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REVIEW 4 major objections 4 minor 57 references

Interplay between altermagnetism and topological superconductivity on an unconventional superconducting platform

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Coupling a d-wave altermagnet to a two-dimensional p-wave superconductor can tune topological phase transitions, producing gapless topological superconductors with Majorana flat edge modes or, for mixed pairing, a hybrid phase with…

desk verdict Model study with a genuinely new hybrid Majorana phase that is backed by numerics but saddled with a wrong analytic spectrum in the helical case. read the letter →

arxiv 2501.05451 v2 pith:C6PGBJNW submitted 2025-01-09 cond-mat.supr-con cond-mat.mes-hall

Pritam Chatterjee , Vladimir Juričić This is my paper · ORCID
classification cond-mat.supr-concond-mat.mes-hall
keywords altermagnetismtopologicalsuperconductivityMajoranaflatedgemodesp-wavesuperconductorchiralhelicalnodal-lineunconventional
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 a d-wave altermagnet, meaning a magnet with zero net magnetization and momentum-dependent spin splitting, can act as a control knob for a two-dimensional p-wave (odd-parity spin-triplet) superconductor, tuning it through topological phase transitions. In a helical p-wave superconductor, increasing the altermagnetic coupling above a critical value drives the system from a gapped topological phase into a gapless topological superconductor whose zero-energy Majorana edge states become flat in momentum. In a chiral p-wave superconductor, the same coupling drives a transition into a gapless nodal-line superconductor, where the quasiparticle gap closes along a line in momentum space. When helical and chiral pairings coexist, the altermagnet produces a hybrid topological phase with dispersive and nearly flat Majorana edge modes at the same edge. A sympathetic reader would care because this is a zero-magnetization route to flat Majorana modes, the edge modes sought for topological quantum computation.

What carries the argument

The load-bearing object is the Bogoliubov–de Gennes Hamiltonian in Eq. (2), the mean-field quasiparticle Hamiltonian that couples a d-wave altermagnetic exchange field $J(k)=2J_A(\cos k_x-\cos k_y)$ oriented along the spin-x direction to helical, chiral, or mixed p-wave pairing on a square lattice. The altermagnetic term is momentum-dependent, spin-splitting, and zero-magnetization, and it explicitly breaks time-reversal symmetry, placing all coupled systems in the tenfold class D, meaning particle-hole symmetry only. The argument proceeds by locating the bulk gap closing at the X point, which gives the critical coupling $J_A^c$, and by a ribbon-geometry edge analysis in which the chosen spin orientation selectively leaves the y-edge gapless, enabling flat Majorana modes in the helical and mixed cases.

What would settle it

Directly diagonalize the lattice BdG Hamiltonian in Eq. (2) in the helical regime with, say, $\mu=2t$, $\Delta_h^p=t$, and $J_A=0.65t$: the claim predicts bulk nodes at an even number of momenta and zero-energy flat edge modes along the y-edge, and if the exact dispersion does not show those flat modes, the gapless-phase claim fails. For the chiral regime, the same numerical check at $J_A=0.8t$ should show the gap closing along a line in momentum space rather than at isolated points.

Watch

Extended reading notes

Core claim

The central claim is that altermagnetic order, despite carrying no net magnetization, can qualitatively change the topology of a 2D p-wave superconductor by coupling through the spin-x exchange term $J(k)\tau_0 s_x$ with $J(k)=2J_A(\cos k_x-\cos k_y)$. For helical pairing, the bulk gap closes at the $X=(\pi,0)$ point when $J_A=J_A^c=\frac{1}{4}|4t-\mu|$, and for $J_A>J_A^c$ the system enters a gapless topological superconductor in symmetry class D, with an even number of nodal points and Majorana flat edge modes in the ribbon geometry. For chiral pairing, the same threshold separates a gapped topological superconductor with Chern number one from a gapless nodal-line superconductor whose edge channels delocalize. For mixed helical–chiral pairing, the overcritical phase is a hybrid topological superconductor in which linearly dispersing and nearly flat Majorana edge states coexist. Particle-hole symmetry alone protects these phases.

Load-bearing premise

The whole phase diagram assumes the altermagnet can be represented by the single spin-x d-wave exchange term $J(k)\tau_0 s_x$ while the p-wave pairing amplitudes stay spatially uniform; if the pairing renormalizes in the presence of the altermagnet or the spin orientation differs, the phase boundaries and the edge-mode localization will shift.

Editorial extensions

If this is right

  • In a helical p-wave superconductor with altermagnetic coupling above $J_A^c$, the system becomes a gapless topological superconductor with zero-energy Majorana flat edge modes along the y-edges, visible as localized zero-energy local density of states.
  • In a chiral p-wave superconductor above $J_A^c$, the bulk gap closes along nodal lines and the edge-localized chiral channels disappear, shifting zero-energy spectral weight into the bulk.
  • With mixed helical and chiral pairing above $J_A^c$, the edge spectrum shows both linearly dispersing Majorana modes and nearly flat Majorana modes on the same edge.
  • Each transition is accompanied by a bulk gap closing, so the topological phase boundaries in the $\mu$-$J_A$ plane follow from the X-point condition $J_A^c=\frac{1}{4}|4t-\mu|$.
  • Because the coupled systems lie in class D, their gapped phases carry a $\mathbb{Z}$ topological invariant, and in the chiral case the Chern number is one.

Reading between the lines

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

  • Because the altermagnet has zero net magnetization, this route to Majorana flat modes avoids the stray-field engineering needed in ferromagnet-based hybrids; the paper notes the zero-magnetization property but does not develop this practical advantage.
  • The paper's own observation that the spin orientation controls edge localization implies a magnetic switch: rotating the altermagnet easy axis could move flat Majorana modes between x- and y-edges.
  • The mechanism may extend to other unconventional pairings, such as d-wave superconductors on altermagnetic substrates, though the paper analyzes only p-wave pairing.
  • A numerical test that uses the full Hamiltonian rather than the analytic spectrum of Eq. (5) is the cleanest way to check whether the predicted node count and flatness survive outside the specific parameter values shown.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a 2D Bogoliubov–de Gennes (BdG) model for a d-wave altermagnet coupled to helical, chiral, and mixed p-wave superconductors. It claims that tuning the altermagnetic amplitude drives a helical topological superconductor into a gapless topological superconductor with Majorana flat edge modes, a chiral superconductor into a gapless nodal-line superconductor, and a mixed-pairing superconductor into a hybrid phase with coexisting dispersive and nearly flat Majorana modes. The evidence presented includes analytic quasiparticle spectra, ribbon-geometry edge spectra, zero-energy LDOS maps, Chern numbers, and bulk gap profiles.

Significance. If the central claims survive correction, the paper would establish a zero-net-magnetization platform for Majorana flat edge modes and for a hybrid topological superconducting phase, with potential relevance to altermagnet/p-wave superconductor heterostructures. The paper has clear strengths: it is based on a concrete lattice model, it reports numerical diagonalization results including LDOS and Chern numbers, and the qualitative phenomenology is sharply formulated and testable. However, the analytic support is currently unreliable, and one model choice appears non-standard for a BdG description of an exchange field.

major comments (4)
  1. [Section II, Eq. (2)] The altermagnetic term is written as J(k) τ0 sx. For a single-particle exchange potential, the BdG representation should contain a Nambu Pauli matrix, e.g. J(k) τz sx, not τ0. With τ0 and a symmetric traceless matrix sx, the term cancels in the second-quantized Hamiltonian: summing 1/2 Φ† (J(k) τ0 sx) Φ over the Brillouin zone gives 1/2 ∑ [c†_k J(k) sx c_k + c_{-k} J(k) sx c†_{-k}] = 1/2 ∑ Tr(J(k) sx) = 0. Thus Eq. (2) as written does not represent an altermagnet, and the J-dependent spectra in Figs. 1, 3, 4, and 6 are not the physical quasiparticle spectra of the model. The authors must either replace τ0 by τz or explicitly justify why a τ0 coupling is physical in their BdG construction.
  2. [Section II, Eq. (5)] The claimed helical spectrum is not the spectrum of hk in Eq. (2). For t=1, μ=2, Δ_h=1, J=0, and k=(π/2,π/2), Eq. (5) gives E=2, whereas direct diagonalization of Eq. (2) gives E=√6 ≈ 2.449. Consequently, Figs. 1(a)–1(d), which are generated from Eq. (5), do not represent the correct band structure, and the statements about the number of gapless nodes and the 'weak topological SC' characterization of the overcritical phase are not established by this analytic expression.
  3. [Section II, Eq. (8)] The critical coupling J_A^c = |4t−μ|/4 is derived from the incorrect spectrum in Eq. (5). At X=(π,0), the pairing terms in Eq. (2) vanish, and the exact gap is √((4t−μ)^2+(4J_A)^2), which never closes for finite J_A. Therefore Eq. (8) is not a legitimate gap-closing condition. The value J_A=0.5t used at μ=2t and Δ=t may still correspond to a transition for that special parameter set, but the general phase boundary in Fig. 6 and the critical-value claims in Sec. III must be re-derived from the correct spectrum.
  4. [Appendix A] The particle-hole symmetry operator P=τx sz K does not satisfy P h(k) P^{-1} = -h(-k) for the Hamiltonian in Eq. (2). For example, in the helical case with J=0 and k=(π/2,π/2), explicit evaluation gives P h(k) P^{-1} ≠ -h(-k); the claim in App. A that the pairing matrices commute with P is incorrect for the τy s0 term, which anticommutes with τx sz. This calls into question the class-D classification in Table I and the statement that the Majorana models are protected by particle-hole symmetry.
minor comments (4)
  1. [Fig. 2 caption] The caption says the band structure is given by Eq. (5), but the chiral spectrum is given by Eq. (7).
  2. [Eq. (8)] The notation J_A^c = ± 1/4 |4t−μ| is ambiguous; it should be written as J_A^c = |4t−μ|/4.
  3. [Appendix A] In the TRS discussion of the chiral case, the pairing term is written with τx sx and τy sx, whereas Eq. (2) uses τx s0 and τy s0; this is a typo that should be corrected.
  4. [Appendix B] The sentence 'the low-energy effective edge Hamiltonian then t takes reduces to' in the paragraph after Eq. (B2) contains a grammatical error and should read 'then takes the reduced form'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central phase diagram and edge-state claims follow from direct diagonalization of the stated BdG Hamiltonian, with no fitted parameter or self-citation chain used to force the result.

full rationale

The paper's central claims are obtained by direct numerical and analytic treatment of the BdG Hamiltonian in Eq. (2), with all parameters (t, mu, Delta_h^p, Delta_c^p, J_A) stated as inputs. The Chern number is computed from the Hamiltonian via the projector formula in Eq. (10), the bulk gap profiles in Fig. 6 are direct spectra, and the ribbon and LDOS results are obtained by numerical diagonalization of the same lattice Hamiltonian. No parameter is fitted to reproduce a targeted phase, and no predicted quantity is equivalent by construction to an input. The self-citations (Refs. 34, 36-38) are used only for computational methodology, such as the LDOS approach, and for contextual comparison; they do not supply a premise that forces the central phase diagram. The analytic spectra in Eqs. (5) and (7) appear not to be the actual eigenvalues of Eq. (2), and Eq. (8) is not a valid gap-closing condition at X=(pi,0), but this is an internal derivation error rather than circularity: the incorrect formula is not assumed as an input and then rediscovered as a predicted output. The model choices, including the d-wave form J(k)=2J_A(cos k_x - cos k_y) and the spin-x orientation, are stated physical inputs, not outputs disguised as inputs. Therefore the derivation chain is self-contained, and no circular step is present.

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

The model depends on the standard BdG and tenfold-way formalism, plus a specific d-wave altermagnet coupling form and the assumption of unrenormalized pairing. No new entities are introduced, and no parameters are fitted to experimental data.

free parameters (3)
  • chemical potential mu = 2t in main figures, varied in Fig. 6
    Chosen by hand to set J_A^c = 0.5t and to access the topological phases; not fitted to experimental data.
  • helical pairing amplitude Delta_h^p = t
    Chosen by hand to produce a sizable gap in the parent superconductor; remains unrenormalized by the altermagnet throughout.
  • chiral pairing amplitude Delta_c^p = t
    Chosen by hand, same rationale as Delta_h^p.
assumptions (4)
  • standard math Tenfold way classification applies to the BdG Hamiltonians considered
    Used in Table I to assign symmetry class D and Z invariant.
  • domain assumption The altermagnet is described by the d-wave exchange coupling J(k)=2 J_A (cos k_x - cos k_y) s_x
    This form is taken from the altermagnetism literature [2,10,13,44] and is the simplest d-wave order; the spin orientation along x is a modelling choice.
  • ad hoc to paper The superconducting pairing potentials Delta_h and Delta_c are spatially uniform and unaffected by the altermagnet
    The paper does not solve the pairing self-consistently; it assumes the parent p-wave pairing survives unchanged when the TRS-breaking altermagnet is introduced.
  • domain assumption The lattice model with nearest-neighbor hopping and no disorder captures the topological properties
    Numerics are performed on a clean square lattice with OBC/PBC; disorder and other bands are not included.

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Cite this review

Pith. "Pith review of Interplay between altermagnetism and topological superconductivity on an unconventional superconducting platform." pith.science (2026). https://pith.science/paper/C6PGBJNW

@misc{pith2026250105451,
  author       = {Pith},
  title        = {Pith review of: Interplay between altermagnetism and topological superconductivity on an unconventional superconducting platform},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C6PGBJNW}},
  note         = {Machine review of arXiv:2501.05451}
}
abstract

We propose a theoretical model to investigate the interplay between altermagnetism and $p$-wave superconductivity, with a particular focus on topological phase transitions in a two-dimensional (2D) $p$-wave superconductor, considering both chiral and helical phases. Our study reveals that the emergence of helical and chiral Majorana states can be tuned by the amplitude of a $d-$wave altermagnetic order parameter, with the outcome depending on the nature of the superconducting state. In the helical superconductor, such an altermagnet can induce a topological phase transition into a gapless topological superconductor hosting Majorana flat edge modes. On the other hand, in the chiral superconductor, the topological transition takes place between a topologically nontrivial gapped phase and a gapless nodal-line superconductor, where the Bogoliubov quasiparticle bands intersect at an isolated line in momentum space. Remarkably, we show that when such an altermagnet is coupled to a mixed-pairing superconductor, with both chiral and helical components, a hybrid topological phase emerges, featuring dispersive Majorana edge modes that coexist with nearly flat Majorana edge states. Our findings therefore establish a novel platform for controlling and manipulating Majorana modes in unconventional superconductors with vanishing total magnetization.

Figures

Figures reproduced from arXiv: 2501.05451 by the authors.

Figure 1
Figure 1. FIG. 1. The band structure for a helical [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Edge states in the ribbon geometry for a helical [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. FIG. 5. Nearly flat band edge mode driven by altermagnet [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figures from the paper (1 more)
Figure 6
Figure 6. Figure 6: FIG. 6. The bulk gap (∆) as a function of the chemical potential ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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