Pith. sign in

REVIEW 4 major objections 5 minor 47 references

Topological Superconductors in Doubly-Coupled Nanowires with Altermagnetism

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read Doubly coupled nanowires with altermagnetism realize four topological superconducting phases—class D, class BDI, and two two-Majorana-per-end phases protected by spin-group and magnetic point-group symmetries—tunable via phase difference…

desk verdict The D/BDI phase diagrams are solid and useful, but the symmetry-protected two-MZM phases are misclassified as BDI when the declared symmetries put them in DIII, and the invariant calculation is not justified as written. read the letter →

arxiv 2608.13265 v1 pith:B2REGTP6 submitted 2026-08-13 cond-mat.mes-hall

Hongfa Pan , Haoyang Wang , Wenguang Zhu , Zhenhua Qiao This is my paper · ORCID
classification cond-mat.mes-hall
keywords topologicalsuperconductivityMajoranazeromodesaltermagnetismcouplednanowiresclassBDIsuperconductorspin-groupsymmetrymagneticpoint-groupJosephsonphasedifference
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 proposes that two parallel semiconductor nanowires, each proximity-coupled to an s-wave superconductor and an altermagnet, can serve as a single tunable platform for topological superconductivity. It claims that this double-wire geometry hosts four distinct topological superconducting phases: a class D phase with one Majorana zero mode per end, a class BDI phase with one or two Majorana zero modes per end, and two phases each carrying two Majorana zero modes per end, protected respectively by a spin-group symmetry and a magnetic point-group symmetry. The claim matters because altermagnetism supplies exchange splitting without net magnetization or strong external magnetic fields, and the additional wire degree of freedom gives experimental knobs—superconducting phase difference and inter-wire coupling—that single-wire platforms lack. If the paper is right, the platform would offer an electrically controlled route to Majorana zero modes whose number, symmetry, and manipulation can be tuned in situ.

What carries the argument

The load-bearing object is the Bogoliubov–de Gennes Hamiltonian of the two coupled wires from Eqs. (1)–(2): each wire has hopping $t$, chemical potential $\mu$, Rashba coupling $\alpha_n$, altermagnetic exchange $2J_n\cos k_x\,\tau_z\sigma_z$ with $d_{x^2-y^2}$ character, and $s$-wave pairing $\Delta_n e^{i\varphi_n}$, joined by inter-wire hopping $t_0$. The argument is carried by rewriting this Hamiltonian in symmetric and antisymmetric wire combinations, which reduces it to a two-band superconductor whose intraband pairing $\Delta_s=\tfrac12|e^{-i\varphi/2}(\Delta_1+\Delta_2 e^{i\varphi})|$ and interband pairing $\Delta_d=\tfrac12|e^{-i\varphi/2}(\Delta_2 e^{i\varphi}-\Delta_1)|$ compete. Depending on parameter choices, the wire-exchange mirror $M=s_x$ combines with particle-hole $P=\tau_x K$ and time-reversal $T=i\sigma_y K$ to build the effective symmetries $T_3=MK$, $T_1=MT$, and $T_2=\tilde M T$, which place the system in class D or BDI and generate unitary symmetries $U_1,U_2$ that block-diagonalize the Hamiltonian so that integer winding numbers can be computed block by block.

What would settle it

Measure the low-energy spectrum of a double-wire device with $J_1=-J_2$, $\alpha_1=-\alpha_2$, $\Delta_1=\Delta_2=0.3$ in units of $t=1$, and phase bias $\varphi=\pi$: the paper predicts two zero-energy Majorana modes per end across a broad range of $\mu$, with a $4\pi$-periodic evolution of the spectrum in $\varphi$; observing only one zero mode, a $2\pi$-periodic spectrum, or zero modes only at isolated points would show the symmetry-protected phase is not realized.

Watch

Extended reading notes

Core claim

The paper argues that coupling two nanowires enlarges the symmetry toolbox available to altermagnet-based Majorana platforms. When the two wires are identical and $\Delta_1=\Delta_2$, the wire-exchange mirror operation $M=s_x$ combines with particle-hole and time-reversal symmetries to produce an effective time-reversal $T_3=MK$ for any phase difference, restoring the BDI class with a $\mathbb{Z}$ winding number; when $\Delta_1\neq\Delta_2$ and $\varphi\neq m\pi$, the system falls into class D. Setting $J_1=-J_2$ with $\varphi=0$ preserves a spin-group symmetry $T_1=MT$ and gives two $T_1$-related Majorana zero modes per end; setting $J_1=-J_2$, $\alpha_1=-\alpha_2$, and $\varphi=\pi$ preserves a magnetic point-group symmetry $T_2=\tilde M T$ with $\tilde M=s_x\tau_z\sigma_z$, again giving two Majorana modes per end and a spectrum that is $4\pi$-periodic in the phase difference. All four phases occupy broad ranges of chemical potential, altermagnetic exchange, and inter-wire coupling, and the phase difference acts as a continuous control that can relax the required altermagnetism strength.

Load-bearing premise

The whole phase diagram rests on the assumption that a nanowire next to an altermagnet simply inherits a directional exchange coupling of the form $2J_n\cos k_x\,\tau_z\sigma_z$ with no additional orbital reconstruction or suppression of the induced superconductivity, and that the two symmetry-protected phases can meet the exact sign requirements $J_1=-J_2$ and, in one phase, $\alpha_1=-\alpha_2$.

Editorial extensions

If this is right

  • A single double-wire device can realize class D and class BDI topological superconductivity without an external Zeeman field, with altermagnetism supplying the magnetism at zero net magnetization.
  • For $\Delta_1=\Delta_2$, the BDI phase with one Majorana zero mode per end survives at generic $\varphi\neq m\pi$, so the fine tuning to $\varphi=0$ or $\pi$ is not required for that phase.
  • The spin-group protected phase keeps two Majorana zero modes per end even when $\varphi$ is nonzero, while the magnetic point-group phase displays a $4\pi$-periodic spectrum in $\varphi$, a concrete experimental signature.
  • Because the topological phase boundaries depend on $\varphi$ and $t_0$, both quantities can serve as continuous in-situ tuning knobs for entering, leaving, and manipulating topological phases.

Reading between the lines

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

  • Beyond the paper's stated results, the phase-difference knob suggests a materials simplification: devices with modest altermagnetic exchange could still reach the topological regime by tuning $\varphi$, so the search for altermagnet partners need not target only the largest exchange splittings.
  • Likewise, the two-Majorana-per-end phases are natural candidates for symmetry-protected fusion or exchange experiments, since the zero modes are related by exact Hamiltonian symmetries; the paper does not itself demonstrate a braiding protocol, but the symmetry structure invites one.
  • A further testable extension is to transfer the same symmetry logic to arrays or networks of coupled wires, using each junction's phase difference as an independent control, and to check whether the predicted $4\pi$-periodic Josephson signature tracks the sign condition $\alpha_1=-\alpha_2$.
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

4 major / 5 minor

Summary. The paper proposes a tight-binding model of two coupled semiconducting nanowires with Rashba spin-orbit coupling, d_x2-y2 altermagnetic exchange, and s-wave pairing with a phase difference φ. Using a Bogoliubov–de Gennes treatment, the authors identify four topological superconducting phases: a class D phase with one Majorana zero mode (MZM) per end, a class BDI phase with one or two MZMs per end, and two phases with two MZMs per end that are claimed to be protected by spin-group symmetry (J1 = -J2, φ = 0) and by magnetic point-group symmetry (J1 = -J2, α1 = -α2, φ = π). The manuscript provides analytic gap-closing conditions, extensive numerical phase diagrams, and finite-wire spectra showing zero-energy states.

Significance. If the symmetry classification is correct, the paper offers a potentially versatile platform for engineering multi-MZM phases using altermagnets, with the superconducting phase difference and inter-wire coupling as tuning knobs. The analytic gap-closing formulas and the systematic phase diagrams are valuable assets, and the D- and BDI-class parts of the paper are built on a standard and largely convincing BdG analysis. The main weakness is that the construction of the topological invariants for the two symmetry-protected phases is only sketched and the notation in Sec. IV is inconsistent; as a result, the central claim of two protected MZMs per end in those phases is not yet fully substantiated and requires a more careful derivation.

major comments (4)
  1. [Sec. IV, first case (spin-group symmetry)] The symmetry operators are reused inconsistently. In Sec. III B, T1 = K and S1 = P T1 = τx, while in Sec. IV the text redefines T1 as M T (with T = iσyK) and then refers to combining "S1 = P T1" with "ST1 = P T1" to form U1 = S1 ST1. With the redefined T1, S1 and ST1 are identical and U1 is the identity, making the claimed block diagonalization empty; with the old T1, ST1 is not defined. The construction of the integer invariant therefore cannot be reproduced as written. Please give explicit matrix definitions of all symmetry operators used at the parameter values of Sec. IV and verify that U1 commutes with both the Hamiltonian and the chiral symmetry entering the winding-number formula.
  2. [Sec. IV, first case, nonzero φ] For J1 = -J2 with a nonzero phase difference, the complex pairing phases break the symmetry K, leaving T1 = M T as the only antiunitary symmetry discussed. Since T1^2 = -1 while P^2 = +1, the system belongs in class DIII in one dimension, whose topological invariant is Z2, not the BDI integer Z computed from the chiral symmetry S1 = τx. The chiral symmetry τx ceases to be a symmetry when sin φ ≠ 0, so the block-diagonalization argument using S1 and the statement in Fig. 4(b) that a nonzero φ does not destroy the topological phase are not supported by the invariant calculation presented. The authors should either restrict the Z = 2 BDI claim to φ = 0 and separately compute the DIII Z2 invariant for φ ≠ 0, or identify a square-+1 symmetry valid for all φ that keeps an integer winding number meaningful.
  3. [Sec. IV, second case (magnetic point-group symmetry)] The definitions of S2 and ST2 are contradictory. The text defines T2 = \tilde M T (with \tilde M = sx τz σz) and then states that "ST2 = P T2 and S2 = P T2 are chiral symmetries with T2 = τzK". As written, ST2 equals S2 and U2 = ST2 S2 is the identity, so the block diagonalization is trivial. The intended construction apparently uses one chiral symmetry built from \tilde M T and one built from τzK, but this is never stated explicitly. Please provide the explicit operator forms, verify that τzK is indeed a symmetry at J1 = -J2, α1 = -α2, φ = π, and show the commutation relations that make U2 a valid unitary symmetry.
  4. [Sec. IV compared with Sec. III B] The manuscript does not establish how the two Sec. IV phases are distinct from the ordinary BDI Z = 2 phase already reported in Sec. III B. Since a BDI Z = 2 phase already hosts two MZMs per end for J1 = J2, the integer Z = 2 alone does not imply a new symmetry-protected phase. The authors should demonstrate that the two MZMs in the Sec. IV phases are pairwise related by the new antiunitary symmetries and that those symmetries protect the degeneracy, for example by a scattering argument or a classification of the extended symmetry group. Without such a demonstration, the claim of two distinct spin-group and magnetic-point-group protected phases is not fully justified.
minor comments (5)
  1. [Abstract] The abstract contains a typo: "alternating?magnetism" should be "altermagnetism".
  2. [Sec. IV and Table II] The notation T1 and T2 is overloaded: T1 = K in Sec. III B but T1 = M T in Sec. IV, and T2 = τzK in Sec. III B but T2 = \tilde M T in Sec. IV. A table summarizing all symmetry operators and the parameter conditions under which each one is valid would greatly improve readability and eliminate the confusion that currently obscures the invariant calculation.
  3. [Appendix A3] The winding-number formula in Eq. (A14) is standard, but the text does not explain how the total invariant is obtained from the two diagonal blocks in Eq. (A15), nor why summing the block invariants gives the protected Z in the presence of the additional antiunitary symmetries. Please add the missing derivation or a reference that covers this block-diagonalization procedure.
  4. [Sec. II, Eq. (1)] The altermagnetic exchange term J_n cos kx τz σz is assumed as a starting point without a microscopic derivation or a discussion of how it arises from proximitizing a nanowire to an altermagnet, including possible orbital effects or suppression of the induced pairing. A brief discussion of this modeling assumption would help assess experimental feasibility.
  5. [Fig. 4 caption] In the caption for Fig. 4(e), the notation "α1,2 = ±0.3" should explicitly specify which wire has which sign; the text states α1 = -α2, but the caption is ambiguous. Similarly, the captions should state explicitly that the invariant calculations in (c) and (g) are performed at φ = 0 and φ = π, respectively.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the phase diagram follows from an explicit tight-binding Hamiltonian and standard symmetry/winding-number computations.

full rationale

The paper starts from an explicitly stated Hamiltonian (Eqs. 1-2) with altermagnetic exchange, Rashba spin-orbit coupling, pairing phases, and inter-wire coupling taken as inputs. All topological phases are derived from this Hamiltonian: gap-closing conditions at kx=0 and kx=pi are computed analytically (Appendix A2), the BDI invariants are obtained from winding numbers of the block-off-diagonal Hamiltonian (Appendix A3), and the spin-group and magnetic point-group phases are obtained by imposing explicit parameter conditions (J1=-J2, phi=0 or phi=pi, alpha1=-alpha2). These conditions are design choices stated in the text, not quantities fitted to the claimed outputs. There is no fitting of parameters to a subset of data that is then renamed a prediction, and no invariant is set equal to its own input. The citations, including Ref. [24] for the dx2-y2 altermagnetic exchange form and Ref. [40] for BDI nanowire symmetry, are external background/model assumptions rather than self-citation chains that carry the derivation. The overloaded use of T1/T2 in Secs. III B and IV and the possible DIII-versus-BDI classification issue in Sec. IV are correctness or consistency concerns, not circularity: they do not make any claimed result equivalent to its inputs by construction. Therefore the derivation chain is self-contained and the circularity score is 0.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The paper is a theory proposal using a standard tight-binding BdG model. There are no new particles, forces, or fields. The free parameters are model inputs chosen to map phase diagrams, not fitted to experimental data. The axioms are standard classification theorems and the domain assumption that the altermagnetic proximity effect has the stated 1D form.

free parameters (7)
  • chemical potential μ = varied, e.g., -2 in Fig. 2b
    Model parameter, not fitted to data; varied to map the phase diagram.
  • nearest-neighbor hopping t = set to 1
    Energy scale; set to 1 in all calculations.
  • Rashba spin-orbit coupling α1, α2 = 0.35 or 0.3 in figures; α1 = ±α2 for symmetric phases
    Model parameter; the magnetic point-group phase requires α1 = -α2.
  • altermagnetic exchange coupling J1, J2 = 0.2 to 0.4 in figures; J1 = -J2 for spin-group phase
    Model parameter; the two-MZM phases require opposite signs.
  • superconducting pairing Δ1, Δ2 = 0.3 and 0.2 in Fig. 3; 0.3 in Fig. 4
    Model parameters; the D-class phase requires Δ1 ≠ Δ2.
  • inter-wire coupling t0 = 0.2 to 0.25
    Model parameter; controls inter-wire coupling and is a tuning knob.
  • superconducting phase difference φ = 0, π, 0.85π in figures
    Model parameter; BDI phases occur at φ = mπ or when Δ1=Δ2.
assumptions (4)
  • domain assumption The tight-binding BdG model captures proximity-induced altermagnetism and superconductivity in the nanowires.
    The altermagnetic term HJn = Jn Σ (c†_{j+1} σz c_j + H.c.) is assumed to describe a 1D wire with dx2-y2 altermagnetic exchange. This is an effective model, not derived from a specific material heterostructure. In Sec II, Eq. (1).
  • standard math The Altland-Zirnbauer classification applies to the quasi-1D BdG Hamiltonian.
    The classification of class D and BDI TSCs in 1D is well established (refs 38-41) and used without modification.
  • standard math The bulk-boundary correspondence holds: zero-energy states in finite chains indicate Majorana zero modes counted by the winding number.
    Standard in the field; used to interpret the numerical spectra in Figs. 2-4.
  • domain assumption The nanowires are clean, identical except for the explicitly tuned parameters, and long enough (Nw=100) that finite-size effects are negligible.
    The model ignores disorder, orbital magnetic fields, and possible suppression of superconductivity by the altermagnet. This is stated implicitly through the clean tight-binding model.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Topological Superconductors in Doubly-Coupled Nanowires with Altermagnetism." pith.science (2026). https://pith.science/paper/B2REGTP6

@misc{pith2026260813265,
  author       = {Pith},
  title        = {Pith review of: Topological Superconductors in Doubly-Coupled Nanowires with Altermagnetism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B2REGTP6}},
  note         = {Machine review of arXiv:2608.13265}
}
read the original abstract

We theoretically investigate the possibility of engineering topological superconductivity in doubly coupled nanowires integrating proximity-induced superconductivity and altermagnetism. By tuning experimentally accessible parameters, we find four distinct topological superconducting phases: class D, class BDI, and two phases hosting two Majorana zero modes per end, protected respectively by spin-group and magnetic point-group symmetries. Beyond inheriting the advantages of alternating?magnetism induced topological superconductivity, our system provides multiple tuning knobs (e.g., superconducting phase difference and inter-wire coupling) to control topological properties.

Figures

Figures reproduced from arXiv: 2608.13265 by the authors.

Figure 2
Figure 2. FIG. 2: (a) Energy eigenvalues as a function of chemical [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (a)-(d) BDI TSCs with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4: (a)-(d) spin-group symmetry protected TSCs: (a) Ener [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 40 canonical work pages

  1. [1]

    Model in the D class and BDI class TSCs For both D class and BDI class TSCs, the Hamilto- nian Hk can be transformed to a new Hamiltonian Hd expressed in the basis {ψ† d1,ψ d1,ψ † d2,ψ d2}, where ψ† d1 = 1 √ 2 {c† 2,↑ −c† 1,↑,c † 2,↓ −c† 1,↓}, ψ† d2 = 1 √ 2 {c† 2,↑ +c† 1,↑,c † 2,↓ +c† 1,↓} (A1) The Hamiltonian Hd can be denoted by Hd(k) = ( H1 H∆12 H † ∆1...

    work page

  2. [2]

    Conditions of D class TSCs The conditions for the topologically nontrivial phase are determined by the closing of the bulk energy gap in the Hamiltonian Hk at kx = 0 or π [ 3, 24]. At kx = 0, the critical exchange coupling for gap clo- sure is given by Jc1 = √ m0 ± √n0 2 √ 2 , (A11) withm0 = 2(2t +µ)2 + ∆2 1 + ∆2 2 + 2t2 0 andn0 = 16t2 0(2t + µ)2 + (∆2 1 ...

    work page

  3. [3]

    In the basis where S0 is diagonal, H0 takes the block-off- diagonal form: H(k) = ( 0 A(k) AT (−k) 0 )

    Calculation of the topological invariant A chiral symmetry S0 is defined as {S0,H 0} = 0. In the basis where S0 is diagonal, H0 takes the block-off- diagonal form: H(k) = ( 0 A(k) AT (−k) 0 ) . (A13) The winding number W is given by W = −i 2π ∫ π −π dz(k) z(k) , (A14) where the complex function z(k) = det[A(k)]/| det[A(k)]|. For TSCs protected by spin-group...

    work page

  4. [4]

    Nayak, S

    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008)

    2008

  5. [5]

    Alicea, Y

    J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Non-Abelian statistics and topological quantum information processing in 1D wire networks , Nat. Phys. 7, 412 (2011)

    work page 2011

  6. [6]

    A. Y. Kitaev, Unpaired Majorana fermions in quantum 8 wires, Phys. Usp. 44, 131 (2001)

    work page 2001

  7. [7]

    Read and D

    N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time- reversal symmetries and the fractional quantum Hall ef- fect, Phys. Rev. B 61, 10267 (2000)

    work page 2000

  8. [8]

    Sato and Y

    M. Sato and Y. Ando, Topological superconductors: a review, Rep. Prog. Phys. 80, 076501 (2017)

    2017

Show all 47 references
  1. [9]

    Gao, Large-scale, ordered and tunable Majorana- zero-mode lattice in iron-based superconductors , Coshare Science 04, 03 (2026)

    H. Gao, Large-scale, ordered and tunable Majorana- zero-mode lattice in iron-based superconductors , Coshare Science 04, 03 (2026)

    2026

  2. [10]

    Sato and S

    M. Sato and S. Fujimoto, Majorana Fermions and Topology in Superconductors , J. Phys. Soc. Jpn. 85, 072001 (2016)

    2016

  3. [11]

    Mourik, K

    V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Signatures of Ma- jorana fermions in hybrid superconductor-semiconductor nanowire devices, Science 336, 1003 (2012)

    2012

  4. [12]

    Kayyalha, D

    M. Kayyalha, D. Xiao, R. Zhang, J. Shin, J. Jiang, F. Wang, Y.-F. Zhao, R. Xiao, L. Zhang, K. M. Fi- jalkowski, P. Mandal, M. Winnerlein, C. Gould, Q. Li, L. W. Molenkamp, M. H. W. Chan, N. Samarth, and C.-Z. Chang, Absence of evidence for chiral Majorana modes in quantum anom...

    2020

  5. [13]

    L. P. Rokhinson, X. Liu, and J. K. Furdyna, The fractional a.c. Josephson effect in a semiconductor- superconductor nanowire as a signature of Majorana par- ticles, Nat. Phys. 8, 795 (2012)

    2012

  6. [14]

    Rainis and D

    D. Rainis and D. Loss, Topological superconductivity and Majorana fermions in one-dimensional systems , Phys. Rev. B 85, 174533 (2012)

    2012

  7. [15]

    Y. Oreg, G. Refael, and F. von Oppen, Helical Liq- uids and Majorana Bound States in Quantum Wires , Phys. Rev. Lett. 105, 177002 (2010)

    2010

  8. [16]

    R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Majorana Fermions and a Topological Phase Transi- tion in Semiconductor-Superconductor Heterostructures , Phys. Rev. Lett. 105, 077001 (2010)

    2010

  9. [17]

    A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikma, Zero-bias peaks and splitting in an Al–InAs nanowire topological superconductor as a signature of Majorana fermions , Nat. Phys. 8, 887 (2012)

    2012

  10. [18]

    H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T. Deng, P. Caroff, H. Q. Xu, and C. M. Marcus, Superconductor-nanowire devices from tunneling to the multichannel regime: Zero-bias oscillations and magnetoconductance crossover , Phys. Rev. B 87, 241401(R) (2013)

    2013

  11. [19]

    H. Pan, W. S. Cole, J. D. Sau, and S. Das Sarma, Generic quantized zero-bias conductance peaks in superconductor-semiconductor hybrid structures , Phys. Rev. B 101, 024506 (2020)

    2020

  12. [20]

    Das Sarma, In search of Majorana , Nat

    S. Das Sarma, In search of Majorana , Nat. Phys. 19, 165 (2023)

    2023

  13. [21]

    P. Yu, J. Shen, M. Gomanko, G. Badawy, E. P. A. M. Bakkers, K. Zuo, V. Mourik, and S. M. Frolov,Nat. Phys. 17, 482 (2021)

    2021

  14. [22]

    H.-Y. Ma, M. Hu, N. Li, J. Liu, W. Yao, J.-F. Jia, and J. Liu, Multifunctional antiferromagnetic materials with giant piezomagnetism and noncollinear spin current , Nat. Commun. 12, 2846 (2021)

    2021

  15. [23]

    M. Hu, X. Cheng, Z. Huang, and J. Liu, Catalog of C-Paired Spin-Momentum Locking in Antiferromagnetic Systems, Phys. Rev. X 15, 021083 (2025)

    2025

  16. [24]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Emerging research landscape of altermagnetism , Phys. Rev. X 12, 040501 (2022)

    2022

  17. [25]

    Zhang, X

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

    2025

  18. [26]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Beyond conven- tional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry , Phys. Rev. X 12, 031042 (2022)

    2022

  19. [27]

    S. A. A. Ghorashi, T. L. Hughes, and E. Rossi, Alter- magnetic Routes to Majorana Modes in Zero Net Mag- netization, Phys. Rev. Lett. 133, 106601 (2024)

    2024

  20. [28]

    Gaidamauskas, J

    E. Gaidamauskas, J. Paaske, and K. Flens- berg, Majorana Bound States in Two-Channel Time-Reversal-Symmetric Nanowire Systems , Phys. Rev. Lett. 112, 126402 (2014)

    2014

  21. [29]

    Zhang, C

    F. Zhang, C. L. Kane, and E. J. Mele, Time-Reversal- Invariant Topological Superconductivity and Majorana Kramers Pairs , Phys. Rev. Lett. 111, 056402 (2013)

    2013

  22. [30]

    C. L. M. Wong and K. T. Law, Majorana Kramers dou- blets in dx2−y2 -wave superconductors with Rashba spin- orbit coupling , Phys. Rev. B 86, 184516 (2012)

    2012

  23. [31]

    Keselman, L

    A. Keselman, L. Fu, A. Stern, and E. Berg, Induc- ing Time-Reversal-Invariant Topological Superconductiv- ity and Fermion Parity Pumping in Quantum Wires , Phys. Rev. Lett. 111, 116402 (2013)

    2013

  24. [32]

    Liu, C.-Y

    T.-T. Liu, C.-Y. Wan, H. Yang, Y.-J. Zhao, B.-J. Xie, W.-Y. Zheng, Z.-X. Yi, D.-D. Guan, S.-Y. Wang, H. Zheng, C.-H. Liu, L. Fu, J.-W. Liu, Y.-Y. Li, and J.- F. Jia, Signatures of hybridization of multiple Majorana zero modes in a vortex , Nature 633 71 (2024)

    2024

  25. [33]

    Kobayashi and A

    S. Kobayashi and A. Furusaki, Double Majorana vortex zero modes in superconducting topological crystalline insulators with surface rotation anomaly , Phys. Rev. B 102, 180505 (2020)

    2020

  26. [34]

    C. Fang, M. J. Gilbert, and B. A. Bernevig, New Class of Topological Superconductors Protected by Magnetic Group Symmetries , Phys. Rev. Lett. 112, 106401 (2014)

    2014

  27. [35]

    Zehetmayer, A review of two-band supercon- ductivity: materials and effects on the thermo- dynamic and reversible mixed-state properties , Supercond

    M. Zehetmayer, A review of two-band supercon- ductivity: materials and effects on the thermo- dynamic and reversible mixed-state properties , Supercond. Sci. Technol. 26 043001 (2013)

    2013

  28. [36]

    Salamone, H

    T. Salamone, H. G. Hugdal, S. H. Jacobsen, and M. Amundsen, High magnetic field su- perconductivity in a two-band superconductor , Phys. Rev. B 107, 174516 (2023)

    2023

  29. [37]

    Alshemi, E

    A. Alshemi, E. M. Forgan , A. Hiess, R. Cubitt, J. S. White, K. Schmalzl, and E. Blackburn, Two Character- istic Contributions to the Superconducting State of 2H- N bSe2, Phys. Rev. Lett. 134, 116001 (2025)

    2025

  30. [38]

    Benfenati, A

    A. Benfenati, A. Samoilenka , and E. Babaev, Boundary effects in two-band superconductors , Phys. Rev. B 103, 144512 (2021)

    2021

  31. [39]

    M. I. Tsindlekht, G. I. Leviev, I. Asulin, A. Sha- roni, O. Millo, I. Felner, Y. B. Paderno, V. B. Filip- pov, and M. A. Belogolovskii, Tunneling and magnetic characteristics of superconducting ZrB 12 single crystals , Phys. Rev. B 69, 212508 (2004) . 9

    2004

  32. [40]

    P. K. Biswas, F. N. Rybakov, R. P. Singh, S. Mukher- jee, N. Parzyk, G. Balakrishnan, M. R. Lees, C. D. De- whurst, E. Babaev, A. D. Hillier, and D. M. Paul, Coex- istence of type-I andtype-II superconductivity signature s in ZrB 12 probed by muonspin rotation measurements , P...

    2020

  33. [41]

    A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insula- tors and superconductors in three spatial dimensions , Phys. Rev. B 78, 195125 (2008)

    2008

  34. [42]

    Kitaev, Periodic table for topological insulators and superconductors, AIP Conf

    A. Kitaev, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134, 22 (2009)

    2009

  35. [43]

    Tewari and J

    S. Tewari and J. D. Sau, Topological Invariants for Spin-Orbit Coupled Superconductor Nanowires , Phys. Rev. Lett. 109, 150408 (2012)

    2012

  36. [44]

    S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, Topological insulators and super- conductors: tenfold way and dimensional hierarchy , New J. Phys. 12, 065010 (2010)

    2010

  37. [45]

    Pientka, A

    F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern, and B. I. Halperin, Topological Su- perconductivity in a Planar Josephson Junction , Phys. Rev. X 7, 021032 (2017)

    2017

  38. [46]

    Liu, J.-Y

    P.-F. Liu, J.-Y. Li, J.-Z. Han, X.-G Wan, and Q.-H. Liu, Spin-Group Symmetry in Magnetic Materials with Negligible Spin-Orbit Coupling , Phys. Rev. X 12, 021016 (2022)

    2022

  39. [47]

    ˇSmejkal, A

    L. ˇSmejkal, A. B. Hellenes, R. Gonz´ alez-Hern´ andez, J. Sinova, and T. Jungwirth, Giant and Tunneling Mag- netoresistance in Unconventional Collinear Antiferro- magnets with Nonrelativistic Spin-Momentum Coupling , Phys. Rev. X 12, 021016 (2022)

    2022

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.