Stability of the coexistence phase of chiral superconductivity and noncollinear spin ordering with a nontrivial topology and strong electron correlations

Anton O. Zlotnikov; Valery V. Val'kov

arxiv: 1907.03418 · v1 · pith:GLSG3DPQnew · submitted 2019-07-08 · ❄️ cond-mat.supr-con

Stability of the coexistence phase of chiral superconductivity and noncollinear spin ordering with a nontrivial topology and strong electron correlations

Valery V. Val'kov , Anton O. Zlotnikov This is my paper

Pith reviewed 2026-05-25 01:11 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con

keywords chiral d+id superconductivity120-degree spin orderingtopological invariant N3triangular latticestrong electron correlationsMajorana modesquantum fluctuations

0 comments

The pith

Quantum charge and spin fluctuations preserve the coexistence of chiral d+id superconductivity and 120-degree spin ordering along with its nontrivial topology.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that quantum charge and spin fluctuations renormalize the magnetic order parameter in a strongly correlated triangular-lattice system but leave the coexistence phase of chiral d+id superconductivity and 120-degree spin ordering intact. This phase retains its nontrivial topology as measured by invariant N3, which continues to support Majorana modes at the sample edges. The fluctuations also move the critical doping levels for topological transitions, while stronger intersite repulsion reduces how many such transitions occur.

Core claim

We show that the quantum charge and spin fluctuations, while sufficiently renormalizing the magnetic order parameter, do not destroy the coexistence phase of chiral d+id superconductivity and 120-degree spin ordering in a strongly correlated 2D system with a triangular lattice. The nontrivial topology characterized by the topological invariant N3 is also preserved. It is shown that the Majorana mode exist among edge states in the topologically nontrivial phase. The spatial structure of such mode is determined. The spin and charge fluctuations shift the critical values of electron density at which quantum topological transitions occur. Increasing intersite Coulomb repulsion leads to decrease

What carries the argument

Coexistence phase of chiral d+id superconductivity and 120-degree spin ordering characterized by topological invariant N3, shown stable under quantum fluctuations in the Hubbard-like model on the triangular lattice.

If this is right

Majorana modes persist among the edge states in the topologically nontrivial phase.
The spatial structure of the Majorana mode can be determined from the model.
Spin and charge fluctuations shift the critical electron densities for quantum topological transitions.
Increasing intersite Coulomb repulsion decreases the number of topological transitions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reported stability suggests the phase could appear in real triangular-lattice compounds at appropriate doping.
Edge spectroscopy experiments could search for the Majorana modes whose structure is fixed by the model.
Tuning intersite repulsion offers a route to simplify the sequence of topological phases with doping.

Load-bearing premise

The mean-field decoupling and fluctuation corrections are performed within a Hubbard-like Hamiltonian on the triangular lattice with interaction parameters and doping chosen to lie inside the coexistence window.

What would settle it

Direct measurement in a doped triangular-lattice material showing whether superconductivity and 120-degree magnetic order coexist at the predicted densities, or whether the predicted Majorana edge modes are absent.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Fluctuations renormalize but preserve the coexistence phase and N3 topology in this specific triangular-lattice Hubbard model; the result is scoped and model-dependent.

read the letter

The paper's central finding is that quantum charge and spin fluctuations, added on top of mean-field, reduce the 120-degree magnetic order parameter but leave the d+id superconducting coexistence intact on the triangular lattice. The N3 invariant stays nonzero, Majorana edge modes survive, and the critical densities for topological transitions shift with fluctuations and intersite repulsion. This is a direct extension of earlier mean-field work on the same setup, now checking stability against those fluctuations in a concrete parameter window where coexistence already exists at mean-field level. They also give the spatial profile of the edge mode. That is the concrete advance. The calculation is scoped to a Hubbard-like Hamiltonian with chosen interactions and doping inside the coexistence region, so the stability claim holds only there. The abstract does not detail the fluctuation method, convergence, or error estimates, which leaves the technical control hard to judge from the summary alone. No internal contradiction appears in the stated claims. This work is for researchers already following topological superconductivity plus magnetism on frustrated lattices. It is narrow but addresses a real question with some calculation, so it is worth sending to referees rather than desk-rejecting.

Referee Report

0 major / 3 minor

Summary. The manuscript examines the stability of the coexistence phase between chiral d+id superconductivity and 120-degree noncollinear spin ordering in a strongly correlated 2D system on the triangular lattice. It reports that quantum charge and spin fluctuations renormalize the magnetic order parameter without destroying the coexistence phase or the nontrivial topology characterized by the invariant N3. The work also identifies Majorana modes among the edge states in the topologically nontrivial phase, determines their spatial structure, and shows that fluctuations shift the critical electron densities for quantum topological transitions while increasing intersite Coulomb repulsion reduces the number of such transitions.

Significance. If the central calculations hold, the result establishes a concrete microscopic example in which a topologically nontrivial superconducting-magnetic coexistence phase remains stable against charge and spin fluctuations within a Hubbard-like model on the triangular lattice. The explicit treatment of edge Majorana modes and the parameter dependence on doping and intersite repulsion provide falsifiable predictions for related 2D materials. The preservation of the N3 invariant under fluctuation corrections is a notable strength of the analysis.

minor comments (3)

[Abstract] The abstract states 'the Majorana mode exist' (grammatical error); correct to 'exists'.
The description of the fluctuation method, convergence checks, and error estimates on the renormalized order parameter should be expanded in the main text to allow independent verification of the reported stability window.
Notation for the topological invariant N3 and the definition of the coexistence window in terms of doping and interaction parameters should be introduced with explicit equations at first use.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and the recommendation for minor revision. The provided summary accurately captures the key results on the stability of the coexistence phase, preservation of the N3 invariant under fluctuations, and the role of Majorana edge modes. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained within stated model

full rationale

The paper starts from a concrete Hubbard-like Hamiltonian on the triangular lattice, performs mean-field decoupling for chiral d+id superconductivity coexisting with 120° spin order, then applies quantum charge and spin fluctuation corrections. The stability of the coexistence phase, preservation of the N3 topological invariant, and existence of edge Majorana modes are obtained as outputs of these equations within a doping and interaction window chosen to support the phase. No quoted step reduces a prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames a known result; the central results follow from the model's equations rather than tautological re-expression of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; the ledger cannot be populated with concrete free parameters, axioms, or invented entities because the underlying Hamiltonian, decoupling scheme, and fluctuation approximation are not specified.

pith-pipeline@v0.9.0 · 5645 in / 1294 out tokens · 24140 ms · 2026-05-25T01:11:27.152718+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

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Introduction Recently, several superconducting systems have been proposed in which the formation of Majorana edge states is possible. Among them are, for example, su- perconductors with chiral p-wave symmetry [1, 2], in- terfaces of a superconductor and a topological insu- lator [3, 4], systems with spin-orbit interaction and proximity-induced superconduc...

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For deﬁniteness, we consider electron-doped systems such as the superconducting NaxCoO2 hydrate [24]

Model To study the coexistence phase of chiral supercon- ductivity and 120 ◦ spin ordering with regard to spin- charge ﬂuctuations and strong electron correlations we use t − J − V model. For deﬁniteness, we consider electron-doped systems such as the superconducting NaxCoO2 hydrate [24]. The Hamiltonian in the atomic representation is determined by the e...

work page
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The excitation spectrum in the noncollinear magnetic phase is gapless on the Fermi contour (the line in the 2D Brillouin zone) at all levels of doping

Gapless excitations in the coexistence phase It is known [15] that the topological transitions oc- cur when the elementary fermion excitations for the sys- tem with periodic boundary conditions become gapless. The excitation spectrum in the noncollinear magnetic phase is gapless on the Fermi contour (the line in the 2D Brillouin zone) at all levels of dop...

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quasiparticle

T opological invariant ˜N3 and Majorana modes To solve the problem of a nontrivial topology of the coexistence phase of superconductivity and noncollinear magnetism at strong electron correlations, we use the method based on the analysis of the integer-valued topo- logical invariant ˜N3 [15]: K K -0.5 0 0.5 1 1.5 2 Fig. 5 Fermi spectrum with regard to the...

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Manifestation of Spin–Nematic Correlations in Spectral Characteristics of the Electronic Structure and Their Inﬂuence on Practical Properties of Cuprate Su- perconductors

Conclusions The results obtained in this work show that strong electron correlations signiﬁcantly renormalize the spin structure parameter but do not destroy the coexistence phase of chiral superconductivity and noncollinear mag- netic ordering. It has been found that a nontrivial topology also holds in this case, which is important for the formation of M...

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Introduction Recently, several superconducting systems have been proposed in which the formation of Majorana edge states is possible. Among them are, for example, su- perconductors with chiral p-wave symmetry [1, 2], in- terfaces of a superconductor and a topological insu- lator [3, 4], systems with spin-orbit interaction and proximity-induced superconduc...

work page

[2] [2]

For deﬁniteness, we consider electron-doped systems such as the superconducting NaxCoO2 hydrate [24]

Model To study the coexistence phase of chiral supercon- ductivity and 120 ◦ spin ordering with regard to spin- charge ﬂuctuations and strong electron correlations we use t − J − V model. For deﬁniteness, we consider electron-doped systems such as the superconducting NaxCoO2 hydrate [24]. The Hamiltonian in the atomic representation is determined by the e...

work page

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The excitation spectrum in the noncollinear magnetic phase is gapless on the Fermi contour (the line in the 2D Brillouin zone) at all levels of doping

Gapless excitations in the coexistence phase It is known [15] that the topological transitions oc- cur when the elementary fermion excitations for the sys- tem with periodic boundary conditions become gapless. The excitation spectrum in the noncollinear magnetic phase is gapless on the Fermi contour (the line in the 2D Brillouin zone) at all levels of dop...

work page

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The electronic ensemble on the triangular lattice is studied in the framework of the Hubbard model with mean- ﬁeld [32] and slave-boson [33] approximations

Renormalization of the magnetic ampli- tude M The theoretical description of magnets with the 120 ◦ ordering of localized spins is carried out in [27–31]. The electronic ensemble on the triangular lattice is studied in the framework of the Hubbard model with mean- ﬁeld [32] and slave-boson [33] approximations. The phase diagrams with diﬀerent spin and cha...

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quasiparticle

T opological invariant ˜N3 and Majorana modes To solve the problem of a nontrivial topology of the coexistence phase of superconductivity and noncollinear magnetism at strong electron correlations, we use the method based on the analysis of the integer-valued topo- logical invariant ˜N3 [15]: K K -0.5 0 0.5 1 1.5 2 Fig. 5 Fermi spectrum with regard to the...

work page

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Manifestation of Spin–Nematic Correlations in Spectral Characteristics of the Electronic Structure and Their Inﬂuence on Practical Properties of Cuprate Su- perconductors

Conclusions The results obtained in this work show that strong electron correlations signiﬁcantly renormalize the spin structure parameter but do not destroy the coexistence phase of chiral superconductivity and noncollinear mag- netic ordering. It has been found that a nontrivial topology also holds in this case, which is important for the formation of M...

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N. Read, D. Green, Phys. Rev. B 61 10267 (2000)

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A. Yu. Kitaev, Physics-Uspekhi, Suppl. 44, 131 (2001)

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L. Fu, C. L. Kane, Phys. Rev. Lett., 100 096407 (2008)

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Snelder, A

M. Snelder, A. A. Golubov, Y. Asano, A. Brinkman, J. Phys.: Condens. Matter 27, 315701 (2015)

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J. D. Sau, R. M. Lutchyn, S. Tewari, S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010)

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A. A. Kopasov, I. M. Khaymovich, A. S. Mel’nikov, Beil- stein J. Nanotechnol. 9, 1184, (2018)

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Zhang, Ch.-X Liu, S

H. Zhang, Ch.-X Liu, S. Gazibegovic, Di Xu, J. A. Lo- gan, G. Wang, N. van Loo, J. D. S. Bommer, M. W. A. de Moor, D. Car, R. L. M. Ophet Veld, P. J. van Veld- hoven, S. Koelling, M. A. Verheijen, M. Pendharkar, D. J. Pennachio, B. Shojaei, J. S. Lee, Ch. J. Palmstrom, E. P. A. M. Bakkers, S. Das Sarma, L. P. Kouwenhoven, Nature 556, 74 (2018)

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Martin, A.F

I. Martin, A.F. Morpurgo, Phys. Rev. B 85, 144505 (2012)

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Y.-M. Lu, Z. Wang, Phys. Rev. Lett. 110, 096403 (2013)

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A. P. Schnyder, S. Ryu, A. Furusaki, A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008)

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Ghosh, J.D

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V. V. Val’kov and A. O. Zlotnikov, JETP Lett. 104, 483 (2016)

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Fidkowski, A

L. Fidkowski, A. Yu. Kitaev, Phys. Rev. B 81, 134509 (2010)

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Morimoto, A

T. Morimoto, A. Furusaki, Ch. Mudry, Phys. Rev. B 92, 125104 (2015)

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K. Ishikawa, T. Matsuyama, Nucl. Phys. B280, 523 (1987)

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Volovik, V.M

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G. E. Volovik, JETP Letters 90, 398 (2009)

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Wang, S.-C

Z. Wang, S.-C. Zhang, Phys. Rev. B 86, 165116 (2012)

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Baskaran, Phys

G. Baskaran, Phys. Rev. Lett. 91, 097003 (2003)

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V. V. Val’kov, T. A. Val’kova, V. A. Mitskan, J. of Magn. Magn. Materials 440, 129 (2017)

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V. V. Val’kov and A. O. Zlotnikov, Phys. Solid State 59, 2120 (2017)

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D. M. Dzebisashvili and A. A. Khudaiberdyev, JETP Lett. 108, 189 (2018)

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Capriotti, A

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S. R. White, A. L. Chernyshev, Phys. Rev. Lett. 99, 127004 (2007)

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Pasrija, S

K. Pasrija, S. Kumar, Phys. Rev. B 93, 195110 (2016)

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Jiang, S

K. Jiang, S. Zhou, Z. Wang, Phys. Rev. B 90, 165135 (2014)

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Weber, A

C. Weber, A. Lauchli, F. Mila, T. Giamarchi, Phys. Rev. B 73, 014519 (2006)

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work page 2004