Altermagnetism Induced Topological Phase Transitions in Kane-Mele Model

Zeyu Li; Zhengtian Li; Zhenhua Qiao

REVIEW 3 major objections 6 minor 75 references

Altermagnetism Induced Topological Phase Transitions in Kane-Mele Model

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper claims that adding d-wave altermagnetism to the Kane-Mele model drives the Z2 topological insulator into a second-order topological insulator and, with Rashba spin-orbit coupling, into quantum anomalous Hall phases whose Chern…

desk verdict A solid model study showing d-wave altermagnetism can drive the Kane-Mele insulator through SOTI and QAHE phases, with the main caveat being the ad hoc sublattice structure of the altermagnetism term. read the letter →

arxiv 2412.20129 v1 pith:AIAIZLKP submitted 2024-12-28 cond-mat.mes-hall

Zhengtian Li , Zeyu Li , Zhenhua Qiao This is my paper · ORCID

classification cond-mat.mes-hall

keywords altermagnetismKane-Melemodelsecond-ordertopologicalinsulatorquantumanomalousHalleffectChernnumberd-wavespinsplittingRashbaspin-orbitcouplingcornerstates

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

The paper is trying to establish that d-wave altermagnetism can act as a symmetry-breaking field that drives the Kane-Mele $\mathbb{Z}_2$ topological insulator through a sequence of topological phases. With the Néel vector in the $x$--$y$ plane, the $\mathbb{Z}_2$ insulator becomes a second-order topological insulator with corner states, and adding intrinsic Rashba spin-orbit coupling turns it first into a quantum anomalous Hall phase with Chern number $C=\pm 1$ and then into a phase with $C=\pm 1$ or $C=\pm 3$. For an out-of-plane Néel vector, Rashba coupling is required to break mirror symmetry, and the system passes through a second-order topological insulator, then a $C=-1$ phase, then a $C=1$ phase whose edge modes have mixed chirality but still carry a net chiral current. If the claim holds, altermagnetism becomes a third magnetic knob, alongside ferromagnetism and antiferromagnetism, for engineering topological phases, with the direction of the Néel vector serving as a control for the Chern number.

What carries the argument

The load-bearing object is the altermagnetism term $H_{\mathrm{AM}}$ with its d-wave angular factor $\Delta=\cos(2\theta)$, implemented as an anisotropic next-nearest-neighbor hopping between same sublattices; this is what breaks time-reversal symmetry without introducing a net magnetization. It is added to the Kane-Mele Hamiltonian (nearest-neighbor hopping plus intrinsic spin-orbit coupling) and, when needed, intrinsic Rashba spin-orbit coupling. The paper's analysis relies on symmetry bookkeeping: which combinations of time reversal $T$, out-of-plane mirror $M_z$, and inversion $I$ are broken by a given Néel-vector orientation decides whether corner states or chiral edge states can appear. The phases are identified by a mirror-graded Zak phase computed with the Wilson loop and by Chern numbers obtained from the same Wilson-loop/Berry-curvature machinery, with an effective-Hamiltonian/local-Chern-number decomposition around the three $M$ points explaining the $\varphi$-dependent plateau values.

What would settle it

A decisive check would be a first-principles calculation of the altermagnetic proximity effect for a concrete honeycomb system: if the induced spin splitting is not of the d-wave $\cos(2\theta)$ form on the next-nearest-neighbor bonds, the band gap closings at the $M$ points would occur at different parameter values or not at all, and the predicted $C=\pm 1$ and $C=\pm 3$ plateaus would not appear. Alternatively, a transport experiment on a candidate altermagnet/Kane-Mele heterostructure measuring the Hall conductance while the altermagnet strength is tuned, for example by temperature or by rotating the Néel vector, should show plateaus at $e^2/h$ and then $3e^2/h$; failure to see the $3e^2/h$ plateau in the predicted orientation windows would falsify the central claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a d-wave altermagnetism term, written as $H_{\mathrm{AM}} = \lambda \sum_{\langle\langle ij\rangle\rangle} c_i^\dagger \Delta \mathbf{s}\cdot\hat{n} c_j$ with $\Delta = \cos(2\theta)$, inserted into the Kane-Mele model produces a well-defined sequence of topological phases. For an in-plane Néel vector the $\mathbb{Z}_2$ topological insulator first turns into a second-order topological insulator (corner states at obtuse angles of a zigzag nanoflake), then after gap closings at the $M_2$ point enters a quantum anomalous Hall phase with $C=\pm 1$, and after further gap closings at $M_1$ and $M_3$ enters a phase with $C=\pm 1$ or $C=\pm 3$ depending on the azimuthal angle of the Néel vector. For an out-of-plane Néel vector, intrinsic Rashba spin-orbit coupling is necessary to break the relevant mirror symmetry, and the sequence is second-order topological insulator, then $C=-1$, then $C=1$, the last one showing three pairs of edge states with mixed chirality and a net chiral current. The Chern-number sectors are tied to local Chern numbers around the three $M$ points, so the Néel-vector orientation effectively switches between $C=1$, $C=-1$, and $C=3$.

Load-bearing premise

The load-bearing premise is that a real d-wave altermagnet coupled to a honeycomb lattice is faithfully described by the single anisotropic next-nearest-neighbor hopping term $H_{\mathrm{AM}}=\lambda \sum c_i^\dagger \cos(2\theta)\,\mathbf{s}\cdot\hat{n}\, c_j$; the paper offers no first-principles derivation or experimental measurement of this coupling, so if the actual altermagnetic exchange has a different site structure or angular dependence the predicted phase sequence and Chern numbers could change.

Editorial extensions

If this is right

A $\mathbb{Z}_2$ topological insulator placed in proximity to an in-plane d-wave altermagnet should show corner-state conduction, a second-order topological insulator, even without Rashba coupling, rather than the original helical edge transport.
With intrinsic Rashba spin-orbit coupling, the same system should exhibit a quantized Hall conductance of $e^2/h$ ($C=\pm 1$) after the first gap reopening and $3e^2/h$ or $e^2/h$ ($C=\pm 3$ or $\pm 1$) after the second, so the altermagnet strength acts as a Chern-number switch.
Rotating the Néel vector in the plane divides the phase diagram into two sectors with opposite Chern numbers in the first QAHE region and six sectors in the second, meaning magnetization direction alone can reverse or change the Hall response.
For an out-of-plane Néel vector, Rashba coupling is essential: without it the system stays a second-order topological insulator, and with it the sequence $C=-1$ then $C=1$ appears, the latter with counter-propagating edge modes that nevertheless yield a net chiral current.

Reading between the lines

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

Beyond the paper, the predicted $C=\pm 3$ plateau is an unusual high-Chern-number state that would be a direct fingerprint of d-wave altermagnetism if observed in a real honeycomb heterostructure, since ferromagnetic proximity typically gives $C=\pm 1$.
The mixed-chirality edge modes (three pairs with counter-propagating parts but a net chiral current) could be distinguished from ordinary chiral edges by nonlocal transport measurements, because part of the edge current flows in the opposite direction at the same boundary.
A natural next step the paper does not take is to identify specific altermagnet/honeycomb-substrate combinations where the $\cos(2\theta)$ hopping dominates, which would make the predicted phase sequence testable; density-functional calculations of the proximity coupling are the obvious way to check the assumed form of $H_{\mathrm{AM}}$.

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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

Quick take: this is a competent tight-binding study with a genuinely new phase diagram—C=±3 QAHE and mixed-chirality edge modes from altermagnetism in the Kane-Mele model. The numerics are standard and the symmetry analysis is careful. The main soft spot is the altermagnetism term itself: Eq. (2) uses a sublattice-uniform d-wave hopping (τ0), while many microscopic altermagnets on bipartite lattices have a staggered (τz) coupling. The paper neither derives the τ0 form nor checks how the phase sequence depends on that choice. That matters because all the claimed phases—SOTI, C=1, C=3—are computed from this single term. If the physical coupling is τz, the inversion parity changes and the phase boundaries may shift or the high-Chern phases may disappear. This is not fatal for a model study, but the authors should either motivate τ0 from a specific material/proximity setup or test τz and report the comparison.

What's new: prior works (including their own Refs. [63,64]) considered ferromagnets and antiferromagnets; this is the first to put d-wave altermagnetism into Kane-Mele and map the φ-dependent phase diagram. The six-section division of the second QAHE phase and the local-Chern-number decomposition around the M points are nice, and the mixed-chirality QAHE with three pairs of edge states but net C=±1 is a real curiosity worth understanding. The corner-state locations for in-plane vs out-of-plane Néel vectors are clearly tied to mirror symmetries.

Soft spots beyond the model term: the Supplemental Material is not included with the arXiv posting, and there is no code or data release. That makes the Wilson-loop numbers and the Zak-phase results hard to verify independently. The effective-Hamiltonian argument is post-hoc, which is fine for explanation, but it doesn't validate the model. The paper also leans on symmetry language (e.g., T⊗Mz⊗I) without fully defining the operations in the text; a referee should ask for explicit matrix forms.

Bottom line: the paper is a good candidate for serious peer review. For a reader working on altermagnetism or topological models, it's worth a close look and probably worth citing as a model prediction. My recommendation: send it to a competent referee, but require the SM and a robustness check or at least a structured discussion of the τ0 vs τz choice.

Referee Report

3 major / 6 minor

Summary. The paper studies a modified Kane-Mele model with an added d-wave altermagnetism term, Eq. (2), of the form H_AM = λ Σ c† Δ s·n c with Δ = cos(2θ). The authors show that for an in-plane Néel vector, the Z2 topological insulator is driven into a second-order topological insulator (SOTI), characterized by mirror-graded Zak phases and corner states. When intrinsic Rashba spin-orbit coupling is included, further transitions take the system into quantum anomalous Hall (QAHE) phases with Chern numbers C = ±1 and then C = ±1 or ±3, with band gap closings at λ = 0.25t and λ = 0.50t. For an out-of-plane Néel vector, a similar sequence of SOTI and QAHE phases is reported, including an unconventional QAHE with mixed-chirality edge states. The topological invariants are computed by Wilson-loop Chern numbers, mirror-graded Zak phases, and Berry curvature distributions, and an effective Hamiltonian around the M points is invoked to explain the φ-dependence of the Chern numbers.

Significance. If the model faithfully captures altermagnetic order in a honeycomb lattice, the paper demonstrates that a single altermagnetic term can generate a rich sequence of topological phases—SOTI and Chern-number-tunable QAHE—from the paradigmatic Kane-Mele Z2 insulator. This goes beyond previous studies using ferromagnetism and antiferromagnetism and introduces a mixed-chirality QAHE with counter-propagating edge modes but net chiral current, which is a novel feature. The calculations are based on standard, presumably reproducible tight-binding diagonalization, Wilson-loop invariants, and symmetry analysis, and the reported gap-closing points are consistent with the phase diagram. The main weaknesses are the unverified form of the altermagnetism term and the reliance on a missing Supplemental Material for several central results.

major comments (3)

[Model Hamiltonian and Symmetry Analysis, Eq. (2)] The d-wave altermagnetism term in Eq. (2) is a real spin-dependent next-nearest-neighbor hopping with a cos(2θ) form factor and equal amplitude on the two sublattices (proportional to τ_0 in sublattice space). The paper does not derive this term from a microscopic altermagnet, and it does not test robustness against the alternate sublattice-staggered (τ_z) form that appears in many bipartite altermagnet models. Because the entire phase diagram and all Chern numbers are computed from this single term, the central claims are only established for this specific choice. The authors should justify the τ_0 structure by symmetry or a lattice derivation, or show that the SOTI and QAHE phase sequence survives a staggered form.
[Quantum Anomalous Hall effect and Ref. [68]] The text repeatedly refers to the Supplemental Material for essential results, including the out-of-plane band structures (Fig. S1), the out-of-plane gap closings at λ = 0.25t and 0.50t (Fig. S2), the armchair ribbon edge states (Fig. S3), the (t_IR, λ) phase diagrams (Fig. S4), and the effective-Hamiltonian local Chern number analysis (Fig. S8). The Supplemental Material is not included with the arXiv submission, as Ref. [68] merely says "See Supplemental Material at url". Consequently, a substantial part of the evidence for the out-of-plane phase sequence and for the φ-dependent Chern number explanation cannot be checked. The authors must provide the Supplemental Material or move the key results into the main text.
[Quantum Anomalous Hall effect, paragraph after Fig. 2(j)] The effective Hamiltonian that is claimed to explain the evolution of the φ-dependent Chern number in the second QAHE phase is not presented anywhere in the manuscript or available Supplemental Material. The text only states that "Based on the effective Hamiltonian, we calculate the local Chern numbers around the three different M points" and then lists the resulting intervals. Without the explicit effective Hamiltonian and the local Chern number calculation, this explanation is not verifiable. The authors should include the effective Hamiltonian and its local Chern number results.

minor comments (6)

[Second-Order Topological Insulator, Eq. (3)] The Zak phase formula in Eq. (3) should specify that the sum runs over occupied bands and that the integral is over the mirror-invariant line k_y = 0; a brief definition of the occupied-band index would improve clarity.
[Second-Order Topological Insulator] The mirror operator is written as M_y = s_x i σ_y, but σ_y is not defined anywhere in the Hamiltonian. Since the Hamiltonian is written in the site basis without explicit sublattice Pauli matrices, a sentence introducing σ as the sublattice Pauli matrices would remove ambiguity.
[Model Hamiltonian and Symmetry Analysis, Eq. (2)] In the definition of Δ = cos(2θ), the azimuthal angle θ is stated to be the angle of d_ij, but the reference axis for θ (presumably the x-axis) is not specified. Please state the reference direction explicitly.
[Model Hamiltonian and Symmetry Analysis, Eq. (1)] The label t_IR for the Rashba spin-orbit coupling is potentially confusing, since 'intrinsic' is also used for t_SO. Consider renaming the Rashba coupling to something like t_R or explicitly defining the abbreviation.
[Fig. 2 caption] The caption is internally inconsistent: it says "(g) and (h) Corresponding Berry curvatures distribution for λ = 0.3t, 0.6t" while (h) is already labeled as a ribbon band structure. Re-label the panels or correct the caption.
[References] Several references contain errors: Ref. [65] is missing the volume number (likely 111) and the year, Ref. [18] contains the extraneous text "M./suppress l", and Ref. [50] is a preprint with no journal or year. These should be corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the topological phases are computed from the stated Hamiltonian; the only self-citation is non-load-bearing background.

full rationale

The paper's derivation chain is: define H = H_KM + H_AM (Eqs. 1 and 2), analyze symmetries, compute bulk and ribbon band structures, and extract Zak phases and Chern numbers via Wilson loops. No parameter is fitted to reproduce the claimed SOTI or QAHE phases; the phase sequence and Chern numbers are direct numerical outputs of the model Hamiltonian. The effective Hamiltonian used in the discussion of the second QAHE phase is derived after the fact from the same model and serves only to rationalize the numerically obtained C = 1 and C = 3 regions; it is not an input used to generate the phase diagram. The one self-citation, Ref. [63], is invoked as a symmetry criterion for when QAHE can appear in a honeycomb lattice, but the present paper independently computes band structures, edge states, and Chern numbers, so the cited result is background rather than load-bearing. The d-wave altermagnetism term in Eq. (2) is an assumed form with cos(2θ) next-nearest-neighbor hopping; whether this faithfully represents a real altermagnet is a modeling/robustness concern, not circularity, because the prediction is derived from the stated Hamiltonian rather than from the conclusion. No equation is equivalent by construction to a fitted input or to a self-citation chain, so the paper is not circular in its central derivation. Minor caveat: several supporting phase diagrams are relegated to the Supplemental Material, which is a completeness issue rather than a circularity issue.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central phase diagram is computed from a tight-binding model whose parameters are chosen by hand; no parameters are fitted to experimental data. The d-wave altermagnetism term is an ad hoc model ingredient. No new particles or conserved quantities are introduced.

free parameters (3)

λ (altermagnetism strength) = varied from 0 to 0.6t
Model parameter controlling H_AM; phase diagram mapped as function of λ. Not fitted to external data.
t_SO (intrinsic spin-orbit coupling) = 0.03t, 0.06t, 0.1t
Chosen representative values; not fitted to data.
t_IR (Rashba spin-orbit coupling) = 0.06t, 0.1t
Chosen representative values; required for out-of-plane Neel vector QAHE.

assumptions (4)

ad hoc to paper The d-wave altermagnetism term in Eq. (2) captures the essential physics of altermagnetic order in a honeycomb lattice.
Introduced specifically for this model; no first-principles derivation from a specific material is given. All subsequent phase predictions depend on this term.
standard math Chern number computed via Wilson loop [60] is the correct invariant for the QAHE phases.
Standard bulk-boundary correspondence in 2D insulators; cited method from Fukui et al.
standard math Mirror-graded Zak phase [71,72] along the y-direction (My-symmetric line) characterizes the second-order TI.
Standard indicator for corner states in higher-order topological insulators.
domain assumption The intrinsic Rashba spin-orbit coupling term in Eq. (1) is the correct form for a low-buckled honeycomb lattice.
Taken from prior work [34,37,63]; no derivation in this paper.

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Pith. "Pith review of Altermagnetism Induced Topological Phase Transitions in Kane-Mele Model." pith.science (2026). https://pith.science/paper/AIAIZLKP

@misc{pith2026241220129,
  author       = {Pith},
  title        = {Pith review of: Altermagnetism Induced Topological Phase Transitions in Kane-Mele Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AIAIZLKP}},
  note         = {Machine review of arXiv:2412.20129}
}

abstract

We theoretically demonstrate that Chern number tunable quantum anomalous Hall effect (QAHE) and second-order topological insulators can be induced in the two-dimensional $\mathbb{Z}_2$ topological insulator (TI), i.e., Kane-Mele model, by applying $d$-wave altermagnetism. When the N\'eel vector of altermagentism lies in the $x-y$ plane, the $\mathbb{Z}_2$ TI is broken and driven into a second-order topological insulator phase, exhibiting the representative corner states at nanoflakes. When the intrinsic Rashba spin-orbit coupling is further included, the second-order TI is further driven into the QAHE phase with various Chern numbers (e.g., $\mathcal{C}=\pm1$ or $\pm3$). When the N\'eel vector is along $z$ direction, the intrinsic Rashba spin-orbit coupling is necessary to break the mirror symmetry to allow a sequential emergence of second-order TI and QAHE along with the increase of altermagentism strength. We also observe the QAHE with mixed-chirality, i.e., there exist counter-propagating edge modes but net chiral current at the ribbon boundary. Our work shows that altermagnetism can play a crucial role in exploring a rich variety of topological phases, just like its counterparts of ferromagnetism and antiferromagnetism.

Figures

Figures reproduced from arXiv: 2412.20129 by the authors.

**Figure 2.** FIG. 2. Evolution of bulk band structures of Kane-Mele model [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) and (d) Band structures of zigzag edge nanorib [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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4 1.6 -0 .3 0.0 0.3 0.4 1.6 E/t (a) (b ) (c) (d ) M M1M3 K K K' K' Γ 2 Number 0.1 -0.1 0.0 FIG. 1. (a-b) Bulk band structures (a) without and (b) with in-plane altermagnetism along y direction. After introduc- ing in-plane altermagnetism, the band structure is split in to spin-down and -up channels labeled by blue and red lines, re- spectively. Inset: The...

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2, 0. 25, 0. 3, 0. 5, 0. 6, respectively. (f) and (h) Band structures of zigzag nanor ibbons with altermagentism strength λ/t =

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6, respectively

3, 0. 6, respectively. Edge states located at opposite boundarie s are highlighted by red and green curves, respectively. (g) and (h) Corresponding Berry curvatures distribution for λ = 0 . 3t, 0. 6t, respectively. (j) Phase diagram as a function of the azimuthal angle φ of the N´ eel vector of altermagnetism withtIR = tSO = 0. 10t. Color bar measures bul...

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