Altermagnetism and Strain Induced Altermagnetic Transition in Cairo Pentagonal Monolayer

Adrian Bahri; Chunjing Jia; Shuyi Li; Xiaoliang Zhang; Yu Zhang

REVIEW 3 major objections 5 minor 49 references

Altermagnetism and Strain Induced Altermagnetic Transition in Cairo Pentagonal Monolayer

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A Cairo pentagonal monolayer is predicted to switch between g-wave and d-wave altermagnetism under diagonal strain.

desk verdict A genuinely new Cairo-pentagon model for g-wave altermagnetism and a strain-driven g-to-d transition, but the DFT strain cell is under-specified and the symmetry link to the model is not verified. read the letter →

arxiv 2412.16857 v1 pith:NTVWUIO3 submitted 2024-12-22 cond-mat.str-el

Shuyi Li , Yu Zhang , Adrian Bahri , Xiaoliang Zhang , Chunjing Jia This is my paper · ORCID

classification cond-mat.str-el

keywords altermagnetismCairopentagonallatticeg-wavespinsplittingd-wavestrainengineeringtight-bindingmodelFeS2monolayerNb2FeB2

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 argues that the Cairo pentagonal lattice, a two-dimensional array of squares and triangles containing both magnetic and non-magnetic atoms, is a new platform for altermagnetism — a magnetic order with zero net moment but spin-split bands. In its unstrained state the model exhibits g-wave altermagnetism, meaning the spin splitting near the Brillouin-zone center is proportional to $k_x k_y (k_x^2 - k_y^2)$. Applying diagonal strain breaks the $C_{4z}$ rotation symmetry and converts the splitting to d-wave form, proportional to $k_x^2 - k_y^2$. Density-functional calculations on monolayer FeS$_2$ and bulk Nb$_2$FeB$_2$ reproduce this g-to-d transition when the lattice angle is sheared from 90° to 88°. If correct, this gives a concrete knob — strain — for switching between distinct altermagnetic orders in real materials.

What carries the argument

The load-bearing object is the Cairo pentagonal lattice with space group $P4/mbm$, containing two magnetic sites with opposite collinear spins and four non-magnetic sites per unit cell. The tight-binding Hamiltonian includes nearest-neighbor magnetic–nonmagnetic hopping $t$, magnetic–magnetic hopping $t_m$, and nonmagnetic–nonmagnetic hoppings $t_{nm,1}$ and $t_{nm,2}$, plus a local exchange coupling $J$ between electrons and the Néel-ordered moments. The paper shows that both the magnetic–nonmagnetic hopping and at least one nonmagnetic–nonmagnetic hopping are indispensable: if either vanishes, the spin splitting disappears and the system becomes a normal Néel antiferromagnet or retains extra degeneracies. Strain is modeled by rescaling hoppings along the $xy$ and $\bar{xy}$ diagonals as $(1+\delta)$ and $(1-\delta)$, which breaks $C_{4z}$ while preserving the combined symmetries $\{C_{2\perp}\|M_{xy}\}$ and $\{C_{2\perp}\|M_{\bar{x}y}\}$; here $C_{2\perp}$ is a 180° rotation about an axis perpendicular to the spins and $M$ are mirror operations. This symmetry change is what reshapes the spin-splitting from $g$-wave to $d$-wave near $\Gamma$.

What would settle it

Spin-resolved photoemission on a strained FeS2 monolayer (or Nb2FeB2) should show the splitting pattern switch from four nodal lines ($k_x=0$, $k_y=0$, $k_x=\pm k_y$) to two ($k_x=\pm k_y$) when the lattice angle is sheared; observing a different nodal-line pattern, or no transition at $\gamma=88^\circ$, would falsify the g-to-d claim.

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Extended reading notes

Core claim

The central claim is that the Cairo pentagonal monolayer, described by a tight-binding model with two antiferromagnetically coupled magnetic sites and four non-magnetic sites per cell, is an altermagnet with $g$-wave spin splitting near the $\Gamma$ point. The spin splitting $\Delta E_n(\mathbf{k}) = E_{n,\uparrow}(\mathbf{k}) - E_{n,\downarrow}(\mathbf{k})$ is proportional to $k_x k_y(k_x^2 - k_y^2)$ for isolated band pairs, with spin-degenerate nodal lines along $k_x=0$, $k_y=0$, $k_x = \pm k_y$. Because the non-magnetic sites break the translation–time-reversal mapping between the two magnetic sublattices, the opposite-spin bands are split at generic momenta while the net magnetization vanishes. When diagonal strain is incorporated as anisotropic hoppings $(1\pm \delta)$ along the two diagonals, the $C_{4z}$ symmetry is lost and the splitting near $\Gamma$ becomes proportional to $k_x^2 - k_y^2$ — a $d$-wave form with only the two nodal lines $k_x = \pm k_y$. The paper reports that density-functional calculations on planar FeS$_2$ and bulk Nb$_2$FeB$_2$ reproduce both the $g$-wave order and its strain-driven transition to $d$-wave order.

Load-bearing premise

The argument assumes that applying a physical shear strain to the lattice changes the electronic structure only through an anisotropic rescaling of the hopping amplitudes — parameterized as $(1\pm\delta)$ — and does not alter the magnetic exchange coupling, the on-site energies, or the directions of the Néel moments.

Editorial extensions

If this is right

Strain becomes a practical control parameter: applying shear strain along either diagonal should toggle the altermagnetic order between $g$- and $d$-wave in a single material.
The presence of non-magnetic sites in a compensated antiferromagnet is sufficient to generate momentum-dependent spin splitting, pointing to a design rule: introduce non-magnetic sublattices with hopping paths that break the magnetic sublattice mapping.
The spin-polarized nodal points carry a $\pi$ Berry phase and can be gapped into Chern bands with $C_\sigma = \mp 2$ by breaking the protecting symmetry, suggesting strain-tunable altermagnetic Chern insulators.
The predicted $g$-to-$d$ transition should show up in spin-resolved photoemission as a change from four nodal lines to two nodal lines crossing $\Gamma$.

Reading between the lines

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

If the strain effect indeed reduces to a rescaling of hopping amplitudes, the same mechanism should generalize to other pentagonal or multi-sublattice magnets: any strain that lowers the point-group symmetry could convert higher-wave altermagnetic splitting to lower-wave forms, possibly realizing $p$- or $f$-wave orders in lattices with different symmetry.
The requirement that non-magnetic hopping $t_{nm}$ be nonzero suggests that strain may act as a switch only in materials where the non-magnetic sublattice has both first and second neighbor paths; this could be tested by comparing iso-structural compounds with different $t_{nm}$ values.
A direct experimental test could use uniaxial pressure or piezo-driven strain on a thin film of FeS$_2$ while measuring the spin texture, looking for the two nodal lines and the $k_x^2-k_y^2$ splitting form.

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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

Colleague,

The paper is worth a look. It builds a tight-binding model for the Cairo pentagonal monolayer with both magnetic and non-magnetic sites and shows it hosts g-wave altermagnetism, with a strain-driven transition to d-wave. That specific model and the g-wave form are new, as far as I can tell. The symmetry analysis is clean, and the Taylor expansions in the SI make the g- and d-wave forms explicit. The identification that the non-magnetic-site hoppings are necessary for the spin splitting is a nice touch. The topological add-on is plausible though a bit disconnected.

The DFT on FeS2 and Nb2FeB2 is the weak point, and it is the load-bearing one. The authors model strain in the tight-binding by rescaling hoppings along the xy and x̄y directions, which preserves the diagonal mirrors. In the DFT, they impose strain by changing the lattice angle γ from 90° to 88° while keeping lattice vector lengths fixed, but they never report the lattice vector orientations or the relaxed atomic positions. In the standard cell choice a=(a,0), b=(b cosγ, b sinγ), that operation breaks the diagonal mirror symmetries that are exactly what enforce the d-wave nodal lines. The degeneracy along Γ-M in the strained DFT would then not be symmetry-protected, and the d-wave assignment is not justified. This is not a fatal flaw yet, but it is a missing check. The authors need to state the strained lattice vectors, show the magnetic space group, and confirm the diagonal mirrors survive.

There are also smaller issues: the model parameters are not fitted to the DFT, so the quantitative match is unverified; the mapping between δ and γ is ad hoc; there are minor inconsistencies between the main text and the SI tables; and no code or data are shipped, only "available upon request." The "first demonstration" claim in the abstract is not supported by a literature search and should be softened.

Overall, the central idea is plausible and the model is solid on its own terms. The paper deserves serious peer review, but it needs revision to make the DFT strain state well-defined and to connect δ and γ quantitatively. I would send it to referees.

Referee Report

3 major / 5 minor

Summary. The paper proposes a tight-binding model for a Cairo pentagonal monolayer with collinear Néel magnetic sites and non-magnetic sites, and shows that the model exhibits g-wave altermagnetism, with spin splitting proportional to kx ky (kx^2 − ky^2) near the Γ point. It then introduces a diagonal-strain term, Eq. (3), that rescales hopping amplitudes along the xy and x̄y directions, and shows that the spin-splitting near Γ becomes d-wave, proportional to kx^2 − ky^2. The paper also analyzes spin-polarized nodal points and their gapping by symmetry-breaking terms, computing Berry curvature and Chern numbers, and presents DFT results for FeS2 and Nb2FeB2 that are argued to reproduce the g-wave and d-wave altermagnetic phases, including the strain-induced transition.

Significance. The symmetry-based analysis is the main strength: the g-wave and d-wave forms follow from the symmetry generators listed in the text and are supported by explicit Taylor-expansion coefficients in SI Tables I and II, not by fitting to a target result. The microscopic mechanism identified—the role of non-magnetic-site hoppings in enabling altermagnetism—is clearly demonstrated by the parameter dependence in Fig. 3. If the DFT strain realization is shown to have the same symmetry reduction as Eq. (3), the work would provide a simple and potentially tunable platform for strain-controlled altermagnetism, with direct relevance for spintronics. However, the materials claim currently rests on an underspecified strain cell and on visual comparison of band structures; the quantitative symmetry connection between the model and the DFT is the key missing piece.

major comments (3)

[Candidate Materials; SI Sec. V; Eq. (3)] The DFT shear strain is described only as changing the angle γ between lattice vectors a and b from 90° to 88° while keeping lengths fixed, but the orientation of the strained cell relative to the mirror planes of the unstrained Cairo lattice is not specified. For an equal-length oblique parametrization, the Bravais lattice is centered-rectangular and its mirror axes rotate away from the original diagonal mirrors unless the cell is oriented appropriately, whereas Eq. (3) preserves Mxy and Mx̄y exactly by construction. The paper does not report the deformed lattice vectors, the relaxed atomic positions, or the magnetic space group of the strained crystals, so the identification of the DFT band structures in Figs. 7e and 8d as d-wave with nodal lines kx = ±ky is not symmetry-enforced as presented. Please provide the strained magnetic space group, or otherwise demonstrate that the γ = 88° cell realizes the same symmetry reduction as Eq. (3), and give the relation between δ and γ if one is intended.
[SI Sec. II; Tables I and II; main-text Fig. 2] The Taylor-expansion coefficients in SI Tables I and II are computed with tm = −0.4, while the band structures in the main text (Figs. 2 and 4) use tm = 0.2. The g-wave and d-wave forms are symmetry-dictated regardless of the sign of tm, but the numerical coefficients in the tables cannot be checked against the plotted dispersions. Please either recompute the tables for the main-text parameter set or explicitly state which parameter set the tables refer to.
[Figs. 7e and 8d; SI Sec. V] The DFT evidence for the g-to-d transition is based solely on visual inspection of degeneracies and inverse spin-splitting along selected paths. The strained Brillouin-zone labels are not defined for the oblique cell, and no quantitative fit of ΔE(k) near Γ to the predicted d-wave form kx^2 − ky^2 is provided. A numerical extraction of ΔE(k) along a few lines in the strained Brillouin zone, compared with the model prediction, would substantiate the claim that the ab initio calculations reproduce the transition rather than merely resembling it.

minor comments (5)

[Fig. 5 caption] The caption for Fig. 5b lists tnm,2 = 0.6 among the fixed parameters even though panel b is specifically for tnm,2 = 0; please correct the caption to avoid confusion.
[References] Reference [17] is a duplicate of reference [16] (both cite the FeSb2 prediction by Mazin et al.); please replace reference [17] with the intended distinct work or remove the duplicate.
[Introduction; Eq. (1)] The symmetry notation {C2⊥||Mx}, {C2⊥||My}, etc., is used without definition; a brief explanation of the double-group notation (rotation about an axis perpendicular to the spins combined with a mirror operation) would make the paper accessible to a wider readership.
[Abstract; main text] The terms 'g-wave' and 'd-wave' are used without defining the angular-momentum classification near Γ; a one-sentence definition of the l = 4 and l = 2 forms would remove ambiguity.
[SI Sec. V] The planar FeS2 monolayer is constructed by manually relocating S atoms from bulk pyrite, and only magnetic relaxation is reported. A brief statement on dynamic or thermal stability, or a citation to prior work establishing it, would strengthen the claim that FeS2 is a realistic realization.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the tight-binding derivation is a symmetry-based model calculation, and the DFT materials results are independent of the model parameters.

full rationale

The paper's tight-binding Hamiltonian (Eq. (1)-(3)) is constructed from the Cairo pentagonal lattice and its magnetic/non-magnetic site arrangement. The g-wave splitting and the strain-induced d-wave form are derived by Taylor expansion of the characteristic equation in the Supplementary Information (Sec. II), not imposed by fitting to a target band splitting. The strain is modeled as anisotropic hopping with parameter δ, which encodes a physical symmetry reduction; the resulting ΔE ∝ kxky(kx^2-ky^2) and ΔE ∝ kx^2-ky^2 follow from the symmetries of the constructed Hamiltonian. This is a deductive model result, not a circular reduction: the model is not defined in terms of the predicted splitting, and no target quantity is used as input. The DFT calculations on FeS2 and Nb2FeB2 are externally computed with PBE(+U), independently relaxed, and compared with model expectations after the fact; no DFT band feature is used as input to the tight-binding model, and no model parameter is fitted to the DFT spin splitting. The paper does not rely on load-bearing self-citations; the cited works on Nb2FeB2 (Refs. [48,49]) and computational methods (Refs. [S1]-[S6]) are by different author groups or standard method papers. There is no renaming of a known result: the Cairo pentagonal altermagnet and the strain-driven g-to-d transition are novel applications of established symmetry principles. One could note that the strain term in Eq. (3) is an ansatz for how strain enters the model, and that the DFT strain cell orientation is under-specified; these are correctness and modeling risks, not circularity under the defined criteria. Hence the circularity score is 0.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claims rest on a symmetrically constructed tight-binding model with hand-chosen parameters (tm, tnm,1, tnm,2, J, d, δ) and on DFT with literature U values. The strain response is modeled only by hopping anisotropy. The Chern number result is derived from a toy model term H2, not from a realistic material. No new particles, forces, or dimensions are introduced; the candidate structures are predicted or existing materials.

free parameters (6)

Model hoppings {tm, tnm,1, tnm,2, t} = tm=0.2, tnm,1=0.9, tnm,2=0.6, t=1
Chosen by hand to produce altermagnetic bands; not fitted to any material.
Exchange coupling J and spin magnitude S = J=1, S=1
Set to integer values for convenience; realistic values would alter quantitative splitting but not symmetry.
On-site energies and chemical potential {εm, εnm, µ} = 0, 0, 0
Set to zero for simplicity.
Non-magnetic site position d = d=0.1a
Chosen within 0<d<a/2; results are argued to be symmetry-determined, but quantitative band structure depends on d.
Strain anisotropy δ = 0.1
Representative value to demonstrate the transition; real strain magnitude in DFT is γ=88° without direct mapping to δ.
DFT Ueff for FeS2 and Nb2FeB2 = 2.0 eV (FeS2), 4.82 eV (Nb2FeB2)
Taken from cited prior work; magnetic ground state and splitting magnitudes depend on U.

assumptions (5)

domain assumption Altermagnetism classification via spin-space-group symmetry (e.g., {C2⊥||M} operators)
The paper relies on the established framework of altermagnetic spin splitting from Refs [1-8] to label g-wave and d-wave phases.
domain assumption The 6-site tight-binding Hamiltonian with the specified hoppings captures the low-energy physics of the Cairo pentagonal monolayer
Invoked in Eq. (1); no ab initio Wannier fitting is performed to validate the model quantitatively.
ad hoc to paper Strain is modeled by (1±δ) rescaling of hopping amplitudes along xy and x̄y directions
Eq. (3) introduces this as the sole effect of diagonal strain; real strain also changes bond angles, on-site energies, and possibly exchange J.
domain assumption GGA+U with the chosen Ueff values predicts the correct magnetic ground state and band structure
Used in the DFT sections; magnetic ordering is not verified experimentally for the planar FeS2 structure.
standard math The Berry phase of each spin-polarized nodal point is quantized to π and the Chern number can be computed by the Fukui discretization
SI Sec. IV; relies on standard lattice gauge theory for Chern numbers.

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Cite this review

Pith. "Pith review of Altermagnetism and Strain Induced Altermagnetic Transition in Cairo Pentagonal Monolayer." pith.science (2026). https://pith.science/paper/NTVWUIO3

@misc{pith2026241216857,
  author       = {Pith},
  title        = {Pith review of: Altermagnetism and Strain Induced Altermagnetic Transition in Cairo Pentagonal Monolayer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NTVWUIO3}},
  note         = {Machine review of arXiv:2412.16857}
}

abstract

Altermagnetism, a recently discovered class of magnetic order characterized by vanishing net magnetization and spin-splitting band structures, has garnered significant research attention. In this work, we introduce a novel two-dimensional system that exhibits $g$-wave altermagnetism and undergoes a strain-induced transition from $g$-wave to $d$-wave altermagnetism. This system can be realized in an unconventional monolayer Cairo pentagonal lattice, for which we present a realistic tight-binding model that incorporates both magnetic and non-magnetic sites. Furthermore, we demonstrate that non-trivial band topology can emerge in this system by breaking the symmetry that protects the spin-polarized nodal points. Finally, \emph{ab initio} calculations on several candidate materials, such as FeS$_2$ and Nb$_2$FeB$_2$, which exhibit symmetry consistent with the proposed tight-binding Hamiltonian, are also presented. These findings open new avenues for exploring spintronic devices based on altermagnetic systems.

Figures

Figures reproduced from arXiv: 2412.16857 by the authors.

**Figure 1.** Cairo pentagonal lattice structure and its electronic energy bands. a A schematics of the pentagonal lattice with space group P4/mbm and the tight-binding model parameters. The magnetic sites with spin-up, spin-down, and non-magnetic (nm) sites are represented by red, blue, and gray colors. t, tm, tnm,1 and tnm,2 represent the hopping strengths between the nearest-neighbor magnetic and nonmagnetic sites, the neares… view at source ↗

**Figure 2.** Band dispersion of pentagonal lattice Hamiltonian with C4z symmetry. a The electronic band structure along the chosen k path in the first Brillouin zone. The parameters used are tm = 0.2, tnm,1 = 0.9, tnm,2 = 0.6, J = 1, ϵm = ϵnm = 0 and µ = 0. The spin up and spin down bands are shown in red and blue, respectively. b and c False color plots of the spin-splitting energy ∆En(k) = En,↑(k)−En,↓(k) for the 4th and 5th… view at source ↗

**Figure 4.** Band dispersion of pentagonal lattice Hamiltonian under xy and xy¯ strains. a The electronic band structure along the chosen k path in the first Brillouin zone. The parameters used are δ = 0.1, tm = 0.2, tnm,1 = 0.9, tnm,2 = 0.6, J = 1, ϵm = ϵnm = 0 and µ = 0. The bands of spin up and down are labeled by red and blue. b and c False color plots of the spin-splitting energy ∆En(k) = En,↑(k) − En,↓(k) for the 4th and … view at source ↗

Figures from the paper (3 more)

**Figure 6.** Figure 6: Band structure and Berry curvature after gapping out the polarized nodal points. a Eight spin polarized nodal points formed by band crossing between the 4th and 5th pairs of bands. b The electronic band structure for the 4th and 5th pairs of bands under inversion symme…

**Figure 7.** Figure 7: Crystal structure and electronic bands of FeS2. a Crystal structure of FeS2. The red and blue colors of Fe atoms indicate the directions of their localized magnetic moments, while the orange arrows represent the direction of diagonal strain. b The electronic band struc…

**Figure 8.** Figure 8: Crystal structure and electronic bands of Nb2FeB2. a Crystal structure of Nb2FeB2 shown from two perspectives. The purple arrows indicate the directions of localized magnetic moments on Fe atoms. b The first Brillouin zone of Nb2FeB2 and high symmetry moment points of …

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Works this paper leans on

49 extracted references · 37 canonical work pages

[1]

Šmejkal, J

L. Šmejkal, J. Sinova, and T. Jungwirth, Beyond conven- tional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, 8 Physical Review X12, 031042 (2022)

work page 2022
[2]

Šmejkal, J

L. Šmejkal, J. Sinova, and T. Jungwirth, Emerging re- search landscape of altermagnetism, Physical Review X 12, 040501 (2022)

2022
[3]

Šmejkal, A

L. Šmejkal, A. H. MacDonald, J. Sinova, S. Nakatsuji, and T. Jungwirth, Anomalous hall antiferromagnets, Na- ture Reviews Materials7, 482 (2022)

work page 2022
[4]

Hayami, Y

S. Hayami, Y. Yanagi, and H. Kusunose, Bottom-up de- sign of spin-split and reshaped electronic band structures in antiferromagnets without spin-orbit coupling: Proce- dure on the basis of augmented multipoles, Physical Re- view B 102, 144441 (2020)

work page 2020
[5]

Hayami, Y

S. Hayami, Y. Yanagi, and H. Kusunose, Momentum- dependent spin splitting by collinear antiferromagnetic ordering, journal of the physical society of japan 88, 123702 (2019)

2019
[6]

González-Hernández, L

R. González-Hernández, L. Šmejkal, K. Výborný, Y. Ya- hagi, J. Sinova, T. c. v. Jungwirth, and J. Železný, Efficient electrical spin splitter based on nonrelativis- tic collinear antiferromagnetism, Phys. Rev. Lett.126, 127701 (2021)

2021
[7]

L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger, Giant momentum-dependent spin splitting in centrosymmetric low-z antiferromagnets, Physical Re- view B 102, 014422 (2020)

work page 2020
[8]

L.-D. Yuan, Z. Wang, J.-W. Luo, and A. Zunger, Predic- tion of low-z collinear and noncollinear antiferromagnetic compounds having momentum-dependent spin splitting even without spin-orbit coupling, Physical Review Mate- rials 5, 014409 (2021)

work page 2021

Show all 49 references

[9]

Z. Feng, X. Zhou, L. Šmejkal, L. Wu, Z. Zhu, H. Guo, R. González-Hernández, X. Wang, H. Yan, P. Qin,et al., An anomalous hall effect in altermagnetic ruthenium dioxide, Nature Electronics5, 735 (2022)

2022
[10]

Fedchenko, J

O. Fedchenko, J. Minár, A. Akashdeep, S. W. D’Souza, D. Vasilyev, O. Tkach, L. Odenbreit, Q. Nguyen, D. Kut- nyakhov, N. Wind, et al. , Observation of time-reversal symmetry breaking in the band structure of altermag- netic ruo2, Science advances10, eadj4883 (2024)

2024
[11]

X. Zhou, W. Feng, R. Zhang, L. Smejkal, J. Sinova, Y. Mokrousov, and Y. Yao, Crystal thermal trans- port in altermagnetic ruo2 (2023), arXiv preprint arXiv:2305.01410

2023 arXiv
[12]

S. Lee, S. Lee, S. Jung, J. Jung, D. Kim, Y. Lee, B. Seok, J. Kim, B. G. Park, L. Šmejkal,et al., Broken kramers degeneracy in altermagnetic mnte, Physical Review Let- ters 132, 036702 (2024)

2024
[13]

Osumi, S

T. Osumi, S. Souma, T. Aoyama, K. Yamauchi, A. Honma, K. Nakayama, T. Takahashi, K. Ohgushi, and T. Sato, Observation of a giant band splitting in alter- magnetic mnte, Physical Review B109, 115102 (2024)

2024
[14]

Krempask` y, L

J. Krempask` y, L. Šmejkal, S. D’souza, M. Hajlaoui, G. Springholz, K. Uhlířová, F. Alarab, P. Constantinou, V. Strocov, D. Usanov, et al. , Altermagnetic lifting of kramers spin degeneracy, Nature626, 517 (2024)

2024
[15]

Z. Liu, M. Ozeki, S. Asai, S. Itoh, and T. Masuda, Chi- ral split magnon in altermagnetic mnte, Physical Review Letters 133, 156702 (2024)

2024
[17]

I. I. Mazin, K. Koepernik, M. D. Johannes, R. González- Hernández, and L. Šmejkal, Prediction of unconventional magnetism in doped fesb2, Proceedings of the National Academy of Sciences118, e2108924118 (2021)

2021
[18]

H. Bai, L. Han, X. Feng, Y. Zhou, R. Su, Q. Wang, L. Liao, W. Zhu, X. Chen, F. Pan,et al., Observation of spin splitting torque in a collinear antiferromagnet ruo 2, Physical Review Letters128, 197202 (2022)

2022
[19]

Karube, T

S. Karube, T. Tanaka, D. Sugawara, N. Kadoguchi, M. Kohda, and J. Nitta, Observation of spin-splitter torque in collinear antiferromagnetic ruo 2, Physical re- view letters 129, 137201 (2022)

2022
[20]

Sumita, M

S. Sumita, M. Naka, and H. Seo, Fulde-ferrell-larkin- ovchinnikov state induced by antiferromagnetic order in κ-type organic conductors, Physical Review Research5, 043171 (2023)

2023
[21]

Zhu, Z.-Y

D. Zhu, Z.-Y. Zhuang, Z. Wu, and Z. Yan, Topological superconductivity in two-dimensional altermagnetic met- als, Physical Review B108, 184505 (2023)

2023
[22]

Brekke, A

B. Brekke, A. Brataas, and A. Sudbø, Two-dimensional altermagnets: Superconductivity in a minimal micro- scopic model, Physical Review B108, 224421 (2023)

2023
[23]

Zhang, L.-H

S.-B. Zhang, L.-H. Hu, and T. Neupert, Finite- momentum cooper pairing in proximitized altermagnets, Nature Communications 15, 1801 (2024)

2024
[24]

Chakraborty and A

D. Chakraborty and A. M. Black-Schaffer, Zero-field finite-momentum and field-induced superconductivity in altermagnets, Physical Review B110, L060508 (2024)

2024
[25]

M. Wei, L. Xiang, F. Xu, L. Zhang, G. Tang, and J. Wang, Gapless superconducting state and mirage gap in altermagnets, Physical Review B109, L201404 (2024)

2024
[26]

Mæland, B

K. Mæland, B. Brekke, and A. Sudbø, Many-body effects on superconductivity mediated by double-magnon pro- cesses in altermagnets, Physical Review B109, 134515 (2024)

2024
[27]

Chourasia, A

S. Chourasia, A. Svetogorov, A. Kamra, and W. Belzig, Thermodynamic properties of a superconductor interfaced with an altermagnet, arXiv preprint arXiv:2403.10456 (2024)

2024 arXiv
[28]

Leiviskä, J

M. Leiviskä, J. Rial, A. Bad’ura, R. L. Seeger, I. Kounta, S. Beckert, D. Kriegner, I. Joumard, E. Schmoranzerová, J. Sinova,et al., Anisotropy of the anomalous hall effect in thin films of the altermagnet candidate mn 5 si 3, Physical Review B109, 224430 (2024)

2024
[30]

Attias, A

L. Attias, A. Levchenko, and M. Khodas, Intrinsic anomalous hall effect in altermagnets, Physical Review B 110, 094425 (2024)

2024
[31]

Liao, Y.-C

C.-T. Liao, Y.-C. Wang, Y.-C. Tien, S.-Y. Huang, and D. Qu, Separation of inverse altermagnetic spin-splitting effect from inverse spin hall effect in ruo 2, Physical Re- view Letters 133, 056701 (2024)

2024
[32]

H. Jin, Z. Tan, Z. Gong, and J. Wang, Anomalous hall effect in two-dimensional vanadium tetrahalogen with altermagnetic phase, Physical Review B 110, 155125 (2024)

2024
[33]

Y.Fang, J.Cano,andS.A.A.Ghorashi,Quantumgeom- etry induced nonlinear transport in altermagnets, Phys- ical Review Letters133, 106701 (2024)

2024
[34]

P. Rao, A. Mook, and J. Knolle, Tunable band topol- ogy and optical conductivity in altermagnets, Physical Review B 110, 024425 (2024)

2024
[35]

Parshukov, R

K. Parshukov, R. Wiedmann, and A. P. Schnyder, Topo- 9 logical responses from gapped weyl points in 2d alter- magnets, arXiv preprint arXiv:2403.09520 (2024)

2024 arXiv
[36]

Tan, Z.-F

C.-Y. Tan, Z.-F. Gao, H.-C. Yang, K. Liu, P.-J. Guo, and Z.-Y. Lu, Bipolarized weyl semimetals and quantum crystal valley hall effect in two-dimensional altermagnetic materials, arXiv preprint arXiv:2406.16603 (2024)

2024 arXiv
[37]

L. Bai, W. Feng, S. Liu, L. Šmejkal, Y. Mokrousov, and Y.Yao,Altermagnetism: Exploringnewfrontiersinmag- netism and spintronics, Advanced Functional Materials , 2409327 (2024)

2024
[38]

Šmejkal, A

L. Šmejkal, A. B. Hellenes, R. González-Hernández, J. Sinova, and T. Jungwirth, Giant and tunneling magne- toresistance in unconventional collinear antiferromagnets with nonrelativistic spin-momentum coupling, Physical Review X 12, 011028 (2022)

2022
[39]

V. Leeb, A. Mook, L. Šmejkal, and J. Knolle, Sponta- neous formation of altermagnetism from orbital ordering, Physical Review Letters132, 236701 (2024)

2024
[40]

M. Roig, A. Kreisel, Y. Yu, B. M. Andersen, and D. F. Agterberg, Minimal models for altermagnetism, Physical Review B 110, 144412 (2024)

2024
[41]

P. Das, V. Leeb, J. Knolle, and M. Knap, Realizing altermagnetism in fermi-hubbard models with ultracold atoms, Physical Review Letters132, 263402 (2024)

2024
[42]

T. Sato, S. Haddad, I. C. Fulga, F. F. Assaad, and J. van den Brink, Altermagnetic anomalous hall effect emerging from electronic correlations, Physical Review Letters 133, 086503 (2024)

2024
[43]

Shen and Q

Y. Shen and Q. Wang, Pentagon-based 2d materials: Classification, properties and applications, Physics Re- ports 964, 1 (2022)

2022
[44]

Y. Wang, Y. Li, and Z. Chen, Not your familiar two di- mensional transition metal disulfide: structural and elec- tronic properties of the pds 2 monolayer, Journal of Ma- terials Chemistry C3, 9603 (2015)

2015
[45]

X. Shao, X. Liu, X. Zhao, J. Wang, X. Zhang, and M. Zhao, Electronic properties of aπ-conjugated cairo pentagonal lattice: Direct band gap, ultrahigh carrier mobility, and slanted dirac cones, Physical Review B98, 085437 (2018)

2018
[46]

P. Li, J. Zhang, C. Zhu, W. Shen, C. Hu, W. Fu, L. Yan, L. Zhou, L. Zheng, H. Lei,et al., Penta-pdpse: a new 2d pentagonal material with highly in-plane optical, elec- tronic, and optoelectronic anisotropy, Advanced Materi- als 33, 2102541 (2021)

2021
[47]

S. Wei, S. Zheng, X. Wen, C. Xie, and J. Liang, A novel antiferromagnetic semiconductor hidden in pyrite, Com- putational Materials Science183, 109852 (2020)

2020
[48]

Hou, H.-C

X.-Y. Hou, H.-C. Yang, Z.-X. Liu, P.-J. Guo, and Z.- Y. Lu, Large intrinsic anomalous hall effect in both nb2feb2 and ta2feb2 with collinear antiferromagnetism, Phys. Rev. B107, L161109 (2023)

2023
[49]

Z.-F. Gao, S. Qu, B. Zeng, Y. Liu, J.-R. Wen, H. Sun, P.-J. Guo, and Z.-Y. Lu, Ai-accelerated discovery of al- termagnetic materials, arXiv preprint arXiv:2311.04418 (2023)

2023 arXiv
[50]

Kresse and J

G. Kresse and J. Hafner, Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium, Phys. Rev. B49, 14251 (1994)

1994
[51]

Altermagnetism and Strain Induced Altermagnetic Transition in Cairo Pentagonal Monolayer

J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996). ACKNOWLEDGEMENTS This work is supported by the Center for Molecular Magnetic Quantum Materials, an Energy Frontier Re- search Center funded by the U.S. D...

1996 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Šmejkal, J

L. Šmejkal, J. Sinova, and T. Jungwirth, Beyond conven- tional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, 8 Physical Review X12, 031042 (2022)

work page 2022

[2] [2]

Šmejkal, J

L. Šmejkal, J. Sinova, and T. Jungwirth, Emerging re- search landscape of altermagnetism, Physical Review X 12, 040501 (2022)

2022

[3] [3]

Šmejkal, A

L. Šmejkal, A. H. MacDonald, J. Sinova, S. Nakatsuji, and T. Jungwirth, Anomalous hall antiferromagnets, Na- ture Reviews Materials7, 482 (2022)

work page 2022

[4] [4]

Hayami, Y

S. Hayami, Y. Yanagi, and H. Kusunose, Bottom-up de- sign of spin-split and reshaped electronic band structures in antiferromagnets without spin-orbit coupling: Proce- dure on the basis of augmented multipoles, Physical Re- view B 102, 144441 (2020)

work page 2020

[5] [5]

Hayami, Y

S. Hayami, Y. Yanagi, and H. Kusunose, Momentum- dependent spin splitting by collinear antiferromagnetic ordering, journal of the physical society of japan 88, 123702 (2019)

2019

[6] [6]

González-Hernández, L

R. González-Hernández, L. Šmejkal, K. Výborný, Y. Ya- hagi, J. Sinova, T. c. v. Jungwirth, and J. Železný, Efficient electrical spin splitter based on nonrelativis- tic collinear antiferromagnetism, Phys. Rev. Lett.126, 127701 (2021)

2021

[7] [7]

L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger, Giant momentum-dependent spin splitting in centrosymmetric low-z antiferromagnets, Physical Re- view B 102, 014422 (2020)

work page 2020

[8] [8]

L.-D. Yuan, Z. Wang, J.-W. Luo, and A. Zunger, Predic- tion of low-z collinear and noncollinear antiferromagnetic compounds having momentum-dependent spin splitting even without spin-orbit coupling, Physical Review Mate- rials 5, 014409 (2021)

work page 2021