Robust chirality-induced spin selectivity in topologically chiral molecular knots

Domain: physics
Published: 2025-12-26
Editorial outcome: major_revision

Abstract
Molecularknotswithtopologicalchiralitydemonstratesuperiorspinselectivitycomparedtoconventionalchiralmaterials,exhibitingultra-highspinpolarizationofnearly90%,conductivityincreasedbytwoordersofmagnitudeandhigh-temperaturestability(upto350°C).However,thephysicaloriginofthisphenomenonremainselusive.Weestablishafundamentaltheoreticalframeworkforthechirality-inducedspinselectivityintopologicallychiralmolecularknotsandidentifyitsessentialconditions.Ourcalculationsshowthatasingletrefoilknotmoleculeexhibitsspinpolarizationexceeding60%alongwithsignificantconductivity,robustnessunaffectedbylatticereductionorstructuralstrain.Importantly,whenthetopologicalknotdegeneratesintoatrivialstructure,accompaniedbythetransitionfromtopologicalchiralitytostructuralchirality,thespinpolarizationsharplydeclines,demonstratingastrongcorrelationbetweentheultrahighspinpolarizationandtheknottopology.Thisworkelucidatesthemechanismofspinselectivityinknotmoleculesandprovidesnewdesignprinciplesforfuturespintronicdevices.

Introduction
Theinterplaybetweenelectronspinandstructuralchiralityleadstoaspin-dependenttransportphenomenonknownaschirality-inducedspinselectivity(CISS)1–6. This effect was first discovered in photoelectron spectroscopy measurements on DNA molecules7 and has since been reported in various chiral systems such as helicenes8,9, chiral crystals10,11, donor-acceptor molecules12, and supramolecular assemblies13–16.AkeysignatureofCISSisthegenerationofhighspinpolarization(upto80%)inchargetransportevenwithoutanyexternalmagneticfield17,18,andthespin-polarizedstatethatcanpersistevenatroomtemperature19.Thiscapabilityformanipulatingspinstatesandgeneratingspincurrentsinnon-magneticmaterialsathightemperaturesmakesCISSpromisingforspintronicdevicedesign20–22,asevidencedbyrecentdemonstrationsofroom-temperaturespin-emittingLEDs23,24 and high-efficiency chiral perovskite photovoltaics25–29. Beyondspin-dependentelectronics,CISSalsoinfluencesenantiospecificchemistry30, catalytic activity31–34, and biological functions35,36. Althoughmultipletheoreticalmodelshavebeenproposedtounderstandthiseffect37–46,theystillexhibitsystematiccontradictionswithorexplanatorylimitationswhenconfrontedbyexperimentalobservations2,andauniversaltheorycapableoffullyexplainingtheCISSeffectinallchiralmaterialsremainselusive47.

ThetraditionalCISSeffectistypicallyrelatedtostructuralchirality, such as helical chains that break the inversion symmetry to produce left- or right-handed chirality (Fig. 1a). However, recent experiments point to a different kind of system where strong CISS ariseswithoutsuchconventionalstereogenicunits,namely,inmoleculartrefoilknots19(Fig.1b).Thesemoleculesderivetheirchiralitynotfromahelicalshapebutfromthetopologicalconfigurationoftheirentanglement,whichisdeterminedbythesequenceandorientationofthe strand crossings (Fig. 1c, d). This form of topological chirality constitutesadistinctlevelofspatialorganization,onethatappearstoconferexceptionalstabilityontheresultingspinselectivity.Experimentally,moleculesthatdisplaythisknot-basedchiralityachievespinpolarization as high as 90%, accompanied by an increase in conductivityofroughlytwoordersofmagnitudecomparedtotraditionalchiralmolecules, and retain performance up to temperatures near 350∘C19,exhibitingvariousadvantagesovertraditionalCISS1–6.TheseexperimentalobservationsestablishtopologicallychiralknotsasadistinctiveplatformforachievingenhancedandrobustCISS.Althoughpioneeringexperimentalworkhaspresentedatheoreticalmodeltoexplainthehighlyspin-polarizedcurrentinthesesystems,themodelitselfremainsgroundedinasimplifiedpicture,thatofauniformhelicalchainwithstructuralchirality19.Thisrepresentation,althoughvaluable,doesnotfullycapturethespatialvariationsandspecificgeometricalarrangementinherentinaknottopology.Asaresult,itofferslimitedinsightintothedistinctivephysicalprocessesthatgovernCISSintheseentangledstructures.Adedicatedtheoreticalframeworkthatdirectlyaddressestopologicalchiralityisstillneededtoclarifytheunderlyingmicroscopicmechanisms.

In this work, we construct a series of spin-dependent transport devicesbasedontrefoilknotmoleculestoestablishatheoreticalframeworkfortheCISSeffectintopologicallycomplexsystems.Acentralchallengeistoclarifytheoriginofspinpolarizationinsuchstructures.Althoughestablishedmechanisms,includingintramolecularspin-orbitcoupling(SOC)37,38,48,49, chiral-induced orbital polarization43, spin-interfaceinteractions50–52,andnon-Hermitianskineffectsunderopenboundaries46,havebeeninvokedtoexplainCISSinotherchiralmolecules, they do not directly account for the role of global molecular topology. Recent studies highlight that SOC in chiral molecules is atomicinnaturebutalsoacomponentrelatedtomoleculargeometryandchirality53–55.Intrefoilknots19,56,thethree-dimensionallycurvedandclosed transport pathways further enrich the SOCmechanism. Here, SOCemergesfromthenon-relativisticlimitoftheDiracequationinacurvedspace,intrinsicallylinkedtothegeometryofthetwistedelectrontransportpaths.ThisformofSOC,termedgeometricSOC57,58,isdifferentfromconventionalRashbaoratomicSOCinitsglobalgeometricoriginandstrongspatialvariationalongtheknottedcontour.Moreover,itsmagnitudecansignificantlyexceedthatofconventionalSOC57,providingaphysicalbasisforultra-highspinpolarization(upto90%),observedexperimentallyintrefoil-knotmolecules19.

Results
TheoreticalmodelwithgeometricSOC
Fromatheoreticalmodelingperspective,thetrefoilknotmoleculecanbesimplifiedasaclosedhelicalconfigurationwithcontinuouslyvaryingcurvature,formingitscharacteristictopologicalknotstructure(Fig.1d).Thethree-dimensionaltwistedpathwaysinthisconfigurationincreasethespatialdegreesoffreedomavailableforelectrontransport,givingrisetocomplexSOCfeatures.Crucially,conventionalSOCmodelsforCISS,oftenderivedfromtheDirac equation in a flat spacetimeelectromagneticfield37,38,48,arefundamentallyinapplicabletothesecontinuouslydeformingtrefoilstructures.Here,wedemonstratethatthetrefoil-knotarrangementofmonomersgeneratesgeometric SOC. Specifically, within the helical structure of trefoil knot molecules, quantum interference fundamentally occurs between direct (i-j) and indirect (i-k-j) paths for electron transmission, introducinganextraspin-dependentBerryphaseduringelectronmotion(see“Method”).ThisphenomenonconstitutesthemicroscopicoriginofgeometricSOC,whoseformisdeterminedbymolecularchirality.LikeaconventionalSOC,thisgeometricSOCcaninducespinflippingduringelectrontransport.Toquantitativelycharacterizethismechanism,weconstructaclosedlattice-knotmodelinwhichthemoleculeisdiscretizedintoNlatticesitesspacedatequalarclength(positionsofsites1andNareshowninFig.2a;seealsoSupplementaryMovie1).Thespatialcoordinatesoftheithsitearegivenby

Ri=Ri(θi)=[rsinθi+Rsin(2θi),ζrcosθi−ζRcos(2θi),hsin(3θi)]T,

where R+r and R−r denote the maximum and minimum radii, respectively,inthetopviewprojectionofthetrefoilknot;ζsymbolizesthechiralityofthemolecular(lefthandedwhenζ=1andrighthandedwhenζ=–1); h represents the molecular height and θ i is the angle coordinateoftheithlattice.TiandNiaredefinedastheunittangentandnormalvectors of the twisted path (SupplementarySection A).

Hgeo=ℏ(a×p)⋅σ/(4mc2)=−Σiℏκi(Ni×pi)⋅σ/(4m)(wherepiistheelectronmomentum,miselectronmass,andσisthePaulivector)inthe equation. Further calculation demonstrates that this geometric SOCisexplicitlygovernedbylattice-dependentcurvatureκi,whichcanbecalculatedbyκi=|R0i×R0i0|/|R0i|3.Tovalidatethis,wecomputethereal-spacedistributionofκiforatrefoilknotwithR:h=2:1(Fig.2b).Furthermore,κiexhibitsstrict2π/3-periodicityversusθiinallmodels(Fig.2c,d),confirmingthatthemagnitudeofκi,directlyreflectingthetwistedgeometryin3DandtherotationalsymmetryoftheknotwithC3,dictatesthemodulationoftheSOCstrengththroughgeometricscaling.

For this trefoil knot model, given that Hgeo is determined, the overall tight-binding Hamiltonian can be formulated as H=Hmol+Hgeo+Hel+Hd, where Hmol+Hel=p2/2m+V(x,y,z). Then, we discretize the Hamiltonian Hmol and Hgeo within the Wannier representation, and the discretization form of the former is given by

Hmol=Σn=1Nεncy ncn+Σn=1N(tncy ncn+1+H:c:)

where cy n=(cy n,↑,cy n,↓) and ε n=⟨Ψn|Hmol|Ψn⟩ represent the creation operator and the onsite energy located at the nth lattice site, tn=⟨Ψn|Hmol|Ψn+1⟩ signifies the nearest-neighbor hopping integral between the nth and (n+1)th lattice sites and can be regarded as a constant.Here,Ψnistheelectronwavefunctionatthenthlatticesite.Similarly,Hgeocanbediscretizedas

Hgeo=iλsoc/2Σn=1N(κn+κn+1)(σn+σn+1)cn+1+H:c:

whereλsoc=ℏ2/16mSwithSbeingthearclengthbetweentwoadjacentlattices(SupplementarySectionA).Withintheframeworkof(Ti,Ni,Bi)andforeachlattice,BiisgivenbyBi=Ti×Ni.Thisunitvectorhasthreecomponents,namelyBi=(Bix,Biy,Biz),andthenthePaulimatrixhasthe relationσi=σxBix+σyBiy+σzBiz,where(σx,σy,σz)denotesthethree dependent Pauli matrix. Unless otherwise specified, we adopt λsoc=0.737t⋅nm to perform all calculations. For example, t= 0.3eV, resultinginλsoc≈0.221(eV⋅nm), satisfies well the basic properties of electronichoppingandthegeometricSOCestablishedhereintrefoilknotmolecules.

FundamentalconditionsforCISSinknotmolecules
Using the Hamiltonian model described above, we identify the key conditionsforspin-polarizedelectrontransportthroughtrefoilknotmolecules.Webeginbyanalyzingatwo-terminalsetup(γ=2)inwhichatrefoilknotmolecule(N=60latticesites)bridgestwonon-magneticmetal electrodes (Fig. 3a, top). Theoretical calculations reveal that when the electrodes are coupled to the lattice sites i=1 and j=31, dividingthemoleculeintotwoperfectlysymmetrichalves,thesystemexhibitshighconductivitybutnospinpolarizationinthetransmittedchargecurrent(Fig.3a,bottom).Thisbehaviorcomesfromthepresenceoftwosymmetrictunnelingchannels.Electronspassingthrough these channels in the curved space acquire Berry phases e±iΦL and e±iΦR (L,Rdenoteleftandrightpathways),whichareequalinmagnitudebutoppositeinsign.Theresultingphasecancellationsuppressesquantum interference, and consequently spin polarization, a mechanismpreviouslyreportedinclosedhelicalmolecularrings59–61, althoughtheoriginsofSOCdiffer.Wefurthertestanopentrefoilknotconfiguration,wheretheknotiscutatonepoint,andelectrodesareplacedattheresultingterminals(Fig.3a,right).Numericalsimulationsshowthatalthoughconductivityremainshigh(SupplementaryFig.2), spinpolarizationremainsnegligible.Theseresultsconfirmthatasingletransportchannelthroughthetrefoilknotisinsufficienttogeneratespin polarization, consistent with an earlier theoretical conclusion drawnforsingle-channelhelicalsystems38.

Whenthetwoelectrodesareattachedtolatticesitesthatbreakthe local left-right symmetry of the trefoil knot, the Berry phases induced by spatial curvature no longer cancel. This leads to the emergenceofspinpolarization,asillustratedinFig.3b.Specifically,in theasymmetrictwo-terminalconfiguration(1,15)60,thesystemmaintains robust conductivity (Supplementary Fig. 2) and exhibits spin polarization of conducting electrons up to 20%. In particular, the mirror-symmetricconfiguration(1,47)60showsspinpolarizationofthesamemagnitudebutoppositesign(Fig.3b,lowerpanel).ThisinversionofthespindirectionuponspatialmirroringreflectstheintrinsicC3symmetryofthetrefoilknotandstronglyconfirmsthatthegeometricSOC term correctly captures the relationship between molecular geometryandspin-dependenttransport.Theseresultsestablishthatasymmetricmultichanneltransportisnecessaryforspinselectivityintrefoilknots,withgeometricSOCprovidingakeymechanismforspinpolarizationinthesetopologicallychiralknotmolecules.

Robustnessofhigh-spinpolarizationinknotmolecules
Tofurtherelucidatetheoriginofhighspinpolarizationintrefoilknotmolecules, we model a four-terminal transport device (γ=4) that mirrorsexperimentalsetupsforchargecurrentmeasurement19.Inthisconfiguration,thethreelowestlatticesitesareconnectedtoametal substrate, while the uppermost site is coupled to an STM probe (Fig.4a).WefirstconsideratrefoilknotwithN=20latticesites,coupledatpositionsi=3,j=6,14,20.ThecalculatedconductancesG↑andG↓revealaspinpolarizationofupto60%(Fig.4b,top).Reversingthemolecularchirality from left- to right-handed also reverses the spin polarizationdirection,confirmingthattheeffectisintrinsicallytiedtothetopologicalchiralityoftheknot.Moreover,thefour-terminalsetupexhibithighelectricalconductivity,especiallyneartheFermilevel, wherefiniteconductanceindicatesmetallictransportbehavior(SupplementaryFig.3).ThiscontrastswithconventionalchiralmaterialssuchasDNA7andhelicenes8,whichtypicallyshowlowconductanceduetosignificantbandgaps.Theobservedmetallicconductionisconsistentwithexperimentalreportsofnearlytwoordersofmagnitudeconductivityenhancementrelativetotraditionalchiralsystems.AdditionalsimulationsforaknotwithN=60latticesites(atconstantmolecularlength)showthatthemaximumspinpolarizationremainsaround60%(Fig.4b,bottom).Inparticular,thisvalueisslightlylowerthanthe~75%polarizationobtainedintheanalogousthree-terminalmodelwiththesamenumberoflatticesites.Thisdecreasecanbeattributedtotheadditionalelectrodeimprovingtheoverallsymmetryofthetransportpathways,consistentwiththeC3rotationalsymmetryofthetrefoilknot,whichpartiallycounteractsthespin-dependentchiralBerryphasegeneratedbyspatialcurvatureandthusmoderatetheresultingspinpolarization.

Inparticular,inthefour-terminaldevicebasedonthetrefoilknot, spin polarization remains stable at approximately 60% as the total latticedensityincreasesfromN=20to100,whiletheoverallmolecularlengthremainsconstant(Fig.4c;SupplementaryMovie1).These results indicate that increasing lattice resolution does not significantly alter the spin polarization. Rather, the polarization is governed primarilybytheintrinsicgeometricconfigurationofthemoleculeandthecouplingscheme,specificallythenumberandplacementofelectrodes, rather than by the total number of discrete lattice sites. This behavior stands in sharp contrast to conventional CISS effects observed in helical molecular chains, where spin polarization often varies considerably, typically increasing with either the number of latticepointsperunitlengthorthetotalchainlength.Together,thesefindingssuggestthattherobustspinselectivityintrefoilknotsstemsfromthemolecule’sintrinsictopologicalcharacter:thestableknottedgeometryenforcesaformoftopologicalinvariance,whichinturnensuresconsistent spin polarization across different lattice discretizations.

TopologicalphasetransitionofCISSinknotmolecules
Toclarifytheroleofknottopologyinstabilizinghighspinpolarization, wesystematicallyvarythestructuralparameterRinatrefoilknotwithN=50 lattice sites. Changing R progressively suppresses the knot geometryandultimatelyremovesitstopologicalcharacter,allowingustotrackthecorrespondingevolutionofspinpolarization.AsRdecreasesfrom2.0to1.1,thecentraltriangularregioncontractswhiletheknottopologyremainsintact(Structure1,Fig.5b;SupplementaryMovie2).Inthisregime,spinpolarizationstaysstablenear60%.AtthecriticalvalueR=1.0,thecentralregioncollapsestoacrossingpoint(Structure 2, Fig. 5b) and the spin polarization drops markedly, although it remains finite. When R is further reduced to 0.9, the topological knot is eliminated: the helical ring opens into a trivial conformation that can be continuously deformed into a simple annulus, while maintaining a fan-like structural chirality (Structures 3–4,Fig.5bandSupplementaryMovie2).Thisstructuralchangecorrespondstoatopologicaltransitioninwhichtopologicalchiralitygiveswaytostructuralchirality.Uponcompletingthistransition,spinpolarizationfallstoapproximately20%,andconductivityissubstantiallyreduced(SupplementaryFig.5).Sequentialsuppressionofspinpolarizationastheknottopologyisremovedprovidesclearevidencethatthepreviouslyobservedhighpolarization(>60%)isprotectedbythenontrivialtopologyofthetrefoilknot.

AstheparameterRgraduallydecreases,thetrefoilknotdeforms intoafan-shapedgeometry.AtR=0,thestructureopenscompletely intoaverticallyorientedcircularring(Structures5and6,topview, Fig.5bandSupplementaryMovie2).Althoughtheknottopologyislost inthislimit,thepersistenceofasymmetrictransportpathswithnon-zero curvature maintains a finite degree of spin polarization in the transmitted electrons.In contrast, reducing the height parameter h flattens the molecular structure (Structures 5-6, side view, Fig. 5b). Whenhreacheszero,thesystemdegeneratesintoacircularplanarringwithuniformcurvature,andthespinpolarizationdecaysasymptoticallytozero.Thismarkeddifferenceinbehaviorrevealsthatspinselectivityinstructurallychiralsystemsishighlysensitivetogeometricdeformation,apropertythatfundamentallydistinguishesitfromtherobustspinpolarizationobservedintopologicallychiralknots.

Methods
ThederivationofgeometricSOCincurvedpathways
ThegeometricSOCinacurvedmolecularstructure,suchasatrefoilknot,canbederivedstartingfromacontinuumHamiltonianoftheform

Hgeo=ℏ(a×p)⋅σ/(4mc2)

wherea=−κc2Nisaneffectivegaugefieldrelatedtocurvature,pisthemomentumoperator,andσdenotesthePaulimatrices.IntheFrenet-Serret frame, the momentum is decomposed into tangential and normalcomponentssuchasp=p∥T+p⊥N.SubstitutingthisexpressionintotheHamiltonianleadsto

Hgeo=ΣiℏmκipkBi⋅σi

whereBiisalocalaxialvectordependingontheFrenetframe,andσi=σxBix+σyBiy+σzBiz.ToensureHermiticity,theproductp∥σiisymmetrized,givingthefollowingform

Hgeo=Σi8ℏmκi(pkσi+σipk)

We then proceed to the second quantization. Using the tangential momentumoperatorp∥=−iℏ∂/∂Tandapplyingadiscretizationoffinitedifferenceonadiscretechainofsites,weapproximatetheactionofp∥onalatticewavefunctionFn as

pkFn=−iℏFn+1−Fn−1/2S

whereSisthelengthofthearcbetweentwoneighboringsites.Afteralgebraicmanipulation,thesecond-quantizedgeometricSOCHamiltoniantake

Hgeo=λsoc/2Σn=1N(κn+κn+1)(σn+σn+1)cn+1+H:c:

Here,cynandcnarethecreationandannihilationoperatorsatthesiten,κnisthelocalcurvature,σndenotesthePaulivectorandλsocistheeffectivecouplingstrengthofgeometricSOC.FurtherdetailsofthederivationareprovidedinSupplementarySectionA.

We describe the conductance and spin polarization by calculating spin-resolvedconductanceusingatwo-terminaldevice(γ=2)asanexample.Thedevicemodelincludesatrefoilknotmoleculecoupledtoleft(L)andright(R)metalelectrodes,withadditionalsemi-infinitevirtualprobesattachedtoeachlatticesitentomodelphase-breakingprocessesandensurecurrentconservation.WithintheLandauer-Büttikerframework,thechargecurrentIν(ν=L,R)flowinginthephysicalelectrodeandthecurrentIninthevirtualelectrodencanbecomputedbelow

Iν=e/ℏΣα,σ∫1−1Tσν,α[fν(E)−fα(E)]dE

In=e/ℏΣα,σ∫1−1Tσn,α[fn(E)−fα(E)]dE

whereTσν,α(E)denotesthespin-dependenttransmissionprobabilityfromelectrodeαtoν,andfμ(E)=[eβμ(E−μμ)+1]−1istheFermi-Diracdistributionintheelectrodeμ,withβμ=(kBTμ)−1andchemicalpotentialμμ.

To determine the local chemical potentials μn in the virtual probes,weimposetheconditionIn=0foralln,leadingtoasystemofnonlinearequations

Σα,σ∫1−1Tσn,αfn(E)dE=Σα,σ∫1−1Tσn,n0fn0(E)dE+Σα,σ∫1−1Tσn,νfν(E)dE

Tosimplifycurrentcalculations,werestrictthecalculationoflocalcurrentInwithinthescopeofthelinearresponseforthismatrixequation.Furthermore,underlow-voltageconditions,wecanemploytheTaylorseriestoexpandtheFermi-Diracdistributionfunctiontothefirstnontrivialorder,therebyfacilitatingtheapproximationofelectronictransportcharacteristics.TheFermi-Diracdistributionfunctioncanbeapproximatelyexpandedas

fα(E,μα)=feq(E,EF)−∂feq/∂E(E,EF)(μα−EF), (μeq=EF)

whereEFistheFermienergy.SettingEF=0andsubstitutingitintothecurrentconservationequationsyieldsalinearsystem

Σα,σ∫1−1Tσn,α(−∂feq/∂E(E,EF))dEΣn∫1−1Tσn,n0(−∂feq/∂E(E,EF))dE=∫1−1(−∂feq/∂E(E,EF))[Tσn,L(E)μL+Tσn,R(E)μR]dE

In this way, the matrix equation comprising n equations can be reformulatedasanequationMb⋅μ=ν,andthesolutionstothismatrixequationcanbeobtainedbyinversionofthematrix.Furthermore,thesolutionofthematrixequationwillallowustodeterminethespecificformofchemicalpotentialforeachlattice.ThematrixMb,inthiscontext,canbeexpressedas

Mb=Σα≠1,ν,σ∫1−1Tσ1,α(E)(−∂feq/∂E(E,EF)) ... Σσ∫1−1Tσ1,n(E)(−∂feq/∂E(E,EF))

Bysolvingthematrixequationmentionedabove,wecandeterminethechemicalpotentialofthevirtualprobeandsubsequentlycalculatethespin-dependentchargecurrentsflowingfromthemetalelectrodeLasfollows

IL,σ=e/ℏ2Σα≠1,ν∫1−1TσL,n(E)[−∂feq/∂E(E,EF)](μL−μR)dE

Withinthelinearresponseregime,theconductivityissimplygivenbyGσ=IL,σ/ΔV,whereΔV=μR−μL.Furthermore,underlow-temperatureconditions,theFermidistributionfunctionreducestotheDiracdeltafunction,andthespinconductivitycanberepresentedasfollows

Gσ=e/ℏ2[TσL,R(EF)+ΣTσL,n(μL−μn)=Δμ]

Thus,wewillderivethespecificexpressionforspinpolarizationPsas

Ps=G↑−G↓/G↑+G↓

Dataavailability
Thedatathatsupporttheplotswithinthispaperandotherfindingsofthisstudyareavailablefromthecorrespondingauthoruponrequest.Sourcedataareprovidedwiththispaper.

Codeavailability
Thecodeusedforthisstudyisavailableat https://github.com/ehehf197907/CISS-Molecular-Knots.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript presents a groundbreaking theoretical framework for understanding the Chiral Induced Spin Selectivity\n(CISS) effect in topological knot molecules. The work is highly original, timely, and scientifically rigorous, offering the first\ncomprehensive model that accounts for the remarkably high spin polarization, enhanced conductivity, and exceptional\nthermal stability observed in recent experimental studies. The introduction of geometric spin-orbit coupling (SOC) as a key\nmechanism distinct form conventional SOC in structurally chiral systems is particularly insightful. The multi-terminal device\nsimulations are well-designed, and the results robustly demonstrate the role of topological chirality and structural asymmetry\nin spin-selectivity transport. This study constitutes a significant advance in the field of chiral spintronics and strongly merits\npublication in Nature Communications, pending minor revisions as outlined below.\n(i) Although the Supplementary Information provides some details on the derivation of Geometric SOC, it would strengthen\nthe manuscript to include a more explicit derivation or intuitive explanation of the geometric SOC term in the main text or\nMethods section.\n(ii) The current theoretical value (~60%) shows a discrepancy with the experimental value (~90%), and the author has\nattributed this gap to the differences between the single-molecule model and the multilayer structure of the thin film. It is\nrecommended to supplement the Discussion section with information on the potential contributions of multilayer effects and\nmany-body interactions (such as electron-electron correlations and electron-phonon coupling) to high polarizability in single\nmolecules.\n(iii) Please ensure consistency in reference formatting, including journal name abbreviations, placement of publication\nyears, and punctuation. Additionally, double-check all in-text citations for accuracy and correspondence with the reference\nlist.", "reviewer_2": "This paper by Sun, et al. establish a theoretical framework to explain the chirality-induced spin selectivity (CISS) observing\nin chiral trefoil knot molecules, thereby propose a new direction “knot-driven spin selectivity” in chiral spintronics. Given that\nthe chiral knot systems are characterized by topological chirality rather than the structural chirality typically associated with\nthe traditional CISS effect, the authors propose an alternative theoretical model. This model combines a curvature-driven\ngeometric spin-orbit coupling (SOC) with a decoherence process to explain why experimental measurements on trefoil knot\nmolecules report such strong spin polarization alongside high conductance.\nAs far as I'm concerned, this manuscript has three main highlights, which can be summarized as follows: (i) The observation\nof the CISS effect in chiral trefoil knot molecular systems marks a significant experimental breakthrough. However, this effect\noriginates from topological chirality, unlike the traditional CISS effect which is grounded in the structural chirality of system\nlike helical chains. This work clearly distinguishes between these two types of chirality and constructs a new theoretical\nframework to elucidate the CISS effect induced by topological chirality. (ii) This work introduces a Berry-phase-related\ngeometric SOC arising from molecular curvature to understand the emergence of spin selectivity and polarization. This\nconcept is novel and provides valuable insights for future studies on irregular chiral systems. (iii) It reveals for the first time\nthe evolution of the CISS effect during the phase transition between topological chirality and structural chirality. The\nnumerical results illustrating this transition process are both compelling and reliable.\nConsidering the above mentions, I support the publication of this manuscript in Nat. Commun. Below is a list of some\nspecific concerns that will be useful for the development on this work.\n(i) This theoretical study primarily draws upon recent experimental research on the CISS effect in trefoil knot molecules (J.\nAm. Chem. Soc. 145, 26791 (2023), e.g., Ref. [66]). As far as I am aware, the experimental work also proposed a theoretical\nmodel to explain the CISS effect in their trefoil-knot molecules (referring to the appendix section of the paper). Why, then, did\nthe authors establish another theoretical framework to study the CISS in this type of system in this study? I recommend that\nthe authors address this point in the revised manuscript.\n(ii) Given that the geometric spin-orbit coupling (SOC) resulting from the spatial curvature of electron transport serves as the\nprimary physical mechanism underlying the spin selectivity in this class of chiral materials, it is recommended that the author\nprovide a detailed discussion and explanation of the main causes and computational methods for generating this type of\nSOC, i.e., Geometric SOC, in the \"Methods\" section of the paper.\n(iii) I recommend that the authors modify and improve some statements in the manuscript to enhance the accuracy of the\narguments. For instance, the authors stated, \"the ultra-high spin polarization in trefoil knot molecules originates from\ntopological protection, exhibiting exceptional robustness.\" However, based on the authors’ computational results, the ultra-\nhigh spin polarization primarily stems from Geometric SOC rather than the topological protection of the topological knot.\nMeanwhile, the topological properties of the knot contribute favorably to enhance the stability of the high spin-polarized\nstates.", "reviewer_3": "The authors claim to present variation of the chiral induced spin selectivity (CISS) effect, that they call \"topological CISS\".\nTheir claim is based on their suggestion that the spin orbit coupling in knot like molecules has three ingredients that they\nclaim do not all exist in the regular spin orbit coupling considered in the CISS. The three ingredients are the regular \"atomic\nlike\" spin orbit coupling (SOC), contribution to the SOC that results from the topology of the system, and the Berry phase that\nresults in interferences that may enhance the SOC.\nAlthough there is a place to discuss the SOC in knot like molecules because of its size, the \"new ingredients\" to the SOC do\nexist in calculations done in the past also for chiral molecules that are not knots. Actually it is well established that there is a\ncomponent that result from the chiral topology that changes sign between the enantiomers. I suggest that the authors will\nlokk at the following references:\n1. T. P. Fay, Chirality-Induced Spin Coherence in Electron Transfer Reactions, J. Phys. Chem. Lett. 12, 1407-1412 (2021).\n2. Fay, T. P.; Limmer, D. T. Origin of chirality induced spin selectivity in photoinduced electron transfer. Nano Lett., 21,\n6696−6702 (2021).\n3. C. Seibel, J. H. Soh, S. Zilberg, A. I. Krylov, Ultrafast Transversal CISS Effect Observed in a Chiral Photoswitching\nMolecule, J. Phys. Chem. Lett., 16, 8514 – 8522 (2025).\nBased on the above I suggest a major revision in which the authors will tune down their claims and will suggest their work\nas a discussion on the SOC in case of knows."}, "author_response": "Significant Changes Made to the Manuscript\n1．The title has been revised to “Robust chirality-induced spin selectivity in topologically chiral\nmolecular knots”, and the term “Topological CISS (TCISS)” has been removed throughout the\ntext to moderate claims. The abstract has been condensed to under 150 words.\n2．Added research background in the second paragraph of Introduction to better justify the proposed\nmodel and method for studying CISS in trefoil knots to explain the reasons why we propose an\nalternative model and method to study CISS in trefoil knot molecules (colored blue).\n3．Expanded the discussion on spin-orbit coupling in CISS by incorporating three suggested three\nreferences (added to the reference) in the third paragraph of the introduction section (colored blue).\n4．Enhanced the Discussion section with analysis of electron-electron and electron-phonon\ninteractions affecting and enhancing spin selectivity (colored blue).\n5．Added and enriched the first subsection in the Method “The derivation of geometric SOC in curved\npathways” (colored blue).\n6．Enriched the Supplementary Information and removed the statements such as “Topological CISS\n(TCISS)” in the Supplementary Information and other files.\n7．Deleted some reference with limited relevance to the research content, thereby restricting the total\nnumber of references to under 70.\n8．Corrected the typographical and language errors throughout the manuscript and the Supplementary\nInformation to improve clarity and accuracy.\nAll changes are highlighted in blue in the revised manuscript.\nReply to Reviewer #1\nWe deeply appreciate the reviewer’s generous assessment of our work as “groundbreaking”, “highly\noriginal”, and a “significant advance”. Their constructive suggestions have significantly enhanced the\nclarity and depth of our manuscript.\nComment (i): \"Although the Supplementary Information provides some details on the derivation of\nGeometric SOC, it would strengthen the manuscript to include a more explicit derivation or intuitive\nexplanation of the geometric SOC term in the main text or Methods section.\"\nResponse:\nWe thank the reviewer for this valuable suggestion. In response, we have substantially expanded the\noriginal subsection titled “Geometric SOC calculation” and retitled it “The derivation of geometric\nSOC in curved pathways” (Page 8, highlighted by blue) to better reflect its content. The enriched\nand revised section now provides the main step-by-step derivation of the geometric SOC term and lists\nthe key expressions to achieve the Hamiltonian of geometric SOC. Note that the Supplementary Section\nA provides more details to derive geometric SOC.\nComment (ii): \"The current theoretical value (~60%) shows a discrepancy with the experimental value\n(~90%), and the author has attributed this gap to the differences between the single-molecule model\nand the multilayer structure of the thin film. It is recommended to supplement the Discussion section\nwith information on the potential contributions of multilayer effects and many-body interactions (such\nas electron-electron correlations and electron-phonon coupling) to high polarizability in single\nmolecules\".\nResponse:\nWe thank the reviewer for this critical insight. We have followed this recommendation and have\nsubstantially expanded the Discussion section (Page 8). We have added a dedicated paragraph that\ndelves into the potential mechanism that could bridge the gap between our single-molecule prediction\nand the experimental thin-film measurements. Specifically, we have enriched the discussion including:\n(1) The possibility of constructive spin filtering through a coherent stack of molecular knots in a\nmultilayer architecture, where the spin polarization could be multiplicatively enhanced.\n(2) The potential role of many-body interactions, such as electron-electron corrections leading to\nenhanced effective SOC, and electron-phonon coupling potentially influence spin relaxation\ndynamics. These factors have contributions to enhance the spin polarization.\nIncorporating these structural and many-body effects into future models is expected to bring theoretical\npredictions closer to the 90% polarization observed experimentally, thus reconciling the current\ndiscrepancy while affirming the physical consistency of the proposed geometric SOC mechanism. We\nmay focus on these topics in our future theoretical investigations.\nComment (iii): \"Please ensure consistency in reference formatting, including journal name\nabbreviations, placement of publication years, and punctuation. Additionally, double-check all in-text\ncitations for accuracy and correspondence with the reference list.”\nResponse:\nWe sincerely apologize for these oversights. We have now conducted a meticulous, line-by-line review\nof the entire reference list and all in-text citations. The references have been reformatted to ensure strict\ncompliance with the Nature Communications style, and we have verified the perfect correspondence\nbetween every citation and its entry in the reference list.\nReply to Reviewer #2\nWe are sincerely grateful to the reviewer for their exceptionally positive and insightful summary of our\nwork's highlights and for their strong support for publication. Their specific concerns have been\ninvaluable in further strengthening the narrative and precision of our manuscript.\nComment (i): “This theoretical study primarily draws upon recent experimental research on the CISS\neffect in trefoil knot molecules (J. Am. Chem. Soc. 145, 26791 (2023), e.g., Ref. [66]). As far as I am\naware, the experimental work also proposed a theoretical model to explain the CISS effect in their\ntrefoil-knot molecules (referring to the appendix section of the paper). Why, then, did the authors\nestablish another theoretical framework to study the CISS in this type of system in this study? I\nrecommend that the authors address this point in the revised manuscript.”\nResponse:\nWe thank the reviewer for raising this important question. Although the original experimental work\n(Ref. [19]) did put forwards a theoretical model and associated methods to explain the observed highly\nspin-polarized current, our work has developed a distinct theoretical framework to analyze the CISS\neffect in this class of systems. The key considerations behind this approach are summarized as follows:\n(1) The original work (Ref. [19]) introduced a qualitative model based on a four-site single helical\nchain, a configuration traditionally employed to represent structural chirality, the experimental\nsystem under investigation actually utilized trefoil-knot molecules. These molecules form a closed\ntopological knot structure, distinct from open chiral chains. As noted by the authors of Ref. [66],\ntheir study addressed topological chirality-induced spin selectivity rather than structural chirality.\nConsequently, applying a structural-chirality model to explore topological chirality may be\ninadequate for revealing the fundamental physical origins of CISS in such knotted systems. This\nlimitation constitutes the primary motivation for developing our alternative theoretical framework\nfocused on topology-induced spin selectivity\n(2) We appreciate the reviewer's comment regarding the theoretical treatment of spin-orbit coupling\n(SOC). In the original model presented in Ref. [19], SOC is incorporated through a next-nearest\nneighbor hopping term within a uniform single helical chain, with the SOC strength parameter λ\ntaken as a fixed constant (λ = 1 meV). While this approach offers a simplified representation, it\ndoes not account for spatial variations in SOC that arise from the specific geometrical arrangement\nof the knot structure. Consequently, such a uniform treatment becomes inadequate for investigating\nthe influence of structural defects on CISS or for capturing potential phase transitions between\ntopological and structural chirality, as the local electronic environment in these systems cannot be\nproperly reflected by a constant SOC value.\n(3) In our theoretical framework, we explicitly introduce an inspired geometric spin-orbit coupling\n(SOC) that stems from the intrinsic curvature of the entire molecular knot, thereby extending\nbeyond the conventional SOC formulation rooted solely in structural chirality. A key feature of this\ngeometric SOC is its direct dependence on the local geometric configuration, being fundamentally\ngoverned by the local curvature of the electron's pathway. Consequently, the strength of this SOC\nis not a fixed parameter but varies with the specific geometrical arrangement of the closed knot\nmolecule. This attribute enables the geometric SOC to more accurately capture the underlying\nphysics of topological knots, offering a clear advantage over the constant SOC approximation\nemployed in the original model.\n(4) Furthermore, our theoretical model offers a quantitative and microscopic perspective on how\ntopological chirality and geometric curvature jointly give rise to the observed high spin polarization\nand conductance. It also enables self-consistent simulation of nonequilibrium quantum transport in\na multi-terminal device geometry, capturing both bulk and interfacial properties in a unified manner.\nTo clarify this important distinction, we have expanded the Introduction (Page 2, Lines 12-19,\nhighlighted by blue) with the following statement: “… Although pioneering experimental work has\npresented a theoretical model to explain the highly spin-polarized current in these systems, the model\nitself remains grounded in a simplified picture, that of a uniform helical chain with structural chirality19.\nThis representation, although valuable, does not fully capture the spatial variations and specific\ngeometrical arrangement inherent in a knot topology. As a result, it offers limited insight into the\ndistinctive physical processes that govern CISS in these entangled structures. A dedicated theoretical\nframework that directly addresses topological chirality is still needed to clarify the underlying\nmicroscopic mechanisms.\"\nComment (ii): “Given that the geometric spin-orbit coupling (SOC) resulting from the spatial\ncurvature of electron transport serves as the primary physical mechanism underlying the spin\nselectivity in this class of chiral materials, it is recommended that the author provide a detailed\ndiscussion and explanation of the main causes and computational methods for generating this type of\nSOC, i.e., Geometric SOC, in the \"Methods\" section of the paper.”\nResponse:\nWe thank the reviewer for reinforcing this point. As detailed in our response to Comment (i) provided\nby Reviewer #1, we have now added and enrich the first subsection in the Method “The derivation of\ngeometric SOC in curved pathways” (Page 8, highlighted by blue). The enriched and revised\nsection now provides the main step-by-step derivation of the geometric SOC term and lists the key\nexpressions to achieve the Hamiltonian of geometric SOC. Note that the Supplementary Section A\nprovides more details to derive geometric SOC.\nComment (iii): “I recommend that the authors modify and improve some statements in the manuscript\nto enhance the accuracy of the arguments. For instance, the authors stated, \"the ultra-high spin\npolarization in trefoil knot molecules originates from topological protection, exhibiting exceptional\nrobustness.\" However, based on the authors’ computational results, the ultra-high spin polarization\nprimarily stems from Geometric SOC rather than the topological protection of the topological knot.\nMeanwhile, the topological properties of the knot contribute favorably to enhance the stability of the\nhigh spin-polarized states.”\nResponse:\nWe are very grateful to the reviewer for this exceptionally precise and correct observation. Our original\nphrasing was indeed inaccurate. We have carefully revised the manuscript to reflect this nuanced\nunderstanding. The specific statement has been revised to (see Page 6, the last sentences of the first\nparagraph, highlighted by blue)\" … Together, these findings suggest that the robust spin selectivity\nin trefoil knots stems from the molecule’s intrinsic topological character: the stable knotted geometry\nenforces a form of topological invariance, which in turn ensures consistent spin polarization across\ndifferent lattice discretizations.\" Similar refinements have been made throughout the text to ensure\nprecise attribution of the effects to their correct physical origins.\nReply to Reviewer #3\nWe are deeply grateful to Reviewer #3 for their thorough assessment and for bringing the relevant\nstudies by Fay et al. and Seibel et al. to our attention. These feedbacks have been instrumental in\nhelping us refine the presentation of our results and more precisely situate the novelty and contribution\nof our work within the broader context of CISS research.\nComments 1: “Although there is a place to discuss the SOC in knot like molecules because of its size,\nthe 'new ingredients' to the SOC do exist in calculations done in the past also for chiral molecules that\nare not knots. Actually it is well established that there is a component that result from the chiral\ntopology that changes sign between the enantiomers. I suggest that the authors will tune down their\nclaims and will suggest their work as a discussion on the SOC in case of knots.”\nResponse:\nWe sincerely thank the reviewer for raising this important point and for pointing out the relevant\nliterature. The reviewer’s insights have helped us to better position our work within the existing field\nand to improve the manuscript significantly. We agree with the reviewer that the components of SOC\nwe discussed, including the contribution from chiral topology and the Berry phase, have indeed been\nexplored in previous studies of chiral molecules, as exemplified by the references provided.\nWe acknowledge that our initial claim of a distinct “topological CISS” effect may have been overstated,\nas the underlying physical mechanisms share commonalities with the established CISS framework.\nFollowing the reviewer’s comment and suggestion, we have made some revisions including:\n(i) We have made major revision to the manuscript to ‘tune down our claims’. Specifically: (1) we\nhave removed the term “topological CISS (TCISS)” throughout the manuscript and reframed\nour work as a detailed investigation into the ‘manifestation and enhancement of the CISS effect\nin molecular knots’; (2) we have revised the article title for greater practical one as “Robust\nchirality-induced spin selectivity in topologically chiral molecular knots”.\n(ii) Following the reviewer’s suggestion, we have made revision to emphasize that the topological\nknot molecules serves as an ideal and pronounced platform to amplify and dissect the various\ncontributions to SOC that are fundamental to the CISS effect, rather than introducing entirely\nnew ingredients. To this end, we have rewritten the third paragraph in the “Introduction”\nsection (Page 2, Second paragraph, highlighted by blue) to address “ … This form of SOC,\ntermed geometric SOC57,58, different from conventional Rashba or atomic SOC in its global\ngeometric origin and strong spatial variation along the knotted contour …”.\n(iii) We have integrated and cited the suggested three references (Fay 2021, Fay & Limmer 2021,\nSeibel et al. 2025) into the Introduction (Page 2, Second paragraph, highlighted by blue) to\nenrich the research background on the spin-orbit coupling, which has been applied to explain\nthe CISS effect and acknowledging the foundational contributions provided by the above three\nliteratures. The above three references, i.e., Ref. [53,54,55] in the main context, are helpful for\nus to introduce the geometric SOC as stressed by “… Recent studies highlight that SOC in\nchiral molecules is atomic in nature but also a component related to molecular geometry and\nchirality53–55…”.\nComments 2: “Based on the above I suggest a major revision in which the authors will tune down\ntheir claims and will suggest their work as a discussion on the SOC in case of knows.”\nResponse:\nWe sincerely thank the reviewer for this suggestion. In response, we have undertaken a major revision\nof the manuscript to moderate our claims and to reframe the work primarily as a discussion of spin-\norbit coupling (SOC) in the context of known systems. Specifically, we have removed phrases such as\n“Topological CISS (TCISS)” and related assertions.\nAlthough the revised version no longer posits a new effect, it underscores a meaningful contribution\nby offering a detailed and quantitative analysis of SOC in a distinctive group of chiral molecules, i.e.,\nmolecular knots. Their intricate and well-defined topology enables clearer separation and a potentially\nstronger manifestation of effects that are subtler in simpler chiral systems. Moreover, our findings\nunderscore the potential of molecular knots as a promising platform for observing pronounced CISS-\nrelated phenomena, owing to their marked topological properties.\nThat is all. We hope that the revised manuscript, with its more measured tone and focused scope, now\naddresses the reviewer’s concerns and meets with their approval."}, {"round": 2, "reviewer_comments": {"reviewer_1": "After reviewing the author's response, I am satisfied and recommend publication as is.", "reviewer_2": "The authors have addressed my concerns, and now the manuscript can be accepted.", "reviewer_3": "The authors responded properly to the reviewers' comments and put their work in the right perspective. The work is\nimportant since it focusses on a special symmetry of molecules that make the molecules very efficient and robust spin filters.\nI recommand publication as is."}, "author_response": "Authors’ Response Letter\nManuscript ID: NCOMMS-25-61102A\nTitle: \"Robust chirality-induced spin selectivity in topologically chiral molecular knots\"\nDear Dr. Tianyu Yang and the esteemed Reviewers,\nWe sincerely appreciate your guidance and the constructive feedback from the three Reviewers. The\ncomments from the all reviewers are copied below:\nComment From Reviewer #1: \"After reviewing the author's response, I am satisfied and recommend\npublication as is.\"\nComment From Reviewer #2: \" The authors have addressed my concerns, and now the manuscript\ncan be accepted.\"\nComment From Reviewer #3: \"The authors responded properly to the reviewers' comments and put\ntheir work in the right perspective. The work is important since it focusses on a special symmetry of\nmolecules that make the molecules very efficient and robust spin filters. I recommand publication as\nis.\"\nReply:\nWe sincerely thank all three Reviewers very much for their positive evaluation and for supporting the\npublication of our manuscript. We sincerely appreciate the time and expertise they dedicated to\nreviewing our work.\nWe are pleased that Reviewer #1 is satisfied and recommends publication as is, that Reviewer #2 finds\ntheir concerns addressed and considers the manuscript ready for acceptance, and that Reviewer #3\nacknowledges the proper response to the comments and recognizes the importance of our work on\nmolecular symmetry for efficient spin filtering.\nIn accordance with the Reviewers' assessments that no further scientific revisions are needed, we are\nsubmitting the final manuscript without changes to its technical content. As requested during the\neditorial check, we have only made the necessary minor formatting adjustments to the manuscript file.\nWe believe the manuscript is now complete and ready for the next steps in the publication process.\nSincerely thank the three Reviewers once again and the editorial team for your guidance.\nYours Sincerely,\nDr. Hua-Hua Fu\nOn behalf of all co-authors.\nSchool of Physics, Huazhong University of Science and Technology,\nWuhan, China."}]

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