arxiv: 2604.24734 · v1 · submitted 2026-04-27 · ❄️ cond-mat.supr-con

Recognition: unknown

Nonintegral Flux Trapping in Frustrated Josephson Networks of Triplet Superconductors

Grayson R. Frazier , Colton Lelievre , Yi Li

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:51 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con

keywords Josephson junction networkstriplet superconductorsd-vector texturesflux trappinggeometric frustrationnonintegral fluxunitary pairingantisymmetric coupling

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The pith

Anisotropic Josephson couplings in triplet superconductors create frustrated d-vector textures that trap nonintegral flux.

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

The paper establishes that Josephson junction networks formed from triplet superconductors, when the couplings between pairing correlations are anisotropic, develop frustrated textures in the d-vector orientation. This frustration prevents simultaneous optimization of the superconducting phase difference and the d-vector alignment, analogous to frustrated spin systems. As a result, the twisting of the internal Cooper pair structure generates an emergent geometric phase that supports spontaneous currents and traps flux in nonintegral amounts. For unitary triplet states, this leads to stable pi-flux trapping once the antisymmetric coupling exceeds a critical value, and the trapped flux is distinct from that in half-quantum vortices.

Core claim

In a Josephson junction network, anisotropic coupling between spin triplet pairing correlations leads to frustrated d vector textures that support spontaneous Josephson currents and nonintegral flux trapping. In analogy to classical spin systems, the presence of geometric frustration and anisotropic couplings means the U(1) phase and d vector orientations cannot be simultaneously optimized. The internal pairing structure twists as Cooper pairs tunnel, and the d vector texture acts as an emergent geometric phase that spontaneously traps fractional flux. For unitary triplet pairing order, pi-flux trapping occurs above a critical antisymmetric Josephson coupling, distinct from usual half-quanta

What carries the argument

The frustrated d-vector texture, which functions as an emergent geometric phase in the superconducting order parameter due to anisotropic Josephson couplings.

If this is right

Networks of triplet superconductors in polycrystals or single crystals can exhibit spontaneous Josephson currents.
Nonintegral amounts of magnetic flux can be trapped without external magnetic fields.
Pi-flux states become stable in unitary triplet superconductors above a critical antisymmetric coupling strength.
This flux trapping is distinct from half-quantum vortex formation.
New routes open to engineer frustrated Josephson networks from magnetic textures combined with spin triplet pairing.

Where Pith is reading between the lines

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

Similar frustration could appear in other superconductors with internal degrees of freedom beyond simple singlet pairing.
Experiments could test this by building artificial junction arrays with controlled anisotropy in known triplet materials.
If present, flux quantization measurements in triplet systems would need to include internal pairing orientation effects.
This mechanism might link to topological superconductivity where d-vector textures influence protected states.

Load-bearing premise

Anisotropic Josephson couplings between spin triplet pairing correlations must be present and dominant in actual materials so that the internal pairing orientation can form a frustrated pattern.

What would settle it

Fabricating a Josephson network from a unitary triplet superconductor and measuring only integer multiples of the flux quantum with no spontaneous currents in zero external field.

Figures

Figures reproduced from arXiv: 2604.24734 by Colton Lelievre, Grayson R. Frazier, Yi Li.

**Figure 1.** Figure 1: FIG. 1. Schematic of Josephson junction network in a spin view at source ↗

**Figure 2.** Figure 2: FIG. 2. Example of minimal-energy view at source ↗

**Figure 3.** Figure 3: FIG. 3. Flux trapping and frustration in three-grain network view at source ↗

read the original abstract

In a Josephson junction network, anisotropic coupling between spin triplet pairing correlations can lead to frustrated $d$ vector textures that support spontaneous Josephson currents and nonintegral flux trapping. Such networks can appear in superconducting polycrystals, as well as single-crystal superconductors. In analogy to classical spin systems, in which the presence of geometric frustration and anisotropic superexchange can lead to nontrivial spin textures, Josephson networks with anisotropic Josephson couplings cannot simultaneously optimize their $\mathrm{U}(1)$ superconducting phase difference and relative $d$ vector orientations. The internal pairing structure of Cooper pairs twists as they tunnel across the Josephson junction, and the $d$ vector texture enters as an emergent geometric phase which can spontaneously trap fractional flux. For unitary triplet pairing order, this mechanism can support $\pi$-flux trapping above a critical value of antisymmetric Josephson coupling, and is distinct from usual half-quantum vortices. The results of this work reveal new routes to engineer frustrated Josephson networks from the interplay of magnetic textures and spin triplet superconducting pairing order.

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

The paper shows that anisotropic triplet Josephson couplings produce frustrated d-vector textures that trap π flux above a critical strength, with the calculation resting on a standard energy functional and explicit minimization for a plaquette.

read the letter

The key point is that this work takes the spin-frustration analogy and applies it to Josephson networks in unitary triplet superconductors, showing that d-vector textures can generate a geometric phase that favors spontaneous π-flux trapping once the antisymmetric coupling exceeds a threshold. The mechanism is presented as distinct from ordinary half-quantum vortices because the flux arises from the relative orientation of the d-vectors rather than from a simple phase winding around a half-quantum core. They construct the effective energy from the usual U(1) phase term plus the d-vector dot-product interaction, compute the geometric-phase contribution on a minimal frustrated plaquette, and obtain the critical coupling by direct minimization. That part is straightforward and internally consistent; no hidden assumptions about boundedness or neglected invariants appear in the derivation. The abstract and stress-test note both confirm the calculation stays within standard terms for these systems. What the paper does cleanly is make the analogy quantitative for a concrete geometry instead of leaving it at the level of qualitative spin analogies. The distinction from half-quantum vortices is drawn by examining the topology of the d-vector texture, which is a useful clarification. Soft spots are modest. The model assumes the anisotropic couplings are present and dominant, which may or may not hold in real polycrystals or single-crystal networks; the paper does not explore how sensitive the critical value is to small isotropic components or to thermal fluctuations. The treatment is limited to the minimal plaquette, so it remains to be seen how the effect scales in larger lattices or whether it survives when the full network topology is included. No numerical simulations or material-specific estimates are given, which keeps the result at the level of a proof-of-principle mechanism. This is the kind of paper that belongs in the reading group for people working on triplet superconductivity and Josephson-junction arrays. A reader who already knows the spin-frustration literature will see the extension immediately and can judge whether the critical-coupling result is worth testing in devices. It is coherent on its own terms, with no load-bearing circularity or fitting disguised as prediction. I would send it to peer review; the derivations are explicit enough that referees can check the energy minimization and the topological distinction without needing extra data.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that anisotropic Josephson couplings between spin-triplet pairing correlations in a Josephson junction network lead to frustrated d-vector textures. These textures produce an emergent geometric phase that supports spontaneous Josephson currents and trapping of π-flux (a nonintegral value) above a critical strength of the antisymmetric coupling. The effect is shown for unitary triplet order, is topologically distinct from conventional half-quantum vortices, and is obtained by direct minimization of an effective energy functional built from standard U(1) phase and d-vector dot-product terms on a minimal frustrated plaquette; the results are argued to apply to polycrystals and single-crystal triplet superconductors.

Significance. If the central mechanism holds, the work identifies a new route to nonintegral flux trapping that arises from the interplay of d-vector texture and anisotropic triplet Josephson coupling, distinct from established half-quantum vortex physics. This could be relevant for flux dynamics in realizable triplet-superconductor networks and suggests engineering possibilities via control of pairing anisotropy. The explicit computation of the geometric-phase contribution to the flux and the parameter-free extraction of the critical coupling by direct minimization are strengths of the derivation.

minor comments (3)

The abstract uses 'nonintegral flux trapping' while the body focuses on π-flux; a parenthetical clarification of the specific fraction would improve immediate readability.
The d-vector texture and the definition of the antisymmetric Josephson coupling term are introduced via the energy functional; an early, self-contained paragraph defining these quantities and their physical origin would aid readers outside the immediate subfield.
The analogy to classical frustrated spin systems is invoked but would benefit from one or two additional specific citations to geometric-frustration models in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of our central results on frustrated d-vector textures and nonintegral flux trapping, and the recommendation for minor revision. The significance statement correctly identifies the distinction from half-quantum vortex physics and the potential relevance to triplet-superconductor networks. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins from a standard Josephson network energy functional built from U(1) phase differences and d-vector dot-product couplings, explicitly computes the geometric-phase flux contribution on a minimal frustrated plaquette, and locates the critical antisymmetric coupling strength by direct minimization of that functional. No step reduces to self-definition of the target quantity, to a fitted parameter renamed as a prediction, or to a load-bearing self-citation whose validity is presupposed. The spin-system analogy supplies only qualitative motivation; the quantitative results follow from the constructed Hamiltonian without circular closure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that anisotropic couplings exist and produce frustration in d-vector textures, with no free parameters, new entities, or additional axioms explicitly introduced in the abstract.

axioms (1)

domain assumption Anisotropic coupling between spin triplet pairing correlations leads to frustrated d-vector textures in Josephson networks.
Invoked to support spontaneous currents and nonintegral flux trapping.

pith-pipeline@v0.9.0 · 5488 in / 1148 out tokens · 51240 ms · 2026-05-07T17:51:12.482995+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

54 extracted references · 46 canonical work pages · 3 internal anchors

[1]

author author G\'erard \ Toulouse ,\ title title Theory of the frustration effect in spin glasses: I , \ @noop journal journal Communications on Physics \ volume 2 ,\ pages 115--119 ( year 1977 ) NoStop

1977
[2]

author author J Vannimenus \ and\ author G Toulouse ,\ title title Theory of the frustration effect. ii. Ising spins on a square lattice , \ 10.1088/0022-3719/10/18/008 journal journal J. Phys. C: Solid State Phys. \ volume 10 ,\ pages L537–L542 ( year 1977 ) NoStop

work page doi:10.1088/0022-3719/10/18/008 1977
[3]

\ Ramirez ,\ title title Geometrical frustration , \ 10.1063/1.2186278 journal journal Phys

author author Roderich \ Moessner \ and\ author Arthur P. \ Ramirez ,\ title title Geometrical frustration , \ 10.1063/1.2186278 journal journal Phys. Today \ volume 59 ,\ pages 24--29 ( year 2006 ) NoStop

work page doi:10.1063/1.2186278 2006
[4]

author author Leon \ Balents ,\ title title Spin liquids in frustrated magnets , \ 10.1038/nature08917 journal journal Nature \ volume 464 ,\ pages 199--208 ( year 2010 ) NoStop

work page doi:10.1038/nature08917 2010
[5]

author author Lucile \ Savary \ and\ author Leon \ Balents ,\ title title Quantum spin liquids: a review , \ 10.1088/0034-4885/80/1/016502 journal journal Rep. Prog. Phys. \ volume 80 ,\ pages 016502 ( year 2016 ) NoStop

work page doi:10.1088/0034-4885/80/1/016502 2016
[6]

Teitel \ and\ author C

author author S. Teitel \ and\ author C. Jayaprakash ,\ title title Phase transitions in frustrated two-dimensional XY models , \ 10.1103/PhysRevB.27.598 journal journal Phys. Rev. B \ volume 27 ,\ pages 598--601 ( year 1983 ) NoStop

work page doi:10.1103/physrevb.27.598 1983
[7]

author author Manfred \ Sigrist \ and\ author T. M. \ Rice ,\ title title Unusual paramagnetic phenomena in granular high-temperature superconductors—a consequence of d -wave pairing? \ 10.1103/revmodphys.67.503 journal journal Rev. Mod. Phys. \ volume 67 ,\ pages 503--513 ( year 1995 ) NoStop

work page doi:10.1103/revmodphys.67.503 1995
[8]

author author R Fazio ,\ title title Quantum phase transitions and vortex dynamics in superconducting networks , \ 10.1016/s0370-1573(01)00022-9 journal journal Phys. Rep. \ volume 355 ,\ pages 235--334 ( year 2001 ) NoStop

work page doi:10.1016/s0370-1573(01)00022-9 2001
[9]

author author J. P. \ Pekola , author J. J. \ Toppari , author M. Aunola , author M. T. \ Savolainen , \ and\ author D. V. \ Averin ,\ title title Adiabatic transport of Cooper pairs in arrays of Josephson junctions , \ 10.1103/PhysRevB.60.R9931 journal journal Phys. Rev. B \ volume 60 ,\ pages R9931--R9934 ( year 1999 ) NoStop

work page doi:10.1103/physrevb.60.r9931 1999
[10]

author author Yuriy \ Makhlin , author Gerd \ Sch\"on , \ and\ author Alexander \ Shnirman ,\ title title Quantum-state engineering with Josephson -junction devices , \ 10.1103/RevModPhys.73.357 journal journal Rev. Mod. Phys. \ volume 73 ,\ pages 357--400 ( year 2001 ) NoStop

work page doi:10.1103/revmodphys.73.357 2001
[11]

Emergent Network of Josephson Junctions in a Kagome Superconductor

author author Tycho J. \ Blom , author Matthijs \ Rog , author Marieke \ Altena , author Andrea Capa \ Salinas , author Stephen D. \ Wilson , author Milan P. \ Allan , author Chuan \ Li , \ and\ author Kaveh \ Lahabi ,\ https://arxiv.org/abs/2510.09777 title Emergent network of J osephson junctions in a kagome superconductor , \ ( year 2026 ),\ http://arx...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[12]

author author K. K. \ Likharev ,\ title title Superconducting weak links , \ 10.1103/revmodphys.51.101 journal journal Rev. Mod. Phys. \ volume 51 ,\ pages 101--159 ( year 1979 ) NoStop

work page doi:10.1103/revmodphys.51.101 1979
[13]

author author Manfred \ Sigrist \ and\ author T. M. \ Rice ,\ title title Paramagnetic effect in high T_c superconductors- a hint for d -wave superconductivity , \ 10.1143/jpsj.61.4283 journal journal J. Phys. Soc. Jpn. \ volume 61 ,\ pages 4283--4286 ( year 1992 ) NoStop

work page doi:10.1143/jpsj.61.4283 1992
[14]

author author L. N. \ Bulaevskii , author V. V. \ Kuzii , \ and\ author A. A. \ Sobyanin ,\ title title Superconducting system with weak coupling to the current in the ground state , \ http://jetpletters.ru/ps/0/article_21163.shtml journal journal JETP Lett. \ volume 25 ,\ pages 314 ( year 1977 ) NoStop

1977
[15]

author author B. I. \ Spivak \ and\ author S. A. \ Kivelson ,\ title title Negative local superfluid densities: The difference between dirty superconductors and dirty bose liquids , \ 10.1103/physrevb.43.3740 journal journal Phys. Rev. B \ volume 43 ,\ pages 3740--3743 ( year 1991 ) NoStop

work page doi:10.1103/physrevb.43.3740 1991
[16]

author author J. R. \ Kirtley , author K. A. \ Moler , \ and\ author D. J. \ Scalapino ,\ title title Spontaneous flux and magnetic-interference patterns in 0- Josephson junctions , \ 10.1103/PhysRevB.56.886 journal journal Phys. Rev. B \ volume 56 ,\ pages 886--891 ( year 1997 ) NoStop

work page doi:10.1103/physrevb.56.886 1997
[17]

author author V. V. \ Ryazanov , author V. A. \ Oboznov , author A. Yu. \ Rusanov , author A. V. \ Veretennikov , author A. A. \ Golubov , \ and\ author J. Aarts ,\ title title Coupling of two superconductors through a ferromagnet: Evidence for a junction , \ 10.1103/PhysRevLett.86.2427 journal journal Phys. Rev. Lett. \ volume 86 ,\ pages 2427--2430 ( ye...

work page doi:10.1103/physrevlett.86.2427 2001
[18]

author author Daniel \ Dom\' nguez , author E. A. \ Jagla , \ and\ author C. A. \ Balseiro ,\ title title Phenomenological theory of the paramagnetic Meissner effect , \ 10.1103/PhysRevLett.72.2773 journal journal Phys. Rev. Lett. \ volume 72 ,\ pages 2773--2776 ( year 1994 ) NoStop

work page doi:10.1103/physrevlett.72.2773 1994
[19]

author author SV Panyukov \ and\ author AD Zaikin ,\ title title Glassy states with frozen-in currents and paramagnetism of granular high temperature superconductors , \ @noop journal journal Physica B \ volume 203 ,\ pages 527--531 ( year 1994 ) NoStop

1994
[20]

author author C. C. \ Tsuei \ and\ author J. R. \ Kirtley ,\ title title Pairing symmetry in cuprate superconductors , \ 10.1103/revmodphys.72.969 journal journal Rev. Mod. Phys. \ volume 72 ,\ pages 969--1016 ( year 2000 ) NoStop

work page doi:10.1103/revmodphys.72.969 2000
[21]

author author C. C. \ Tsuei , author J. R. \ Kirtley , author C. C. \ Chi , author Lock See \ Yu-Jahnes , author A. Gupta , author T. Shaw , author J. Z. \ Sun , \ and\ author M. B. \ Ketchen ,\ title title Pairing symmetry and flux quantization in a tricrystal superconducting ring of YBa _2 Cu _3 O _ 7- , \ 10.1103/physrevlett.73.593 journal journal Phys...

work page doi:10.1103/physrevlett.73.593 1994
[22]

\ Kirtley , author C.C

author author J.R. \ Kirtley , author C.C. \ Tsuei , author H Raffy , author Z.Z. \ Li , author A Gupta , author J.Z. \ Sun , \ and\ author S Megtert ,\ title title Half-integer flux quantum effect in tricrystal Bi _2 Sr _2 CaCu _2 O _ 8+ , \ @noop journal journal Europhys. Lett. \ volume 36 ,\ pages 707--712 ( year 1996 ) NoStop

1996
[23]

author author Yufan \ Li , author Xiaoying \ Xu , author M-H \ Lee , author M-W \ Chu , \ and\ author C L \ Chien ,\ title title Observation of half-quantum flux in the unconventional superconductor -Bi _2 Pd , \ 10.1126/science.aau6539 journal journal Science \ volume 366 ,\ pages 238--241 ( year 2019 ) NoStop

work page doi:10.1126/science.aau6539 2019
[24]

author author Xiaoying \ Xu , author Yufan \ Li , \ and\ author C. L. \ Chien ,\ title title Spin-triplet pairing state evidenced by half-quantum flux in a noncentrosymmetric superconductor , \ 10.1103/PhysRevLett.124.167001 journal journal Phys. Rev. Lett. \ volume 124 ,\ pages 167001 ( year 2020 ) NoStop

work page doi:10.1103/physrevlett.124.167001 2020
[25]

author author Junyi \ Zhang \ and\ author Yi Li ,\ https://arxiv.org/abs/2509.24581 title Spontaneous flux trapping in granular rings of unconventional superconductors , \ ( year 2025 ),\ http://arxiv.org/abs/2509.24581 arXiv:2509.24581 NoStop

work page arXiv 2025
[26]

author author Grayson R. \ Frazier , author Junyi \ Zhang , \ and\ author Yi Li ,\ title title Anisotropic Josephson coupling of d -vectors in triplet superconductors arising from frustrated spin textures , \ https://arxiv.org/abs/2506.15661 journal journal arXiv:2506.15661 \ ( year 2025 ) NoStop

work page internal anchor Pith review Pith/arXiv arXiv 2025
[27]

author author Grayson R. \ Frazier \ and\ author Yi Li ,\ https://arxiv.org/abs/2510.25756 title Spatially inhomogeneous triplet pairing order and J osephson diode effect induced by frustrated spin textures , \ ( year 2025 ),\ http://arxiv.org/abs/2510.25756 arXiv:2510.25756 NoStop

work page internal anchor Pith review Pith/arXiv arXiv 2025
[28]

Balian \ and\ author N

author author R. Balian \ and\ author N. R. \ Werthamer ,\ title title Superconductivity with pairs in a relative p wave , \ 10.1103/physrev.131.1553 journal journal Phys. Rev. \ volume 131 ,\ pages 1553--1564 ( year 1963 ) NoStop

work page doi:10.1103/physrev.131.1553 1963
[29]

\ Leggett ,\ title title A theoretical description of the new phases of liquid ^3 He , \ 10.1103/revmodphys.47.331 journal journal Rev

author author Anthony J. \ Leggett ,\ title title A theoretical description of the new phases of liquid ^3 He , \ 10.1103/revmodphys.47.331 journal journal Rev. Mod. Phys. \ volume 47 ,\ pages 331--414 ( year 1975 ) NoStop

work page doi:10.1103/revmodphys.47.331 1975
[30]

Laing and G

author author Grigory E. \ Volovik ,\ 10.1093/acprof:oso/9780199564842.001.0001 title The Universe in a Helium Droplet \ ( publisher Oxford University PressOxford ,\ year 2009 ) NoStop

work page doi:10.1093/acprof:oso/9780199564842.001.0001 2009
[31]

author author Andrew Peter \ Mackenzie \ and\ author Yoshiteru \ Maeno ,\ title title The superconductivity of Sr _2 RuO _4 and the physics of spin-triplet pairing , \ 10.1103/revmodphys.75.657 journal journal Rev. Mod. Phys. \ volume 75 ,\ pages 657--712 ( year 2003 ) NoStop

work page doi:10.1103/revmodphys.75.657 2003
[32]

author author Vinay \ Ambegaokar \ and\ author Alexis \ Baratoff ,\ title title Tunneling between superconductors , \ 10.1103/physrevlett.10.486 journal journal Phys. Rev. Lett. \ volume 10 ,\ pages 486--489 ( year 1963 ) NoStop

work page doi:10.1103/physrevlett.10.486 1963
[33]

\ Larkin ,\ title title The Josephson effect in superconductors with heavy fermions , \ http://jetpletters.ru/ps/1404/article_21329.shtml journal journal J

author author VB Geshkenbein \ and\ author A.I. \ Larkin ,\ title title The Josephson effect in superconductors with heavy fermions , \ http://jetpletters.ru/ps/1404/article_21329.shtml journal journal J. Exp. Theor. Phys. Lett. \ volume 43 ,\ pages 395 ( year 1986 ) NoStop

1986
[34]

author author Manfred \ Sigrist \ and\ author Kazuo \ Ueda ,\ title title Phenomenological theory of unconventional superconductivity , \ 10.1103/RevModPhys.63.239 journal journal Rev. Mod. Phys. \ volume 63 ,\ pages 239--311 ( year 1991 ) NoStop

work page doi:10.1103/revmodphys.63.239 1991
[35]

author author Grayson R. \ Frazier , author Junjia \ Zhang , author Junyi \ Zhang , author Xinyu \ Sun , \ and\ author Yi Li ,\ title title Designing phase sensitive probes of monopole superconducting order , \ 10.1103/PhysRevResearch.6.043189 journal journal Phys. Rev. Res. \ volume 6 ,\ pages 043189 ( year 2024 ) NoStop

work page doi:10.1103/physrevresearch.6.043189 2024
[36]

Buzdin ,\ title title Direct coupling between magnetism and superconducting current in the Josephson _ 0 junction , \ 10.1103/PhysRevLett.101.107005 journal journal Phys

author author A. Buzdin ,\ title title Direct coupling between magnetism and superconducting current in the Josephson _ 0 junction , \ 10.1103/PhysRevLett.101.107005 journal journal Phys. Rev. Lett. \ volume 101 ,\ pages 107005 ( year 2008 ) NoStop

work page doi:10.1103/physrevlett.101.107005 2008
[37]

author author A. A. \ Reynoso , author Gonzalo \ Usaj , author C. A. \ Balseiro , author D. Feinberg , \ and\ author M. Avignon ,\ title title Anomalous Josephson current in junctions with spin polarizing quantum point contacts , \ 10.1103/PhysRevLett.101.107001 journal journal Phys. Rev. Lett. \ volume 101 ,\ pages 107001 ( year 2008 ) NoStop

work page doi:10.1103/physrevlett.101.107001 2008
[38]

author author D. B. \ Szombati , author S. Nadj-Perge , author D. Car , author S. R. \ Plissard , author E. P. A. M. \ Bakkers , \ and\ author L. P. \ Kouwenhoven ,\ title title Josephson _0 -junction in nanowire quantum dots , \ 10.1038/nphys3742 journal journal Nature Physics \ volume 12 ,\ pages 568--572 ( year 2016 ) NoStop

work page doi:10.1038/nphys3742 2016
[39]

author author Morten \ Amundsen , author Jacob \ Linder , author Jason W. A. \ Robinson , author Igor \ Z Z uti c \' c , \ and\ author Niladri \ Banerjee ,\ title title Colloquium: Spin-orbit effects in superconducting hybrid structures , \ 10.1103/RevModPhys.96.021003 journal journal Rev. Mod. Phys. \ volume 96 ,\ pages 021003 ( year 2024 ) NoStop

work page doi:10.1103/revmodphys.96.021003 2024
[40]

author author M. M. \ Salomaa \ and\ author G. E. \ Volovik ,\ title title Quantized vortices in superfluid ^3 He , \ 10.1103/revmodphys.59.533 journal journal Rev. Mod. Phys. \ volume 59 ,\ pages 533--613 ( year 1987 ) NoStop

work page doi:10.1103/revmodphys.59.533 1987
[41]

This includes (i) derivation of the critical value for anisotropic coupling in the three-grain system and (ii) more details on the minimal energy d vector configurations

@noop note See Supplemental Material for additional details. This includes (i) derivation of the critical value for anisotropic coupling in the three-grain system and (ii) more details on the minimal energy d vector configurations. Stop
[42]

author author A. B. \ Harris , author C. Kallin , \ and\ author A. J. \ Berlinsky ,\ title title Possible N \'eel orderings of the kagom\'e antiferromagnet , \ 10.1103/PhysRevB.45.2899 journal journal Phys. Rev. B \ volume 45 ,\ pages 2899--2919 ( year 1992 ) NoStop

work page doi:10.1103/physrevb.45.2899 1992
[43]

author author MF Collins \ and\ author OA Petrenko ,\ title title Review/synth \`e se: triangular antiferromagnets , \ @noop journal journal Canadian journal of physics \ volume 75 ,\ pages 605--655 ( year 1997 ) NoStop

1997
[44]

\ Gor'kov \ and\ author Emmanuel I

author author Lev P. \ Gor'kov \ and\ author Emmanuel I. \ Rashba ,\ title title Superconducting 2d system with lifted spin degeneracy: Mixed singlet-triplet state , \ 10.1103/PhysRevLett.87.037004 journal journal Phys. Rev. Lett. \ volume 87 ,\ pages 037004 ( year 2001 ) NoStop

work page doi:10.1103/physrevlett.87.037004 2001
[45]

Smidman, M

author author M. Smidman , author M. B. \ Salamon , author H. Q. \ Yuan , \ and\ author D. F. \ Agterberg ,\ title title Superconductivity and spin–orbit coupling in non-centrosymmetric materials: a review , \ 10.1088/1361-6633/80/3/036501 journal journal Rep. Prog. Phys. \ volume 80 ,\ pages 036501 ( year 2017 ) NoStop

work page doi:10.1088/1361-6633/80/3/036501 2017
[46]

author author G. E. \ Volovik \ and\ author V. P. \ Mineev ,\ title title Line and point singularities in superfluid He ^3 , \ http://jetpletters.ru/ps/0/article_27785.shtml journal journal JETP Lett. \ volume 24 ,\ pages 605 ( year 1976 ) NoStop

1976
[47]

author author D. A. \ Ivanov ,\ title title Non-abelian statistics of half-quantum vortices in p -wave superconductors , \ 10.1103/physrevlett.86.268 journal journal Phys. Rev. Lett. \ volume 86 ,\ pages 268--271 ( year 2001 ) NoStop

work page doi:10.1103/physrevlett.86.268 2001
[48]

author author Jason \ Alicea ,\ title title New directions in the pursuit of M ajorana fermions in solid state systems , \ 10.1088/0034-4885/75/7/076501 journal journal Rep. Prog. Phys. \ volume 75 ,\ pages 076501 ( year 2012 ) NoStop

work page doi:10.1088/0034-4885/75/7/076501 2012
[49]

\ Meyerheim , author Takis \ Kontos , \ and\ author Stuart S

author author Kun-Rok \ Jeon , author Binoy Krishna \ Hazra , author Kyungjune \ Cho , author Anirban \ Chakraborty , author Jae-Chun \ Jeon , author Hyeon \ Han , author Holger L. \ Meyerheim , author Takis \ Kontos , \ and\ author Stuart S. P. \ Parkin ,\ title title Long-range supercurrents through a chiral non-collinear antiferromagnet in lateral Jose...

work page doi:10.1038/s41563-021-01061-9 2021
[50]

\ Meyerheim , author Takis \ Kontos , author Audrey \ Cottet , \ and\ author Stuart S

author author Kun-Rok \ Jeon , author Binoy Krishna \ Hazra , author Jae-Keun \ Kim , author Jae-Chun \ Jeon , author Hyeon \ Han , author Holger L. \ Meyerheim , author Takis \ Kontos , author Audrey \ Cottet , \ and\ author Stuart S. P. \ Parkin ,\ title title Chiral antiferromagnetic Josephson junctions as spin-triplet supercurrent spin valves and d.c....

work page doi:10.1038/s41565-023-01336-z 2023
[51]

Ribak , author I

author author A. Ribak , author I. Silber , author C. Baines , author K. Chashka , author Z. Salman , author Y. Dagan , \ and\ author A. Kanigel ,\ title title Gapless excitations in the ground state of 1T -TaS _2 , \ 10.1103/physrevb.96.195131 journal journal Phys. Rev. B \ volume 96 ,\ pages 195131 ( year 2017 ) NoStop

work page doi:10.1103/physrevb.96.195131 2017
[52]

author author Eylon \ Persky , author Anders V. \ Bjørlig , author Irena \ Feldman , author Avior \ Almoalem , author Ehud \ Altman , author Erez \ Berg , author Itamar \ Kimchi , author Jonathan \ Ruhman , author Amit \ Kanigel , \ and\ author Beena \ Kalisky ,\ title title Magnetic memory and spontaneous vortices in a van der W aals superconductor , \ 1...

work page doi:10.1038/s41586-022-04855-2 2022
[53]

Silber , author S

author author I. Silber , author S. Mathimalar , author I. Mangel , author A. K. \ Nayak , author O. Green , author N. Avraham , author H. Beidenkopf , author I. Feldman , author A. Kanigel , author A. Klein , author M. Goldstein , author A. Banerjee , author E. Sela , \ and\ author Y. Dagan ,\ title title Two-component nematic superconductivity in 4Hb-Ta...

work page doi:10.1038/s41467-024-45169-3 2024
[54]

author author Jacques \ Villain ,\ title title Spin glass with non-random interactions , \ 10.1088/0022-3719/10/10/014 journal journal Journal of Physics C: Solid State Physics \ volume 10 ,\ pages 1717--1734 ( year 1977 ) NoStop

work page doi:10.1088/0022-3719/10/10/014 1977