REVIEW 4 major objections 7 minor 45 references
Topological Majorana zero modes and the superconducting diode effect driven by Fulde-Ferrell-Larkin-Ovchinnikov pairing in a helical Shiba chain
T0 review · 4 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read A helical Shiba chain under an out-of-plane field develops finite-momentum FFLO pairing that supports Majorana zero modes and makes the supercurrent non-reciprocal, yielding a superconducting diode effect.
desk verdict Self-consistent FFLO in a helical Shiba chain is an interesting proposal, but the proximity-effect assumption and a missing q0-vs-topology cross-check make the quantitative claims provisional. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The argument runs through a self-consistent Bogoliubov-de Gennes (BdG) mean-field calculation. A unitary transformation removes the position dependence of the spin spiral and converts it into an effective spin-orbit coupling plus exchange field, while the out-of-plane Zeeman field shifts the BdG bands and favors pairing at finite momentum. The FFLO order parameter $\Delta(q)$ is obtained by minimizing the condensation energy $\Omega(q,\Delta)$, and the equilibrium momentum $q_0$ is fixed by the stationarity condition $\delta\Omega/\delta q=0$. The topological signature is the bulk dipole moment $P_x$, which equals $0.5$ in the Majorana phase, and the diode effect appears as asymmetry in the supercurrent $j(q)=-2e\,\partial\Omega/\partial q$.
What would settle it
Measure the opposite-direction critical currents of a helical Shiba chain under an out-of-plane field: equal magnitudes for non-zero field would contradict the predicted diode effect. A more direct test is to resolve the Cooper pair momentum $q_0$ (for example through the field-tuned modulation of the gap in a Josephson junction) and check for the linear dependence on the out-of-plane field.
Extended reading notes
Core claim
The central claim is that a helical Shiba chain with an out-of-plane magnetic field hosts an intrinsic FFLO superconducting ground state with equilibrium Cooper pair momentum $q_0$, and that this state simultaneously presents topological Majorana zero modes and a superconducting diode effect. The authors derive a self-consistent FFLO gap $\Delta(q)$, find the ground-state momentum by minimizing the condensation energy, and show in a finite lattice that zero-energy end states appear with bulk dipole polarization $P_x=0.5$. They further show the supercurrent $j(q)$ loses its symmetry under $q\to -q$ when the field is on, giving unequal critical currents and a diode efficiency up to about 60 percent.
Load-bearing premise
The assumption that the pairing induced in the one-dimensional Shiba chain obeys the same self-consistency condition as the bulk superconductor, so that the FFLO gap and its ground-state momentum are well defined; if that fails, the finite $q_0$ and the diode effect could disappear.
Editorial extensions
If this is right
- A helical Shiba chain under an out-of-plane field is predicted to be simultaneously a topological superconductor and an intrinsic superconducting diode, so both functionalities come from the same finite Cooper pair momentum $q_0$.
- The equilibrium momentum $q_0$ grows linearly with the out-of-plane field, giving a field-tunable knob for the diode asymmetry.
- For trivial spin textures (ferromagnetic or antiferromagnetic, $g=0$ or $\pi$), no FFLO pairing and no diode effect appear, so the helical texture is essential.
- The diode efficiency peaks when the renormalized chemical potential vanishes ($\mu\sim g^2/2$), matching expectations for finite-momentum superconductors.
Reading between the lines
- If the proximity-induced gap follows bulk self-consistency, the finite $q_0$ should also be visible as a field-tunable phase modulation of the gap along the chain, which a scanning Josephson probe could test.
- The same mechanism likely applies to other one-dimensional proximitized systems with non-collinear magnetic order, suggesting a general route to diode behavior without Rashba spin-orbit coupling.
- Because the Majorana modes and the diode effect both derive from the same $q_0$, a device that tunes the field to optimize the diode efficiency may simultaneously be tuning the topological gap; the two effects may not be independently optimizable.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript studies a one-dimensional helical Shiba chain deposited on an s-wave superconductor, with an out-of-plane Zeeman field, and analyzes it with a self-consistent BdG mean-field approach in which the attractive Hubbard interaction is decoupled in the s-wave FFLO channel. The authors derive a finite equilibrium Cooper-pair momentum q0 from the minimization of the condensation energy, show that the resulting FFLO state supports topological Majorana zero modes (characterized by bulk dipole moment Px = 0.5), and demonstrate nonreciprocal critical currents with a diode efficiency up to about 60%. The central claims are that the FFLO pairing is intrinsic to this setup when Bz is nonzero and that it simultaneously yields topological MZMs and an intrinsic superconducting diode effect. Results are presented for both a continuum model (Eqs. (1)-(4)) and a lattice model with open boundary conditions (Eq. (5)), with the uniform-gap comparison relegated to the Supplemental Material.
Significance. If the central assumption is accepted, the paper offers a coherent and interesting proposal that connects finite-momentum FFLO pairing, topological MZMs, and nonreciprocal critical currents in a single experimentally relevant platform. The explicit symmetry analysis, the self-consistent treatment in both continuum and lattice formulations, the use of the bulk dipole moment Px as the topological invariant, and the SM comparison with a constant superconducting gap are all strengths. The reported diode efficiencies are high, which makes the system potentially attractive for applications. However, the validity of the load-bearing assumption about proximity-induced pairing, and the connection between the equilibrium q0 and the topological regime, are not yet demonstrated; the significance is therefore conditional on these points being resolved.
major comments (4)
- [Model hamiltonian and mechanism of FFLO pairing, after Eq. (1)] The derivation of the finite equilibrium momentum q0 via Eq. (4) assumes that the proximity-induced pairing in the 1D Shiba chain obeys the same self-consistency condition as the bulk superconductor. This is stated as a 'zeroth order approximation', but it is load-bearing: the diode effect computed from Eq. (7) and shown in Fig. 4 is generated by the q0 obtained from minimizing the chain's condensation energy alone, while the parent s-wave superconductor would favor q=0 in equilibrium absent an applied supercurrent. The paper should either justify this approximation quantitatively (for example, by estimating the parent-SC gradient and condensation energy cost against the chain's FFLO gain) or model the proximitized system including the parent superconductor. Without such a justification, q0 is effectively an input rather than a derived equilibrium property, and the SDE claim is not established.
- [Signatures of topologically protected MZMs, Fig. 3] The topological calculation treats q as an independent parameter, but the paper never reports the equilibrium q0 obtained from Eq. (12) for the lattice parameters used in Fig. 3 (Bz/Δ0 = 0.3, J/Δ0 = 0.65, U = 2.78 meV, t = 0.5), nor does it overlay q0 on the minigap and Px phase diagrams in Figs. 3(c,d). Without this overlay, the claim that the FFLO ground state supports MZMs is incomplete: the topological regime might occur at q values that are not the equilibrium ones. The continuum q0 from Fig. 2 cannot be used because the parameter sets differ (for example, U is 0.358 meV in Fig. 2 but 2.78 meV in Fig. 3), so the equilibrium state is not shown to be topological even within the assumed self-consistency.
- [Topological characterization and SM S1] The lattice self-consistent calculation is presented for a single 400-site chain with no finite-size scaling or convergence analysis. Since Px is computed from the occupied eigenstates of the BdG Hamiltonian at fixed L, a demonstration that Px = 0.5 persists with increasing L, and that the zero-energy level splitting decays systematically (ideally exponentially), is needed to support the topological MZM claim. As written, the 'topological phase' could be affected by finite-size artifacts or by the details of the self-consistency iteration, which is not described.
- [Realizing non-reciprocal charge transport, Fig. 4] The diode efficiency η in Fig. 4(b) is a headline quantitative result (up to about 60%) but is computed from the continuum free energy, while the lattice model in SM S1 uses different parameters and is presented only as qualitative agreement for q0. Since the SDE is a central claim, the paper should either provide the lattice-model η for the same parameter regime or explain why the continuum result is quantitatively reliable. As it stands, the 60% efficiency is not backed by the same level of numerical evidence as the topological signatures.
minor comments (7)
- [Section heading] The heading 'Signatures of toplogically protected MZMs' contains a typo; it should be 'topologically'.
- [Before Eq. (7)] The phrase 'computed form the condensation energy' should read 'computed from the condensation energy'.
- [Fig. 2 caption] Panel (b) shows q0 for 'various values of J', but the J/Δ0 values are not listed; please add them to the caption.
- [Reference [42]] The Supplemental Material reference is left as 'XXXXXXXXXXX'; the actual DOI or arXiv link should be provided.
- [After Eq. (6)] The text uses 'px = 0.5' with a lower-case p; this should be Px to match the definition in Eq. (6).
- [Fig. 4(a) caption] The normalization 'j0 ≡ jc(Bz = 0, J/Δ0 = 0.4)' is confusing because panel (a) is computed with J/Δ0 = 4; please clarify.
- [SM S2 and main text] Several typographical errors remain, including 'fucntion' and 'undergoes' in the SM and 'parammeters' in the main text Summary; a careful proofread is needed.
Circularity Check
No significant circularity: q0, MZMs, and diode efficiency are computed from the stated mean-field Hamiltonian, not imposed as inputs.
full rationale
The derivation chain is self-contained. The FFLO order parameter is introduced as the s-wave ansatz Δ(x)=Δe^{iqx} in Eq. (1), but the ground-state momentum q0 is not fixed by that ansatz; it is obtained by minimizing the computed free-energy density Ω(q,Δ) via Eq. (4), and Fig. 2(b) reports a finite q0 only for nonzero Bz. The topological claim is computed by diagonalizing the lattice BdG Hamiltonian Eq. (5) under OBC and evaluating the bulk dipole moment Px, Eq. (6); the Px=0.5 region in Fig. 3(d) is a calculated phase boundary, not an input. The SDE is likewise derived: j(q) is the derivative −2e ∂Ω/∂q of the same free energy (Eq. 7), and η is defined by Eq. (8) from the computed unequal critical currents; the asymmetry j(q)≠−j(−q) appears only when the self-consistent Δ(q) becomes asymmetric under Bz (Fig. 2a). No parameter is fitted to the target result and then renamed as a prediction. The paper's own self-consistency assumption for the proximity-induced pairing (stated after Eq. 1 as "as a zeroth order approximation, we assume that the self-consistency condition applies similarly") is a modeling assumption with cited justification [31,37], not a restatement of the claimed MZM or diode results; it is a physical fragility but not circular. The self-citation [37] is used for the mean-field/proximity framework, but the quantitative results (q0, Px, η) are computed in the present work. A remaining gap is that the equilibrium q0 is not overlaid on the Px=0.5 region, so the paper does not explicitly demonstrate that the FFLO ground state lies in the topological phase; this is an omitted verification, not circularity.
Assumptions & free parameters
free parameters (8)
- U (attractive Hubbard interaction) =
0.358 meV, 2.65 meV, or 2.78 meV depending on figure
- Delta0 (BCS gap scale) =
1 meV (set for entire analysis)
- t (hopping amplitude) =
0.5 Delta0 in the lattice model
- mu (chemical potential) =
1.0 Delta0 in most plots; varied in Fig. 4(c)
- J (exchange coupling between electron spin and magnetic moments) =
varied from 0.4 to 4 Delta0
- Bz (out-of-plane Zeeman field) =
varied from 0 to about 0.6 Delta0
- g (spin rotation angle between adjacent moments) =
pi/2 or 2pi/3
- beta^-1 (temperature) =
0.01 meV or 0.1 meV
assumptions (6)
- domain assumption BdG mean-field decoupling of the attractive Hubbard interaction in the s-wave FFLO pairing channel
- ad hoc to paper The bulk superconductor's self-consistency condition applies to the proximity-induced pairing in the 1D Shiba chain
- domain assumption The magnetic moments are classical spins with a fixed spin-spiral texture
- ad hoc to paper Precession of the classical spins under Bz is negligible
- domain assumption A uniform FFLO order parameter with a single wavevector q
- standard math Thermodynamic minimization of the free energy density in the grand canonical ensemble
Cite this review
Pith. "Pith review of Topological Majorana zero modes and the superconducting diode effect driven by Fulde-Ferrell-Larkin-Ovchinnikov pairing in a helical Shiba chain." pith.science (2026). https://pith.science/paper/A3TZVT3P
@misc{pith2026241211784,
author = {Pith},
title = {Pith review of: Topological Majorana zero modes and the superconducting diode effect driven by Fulde-Ferrell-Larkin-Ovchinnikov pairing in a helical Shiba chain},
year = {2026},
howpublished = {\url{https://pith.science/paper/A3TZVT3P}},
note = {Machine review of arXiv:2412.11784}
}
abstract
We propose a theoretical framework for the realization of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing in a helical Shiba chain subjected to an out-of-plane Zeeman field, analyzed through a self-consistent Bogoliubov-de-Gennes (BdG) mean-field formalism approach. A chain of magnetic adatoms with helical spin texture deposited on the surface of a common $s$-wave superconductor, has emerged as a pivotal platform for realizing topological Majorana zero modes (MZMs). Our study reveals the crucial role of finite momentum pairing of Cooper pairs in the form of FFLO state which also supports topological MZMs at the ends of the chain. Interestingly, we demonstrate that FFLO pairing facilitates non-reciprocal charge transport, giving rise to superconducting diode effect in our system where both time-reversal and inversion symmetries are broken. Such diode effect stems directly from the presence of finite Cooper pair momentum of the FFLO ground state. Our comprehensive analysis highlights the intricate interplay between the richness of helical Shiba chain, the out-of-plane Zeeman field, and FFLO pairing in the emergence of MZMs and driving the superconducting diode effect. These findings offer valuable insights into the design and realization of topological superconducting devices with diode-like properties, potentially advancing technological applications in quantum computing and superconducting electronics.
Figures
Figures from the paper (5 more)
Reference graph
Works this paper leans on
-
[1]
Unpaired majorana fermions in quan- tumwires,
A Yu Kitaev, “Unpaired majorana fermions in quan- tumwires,” Physics-Uspekhi 44, 131 (2001)
work page 2001
-
[2]
Introduction to topological superconductivity and majorana fermions,
Martin Leijnse and Karsten Flensberg, “Introduction to topological superconductivity and majorana fermions,” Semiconductor Science and Technology 27, 124003 (2012)
work page 2012
-
[3]
Topological in- sulators and superconductors,
Xiao-Liang Qi and Shou-Cheng Zhang, “Topological in- sulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011)
work page 2011
-
[4]
New directions in the pursuit of majorana fermions in solid state systems,
Jason Alicea, “New directions in the pursuit of majorana fermions in solid state systems,” Reports on Progress in Physics 75, 076501 (2012)
2012
-
[5]
Helical liquids and majorana bound states in quantum wires,
Yuval Oreg, Gil Refael, and Felix von Oppen, “Helical liquids and majorana bound states in quantum wires,” Phys. Rev. Lett. 105, 177002 (2010)
work page 2010
-
[6]
Roman M. Lutchyn, Jay D. Sau, and S. Das Sarma, “Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures,” Phys. Rev. Lett. 105, 077001 (2010)
work page 2010
-
[7]
Majorana fermions in superconduc- tors,
C.W.J. Beenakker, “Majorana fermions in superconduc- tors,” Annual Review of Condensed Matter Physics 4, 113–136 (2013)
work page 2013
-
[8]
Non-abelian statistics of half-quantum vortices in p-wave superconductors,
D. A. Ivanov, “Non-abelian statistics of half-quantum vortices in p-wave superconductors,” Phys. Rev. Lett.86, 268–271 (2001)
2001
Show all 45 references
-
[9]
Fault-tolerant quantum computation by anyons,
A.Yu. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics 303, 2–30 (2003)
2003
-
[10]
Non-abelian states of matter,
Ady Stern, “Non-abelian states of matter,” Nature 464, 187–193 (2010)
2010
-
[11]
Non-abelian anyons and topological quantum computation,
Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008)
2008
-
[12]
Periodic table for topological insula- tors and superconductors,
Alexei Kitaev, “Periodic table for topological insula- tors and superconductors,” AIP Conference Proceedings 1134, 22–30 (2009)
2009
-
[13]
Majorana quasiparticles in condensed matter,
Ram´ on Aguado, “Majorana quasiparticles in condensed matter,” La Rivista del Nuovo Cimento 40, 523–593 (2017)
2017
-
[14]
Sig- natures of majorana fermions in hybrid superconductor- semiconductor nanowire devices,
V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, “Sig- natures of majorana fermions in hybrid superconductor- semiconductor nanowire devices,” Science 336, 1003– 1007 (2012)
2012
-
[15]
Proposal for realizing majorana fermions in chains of magnetic atoms on a superconductor,
S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and Ali Yazdani, “Proposal for realizing majorana fermions in chains of magnetic atoms on a superconductor,” Phys. Rev. B 88, 020407 (2013)
2013
-
[16]
Topological superconducting phase in helical shiba chains,
Falko Pientka, Leonid I. Glazman, and Felix von Op- pen, “Topological superconducting phase in helical shiba chains,” Phys. Rev. B 88, 155420 (2013)
2013
-
[17]
Topological superconductivity and majo- rana fermions in rkky systems,
Jelena Klinovaja, Peter Stano, Ali Yazdani, and Daniel Loss, “Topological superconductivity and majo- rana fermions in rkky systems,” Phys. Rev. Lett. 111, 186805 (2013)
2013
-
[18]
Topological super- conductivity and high chern numbers in 2d ferromagnetic shiba lattices,
Joel R¨ ontynen and Teemu Ojanen, “Topological super- conductivity and high chern numbers in 2d ferromagnetic shiba lattices,” Phys. Rev. Lett. 114, 236803 (2015)
2015
-
[19]
Prevalence of trivial zero-energy subgap states in nonuniform helical spin chains on the surface of superconductors,
Richard Hess, Henry F. Legg, Daniel Loss, and Je- lena Klinovaja, “Prevalence of trivial zero-energy subgap states in nonuniform helical spin chains on the surface of superconductors,” Phys. Rev. B 106, 104503 (2022)
2022
-
[20]
Observation of majo- rana fermions in ferromagnetic atomic chains on a super- conductor,
Stevan Nadj-Perge, Ilya K. Drozdov, Jian Li, Hua Chen, Sangjun Jeon, Jungpil Seo, Allan H. MacDonald, B. An- drei Bernevig, and Ali Yazdani, “Observation of majo- rana fermions in ferromagnetic atomic chains on a super- conductor,” Science 346, 602–607 (2014)
2014
-
[21]
To- ward tailoring majorana bound states in artificially con- structed magnetic atom chains on elemental supercon- ductors,
Howon Kim, Alexandra Palacio-Morales, Thore Posske, Levente R´ ozsa, Kriszti´ an Palot´ as, L´ aszl´ o Szunyogh, Michael Thorwart, and Roland Wiesendanger, “To- ward tailoring majorana bound states in artificially con- structed magnetic atom chains on elemental supercon- ducto...
2018
-
[22]
Topological shiba bands in ar- tificial spin chains on superconductors,
Lucas Schneider, Philip Beck, Thore Posske, Daniel Crawford, Eric Mascot, Stephan Rachel, Roland Wiesen- danger, and Jens Wiebe, “Topological shiba bands in ar- tificial spin chains on superconductors,” Nature Physics 17, 943–948 (2021)
2021
-
[23]
Majo- rana modes with side features in magnet-superconductor hybrid systems,
Daniel Crawford, Eric Mascot, Makoto Shimizu, Philip Beck, Jens Wiebe, Roland Wiesendanger, Harald O. Jeschke, Dirk K. Morr, and Stephan Rachel, “Majo- rana modes with side features in magnet-superconductor hybrid systems,” npj Quantum Materials 7, 117 (2022)
2022
-
[24]
Precursors of majorana modes and their length-dependent energy oscillations probed at both ends of atomic shiba chains,
Lucas Schneider, Philip Beck, Jannis Neuhaus- Steinmetz, Levente R´ ozsa, Thore Posske, Jens Wiebe, and Roland Wiesendanger, “Precursors of majorana modes and their length-dependent energy oscillations probed at both ends of atomic shiba chains,” Nature Nanotechnology 17, 384–...
2022
-
[25]
Spin-orbit coupling induced splitting of yu-shiba-rusinov states in antiferromagnetic dimers,
Philip Beck, Lucas Schneider, Levente R´ ozsa, Kriszti´ an Palot´ as, Andr´ as L´ aszl´ offy, L´ aszl´ o Szunyogh, Jens Wiebe, and Roland Wiesendanger, “Spin-orbit coupling induced splitting of yu-shiba-rusinov states in antiferromagnetic dimers,” Nature Communications 12, 2040 (2021)
2021
-
[26]
Superconductivity in a strong spin-exchange field,
Peter Fulde and Richard A. Ferrell, “Superconductivity in a strong spin-exchange field,” Phys. Rev. 135, A550– A563 (1964)
1964
-
[27]
Nonuniform state of superconductors,
A. I. Larkin and Y. N. Ovchinnikov, “Nonuniform state of superconductors,” Zh. Eksp. Teor. Fiz. 47, 1136–1146 (1964)
1964
-
[28]
Supercurrent diode effect and finite-momentum superconductors,
Noah F. Q. Yuan and Liang Fu, “Supercurrent diode effect and finite-momentum superconductors,” Pro- ceedings of the National Academy of Sciences 119, e2119548119 (2022)
2022
-
[29]
Theory of the supercurrent diode effect in rashba superconductors with arbitrary dis- order,
S. Ili´ c and F. S. Bergeret, “Theory of the supercurrent diode effect in rashba superconductors with arbitrary dis- order,” Phys. Rev. Lett. 128, 177001 (2022)
2022
-
[30]
Superconducting diode effect and nonreciprocal transition lines,
Akito Daido and Youichi Yanase, “Superconducting diode effect and nonreciprocal transition lines,” Phys. Rev. B 106, 205206 (2022)
2022
-
[31]
Superconducting diode effect in quasi-one-dimensional systems,
Tatiana de Picoli, Zane Blood, Yuli Lyanda-Geller, and Jukka I. V¨ ayrynen, “Superconducting diode effect in quasi-one-dimensional systems,” Phys. Rev. B 107, 224518 (2023). 7
2023
-
[32]
A phenomenological theory of superconductor diodes,
James Jun He, Yukio Tanaka, and Naoto Nagaosa, “A phenomenological theory of superconductor diodes,” New Journal of Physics 24, 053014 (2022)
2022
-
[33]
Topo- logical superfluids with finite-momentum pairing and majorana fermions,
Chunlei Qu, Zhen Zheng, Ming Gong, Yong Xu, Li Mao, Xubo Zou, Guangcan Guo, and Chuanwei Zhang, “Topo- logical superfluids with finite-momentum pairing and majorana fermions,” Nature Communications 4, 2710 (2013)
2013
-
[34]
The superconducting diode effect,
Muhammad Nadeem, Michael S Fuhrer, and Xiaolin Wang, “The superconducting diode effect,” Nature Re- views Physics 5, 558–577 (2023)
2023
-
[35]
Superconducting diode effect due to magne- tochiral anisotropy in topological insulators and rashba nanowires,
Henry F. Legg, Daniel Loss, and Jelena Klino- vaja, “Superconducting diode effect due to magne- tochiral anisotropy in topological insulators and rashba nanowires,” Phys. Rev. B 106, 104501 (2022)
2022
-
[36]
Intrin- sic superconducting diode effect,
Akito Daido, Yuhei Ikeda, and Youichi Yanase, “Intrin- sic superconducting diode effect,” Phys. Rev. Lett. 128, 037001 (2022)
2022
-
[37]
Optimiz- ing one dimensional superconducting diodes: Inter- play of rashba spin-orbit coupling and magnetic fields,
Sayak Bhowmik, Dibyendu Samanta, Ashis K. Nandy, Arijit Saha, and Sudeep Kumar Ghosh, “Optimiz- ing one dimensional superconducting diodes: Inter- play of rashba spin-orbit coupling and magnetic fields,” arXiv:2407.12455 [cond-mat.supr-con]
-
[38]
Interlayer exchange coupling: A general scheme turning chiral magnets into magnetic multilayers car- rying atomic-scale skyrmions,
Ashis Kumar Nandy, Nikolai S. Kiselev, and Stefan Bl¨ ugel, “Interlayer exchange coupling: A general scheme turning chiral magnets into magnetic multilayers car- rying atomic-scale skyrmions,” Phys. Rev. Lett. 116, 177202 (2016)
2016
-
[39]
Tailoring the phase transition from topological superconductor to trivial superconductor in- duced by magnetic textures of a spin chain on a p-wave superconductor,
Pritam Chatterjee, Saurabh Pradhan, Ashis K. Nandy, and Arijit Saha, “Tailoring the phase transition from topological superconductor to trivial superconductor in- duced by magnetic textures of a spin chain on a p-wave superconductor,” Phys. Rev. B 107, 085423 (2023)
2023
-
[40]
Topological super- conductivity by engineering noncollinear magnetism in magnet/superconductor heterostructures: A realistic prescription for the two-dimensional kitaev model,
Pritam Chatterjee, Sayan Banik, Sandip Bera, Arnob Kumar Ghosh, Saurabh Pradhan, Arijit Saha, and Ashis K. Nandy, “Topological super- conductivity by engineering noncollinear magnetism in magnet/superconductor heterostructures: A realistic prescription for the two-dimensional ...
2024
-
[41]
Second-order topological su- perconductor via noncollinear magnetic texture,
Pritam Chatterjee, Arnob Kumar Ghosh, Ashis K. Nandy, and Arijit Saha, “Second-order topological su- perconductor via noncollinear magnetic texture,” Phys. Rev. B 109, L041409 (2024)
2024
-
[42]
See the Supplementary Material at XXXXXXXXXXX for the details of self-consistent analysis in real space and topological superconductivity considering uniform super- conducting gap
-
[43]
Quantum-mechanical position operator in extended systems,
Raffaele Resta, “Quantum-mechanical position operator in extended systems,” Phys. Rev. Lett. 80, 1800–1803 (1998)
1998
-
[44]
Many-body electric multipole operators in ex- tended systems,
William A. Wheeler, Lucas K. Wagner, and Taylor L. Hughes, “Many-body electric multipole operators in ex- tended systems,” Phys. Rev. B 100, 245135 (2019)
2019
-
[45]
Many-body order parameters for multipoles in solids,
Byungmin Kang, Ken Shiozaki, and Gil Young Cho, “Many-body order parameters for multipoles in solids,” Phys. Rev. B 100, 245134 (2019). 8 Supplemental material for “Topological Majorana zero modes and the superconducting diode effect driven by Fulde-Ferrell-Larkin-Ovchinnikov ...
2019
Reviewed August 11, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.