Pith. sign in

REVIEW 3 major objections 5 minor 89 references

Spontaneous emergence of altermagnetism in the single-orbital extended Hubbard model

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read This paper shows that $d$-wave altermagnetism can emerge spontaneously in the single-orbital extended Hubbard model on a square lattice, without crystallographic anisotropy or multi-orbital physics, through coexisting antiferromagnetism…

desk verdict A credible new single-orbital route to altermagnetism, but the 'wide region' claim hangs on a Hartree-term omission the authors themselves admit shrinks the phase. read the letter →

arxiv 2507.00837 v2 pith:M6G3WLER submitted 2025-07-01 cond-mat.str-el

Jin-Wei Dong , Yu-Han Lin , Ruiqing Fu , Xianxin Wu , Gang Su , Ziqiang Wang , Sen Zhou This is my paper · ORCID
classification cond-mat.str-el PACS 71.10.Fd75.10.-b
keywords altermagnetismextendedHubbardmodelt-U-Vmean-fieldtheoryd-wavespinbondordercurrentconductivitysquarelattice
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 establishes a new microscopic route to altermagnetism: a $d$-wave altermagnet can emerge spontaneously in the single-orbital extended Hubbard model ($t$-$U$-$V$) on the square lattice, without any crystallographic anisotropy or orbital degrees of freedom. Working at the mean-field level and doping the system 8% away from half-filling, the authors show that the ground state in a wide region of the $U$-$V$ plane combines onsite collinear antiferromagnetism with complex $d$-wave nearest-neighbor spin bond orders. The coexistence produces spin-split electronic bands with compensated zero net magnetization — the defining signature of altermagnetism — with the spin polarization oriented perpendicular to the ordered moments. It suggests that altermagnetism could be a generic property of doped Mott-Hubbard systems rather than something requiring special crystal or orbital ingredients, and it predicts spin-transport behavior that reverses with carrier doping.

What carries the argument

The machinery is a mean-field decoupling of the $t$-$U$-$V$ Hamiltonian in terms of local magnetic moments and nearest-neighbor spin/charge bond operators, solved self-consistently on a $\sqrt{2}\times\sqrt{2}$ unit cell. The central object is the coexistent order-parameter set: collinear AFM moments $m_i=\gamma_i m \mathbf{e}_{AFM}$ with staggered sign $\gamma_i$, plus complex $d$-wave spin bond orders $\chi^\mu_{\langle ij\rangle}=\eta_{ij}(\chi'_{\mu d}+i\gamma_i\chi''_{\mu d})$, whose real part is the $d$-wave spin bond ($d$SB) order and imaginary part the $d$-wave spin current ($d$SC) order. The spin directions of the AFM, $d$SB, and $d$SC orders are mutually perpendicular and form a right-handed chirality. At the $X$ point the four eigenenergies are $E_{\tau\tau'}=\tau Um/2+\tau' 2V(\chi_{dSC}-\tau\Omega \chi_{dSB})$, so the band splitting is highly uneven; for hole doping one band of the lower doublet is pushed below the Fermi level, stabilizing the state. The same self-consistent machinery produces the $U$-$V$ phase diagram and the doping evolution.

What would settle it

Run a self-consistent mean-field calculation that keeps the direct Hartree terms of $U$ and $V$ on the same $t$-$U$-$V$ model at 8% hole doping and $(U,V)=(3,1.5)$; if the lowest-energy state is a $(\pi,\pi)$ charge-density wave or a paramagnet with no spin-split bands, the claim that $d$-wave AM is the spontaneous ground state is falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that a $d$-wave altermagnetic metal is the mean-field ground state of the single-orbital $t$-$U$-$V$ model on a square lattice at moderate interactions and 8% hole doping. The state is realized by the simultaneous development of onsite antiferromagnetic order (driven by $U$) and complex $d$-wave spin bond orders on nearest-neighbor bonds (driven by $V$): a real $d$-wave spin bond order and an imaginary $d$-wave spin current order, with spin directions mutually perpendicular and forming a right-handed chirality. These orders lift Kramers degeneracy without spin-orbit coupling, producing momentum-dependent spin splitting of $d$-wave symmetry while keeping net magnetization zero. The paper further shows that doping evolution tunes the orders, that the spin splitting is partially spin-polarized along a direction perpendicular to the ordered moments, and that electron versus hole doping reverses the chirality and the longitudinal spin conductivity.

Load-bearing premise

The wide $d$-wave AM region rests on leaving out the direct Hartree terms of $U$ and $V$ to avoid double-counting with density-functional theory, and if those terms are restored a competing $(\pi,\pi)$ charge-density wave takes over and the $d$-wave AM region shrinks significantly.

Editorial extensions

If this is right

  • A single-orbital correlated material with strong nearest-neighbor Coulomb repulsion, doped away from half-filling, can display altermagnetic spin splitting without any structural anisotropy or orbital ordering.
  • The $d$-wave AM state is the mean-field ground state over a wide $U$-$V$ region at 8% hole doping, so the phase diagram indicates where to look for such a state experimentally.
  • The spin-split bands are partially polarized perpendicular to the ordered moments, so spin-resolved photoemission should see a $d$-wave spin texture rather than polarization along the Néel vector.
  • Electron doping reverses the handedness of the three-order complex and flips the sign of the longitudinal spin conductivity, giving a doping- or gate-controlled spin-current switch.

Reading between the lines

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

  • If the mean-field result survives in more accurate many-body treatments, altermagnetism could be a generic feature of doped square-lattice Mott insulators, making cuprate-like compounds natural places to hunt for spin-split Fermi surfaces without spin-orbit coupling.
  • The particle-hole asymmetry in spin conductivity suggests an all-electric spintronic switch: reversing the sign of doping or gate voltage reverses the spin-current direction while the electric field stays fixed, an application the paper only hints at.
  • The sensitivity to the Hartree channel means the phase boundary in the $U$-$V$ plane is not the final word; a systematic head-to-head comparison between Hartree-Fock and slave-boson treatments at the same parameters would tell whether the wide region is an artifact or a stable feature.
  • A targeted experiment, spin- and angle-resolved photoemission on lightly doped Sr$_2$IrO$_4$, could test the predicted perpendicular spin polarization even before transport measurements are available.
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.

Referee Report

3 major / 5 minor

Summary. The manuscript studies the single-orbital extended Hubbard model (t-U-V model) on the square lattice within a self-consistent mean-field approximation. At 8% hole doping and moderate U and V, the authors find a mean-field ground state that they identify as a d-wave altermagnet: coexisting collinear AFM order, real d-wave spin bond order (dSB), and imaginary d-wave spin current order (dSC), with compensated magnetization and momentum-dependent, partially spin-polarized band splitting. The paper presents the U-V phase diagram, a symmetry analysis classifying the state as an S-type strong altermagnet, the doping evolution of the order parameters, and a Kubo-formula spin conductivity calculation. The central claim is that d-wave altermagnetism can emerge spontaneously in a single-orbital model without crystallographic anisotropy or multi-orbital physics, and that the spin-splitting direction is perpendicular to the ordered moment, with spin current reversed between electron- and hole-doping.

Significance. If the central claim holds, the paper identifies an original and potentially important mechanism: interaction-driven bond order in a single-orbital correlated model generates non-relativistic spin splitting and tunable spin transport without invoking structural anisotropy or orbital degrees of freedom. The symmetry analysis is careful, and the self-consistent energy-minimization approach is a standard and internally consistent tool. The predicted partial spin polarization perpendicular to the ordered moments, and its reversal between electron and hole doping, are concrete and falsifiable statements. However, the significance is currently limited by an admitted sensitivity: the direct Hartree terms of U and V are omitted from the mean-field decoupling, and the manuscript itself states that including them promotes a charge-density-wave instability and significantly shrinks the d-wave AM region. Because that supporting material is not available in the manuscript, the 'wide region' claim in the abstract and Fig. 1(a) is not yet established for the model written in Eq. (1).

major comments (3)
  1. [Eq. (2) and Summary/Discussion] The mean-field Hamiltonian in Eq. (2) neglects the direct Hartree terms of U and V, justified by saying that their contributions are already considered in density functional theory. This justification does not apply to the standalone model Hamiltonian in Eq. (1), whose mean-field treatment should contain all Hartree and Fock channels. The Summary explicitly concedes that including the Hartree channel promotes a (pi,pi) charge-density-wave instability and 'significantly reduces the parameter region occupied by the d-wave AM phase.' Since this result is only stated to be in the Supplemental Materials, the 'wide region' of d-wave AM in Fig. 1(a) is not demonstrated for Eq. (1). Please include the Hartree-included mean-field phase diagram in the main text, or clearly reframe the abstract and conclusions so that the claim refers to the specific mean-field Hamiltonian actually solved.
  2. [Phase diagram and d-wave AM, Fig. 1(a)] The phase diagram in Fig. 1(a) is computed with the Hartree terms omitted, and the paper states that the main findings survive in a strong-coupling slave-boson treatment only in the Supplemental Materials. Neither the Hartree-included results nor the slave-boson results are present in the manuscript. Because these are the two pieces of evidence that would show the d-wave AM region is not an artifact of the truncated mean-field decoupling, the main text should at least summarize the quantitative extent of the AM region after Hartree inclusion, rather than leaving this load-bearing check in an unavailable supplement.
  3. [Summary and discussions, last paragraph] The final paragraph acknowledges that 'mean-field theory is an uncontrolled variational approximation' and that 'competing fluctuations and alternative ordering tendencies beyond the chosen mean-field ansatz may substantially modify the phase diagram.' This is an appropriate caveat, but it sits in tension with the abstract's unqualified 'demonstrate' and 'wide region' language. The authors should either provide the Hartree-included phase diagram that supports the qualitative claim, or soften the abstract and the opening claim of the Introduction so that 'possible route' rather than 'demonstrated wide region' is the operative statement.
minor comments (5)
  1. [Introduction, first paragraph] The phrase 'the hall mark of AFM' contains a typo; it should be 'the hallmark of AFM.'
  2. [Model and mean-field theory, Eq. (2)] The notation 'm^i' and 'chi^nu_ij' is introduced in the text but the equations would benefit from explicitly defining the Pauli-matrix indices and the sublattice factor gamma_i before Eq. (2) is used.
  3. [Symmetry analyses of d-wave AM] The symmetry discussion is dense and would be more accessible if the key composite symmetries were listed in a compact table with their explicit action on momenta and spin, rather than only in the text.
  4. [Doping evolution and spin conductivity] The Kubo formula for spin conductivity uses a constant scattering rate Gamma=0.02, but the units and normalization of sigma_ab are not specified; please state the expected order of magnitude or the unit (e.g., e/2 in appropriate lattice units) so that the magnitude in Fig. 3(c) is interpretable.
  5. [References] Reference [2] lists 'Proc. Nati. Acad. Sci.'; the standard abbreviation is 'Proc. Natl. Acad. Sci.'

Circularity Check

0 steps flagged · score 0.0 of 10

The mean-field derivation is self-contained; the d-wave AM state is a numerical energy minimum, and the admitted Hartree sensitivity is a correctness limitation rather than a circular construction.

full rationale

The claimed derivation is not circular. The d-wave AM state is obtained by numerically solving the self-consistent mean-field equations (Eq. 2) for the stated t-U-V Hamiltonian (Eq. 1), starting from multiple initial conditions and retaining the lowest-energy converged solution; the spin-split bands, Fermi surface, and spin conductivity are then computed from the resulting band structure. No order parameter or spin-splitting is imposed by construction: the AFM, dSB, and dSC amplitudes (m=0.269, chi_dSC=0.083, chi_dSB=0.062 at (U,V)=(3,1.5)) are self-consistently determined, and the phase diagram follows from energy comparison. The dSC/dCC degeneracy invoked for the cyan region is supported by an independent prior work (Raghu et al., 2008) in addition to the authors' own Ref. [65], and does not fix the central AM state. The one substantive concern is the deliberate omission of the direct Hartree terms in Eq. (2), justified by citing DFT double-counting ([63]); the Summary itself concedes that restoring this channel promotes a (pi,pi) CDW and 'significantly reduces the parameter region occupied by the d-wave AM phase.' This is an admitted approximation sensitivity and a possible correctness/overclaim issue for the abstract's 'wide region' wording, but it is not circular: the calculation is not defined in terms of its conclusion, no equation reduces to another by construction, and no fitted parameter is relabeled as a prediction. Therefore the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model parameters U and V are scanned interaction strengths, not fitted to data; they are listed as free parameters only because the claimed AM region is selected by their values. The spectral broadening Gamma is a numerical choice in the Kubo calculation. The load-bearing modeling choices are the mean-field decoupling, the omission of Hartree terms, and the sqrt(2) by sqrt(2) unit cell ansatz. No new particles, forces, or dimensions are introduced.

free parameters (3)
  • U (onsite Coulomb interaction) = scanned, not fitted; representative U=3
    Model input in the t-U-V Hamiltonian, not fitted to data. The claimed d-wave AM region spans moderate U values.
  • V (nearest-neighbor Coulomb interaction) = scanned, not fitted; representative V=1.5
    Model input that drives d-wave spin bond orders; not fitted to data.
  • Gamma (spectral broadening) = 0.02
    Chosen constant in the Kubo formula for spin conductivity; affects the numerical value of the conductivity but not the existence of the AM state.
assumptions (5)
  • domain assumption Mean-field decoupling of the onsite and nearest-neighbor interactions (Eq. 2) captures the correct ordering tendencies of the t-U-V model.
    The paper calls mean-field theory an uncontrolled variational approximation, and the entire phase diagram is obtained within this approximation.
  • ad hoc to paper Direct Hartree terms of U and V can be omitted from the mean-field Hamiltonian without changing the qualitative ground state.
    Omission is justified 'to avoid double counting' with DFT (Ref. 63), but the Summary states that including the Hartree channel promotes a (pi,pi) CDW and significantly reduces the d-wave AM region.
  • domain assumption Restricting candidate states to a sqrt(2) by sqrt(2) unit cell with onsite moments and nearest-neighbor bond order parameters is sufficient to find the lowest-energy mean-field state.
    The text says 'We consider the quantum states to be periodic with an enlarged sqrt(2) by sqrt(2) unit cell'; stripes, incommensurate order, and other channels are excluded, and the Summary says competing fluctuations may modify the phase diagram.
  • standard math Particle-hole symmetry of the nearest-neighbor-hopping t-U-V model maps hole-doped states to electron-doped states with reversed spin directions.
    Used to obtain electron-doped behavior and reversed spin conductivity without a separate calculation.
  • domain assumption The single-orbital t-U-V model captures relevant low-energy physics of materials such as cuprates and iridates.
    Used in the Summary to suggest Sr2IrO4 as a candidate and to claim the finding expands the range of AM materials.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Spontaneous emergence of altermagnetism in the single-orbital extended Hubbard model." pith.science (2026). https://pith.science/paper/M6G3WLER

@misc{pith2026250700837,
  author       = {Pith},
  title        = {Pith review of: Spontaneous emergence of altermagnetism in the single-orbital extended Hubbard model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M6G3WLER}},
  note         = {Machine review of arXiv:2507.00837}
}
abstract

Altermagnetism (AM), the recently discovered third class of collinear magnetic order, is characterized by non-relativistic momentum-dependent spin-split electronic structure with compensated zero net magnetization. It can arise from the conventional antiferromagnetism by introducing local anisotropy on the two opposite-spin sublattices, either through structural changes in local crystallographic symmetry or spontaneous emergence of local staggered orbital order from electron correlations in multi-orbital systems. Here, we demonstrate on the two-dimensional square lattice that a $d$-wave AM can emerge spontaneously in the single-orbital extended Hubbard model, without invoking crystallographic anisotropy and multi-orbital physics. We carry out mean-field studies on the concrete single-orbital $t$-$U$-$V$ model with $U$ and $V$ the onsite and nearest-neighbor Coulomb interactions, obtaining the mean-field ground states, analyzing their properties, and determining the phase diagram in the $U$-$V$ plane. The $d$-wave AM with novel spin-transport behavior is found to be stabilized in a wide region of the phase diagram when the system is doped away from half-filling, actualized by the coexistence of onsite antiferromagnetic order and complex $d$-wave nearest-neighbor spin bond orders. Our findings provide an alternative route to achieve AM and substantially expand the range of candidate AM materials.

Figures

Figures reproduced from arXiv: 2507.00837 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Phase diagram of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematics of (a) AFM, (b) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. c as a function of hole doping x. Clearly, the longi￾tudinal spin conductivity is zero at exactly half filling x = 0, becomes nonzero and enhances gradually upon hole doping. It reaches its maximum value at xc and then drops abruptly to zero, as the mean-field ground state undergoes a first order transition from d-wave AM to PM phase. The states on the electron-doped side can be readily ob￾tained from those on the h… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

89 extracted references · 62 canonical work pages

  1. [1]

    ˇSmejkal, R

    L. ˇSmejkal, R. Gonz ´alez-Hern´andez, T. Jungwirth, and J. Sinova, Crystal time-reversal symmetry breaking and spon- taneous Hall effect in collinear antiferromagnets, Sci. Adv.6, eaaz8809 (2020)

    2020

  2. [2]

    I. I. Mazin, K. Koepernik, M. D. Johannes, R. Gonz ´alez- Hern´andez, and L. ˇSmejkal, Prediction of unconventional magnetism in doped FeSb 2, Proc. Nati. Acad. Sci.118, e2108924118 (2021)

    2021

  3. [3]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Beyond Conventional Ferromagnetism and Antiferromagnetism: A Phase with Non- relativistic Spin and Crystal Rotation Symmetry, Phys. Rev. X 12, 031042 (2022)

    2022

  4. [4]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Emerging Research Landscape of Altermagnetism, Phys. Rev. X12, 040501 (2022)

    2022

  5. [5]

    Mazin (The PRX Editors), Editorial: Altermagnetism—A New Punch Line of Fundamental Magnetism, Phys

    I. Mazin (The PRX Editors), Editorial: Altermagnetism—A New Punch Line of Fundamental Magnetism, Phys. Rev. X12, 040002 (2022)

    2022

  6. [6]

    L. Bai, W. Feng, S. Liu, L. ˇSmejkal, Y . Mokrousov, and Y . Yao, Altermagnetism: Exploring new frontiers in magnetism and spintronics, Adv. Funct. Mater.34, 2409327 (2024)

    2024

  7. [7]

    P. Liu, J. Li, J. Han, X. Wan, and Q. Liu, Spin-Group Symme- try in Magnetic Materials with Negligible Spin-Orbit Coupling, Phys. Rev. X12, 021016 (2022)

    2022

  8. [8]

    X. Chen, J. Ren, Y . Zhu, Y . Yu, A. Zhang, P. Liu, J. Li, Y . Liu, C. Li, and Q. Liu, Enumeration and Representation Theory of Spin Space Groups, Phys. Rev. X14, 031038 (2024)

    2024

Show all 89 references
  1. [9]

    Jiang, Z

    Y . Jiang, Z. Song, T. Zhu, Z. Fang, H. Weng, Z.-X. Liu, J. Yang, and C. Fang, Enumeration of Spin-Space Groups: To- ward a Complete Description of Symmetries of Magnetic Or- ders, Phys. Rev. X14, 031039 (2024)

    2024

  2. [10]

    Z. Xiao, J. Zhao, Y . Li, R. Shindou, and Z.-D. Song, Spin Space Groups: Full Classification and Applications, Phys. Rev. X14, 031037 (2024)

    2024

  3. [11]

    Zeng and Y .-J

    S. Zeng and Y .-J. Zhao, Description of two-dimensional alter- magnetism: Categorization using spin group theory, Phys. Rev. B110, 054406 (2024)

    2024

  4. [12]

    C. Wu, K. Sun, E. Fradkin, and S.-C. Zhang, Fermi liquid in- stabilities in the spin channel, Phys. Rev. B75, 115103 (2007)

    2007

  5. [13]

    Gonz ´alez-Hern´andez, L

    R. Gonz ´alez-Hern´andez, L. ˇSmejkal, K. V ´yborn´y, Y . Yahagi, J. Sinova, T. c. v. Jungwirth, and J. ˇZelezn´y, Efficient Electrical Spin Splitter Based on Nonrelativistic Collinear Antiferromag- netism, Phys. Rev. Lett.126, 127701 (2021)

    2021

  6. [14]

    ˇSmejkal, A

    L. ˇSmejkal, A. B. Hellenes, R. Gonz´alez-Hern´andez, J. Sinova, and T. Jungwirth, Giant and Tunneling Magnetoresistance in Unconventional Collinear Antiferromagnets with Nonrelativis- tic Spin-Momentum Coupling, Phys. Rev. X12, 011028 (2022). 6

    2022

  7. [15]

    Shao, S.-H

    D.-F. Shao, S.-H. Zhang, M. Li, C.-B. Eom, and E. Y . Tsymbal, Spin-neutral currents for spintronics, Nat. Commun.12, 7061 (2021)

    2021

  8. [16]

    Hoyer, R

    R. Hoyer, R. Jaeschke-Ubiergo, K.-H. Ahn, L. ˇSmejkal, and A. Mook, Spontaneous crystal thermal Hall effect in insulating altermagnets, Phys. Rev. B111, L020412 (2025)

    2025

  9. [17]

    R. He, D. Wang, N. Luo, J. Zeng, K.-Q. Chen, and L.-M. Tang, Nonrelativistic Spin-Momentum Coupling in Antifer- romagnetic Twisted Bilayers, Phys. Rev. Lett.130, 046401 (2023)

    2023

  10. [18]

    X. Zhou, W. Feng, R.-W. Zhang, L. ˇSmejkal, J. Sinova, Y . Mokrousov, and Y . Yao, Crystal Thermal Transport in Al- termagnetic RuO2, Phys. Rev. Lett.132, 056701 (2024)

    2024

  11. [19]

    J. K. Dai, L. Han, W. Q. He, Y . X. Zhu, H. Bai, W. X. Zhu, S. X. Liang, Y . C. Zhang, Y . Z. Cao, J. W. Liu, F. Pan, and C. Song, Thermal-modulation spin-orbit torque for 180 ◦ N´eel vector switching, Phys. Rev. Mater.8, 074406 (2024)

    2024

  12. [20]

    Z. Feng, X. Zhou, L. ˇSmejkal, L. Wu, Z. Zhu, H. Guo, R. Gonz ´alez-Hern´andez, X. Wang, H. Yan, P. Qin,et al., An anomalous Hall effect in altermagnetic ruthenium dioxide, Na- ture Electronics5, 735 (2022)

    2022

  13. [21]

    Bhowal and N

    S. Bhowal and N. A. Spaldin, Ferroically Ordered Magnetic Octupoles ind-Wave Altermagnets, Phys. Rev. X14, 011019 (2024)

    2024

  14. [22]

    D. S. Antonenko, R. M. Fernandes, and J. W. F. Venderbos, Mirror Chern Bands and Weyl Nodal Loops in Altermagnets, Phys. Rev. Lett.134, 096703 (2025)

    2025

  15. [23]

    X. Duan, J. Zhang, Z. Zhu, Y . Liu, Z. Zhang, I. ˇZuti´c, and T. Zhou, Antiferroelectric Altermagnets: Antiferroelectricity Alters Magnets, Phys. Rev. Lett.134, 106801 (2025)

    2025

  16. [24]

    M. Gu, Y . Liu, H. Zhu, K. Yananose, X. Chen, Y . Hu, A. Stroppa, and Q. Liu, Ferroelectric Switchable Altermag- netism, Phys. Rev. Lett.134, 106802 (2025)

    2025

  17. [25]

    Takahashi, C

    K. Takahashi, C. R. W. Steward, M. Ogata, R. M. Fernandes, and J. Schmalian, Elasto-Hall conductivity and the anomalous Hall effect in altermagnets, Phys. Rev. B111, 184408 (2025)

    2025

  18. [26]

    ˇSmejkal, A

    L. ˇSmejkal, A. Marmodoro, K.-H. Ahn, R. Gonz ´alez- Hern´andez, I. Turek, S. Mankovsky, H. Ebert, S. W. D’Souza, O. c. v. ˇSipr, J. Sinova, and T. c. v. Jungwirth, Chiral Magnons in Altermagnetic RuO2, Phys. Rev. Lett.131, 256703 (2023)

    2023

  19. [27]

    Kriegner, K

    D. Kriegner, K. V `yborn`y, K. Olejn´ık, H. Reichlov´a, V . Nov´ak, X. Marti, J. Gazquez, V . Saidl, P. N ˇemec, V . V olobuev,et al., Multiple-stable anisotropic magnetoresistance memory in anti- ferromagnetic MnTe, Nat. Commun.7, 11623 (2016)

    2016

  20. [28]

    I. I. Mazin, Notes on altermagnetism and superconductivity, AAPPS Bull.35, 18 (2025)

    2025

  21. [29]

    A. Bose, S. Vadnais, and A. Paramekanti, Altermagnetism and superconductivity in a multiorbitalt-Jmodel, Phys. Rev. B110, 205120 (2024)

    2024

  22. [30]

    Parthenios, P

    N. Parthenios, P. M. Bonetti, R. Gonz ´alez-Hern´andez, W. H. Campos, L. ˇSmejkal, and L. Classen, Spin and pair density waves in two-dimensional altermagnetic metals, Phys. Rev. B 112, 214410 (2025)

    2025

  23. [31]

    M. Roig, A. Kreisel, Y . Yu, B. M. Andersen, and D. F. Agter- berg, Minimal models for altermagnetism, Phys. Rev. B110, 144412 (2024)

    2024

  24. [32]

    M. Hu, X. Cheng, Z. Huang, and J. Liu, Catalog ofC-Paired Spin-Momentum Locking in Antiferromagnetic Systems, Phys. Rev. X15, 021083 (2025)

    2025

  25. [33]

    Y . Yu, H. G. Suh, M. Roig, and D. F. Agterberg, Altermag- netism from coincident Van Hove singularities: application to κ-Cl, Nat. Commun.16, 2950 (2025)

    2025

  26. [34]

    L. V . Pupim and M. S. Scheurer, Adatom Engineering Mag- netic Order in Superconductors: Applications to Altermagnetic Superconductivity, Phys. Rev. Lett.134, 146001 (2025)

    2025

  27. [35]

    Krempask ´y, L

    J. Krempask ´y, L. ˇSmejkal, S. W. D’Souza, M. Hajlaoui, g. Springholz, K. Uhl ´ıˇrov´a, F. Alarab, P. C. Constantinou, V . Strocov, D. Usanov, W. R. Pudelko, R. Gonz´alez-Hern´andez, A. B. Hellenes, Z. Jansa, H. Reichlov´a, Z. ˇSob´aˇn, R. D. G. Be- tancourt, P. Wadley, J. Si...

    2024

  28. [36]

    Y .-P. Zhu, X. Chen, X.-R. Liu, Y . Liu, P. Liu, H. Zha, G. Qu, C. Hong, J. Li, Z. Jiang, X.-M. Ma, Y .-J. Hao, M.-Y . Zhu, W. Liu, M. Zeng, S. Jayaram, M. Lenger, J. Ding, S. Mo, K. Tanaka, M. Arita, Z. Liu, M. Ye, D. Shen, J. Wrachtrup, Y . Huang, R.-H. He, S. Qiao, Q. Liu, ...

    2024

  29. [37]

    S. Lee, S. Lee, S. Jung, J. Jung, D. Kim, Y . Lee, B. Seok, J. Kim, B. G. Park, L. ˇSmejkal, C.-J. Kang, and C. Kim, Broken Kramers Degeneracy in Altermagnetic MnTe, Phys. Rev. Lett. 132, 036702 (2024)

    2024

  30. [38]

    Osumi, S

    T. Osumi, S. Souma, T. Aoyama, K. Yamauchi, A. Honma, K. Nakayama, T. Takahashi, K. Ohgushi, and T. Sato, Obser- vation of a giant band splitting in altermagnetic MnTe, Phys. Rev. B109, 115102 (2024)

    2024

  31. [39]

    Fedchenko, J

    O. Fedchenko, J. Min´ar, A. Akashdeep, S. W. D’Souza, D. Vasi- lyev, O. Tkach, L. Odenbreit, Q. Nguyen, D. Kutnyakhov, N. Wind, L. Wenthaus, M. Scholz, K. Rossnagel, M. Hoesch, M. Aeschlimann, B. Stadtm ¨uller, M. Kl ¨aui, G. Sch ¨onhense, T. Jungwirth, A. B. Hellenes, G. Jako...

    2024

  32. [40]

    Reimers, L

    S. Reimers, L. Odenbreit, L. ˇSmejkal, V . N. Strocov, P. Con- stantinou, A. B. Hellenes, R. J. Ubiergo, W. H. Campos, V . K. Bharadwaj, A. Chakraborty, T. Denneulin, W. Shi, R. E. Dunin- Borkowski, S. Das, M. Kl¨aui, J. Sinova, and M. Jourdan, Direct observation of altermagne...

    2024

  33. [41]

    Reichlova, R

    H. Reichlova, R. L. Seeger, R. Gonz´alez-Hern´andez, I. Kounta, R. Schlitz, D. Kriegner, P. Ritzinger, M. Lammel, M. Leivisk¨a, A. B. Hellenes, K. Olejn ´ık, V . Petˇriˇcek, P. Dole ˇzal, L. Ho- rak, E. Schmoranzerova, A. Badura, S. Bertaina, A. Thomas, V . Baltz, L. Michez, J...

    2024

  34. [42]

    Z. Lin, D. Chen, W. Lu, X. Liang, S. Feng, K. Yamagami, J. Osiecki, M. Leandersson, B. Thiagarajan, J. Liu, C. Felser, and J. Ma, Observation of Giant Spin Splitting andd-wave Spin Texture in Room Temperature Altermagnet RuO 2, arXiv preprint arxiv:2402.04995 (2024)

    2024 arXiv

  35. [43]

    G. Yang, Z. Li, S. Yang, J. Li, H. Zheng, W. Zhu, Z. Pan, Y . Xu, S. Cao, W. Zhao, A. Jana, J. Zhang, M. Ye, Y . Song, L.-H. Hu, L. Yang, J. Fujii, I. V obornik, M. Shi, H. Yuan, Y . Zhang, Y . Xu, and Y . Liu, Three-dimensional mapping and electronic origin of large altermagn...

    2025

  36. [44]

    Zeng, M.-Y

    M. Zeng, M.-Y . Zhu, Y .-P. Zhu, X.-R. Liu, X.-M. Ma, Y .-J. Hao, P. Liu, G. Qu, Y . Yang, Z. Jiang, K. Yamagami, M. Arita, X. Zhang, T.-H. Shao, Y . Dai, K. Shimada, Z. Liu, M. Ye, Y . Huang, Q. Liu, and C. Liu, Observation of Spin Splitting in Room-Temperature Metallic Antif...

    2024

  37. [45]

    R. B. Regmi, H. Bhandari, B. Thapa, Y . Hao, N. Sharma, J. McKenzie, X. Chen, A. Nayak, M. E. Gazzah, B. G. M´arkus, 7 L. Forr ´o, X. Liu, H. Cao, J. F. Mitchell, I. I. Mazin, and N. J. Ghimire, Altermagnetism in the layered intercalated tran- sition metal dichalcogenide CoNb4...

    2025

  38. [46]

    Zhang, X

    F. Zhang, X. Cheng, Z. Yin, C. Liu, L. Deng, Y . Qiao, Z. Shi, S. Zhang, J. Lin, Z. Liu, M. Ye, Y . Huang, X. Meng, C. Zhang, T. Okuda, K. Shimada, S. Cui, Y . Zhao, G.-H. Cao, S. Qiao, J. Liu, and C. Chen, Crystal-symmetry-paired spin-valley lock- ing in a layered room-temper...

    2025

  39. [47]

    Candelora, M

    C. Candelora, M. Xu, S. Cheng, A. D. Vita, D. Romanin, C. Bigi, M. B. Petersen, A. LaFleur, M. Calandra, J. Miwa, Y . Hwang, Z. Wang, F. Mazzola, and I. Zeljkovic, Discovery of magnetic-field-tunable density modulations and spin tilting in a layered altermagnet, Commun. Mater....

    2026

  40. [48]

    S. G. Jeong, I. H. Choi, S. Nair, L. Buiarelli, B. Pourbahari, J. Y . Oh, N. Bassim, D. Hirai, A. Seo, W. S. Choi, R. M. Fernandes, T. Birol, L. Zhao, J. S. Lee, and B. Jalan, Altermagnetic polar metallic phase in ultrathin epitaxially strained RuO2 films, Proc. Nati. Acad. Sc...

    2026

  41. [49]

    O. E. Parfenov, D. V . Averyanov, I. S. Sokolov, A. N. Mihalyuk, O. A. Kondratev, A. N. Taldenkov, A. M. Tokmachev, and V . G. Storchak, Pushing an Altermagnet to the Ultimate 2D Limit: Symmetry Breaking in Monolayers of GdAlSi, J. Am. Chem. Soc.147, 5911 (2025)

    2025

  42. [50]

    Jiang, M

    B. Jiang, M. Hu, J. Bai, Z. Song, C. Wu, G. Qu, W. Li, W. Zhu, H. Pi, Z. Wei, Y .-J. Su, Y . Huang, X. Zheng, Y . Peng, L. He, S. Li, J. Luo, Z. Li, G. Chen, H. Li, H. Weng, and T. Qian, A metallic room-temperatured-wave altermagnet, Nat. Phys.21, 754 (2025)

    2025

  43. [51]

    T. A. Maier and S. Okamoto, Weak-coupling theory of neu- tron scattering as a probe of altermagnetism, Phys. Rev. B108, L100402 (2023)

    2023

  44. [52]

    Brekke, A

    B. Brekke, A. Brataas, and A. Sudbø, Two-dimensional alter- magnets: Superconductivity in a minimal microscopic model, Phys. Rev. B108, 224421 (2023)

    2023

  45. [53]

    R. M. Fernandes, V . S. de Carvalho, T. Birol, and R. G. Pereira, Topological transition from nodal to nodeless Zeeman splitting in altermagnets, Phys. Rev. B109, 024404 (2024)

    2024

  46. [54]

    P. Das, V . Leeb, J. Knolle, and M. Knap, Realizing Altermag- netism in Fermi-Hubbard Models with Ultracold Atoms, Phys. Rev. Lett.132, 263402 (2024)

    2024

  47. [55]

    Y . Guo, H. Liu, O. Janson, I. C. Fulga, J. van den Brink, and J. I. Facio, Spin-split collinear antiferromagnets: A large-scale ab-initio study, Mater. Today Phys.32, 100991 (2023)

    2023

  48. [56]

    D ¨urrnagel, H

    M. D ¨urrnagel, H. Hohmann, A. Maity, J. Seufert, M. Klett, L. Klebl, and R. Thomale, Altermagnetic Phase Transition in a Lieb Metal, Phys. Rev. Lett.135, 036502 (2025)

    2025

  49. [57]

    Ferrari and R

    F. Ferrari and R. Valent ´ı, Altermagnetism on the Shastry- Sutherland lattice, Phys. Rev. B110, 205140 (2024)

    2024

  50. [58]

    X. Zhu, X. Huo, S. Feng, S.-B. Zhang, S. A. Yang, and H. Guo, Design of Altermagnetic Models from Spin Clusters, Phys. Rev. Lett.134, 166701 (2025)

    2025

  51. [59]

    V . Leeb, A. Mook, L.ˇSmejkal, and J. Knolle, Spontaneous For- mation of Altermagnetism from Orbital Ordering, Phys. Rev. Lett.132, 236701 (2024)

    2024

  52. [60]

    Giuli, C

    S. Giuli, C. Mejuto-Zaera, and M. Capone, Altermagnetism from interaction-driven itinerant magnetism, Phys. Rev. B111, L020401 (2025)

    2025

  53. [61]

    Y . Li, V . Leeb, K. Wohlfeld, R. Valent´ı, and J. Knolle, Explor- ingd-wave magnetism in cuprates from oxygen moments, Phys. Rev. B112, 125139 (2025)

    2025

  54. [62]

    Camerano, A

    L. Camerano, A. O. Fumega, J. L. Lado, A. Stroppa, and G. Pro- feta, Multiferroic nematicd-wave altermagnetism driven by orbital-order on the honeycomb lattice, npj 2D Mater. Appl.9, 75 (2025)

    2025

  55. [63]

    Jiang, J

    K. Jiang, J. Hu, H. Ding, and Z. Wang, Interatomic Coulomb interaction and electron nematic bond order in FeSe, Phys. Rev. B93, 115138 (2016)

    2016

  56. [64]

    See Supplemental Materials, which includes Refs. [3, 4, 6– 10, 63, 65, 66], for (i) detailed spin space group symmetry analyses, (ii) additional information about the effects of an- gles between spin directions of AFM,dSC, anddSB orders, (iii) additional Hartree-Fock mean-fiel...

  57. [65]

    Dong, Y .-P

    J.-W. Dong, Y .-P. Huang, Z. Wang, and S. Zhou, Micro- scopic model realization ofd-wave pseudospin current order in Sr2IrO4, Phys. Rev. B109, 045121 (2024)

    2024

  58. [66]

    Raghu, X.-L

    S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang, Topolog- ical Mott Insulators, Phys. Rev. Lett.100, 156401 (2008)

    2008

  59. [67]

    Affleck and J

    I. Affleck and J. B. Marston, Large-n limit of the Heisenberg- Hubbard model: Implications for high-T c superconductors, Phys. Rev. B37, 3774 (1988)

    1988

  60. [68]

    C. M. Varma, Non-Fermi-liquid states and pairing instability of a general model of copper oxide metals, Phys. Rev. B55, 14554 (1997)

    1997

  61. [69]

    X. Feng, K. Jiang, Z. Wang, and J. Hu, Chiral flux phase in the Kagome superconductorAV 3Sb5, Sci. Bull.66, 1384 (2021)

    2021

  62. [70]

    Zhou and Z

    S. Zhou and Z. Wang, Chern Fermi pocket, topological pair density wave, and charge-4e and charge-6e superconductivity in kagom´e superconductors, Nat. Commun.13, 7288 (2022)

    2022

  63. [71]

    J.-W. Dong, Z. Wang, and S. Zhou, Loop-current charge density wave driven by long-range Coulomb repulsion on the kagom ´e lattice, Phys. Rev. B107, 045127 (2023)

    2023

  64. [72]

    Cheong and F.-T

    S.-W. Cheong and F.-T. Huang, Altermagnetism classification, npj Quant. Mater.10, 38 (2025)

    2025

  65. [73]

    Yao, J.-X

    Z.-J. Yao, J.-X. Li, and Z. D. Wang, Extended Hubbard model of superconductivity and charge-density-waves in the layered 2Htransition metal dichalcogenides, Phys. Rev. B74, 212507 (2006)

    2006

  66. [74]

    F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hub- bard Model Physics in Transition Metal Dichalcogenide Moir ´e Bands, Phys. Rev. Lett.121, 026402 (2018)

    2018

  67. [75]

    Gneist, L

    N. Gneist, L. Classen, and M. M. Scherer, Competing instabil- ities of the extended Hubbard model on the triangular lattice: Truncated-unity functional renormalization group and applica- tion to moir´e materials, Phys. Rev. B106, 125141 (2022)

    2022

  68. [76]

    Z. Song, U. F. P. Seifert, Z.-X. Luo, and L. Balents, Mott insula- tors in moir´e transition metal dichalcogenides at fractional fill- ings: Slave-rotor mean-field theory, Phys. Rev. B108, 155109 (2023)

    2023

  69. [77]

    Yamaguchi, N

    K. Yamaguchi, N. M. H. Nakano, and T. Fueno, Extended Hub- bard Models for Transition Metal Oxides and Halides: Impor- tance of Spin and Charge Fluctuations in Charge Transfer Met- als, Jpn. J. Appl. Phys.27, L1835 (1988)

    1988

  70. [78]

    I. R. Reddy, C.-J. Kang, S. Kim, and B. Kim, Exploring the role of nonlocal Coulomb interactions in perovskite transition metal oxides, npj Comput. Mater.10, 286 (2024)

    2024

  71. [79]

    Wang, Y .-M

    H.-X. Wang, Y .-M. Wu, Y .-F. Jiang, and H. Yao, Spectral properties of a one-dimensional extended Hubbard model from bosonization and time-dependent variational principle: Appli- cations to one-dimensional cuprates, Phys. Rev. B109, 045102 (2024)

    2024

  72. [80]

    Sun and H.-Q

    Z. Sun and H.-Q. Lin, Exploring high-temperature supercon- ductivity in the extended Hubbard model with antiferromag- netic tendencies, Phys. Rev. B109, 035107 (2024)

    2024

  73. [81]

    La Magna and R

    A. La Magna and R. Pucci, Effective electron Hamiltonian 8 for a quasi one-dimensional system. The (TMTSF)2X and the (TMTTF)2X systems, Eur. Phys. J. B4, 421–429 (1998)

    1998

  74. [82]

    de la Torre, S

    A. de la Torre, S. McKeown Walker, F. Y . Bruno, S. Ricc ´o, Z. Wang, I. Gutierrez Lezama, G. Scheerer, G. Giriat, D. Jac- card, C. Berthod, T. K. Kim, M. Hoesch, E. C. Hunter, R. S. Perry, A. Tamai, and F. Baumberger, Collapse of the Mott Gap and Emergence of a Nodal Liquid i...

    2015

  75. [83]

    L. Zhao, D. H. Torchinsky, H. Chu, V . Ivanov, R. Lifshitz, R. Flint, T. Qi, G. Cao, and D. Hsieh, Evidence of an odd- parity hidden order in a spin–orbit coupled correlated iridate, Nat. Phys.12, 32 (2016)

    2016

  76. [84]

    S. Zhou, K. Jiang, H. Chen, and Z. Wang, Correlation Effects and Hidden Spin-Orbit Entangled Electronic Order in Parent and Electron-Doped Iridates Sr 2IrO4, Phys. Rev. X7, 041018 (2017)

    2017

  77. [85]

    M. Kato, K. Machida, H. Nakanishi, and M. Fujita, Soliton lat- tice modulation of incommensurate spin density wave in two dimensional Hubbard model-a mean field study, J. Phys. Soc. Jpn.59, 1047 (1990)

    1990

  78. [86]

    Yamase, A

    H. Yamase, A. Eberlein, and W. Metzner, Coexistence of In- commensurate Magnetism and Superconductivity in the Two- Dimensional Hubbard Model, Phys. Rev. Lett.116, 096402 (2016)

    2016

  79. [87]

    Zheng, C.-M

    B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang, and G. K.-L. Chan, Stripe order in the underdoped region of the two-dimensional Hubbard model, Science358, 1155 (2017)

    2017

  80. [88]

    H. Xu, H. Shi, E. Vitali, M. Qin, and S. Zhang, Stripes and spin- density waves in the doped two-dimensional Hubbard model: Ground state phase diagram, Phys. Rev. Res.4, 013239 (2022)

    2022

  81. [89]

    Xu, C.-M

    H. Xu, C.-M. Chung, M. Qin, U. Schollw ¨ock, S. R. White, and S. Zhang, Coexistence of superconductivity with partially filled stripes in the Hubbard model, Science384, 7691 (2024)

    2024

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.