kune\v{s}, Kyo-Hoon Ahn.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Crystal structure of RuO [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Leading eigenvalue Λ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The position of the instability along the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The temperature dependence of the largest eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Fermi surfaces of antiferromagnetic RuO [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The cuts of [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Orbital-resolved density of states in the NM phase [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (letf) Temperature dependence of the local magnetic [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Site-resolved effect of the staggered potential ∆ on [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. The temperature dependence of the smallest eigenvalues Λ [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The temperature dependence of the smallest eigenvalues Λ [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The absolute values of the overlaps of the eigenvectors [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Momentum-resolved contribution to the leading eigenvalue of static bubble ˜χ [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Imaginary part of the [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Imaginary part of the (a), (c), (e), (g) [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Imaginary part of the (a), (c), (e), (g) [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
read the original abstract

We study the instability in RuO$_2$ using the Hartree-Fock approximation followed by the random phase approximation. We employ a three-orbital Hubbard model without spin-orbit coupling. An analysis of the eigenvalues and eigenvectors of the static susceptibility in the non-magnetic phase for various local interaction parameters $U$, $J_H$, and hole doping $n$ shows that the spin susceptibility is the dominant response channel. In the stoichiometric system without spin-orbit coupling, commensurate altermagnetic order is identified as the leading instability at sufficiently low temperatures, whereas at higher temperatures or finite hole doping, incommensurate wave vectors emerge. To elucidate the origin of the magnetic instability, we analyze the band spitting by the staggered Weiss field and discuss the qualitative difference between altermagnets and antiferromagnets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript studies magnetic instabilities in RuO2 via a three-orbital Hubbard model (no SOC) treated in Hartree-Fock followed by RPA. Analysis of eigenvalues and eigenvectors of the static susceptibility in the non-magnetic phase, for varying U, JH and hole doping n, shows spin susceptibility dominates; in the stoichiometric limit the leading instability at low T is identified as commensurate altermagnetic order, while incommensurate vectors appear at higher T or finite doping. Band splitting under a staggered Weiss field is used to contrast altermagnets with conventional antiferromagnets.

Significance. If the model faithfully represents the dominant response, the work supplies a concrete microscopic route to altermagnetic order in RuO2 and clarifies how the staggered Weiss field splits bands differently from antiferromagnets. The RPA eigenvalue analysis offers a systematic, parameter-dependent way to rank competing instabilities, which is useful for materials where altermagnetism is conjectured but not yet confirmed by direct experiment.

major comments (3)
  1. [Abstract and Results] The central identification of commensurate altermagnetic order as the leading instability (abstract and results) rests on the largest eigenvalue of the RPA susceptibility matrix occurring at the commensurate wavevector and diverging first upon lowering T. No numerical values are supplied for the critical U/JH ratio, the temperature scale, or the magnitude of the eigenvalue divergence, nor are convergence checks with respect to k-mesh density or interaction cutoff reported; without these the claim that the instability is 'at sufficiently low temperatures' remains qualitative and difficult to assess for robustness.
  2. [Model Hamiltonian] The three-orbital Hubbard model without SOC is adopted as the starting point (model section). Because real RuO2 possesses non-negligible spin-orbit coupling that can shift nesting vectors and mix orbital characters, the paper should demonstrate that the altermagnetic peak remains the largest eigenvalue when a small SOC term is added or when the tight-binding parameters are taken from a full DFT downfolding that includes all t2g and eg states; otherwise the reported leading instability risks being an artifact of the orbital truncation and SOC omission.
  3. [Discussion] The distinction between altermagnets and antiferromagnets is drawn from the band splitting induced by the staggered Weiss field (discussion). A quantitative plot or table comparing the RPA susceptibility intensity at the commensurate altermagnetic vector versus nearby incommensurate vectors, for the same set of U, JH and n values, would be required to substantiate that the altermagnetic channel is unambiguously dominant rather than merely competitive.
minor comments (2)
  1. The manuscript would benefit from explicit statements of the numerical values of U and JH (in eV) used for the stoichiometric plots and from a brief description of the k-point sampling employed in the susceptibility calculations.
  2. Figure captions should specify the temperature or interaction parameters at which each susceptibility map is evaluated so that readers can directly connect the panels to the claims about low-T versus high-T behavior.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and Results] The central identification of commensurate altermagnetic order as the leading instability (abstract and results) rests on the largest eigenvalue of the RPA susceptibility matrix occurring at the commensurate wavevector and diverging first upon lowering T. No numerical values are supplied for the critical U/JH ratio, the temperature scale, or the magnitude of the eigenvalue divergence, nor are convergence checks with respect to k-mesh density or interaction cutoff reported; without these the claim that the instability is 'at sufficiently low temperatures' remains qualitative and difficult to assess for robustness.

    Authors: We agree that quantitative details would make the identification more robust. In the revised manuscript we will report the critical U/JH values at which the leading eigenvalue diverges, provide an estimate of the temperature scale from the T-dependence of the susceptibility, and include convergence tests with respect to k-mesh density. revision: yes

  2. Referee: [Model Hamiltonian] The three-orbital Hubbard model without SOC is adopted as the starting point (model section). Because real RuO2 possesses non-negligible spin-orbit coupling that can shift nesting vectors and mix orbital characters, the paper should demonstrate that the altermagnetic peak remains the largest eigenvalue when a small SOC term is added or when the tight-binding parameters are taken from a full DFT downfolding that includes all t2g and eg states; otherwise the reported leading instability risks being an artifact of the orbital truncation and SOC omission.

    Authors: The three-orbital model without SOC is chosen to isolate the correlation-driven altermagnetic instability within the t2g manifold. We will add a symmetry-based discussion showing that a small SOC term is not expected to alter the leading character of the instability. A full numerical demonstration with SOC or a complete DFT downfolding lies outside the present scope. revision: partial

  3. Referee: [Discussion] The distinction between altermagnets and antiferromagnets is drawn from the band splitting induced by the staggered Weiss field (discussion). A quantitative plot or table comparing the RPA susceptibility intensity at the commensurate altermagnetic vector versus nearby incommensurate vectors, for the same set of U, JH and n values, would be required to substantiate that the altermagnetic channel is unambiguously dominant rather than merely competitive.

    Authors: We agree that an explicit comparison is useful. The revised manuscript will include a new figure (or table) that directly compares the largest susceptibility eigenvalues at the commensurate altermagnetic vector with those at nearby incommensurate vectors for representative values of U, JH and n. revision: yes

standing simulated objections not resolved
  • A complete numerical verification of the leading instability after inclusion of SOC or using a full DFT downfolding that incorporates all t2g and eg states.

Circularity Check

0 steps flagged

No significant circularity; susceptibility eigenvalues computed directly from model

full rationale

The paper constructs a three-orbital Hubbard Hamiltonian, solves it in the non-magnetic Hartree-Fock approximation, and then evaluates the static spin susceptibility matrix via RPA. The leading instability is identified by inspecting the largest eigenvalue and its wavevector for given U, JH, and filling; this identification is a direct numerical output of the chosen model parameters and does not reduce to a fit, a self-citation, or a definitional tautology. No load-bearing step relies on prior results by the same authors or on an ansatz smuggled through citation. The derivation chain is therefore self-contained within the stated model and approximation.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The central claim rests on the validity of the three-orbital Hubbard model, the Hartree-Fock mean-field treatment of the non-magnetic phase, and the random-phase-approximation evaluation of susceptibility; interaction strengths U and JH and doping n are treated as tunable parameters.

free parameters (3)
  • U
    Local Coulomb interaction strength varied across a range to locate the instability
  • JH
    Hund's coupling varied to locate the instability
  • n
    Hole doping level varied to examine doping dependence
axioms (3)
  • domain assumption Three-orbital Hubbard model without spin-orbit coupling
    Used to represent the electronic structure of RuO2
  • standard math Hartree-Fock approximation for the non-magnetic phase
    Mean-field decoupling employed to obtain the reference state
  • standard math Random phase approximation for static susceptibility
    Linear response theory used to extract instabilities from eigenvalues

pith-pipeline@v0.9.0 · 5445 in / 1400 out tokens · 56405 ms · 2026-05-14T21:44:41.794697+00:00 · methodology

discussion (0)

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Reference graph

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