Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Quantum critical scaling of altermagnetism

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The long-wavelength theory of an altermagnet is a nonlinear sigma model with a Berry-phase term that one-loop RG ignores, while metallic Dirac fermions set $z=1+\gamma_1$.

desk verdict The NLSM derivation and RG result are solid, but the fermionic Dirac section is built on a miscomputed eigenvalue: Eq. (29) does not follow from Eq. (28), so the nodes, velocities, and z result are unproven as written. read the letter →

arxiv 2505.04683 v2 pith:EINLXXLO submitted 2025-05-07 cond-mat.str-el cond-mat.stat-mech

Sondre Duna Lundemo , Flavio S. Nogueira , Asle Sudb{o} This is my paper · ORCID
classification cond-mat.str-elcond-mat.stat-mech
keywords altermagnetismnonlinearsigmamodelNéelquantumcriticalpointBerryphaserenormalizationgroupDiracfermionsdynamicalexponentDyson-Schwingerequations
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

Starting from a minimal microscopic model of a checkerboard altermagnet, this paper derives the long-wavelength theory of the staggered magnetization as a nonlinear $\sigma$ model whose only new term is a Berry-phase functional $W[n]$. That term produces a d-wave splitting of the spin-wave dispersions, $\omega_+(k)-\omega_-(k)=2\theta g c^2 k_x k_y$. The paper then shows that at one loop in $d=2+1$ this $\theta$ term does not enter the renormalization group flow, so to leading order the altermagnetic N\'eel critical point is not a new universality class. Extending the theory to metallic altermagnets, it obtains four flavors of d-wave spin-split Dirac fermions and finds, via Dyson-Schwinger equations, that magnetic fluctuations push the dynamical critical exponent $z$ above 1 while the Coulomb interaction pulls it below 1, with a critical coupling at which $z=1$ is restored.

What carries the argument

The load-bearing object is the altermagnetic nonlinear $\sigma$ model, Eq. (11): the standard O(3) NLSM for the N\'eel vector $n$ plus the Berry-phase functional $W[n]=\int\epsilon_{abc}n_a\partial_\tau n_b\partial_x\partial_y n_c$, which survives the gradient expansion because the staggered next-nearest-neighbor exchange contributes a term linear in the ferromagnetic fluctuation $l$. Its role is to encode the altermagnetic multipolar character in the bosonic sector and to produce the d-wave spin-wave splitting. In the fermionic sector the equivalent machinery is the Dirac-fermion representation of the eight nodes, Eq. (36), together with the Kondo vertex $\kappa\pi^2\sum_s(-1)^s\bar\psi_s(\rho^0\otimes i\tau^1)\psi_s$; the algebraic structure of that vertex is what forces $\gamma_1\neq\gamma_2$ and hence the anisotropic scaling $\Delta\neq 0$.

What would settle it

Run an unbiased numerical simulation (quantum Monte Carlo or tensor-network) of the checkerboard spin model in Eq. (2) at its N\'eel quantum critical point and extract the correlation-length and dynamical critical exponents: statistically significant deviations from the O(3) nonlinear-$\sigma$-model values would falsify the claim that the Berry-phase $\theta$ term leaves the altermagnetic critical point in the same universality class at leading order.

Watch

Extended reading notes

Core claim

The paper's central claim is that the long-wavelength effective field theory of a checkerboard altermagnet is the nonlinear $\sigma$ model $S_{\rm am}[n]=\frac{1}{2g}\int d^2x\,d\tau\,(\partial_\mu n)^2+i\theta W[n]$ with $W[n]=\int\epsilon_{abc}n_a\partial_\tau n_b\partial_x\partial_y n_c$, obtained by integrating out the ferromagnetic fluctuation field $l$ from the microscopic spin coherent-state action. The $\theta$ term is not an ad hoc addition: it comes from the cross term of the spin Berry phase with the staggered next-nearest-neighbor exchange, and its coefficient is fixed by microscopic parameters, $\theta=S\alpha\gamma$. It yields nondegenerate spin-wave dispersions with a d-wave splitting, Eq. (15). A one-loop RG calculation in $d=2+1$ gives the standard NLSM $\beta$ functions with $\theta$ perturbatively irrelevant, so at leading order the altermagnetic N\'eel quantum critical point is in the same universality class as the collinear antiferromagnet. In the metallic case the fermionic sector realizes four flavors of Dirac fermions with d-wave spin splitting, and the Dyson-Schwinger analysis gives $z=1+\gamma_1$, with $\gamma_1$ determined by competing magnetic ($u$) and Coulomb ($\lambda$) fluctuations, together with an anisotropic scaling exponent $\Delta=\gamma_1-\gamma_2$ that is insensitive to the Coulomb interaction.

Load-bearing premise

The fermionic part of the model is an effective ansatz rather than a derived consequence: it assumes no diagonal next-nearest-neighbor hopping and a staggered diagonal hopping strong enough that eight Dirac nodes sit exactly on the Fermi surface, so outside that parameter window the predicted Coulomb-versus-magnetic competition for $z$ does not apply.

Editorial extensions

If this is right

  • Spin waves in a checkerboard altermagnet are split according to $\omega_+(k)-\omega_-(k)=2\theta g c^2 k_x k_y$, a d-wave signature that vanishes on the lines $k_x=0$ and $k_y=0$ and is directly measurable.
  • At one loop the $\theta$ term is perturbatively irrelevant, so the altermagnetic N\'eel quantum critical point shares the universality class of the standard collinear antiferromagnet; any distinct critical behavior must come from higher loops or nonperturbative physics.
  • In the metallic regime, magnetic fluctuations increase the dynamical critical exponent ($z=1+\gamma_1$) while the Coulomb interaction decreases it, and at the critical coupling $(\kappa g)^2_\ast$ the mean-field value $z=1$ is restored.
  • Quantum fluctuations induce an anisotropic scaling of momenta, $\Delta=\gamma_1-\gamma_2\neq 0$, which is a direct consequence of the altermagnetic Kondo vertex and is not affected by the Coulomb interaction.
  • The sublattice-magnetization fluctuations do not spontaneously gap the Dirac nodes for realistic parameters, since mass generation would require $N>N_c=1+4\pi^2/(\kappa g)^2$.

Reading between the lines

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

  • Beyond the paper: if the one-loop irrelevance of $\theta$ survives nonperturbatively, the altermagnet-versus-antiferromagnet distinction should show up mainly in magnon transport or finite-size responses tied to the d-wave splitting, rather than in universal critical exponents.
  • Beyond the paper: the fermionic $z$ result could be tested by tuning the ratio $J_K/J$ through the critical value $(\kappa g)^2_\ast$ in cold-atom or Lieb-lattice materials, producing a sharp crossover of $z$ across 1.
  • Beyond the paper: the derivation strategy, integrating out ferromagnetic fluctuations against the Berry phase, should generalize to any collinear spin system with staggered higher-multipole order, so the same $\theta W[n]$ structure may appear in other frustrated lattices such as Shastry-Sutherland.
  • Beyond the paper: because $\Delta=\gamma_1-\gamma_2$ is insensitive to the Coulomb coupling, the anisotropy in the fermion velocity renormalization is a cleaner diagnostic of magnetic fluctuations than $z$ itself.
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

3 major / 4 minor

Summary. The manuscript derives a low-energy nonlinear sigma model for the staggered magnetization of a checkerboard-lattice altermagnet starting from a microscopic exchange Hamiltonian. It obtains a Berry-phase-origin theta term W[n] = integral epsilon_abc n_a d_tau n_b d_x d_y n_c, computes the resulting spin-wave splitting, and performs a one-loop RG analysis in 2+1 dimensions, finding that theta is perturbatively irrelevant and that the standard NLSM beta functions are recovered. The paper then introduces an itinerant fermion model whose low-energy theory is four flavors of anisotropic Dirac fermions, couples them to the NLSM fluctuations and to the Coulomb interaction, and uses Dyson-Schwinger equations to obtain anomalous velocity dimensions gamma_1, gamma_2 and a dynamical critical exponent z = 1 + gamma_1 that depends on the competition between magnetic and Coulomb fluctuations.

Significance. The microscopic derivation of the theta term from the Berry phase (Sec. II) and the one-loop NLSM RG analysis (Sec. III) are valuable and internally coherent; the spin-wave splitting in Eq. (15) is a concrete falsifiable prediction, and the beta functions reduce to the standard NLSM result at theta = 0. If the fermionic calculation were correct, the predicted competition between magnetic and Coulomb fluctuations in the dynamical exponent would be a useful contribution to the altermagnetism literature. However, the eigenvalue error in Sec. IV affects the quantitative z prediction, so the paper's ultimate significance will depend on whether that section can be corrected.

major comments (3)
  1. [Sec. IV.A, Eqs. (28)-(29)] Equation (29) is not the spectrum of the Hamiltonian in Eq. (28). For the 2x2 matrix with diagonal entries A = -Gamma_bullet - mu + alpha J_KS and D = -Gamma_circle - mu - alpha J_KS, the eigenvalues are E_plusminus = -mu - (Gamma_bullet + Gamma_circle)/2 plusminus sqrt(gamma^2 + [alpha J_KS + (Gamma_circle - Gamma_bullet)/2]^2), whereas Eq. (29) uses -mu - (Gamma_circle + Gamma_bullet) for the trace part and [Gamma_bullet - Gamma_circle + alpha J_KS]^2 inside the square root. The trace term is off by a factor of two and the off-diagonal combination has the wrong factor and sign. This is load-bearing because the node positions, velocities, and all subsequent Dyson-Schwinger results are calculated from Eq. (29).
  2. [Sec. IV.B, Eq. (31) and Fig. 3] The node condition in Eq. (31) is inconsistent with both the correct eigenvalue formula and the stated spin convention. On the line ky = pi - kx with t' = 0, gamma = 0 and Gamma_circle = -Gamma_bullet, so the zero-gap condition for alpha = +1 is Gamma_bullet - Gamma_circle = 2 J_KS, which by Eq. (30) gives sin(kx) sin(ky) = J_KS/(2 delta t); Eq. (31) instead yields sin^2(k*) = J_KS/(4 delta t). Moreover, if Eq. (29) is used literally, the nodes on this line occur for alpha = -1, not for alpha = +1 as claimed for the up-spin sector. Consequently, the node positions and their spin assignment in Fig. 3 do not follow from the Hamiltonian given.
  3. [Sec. IV.D, Eqs. (33), (41)-(43)] Because the velocities v1 and v2 in Eq. (33) are obtained by expanding around the incorrect nodes of Eq. (31), the anomalous dimensions in Eq. (41) and the resulting z = 1 + gamma_1 in Eq. (42) are not established for the model defined by Eq. (28). In addition, the Dyson-Schwinger calculation starts from the assumption |v1| = |v2| = v; even if the qualitative competition between magnetic and Coulomb fluctuations survives a factor-of-two correction in the velocities, the quantitative critical coupling in Eq. (43) and Fig. 5 need to be recomputed after the diagonalization is corrected, and the parameter regime satisfying |v1| = |v2| should be stated explicitly.
minor comments (4)
  1. [Sec. IV.A, Eq. (30) and Fig. 1] Please state explicitly whether the diagonal vectors u and v in the fermion Hamiltonian are a(xhat + yhat) or the unit vectors a(xhat + yhat)/sqrt(2); the identity Eq. (30) assumes the former convention while Sec. II uses the latter.
  2. [Sec. IV.B, Eq. (33) and Sec. IV.D, Eq. (41)] The symbol vhat is used both for a spatial direction vector in the linearized Hamiltonian and for the dimensionless velocity ratio in the Dyson-Schwinger analysis; these two meanings should be disambiguated.
  3. [Sec. IV.C, footnote on p. 6] The footnote states that neglected Kondo-induced couplings contain additional derivatives and are therefore unimportant, but an explicit power-counting argument would be preferable because the pi field is gapless at the critical point.
  4. [Fig. 3 caption] The caption states that the plot shows the dispersion in Eq. (29), but Eq. (29) is not the eigenvalue of Eq. (28); the figure and caption should be updated after the spectrum is corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central NLSM and scaling results are derived from a stated microscopic model, and self-citations are provenance rather than load-bearing inputs.

full rationale

The paper's central derivation is self-contained. The altermagnetic NLSM action in Eq. (11) is obtained by an explicit gradient expansion from the microscopic Hamiltonian in Eq. (2), with the Berry phase handled in Eqs. (4)-(5) and the ferromagnetic fluctuation field l integrated out to produce Eq. (10). The theta term is then used to derive the spin-wave dispersion Eq. (15) by solving the linearized equations of motion, so the nondegenerate spectrum is a consequence of the microscopic input rather than an input renamed as a prediction. The one-loop RG beta functions in Eqs. (22) are computed in Appendix A and correctly reduce to the standard NLSM results for theta = 0; no parameter is fitted to force the claimed irrelevance of theta. The fermionic sector is explicitly presented as an effective modeling choice, with the paper stating that the authors "choose the hopping parameters in a way that imprints the symmetries of the lattice," and the Dirac-node regime is obtained under clearly stated conditions t' = 0 and delta t > J_KS/4. The Dyson-Schwinger calculation assumes |v1| = |v2| as an initial simplification and derives z = 1 + gamma_1 from the resulting self-energies; the u -> 0 limit reproduces the independent graphene result of Son, which acts as an external consistency check. The self-citations to Refs. [23] and [63] supply the starting model and an integral evaluation, respectively, but neither imports the paper's central conclusions, and no uniqueness theorem or fitted parameter is used to close the argument. A possible algebraic mismatch between Eq. (29) and the eigenvalues of Eq. (28) would be a correctness concern about the low-energy fermion theory, not a circularity, because the derivation chain is not closed by assuming the target result.

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

No parameters are fitted to data: all couplings (J, J', gamma, S, t, delta t, J_KS, e) are inputs of the microscopic model and standard field theory. The calculation uses standard methods and reproduces the known graphene limit when u goes to 0.

assumptions (5)
  • domain assumption The checkerboard Lieb-lattice model of Eq (2) is a valid minimal microscopic model of altermagnetism.
    The entire NLSM derivation starts from this model, originally proposed in Ref [23]. If real altermagnets have different microscopic symmetries, the effective theory changes.
  • standard math The spin coherent state path integral and gradient expansion to leading order in lattice spacing are valid for the long-wavelength staggered magnetization.
    Used in Sec II following Refs [41,42]; this is the standard treatment of quantum antiferromagnets.
  • domain assumption The Kondo coupling form Eq (27) with sublattice Pauli tau^3 captures the coupling between fermions and smooth n fluctuations.
    Its mean-field part produces the d-wave splitting, and its fluctuating part yields Eq (39). The form is chosen by symmetry, not derived from the microscopic model.
  • domain assumption The parameter regime t'=0 and delta t > J_KS/4 puts the fermion nodes exactly at epsilon(k)=0.
    Required for the low-energy Dirac theory in Sec IV.B. If this regime fails, the DS calculation and the z prediction do not apply.
  • domain assumption Derivative-suppressed Kondo couplings to l, pi self-interactions, and vertex corrections are neglected in the DS calculation.
    Stated in footnote 1 and Sec IV.D. These are standard leading-order truncations, but their quantitative effect on the anomalous dimensions (41) is not bounded.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum critical scaling of altermagnetism." pith.science (2026). https://pith.science/paper/EINLXXLO

@misc{pith2026250504683,
  author       = {Pith},
  title        = {Pith review of: Quantum critical scaling of altermagnetism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EINLXXLO}},
  note         = {Machine review of arXiv:2505.04683}
}
abstract

The term altermagnetism has recently been introduced to describe the N\'eel order of a class of materials whose magnetic sublattices are neither related by translation nor inversion. While these materials arguably have large technological potential, little effort has been devoted to studying the universal distinction of this phase of matter compared to collinear antiferromagnetism. Employing a recently proposed minimal microscopic model, we explicitly derive a nonlinear sigma model describing long-wavelength fluctuations of the staggered magnetization in this system, including quantum effects to leading order. The term that distinguishes the altermagnetic nonlinear sigma model from its antiferromagnetic counterpart is an interaction term that derives directly from the Berry phase of the microscopic spin degrees of freedom. Its effects on the one-loop renormalization group flow in $d=2+1$ dimensions are examined. Extending the theory to describe the fermionic excitations of the metallic altermagnet, we find an effective low-energy model of $d$-wave spin-split Dirac fermions interacting with the magnetic fluctuations. Using a Dyson-Schwinger approach, we derive the many-body effects on the dynamical critical scaling due to the competition between the long-range Coulomb interaction and the fluctuations of the staggered magnetization.

Figures

Figures reproduced from arXiv: 2505.04683 by the authors.

Figure 1
Figure 1. FIG. 1. The Lieb lattice of the microscopic model, with the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. One-particle irreducible diagrams contributing to the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Plot of dispersion in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Fermionic self energies due to the spin fluctuations [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The functions [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Nanoscale Imaging of Strain-Controlled Altermagnetic Domains in {\alpha}-MnTe

    cond-mat.mtrl-sci 2026-07 conditional novelty 7.0 of 10

    In alpha-MnTe, compression makes magnetic domains grow by merging, and unloading leaves them fragmented in a different, metastable pattern, so the material remembers the strain history.

Reference graph

Works this paper leans on

65 extracted references · 33 canonical work pages · cited by 1 Pith paper

  1. [1]

    Chakravarty, B

    S. Chakravarty, B. I. Halperin, and D. R. Nelson, Low- temperature behavior of two-dimensional quantum anti- ferromagnets, Phys. Rev. Lett. 60, 1057 (1988)

    work page 1988

  2. [2]

    Chakravarty, B

    S. Chakravarty, B. I. Halperin, and D. R. Nelson, Two- dimensional quantum Heisenberg antiferromagnet at low temperatures, Phys. Rev. B 39, 2344 (1989)

    1989

  3. [3]

    Auerbach, Interacting Electrons and Quantum Mag- netism, Graduate Texts in Contemporary Physics (Springer New York, New York, NY, 1994)

    A. Auerbach, Interacting Electrons and Quantum Mag- netism, Graduate Texts in Contemporary Physics (Springer New York, New York, NY, 1994)

    work page 1994

  4. [4]

    P. W. Anderson, The Resonating Valence Bond State in La2CuO4 and Superconductivity, Science 235, 1196 (1987)

    1987

  5. [5]

    Affleck, Z

    I. Affleck, Z. Zou, T. Hsu, and P. W. Anderson, SU(2) gauge symmetry of the large- U limit of the Hubbard model, Phys. Rev. B 38, 745 (1988)

    work page 1988

  6. [6]

    Baltz, A

    V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Antiferromagnetic spintronics, Rev. Mod. Phys. 90, 015005 (2018)

    2018

  7. [7]

    Sachdev and R

    S. Sachdev and R. Jalabert, Effective lattice models for two-dimensional quantum antiferromagnets, Mod. Phys. Lett. B 04, 1043 (1990)

    work page 1990

  8. [8]

    Mudry and E

    C. Mudry and E. Fradkin, Separation of spin and charge quantum numbers in strongly correlated systems, Phys. Rev. B 49, 5200 (1994)

    work page 1994

Show all 65 references
  1. [9]

    Wen, Quantum orders and symmetric spin liquids, Phys

    X.-G. Wen, Quantum orders and symmetric spin liquids, Phys. Rev. B 65, 165113 (2002)

    2002

  2. [10]

    J. M. Kosterlitz and D. J. Thouless, Ordering, metasta- bility and phase transitions in two-dimensional systems, J. Phys. C: Solid State Phys. 6, 1181 (1973)

    1973

  3. [11]

    D. R. Nelson and J. M. Kosterlitz, Universal Jump in the Superfluid Density of Two-Dimensional Superfluids, Phys. Rev. Lett. 39, 1201 (1977)

    1977

  4. [12]

    F. D. M. Haldane, O(3) nonlinear σ model and the topo- logical distinction between integer- and half-integer-spin antiferromagnets in two dimensions, Phys. Rev. Lett.61, 1029 (1988)

    1988

  5. [13]

    Read and S

    N. Read and S. Sachdev, Valence-bond and spin-Peierls ground states of low-dimensional quantum antiferromag- nets, Phys. Rev. Lett. 62, 1694 (1989)

    1989

  6. [14]

    Senthil, A

    T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Deconfined Quantum Critical Points, Science 303, 1490 (2004)

    2004

  7. [15]

    Senthil, L

    T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Quantum criticality beyond the Landau- Ginzburg-Wilson paradigm, Phys. Rev. B 70, 144407 (2004)

    2004

  8. [16]

    Senthil, L

    T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Deconfined Criticality Critically Defined, J. Phys. Soc. Jpn. 74, 1 (2005)

    2005

  9. [17]

    Kragset, E

    S. Kragset, E. Smørgrav, J. Hove, F. S. Nogueira, and A. Sudbø, First-Order Phase Transition in Easy-Plane Quantum Antiferromagnets, Phys. Rev. Lett. 97, 247201 (2006)

    2006

  10. [18]

    A. B. Kuklov, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, Deconfined criticality, runaway flow in the two-component scalar electrodynamics and weak first- order superfluid-solid transitions, Annals of Physics July 2006 Special Issue, 321, 1602 (2006)

    2006

  11. [19]

    A. B. Kuklov, M. Matsumoto, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, Deconfined Criticality: Generic First-Order Transition in the SU(2) Symmetry Case, Phys. Rev. Lett. 101, 050405 (2008)

    2008

  12. [20]

    E. V. Herland, T. A. Bojesen, E. Babaev, and A. Sudbø, Phase structure and phase transitions in a three- dimensional SU (2) superconductor, Phys. Rev. B 87, 134503 (2013)

    2013

  13. [21]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Emerging Re- search Landscape of Altermagnetism, Phys. Rev. X 12, 040501 (2022)

    2022

  14. [22]

    ˇSmejkal, J

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

    2022

  15. [23]

    Brekke, A

    B. Brekke, A. Brataas, and A. Sudbø, Two-dimensional altermagnets: Superconductivity in a minimal micro- scopic model, Phys. Rev. B 108, 224421 (2023)

    2023

  16. [24]

    Mæland, B

    K. Mæland, B. Brekke, and A. Sudbø, Many-body effects on superconductivity mediated by double-magnon pro- cesses in altermagnets, Phys. Rev. B 109, 134515 (2024)

    2024

  17. [25]

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

    2024

  18. [26]

    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

  19. [27]

    Krempask´ y, L

    J. Krempask´ y, L. ˇSmejkal, S. W. D’Souza, M. Ha- jlaoui, G. Springholz, K. Uhl´ ıˇ rov´ a, F. Alarab, P. C. Constantinou, V. Strocov, D. Usanov, W. R. Pudelko, R. Gonz´ alez-Hern´ andez, A. Birk Hellenes, Z. Jansa, H. Reichlov´ a, Z. ˇSob´ aˇ n, R. D. Gonzalez Betancourt, P....

    2024

  20. [28]

    Reimers, L

    S. Reimers, L. Odenbreit, L. ˇSmejkal, V. N. Strocov, P. Constantinou, A. B. Hellenes, R. Jaeschke 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 altermag...

    2024

  21. [29]

    J. A. Hertz, Quantum critical phenomena, Phys. Rev. B 14, 1165 (1976)

    1976

  22. [30]

    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 (2024), arXiv:2412.14251 [cond-mat]

    2024

  23. [31]

    S. He, J. Zhao, H.-G. Luo, and S. Hu, Altermagnetism and beyond in the t-t′-δ fermi-hubbard model (2025), arXiv:2503.08362 [cond-mat]. 14

    2025 arXiv

  24. [32]

    H. Chen, G. Duan, C. Liu, Y. Cui, W. Yu, Z. Y. Xie, and R. Yu, Spin excitations of the Shastry-Sutherland model – altermagnetism and proximate deconfined quan- tum criticality (2024), arXiv:2411.00301 [cond-mat]

    2024

  25. [33]

    Ferrari and R

    F. Ferrari and R. Valent´ ı, Altermagnetism on the Shastry-Sutherland lattice, Phys. Rev. B 110, 205140 (2024)

    2024

  26. [34]

    Giuli, C

    S. Giuli, C. Mejuto-Zaera, and M. Capone, Alter- magnetism from interaction-driven itinerant magnetism, Phys. Rev. B 111, L020401 (2025)

    2025

  27. [35]

    L. D. Re, Dirac points and topological phases in corre- lated altermagnets (2024), arXiv:2408.14288 [cond-mat]

    2024 arXiv

  28. [36]

    Kaushal and M

    N. Kaushal and M. Franz, Altermagnetism in modified Lieb lattice Hubbard model (2024), arXiv:2412.16421 [cond-mat]

    2024 arXiv

  29. [37]

    K. V. Yershov, V. P. Kravchuk, M. Daghofer, and J. van den Brink, Fluctuation-induced piezomagnetism in local moment altermagnets, Phys. Rev. B 110, 144421 (2024)

    2024

  30. [38]

    Gomonay, V

    O. Gomonay, V. P. Kravchuk, R. Jaeschke-Ubiergo, K. V. Yershov, T. Jungwirth, L. ˇSmejkal, J. van den Brink, and J. Sinova, Structure, control, and dynamics of altermag- netic textures, npj Spintronics 2, 1 (2024)

    2024

  31. [39]

    P. M. Cˆ onsoli and M. Vojta, SU( N) Altermagnetism: Lattice Models, Magnon Modes, and Flavor-Split Bands, Physical Review Letters 134, 196701 (2025)

    2025

  32. [40]

    Canals, From the square lattice to the checkerboard lattice: Spin-wave and large-n limit analysis, Phys

    B. Canals, From the square lattice to the checkerboard lattice: Spin-wave and large-n limit analysis, Phys. Rev. B 65, 184408 (2002)

    2002

  33. [41]

    Affleck, Quantum spin chains and the Haldane gap, J

    I. Affleck, Quantum spin chains and the Haldane gap, J. Phys.: Condens. Matter 1, 3047 (1989)

    1989

  34. [42]

    Abanov, Topology, geometry and quantum interfer- ence in condensed matter physics, in Topology and Con- densed Matter Physics , edited by S

    A. Abanov, Topology, geometry and quantum interfer- ence in condensed matter physics, in Topology and Con- densed Matter Physics , edited by S. M. Bhattacharjee, M. Mj, and A. Bandyopadhyay (Springer, Singapore,

  35. [43]

    Bhowal and N

    S. Bhowal and N. A. Spaldin, Ferroically Ordered Mag- netic Octupoles in d-Wave Altermagnets, Phys. Rev. X 14, 011019 (2024)

    2024

  36. [44]

    P. A. McClarty and J. G. Rau, Landau Theory of Al- termagnetism, Phys. Rev. Lett. 132, 176702 (2024), arXiv:2308.04484 [cond-mat]

    2024 arXiv

  37. [45]

    P. A. McClarty, A. Gukasov, and J. G. Rau, Observing altermagnetism using polarized neutrons, Phys. Rev. B 111, L060405 (2025)

    2025

  38. [46]

    Witten, Non-abelian bosonization in two dimensions, Commun.Math

    E. Witten, Non-abelian bosonization in two dimensions, Commun.Math. Phys. 92, 455 (1984)

    1984

  39. [47]

    Zinn-Justin, Quantum Field Theory and Critical Phe- nomena: Fifth Edition , International Series of Mono- graphs on Physics (Oxford University Press, 2021)

    J. Zinn-Justin, Quantum Field Theory and Critical Phe- nomena: Fifth Edition , International Series of Mono- graphs on Physics (Oxford University Press, 2021)

    2021

  40. [48]

    D. R. Nelson and R. A. Pelcovits, Momentum-shell re- cursion relations, anisotropic spins, and liquid crystals in 2 +ϵ dimensions, Phys. Rev. B 16, 2191 (1977)

    1977

  41. [49]

    Sachdev, Quantum Phase Transitions , 2nd ed

    S. Sachdev, Quantum Phase Transitions , 2nd ed. (Cam- bridge University Press, Cambridge, 2011)

    2011

  42. [50]

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

    2024

  43. [51]

    A. Bose, S. Vadnais, and A. Paramekanti, Altermag- netism and superconductivity in a multiorbital t − J model, Phys. Rev. B 110, 205120 (2024)

    2024

  44. [52]

    Meng-Han, G

    Z. Meng-Han, G. Xuan, and D.-X. Yao, Dirac points and Weyl phase in a honeycomb altermagnet (2024), arXiv:2412.03657 [cond-mat]

    2024

  45. [53]

    D. S. Antonenko, R. M. Fernandes, and J. W. F. Vender- bos, Mirror Chern Bands and Weyl Nodal Loops in Al- termagnets, Physical Review Letters 134, 096703 (2025)

    2025

  46. [54]

    Abanov, A

    Ar. Abanov, A. V. Chubukov, and J. Schmalian, Quantum-critical theory of the spin-fermion model and its application to cuprates: Normal state analysis, Ad- vances in Physics 52, 119 (2003)

    2003

  47. [55]

    M. A. Metlitski and S. Sachdev, Quantum phase transi- tions of metals in two spatial dimensions. II. Spin density wave order, Phys. Rev. B 82, 075128 (2010)

    2010

  48. [56]

    Lee, Recent Developments in Non-Fermi Liquid Theory, Annual Review of Condensed Matter Physics 9, 227 (2018)

    S.-S. Lee, Recent Developments in Non-Fermi Liquid Theory, Annual Review of Condensed Matter Physics 9, 227 (2018)

    2018

  49. [57]

    I. F. Herbut, Quantum critical points with the coulomb interaction and the dynamical exponent: When and why z = 1, Phys. Rev. Lett. 87, 137004 (2001)

    2001

  50. [58]

    I. F. Herbut, Interactions and Phase Transitions on Graphene’s Honeycomb Lattice, Phys. Rev. Lett. 97, 146401 (2006)

    2006

  51. [59]

    D. T. Son, Quantum critical point in graphene ap- proached in the limit of infinitely strong Coulomb in- teraction, Phys. Rev. B 75, 235423 (2007)

    2007

  52. [60]

    Hsiao and D

    W.-H. Hsiao and D. T. Son, Duality and universal trans- port in mixed-dimension electrodynamics, Physical Re- view B 96, 075127 (2017)

    2017

  53. [61]

    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. Shi- mada, S. Cui, Y. Zhao, G.-H. Cao, S. Qiao, J. Liu, and C. Chen, Crystal-symmetry-paired spin-valley locking in a layered room-temperatu...

    2024

  54. [62]

    Bauer, A

    C. Bauer, A. R¨ uckriegel, A. Sharma, and P. Kopietz, Nonperturbative renormalization group calculation of quasiparticle velocity and dielectric function of graphene, Phys. Rev. B 92, 121409 (2015)

    2015

  55. [63]

    S. D. Lundemo, F. S. Nogueira, and A. Sudbø, Topology- driven deconfined quantum criticality in magnetic bilay- ers, Phys. Rev. B 111, 075158 (2025)

    2025

  56. [64]

    Appelquist, D

    T. Appelquist, D. Nash, and L. C. R. Wijewardhana, Critical Behavior in (2+1)-Dimensional QED, Phys. Rev. Lett. 60, 2575 (1988)

    1988

  57. [65]

    T. W. Appelquist, M. Bowick, D. Karabali, and L. C. R. Wijewardhana, Spontaneous chiral-symmetry breaking in three-dimensional QED, Phys. Rev. D33, 3704 (1986)

    1986

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.