arXiv:0903.3699v2 [cond-mat.str-el] 29 May 2009Spinresonanceinachiral helimagnetJun-ichiro KishineDepartment of BasicSciences, KyushuInstituteof Technology, Kitakyushu804-8550, JapanA. S. OvchinnikovDepartment of Physics, Ural State University, Ekaterinburg 620083, RussiaIt is suggested that marked features of symmetry breaking mechanism and elementary excitationsinchiral helimagnet come upas visible eects inelectronspinresonance (ESR) prole. Underthemagneticeldappliedparallelandperpendiculartothehelicalaxis,elementaryexcitationsarerespectivelydescribedbythehelimagnonassociatedwithrotational symmetrybreakingandthemagnetickinkcrystal phononassociatedwithtranslational symmetrybreaking. WedemonstratehowtheESRspectradistinguishtheseexcitations.In magnetism, chirality means the left- or right-handedness associated with the helical order of mag-neticmoments. Helimagneticordercanarisefromspon-taneoussymmetrybreakinginsystemswithcompetingexchange interactions[1] (symmetrichelimagnets), or itcanbestabilizedbytheDzyaloshinkii-Moriya(DM)an-tisymmetric exchange interaction[2, 3], which is real-ized in crystals lacking rotoinversionsymmetry (chi-ralhelimagnets). To clarify physical outcome of thechi-ral spin modulation is of great interest, especially in con-nectionwiththesymmetry breaking mechanism andthespectrumofelementaryexcitationswhicharequitesen-sitivetothedirectionoftheappliedmagneticeld.Inachiral helimagnet, underthemagneticeldpar-alleltothehelicalaxis,thegroundstate(GS)generallychangesfromplanarspiral toconical states[Fig. 1(a)].Theincommensuratemodulationperiod2/Q0isxedthroughQ0= tan1(D/J), whereDandJarenearest-neighbor DM interaction and ferromagnetic exchange in-teractionstrengths[4, 5]. The GS has innite degen-eracyassociatedwitharbitrarychoice of the originofthephaseangle0. Consequently, therotational sym-metryaroundthehelical axisis spontaneouslybroken.Then,thereappearshelimagneticspin-wave(chiralheli-magnon)mode[6] astheNambu-Goldstone(NG)mode,which is well described in conventional spin wave picture.The chiral helimagnon has been studied in the context ofcubicmagnetMnSi[7, 8] anditspeculiarnaturehasat-tractedmuchattentioninitsownright[9].Ontheotherhand, underthemagneticeldappliedperpendicular to the helical axis, the GS possesses a peri-odic array of the commensurate (C) and incommensurate(IC) domains partitioned by discommensurations (DCs),i.e. the internal lattice whichis calledmagnetic kinkcrystal(MKC)orsometimesreferredtoaschiralsolitonlattice[4,5]isstabilizedasshowninFig.1(b). Actually,formationoftheMKCstateisreportedinCuB2O4[10].Thisstateisalso regarded asnon-trivial topological GS.The topological GS in chiral magnet has attracted activeattentionfromvariousviewpoints[11]. Asthemagneticeldstrengthincreases, thespatial periodof MKClat-tice, Lkink, increasesandnallygoestoinnityat thecriticaleldstrength. Recently, weshowedthatthisin-ternal latticeexhibitsmutual slidingwhichmaybeex-perimentallydetectable[12,13]. Inthiscase,theGShasinnite degeneracyassociatedwitharbitrarychoice ofthecenterof massposition. Consequently, thetransla-tional symmetryalongthehelical axisisspontaneouslybroken. Then, theelementaryexcitationsaredescribedbyphononmodeofcorrelatedkinks. Whatisinterest-ingisthatwecancontrol thesizeof therstBrillouinzone of the MKC lattice upon changing the magnetic eldstrength.FIG. 1: (a) Conical and(b) magnetic kinkcrystal (MKC)states. Thehelicalaxisisz-axis.Theelementaryexcitations inthe kinkcrystal statewasrstinvestigatedbySutherland[14]. Heconsideredthe sine-Gordonmodel for a single scalar eld corre-sponding to the tangential -mode of the planar XY spinsandfoundthattheelementaryexcitationsconsistoftheacousticandopticalbandsseparatedbytheenergygap.Theacousticbandis formedout of correlatedtransla-tionsoftheindividual kinksandcorrespondstogaplessNGbosons. Theoptical bandcorrespondstorenormal-ized Klein-Gordon bosons. In chiral helimagnet, we needtotakeaccountofnotonlythe-modebutthelongitu-dinal-mode(isananglebetweenthespinvectorandthehelical axis). Inpreviousworks[12], wepointedoutthat the-mode acquires an energy gap originating fromtheDMinteraction.Then, natural questionarisesastowhethertheheli-2magnon and MKC phonon have observable consequencesforthemagneticresponseusingESRtechnique. Inthispaper,wedemonstratehowthesymmetrybreakingpat-terns and the elementary excitations come up in the ESRsignals.IntheESRexperiment, thestaticmagneticeldH0isappliedtocauseLarmorprecessionofmagneticspins.Thensupplying electromagneticenergycarriedbymi-crowave radiation, resonant absorption occurs at the pre-cession frequency. The microwave is described as the uni-formoscillatingmagneticeld(rf eld) h(t) polarizedinthedirectionperpendiculartoH0(Faradaycongu-ration). Therf eldgivesrisetotheZeemancouplingwithspin, HZ= H(t)S0, where H(t) =geBh(t)(geis theelectrons g-factor andBis theBohr mag-neton) and S0is the uniform(q =0) component ofthespinvariable. For H(t) =Hecos (t), theESRspectrum(absorbedenergyper unit time) is givenby,Q() =H2 () /2, whereewith=x, y, zde-notestheunitvectoralongx, y, andzaxis[Fig. 1], re-spectively, andisamicrowavefrequency. Theimag-inary part of the dynamical susceptibility, () =_1 e/kBT_C () /2, is relatedtothe correlationfunctionC ()= S0() S0)throughtheuctuation-dissipationtheorem. Inquantummechanical language,the Lamor precession corresponds to equally spaced Zee-mansplittingoftheenergylevels. Becauseoftheequalspacing of the quantumenergy levels, the quantum-classical correspondenceexactlyholdsandtheclassicalfrequencyisequaltoquantumoneasfarasweconsiderGaussianuctuations.First, weconsider thecasewherethemagneticeldis appliedparallel to the helical axis (z-axis) andtherf eld is polarized along the y-axis. Then, the ele-mentary excitations are described as spin waves overtheconical magneticstructure. Aquantizedspinwaveis helimagnon. Then, the ESRspectrumis givenbyQhmag() = H2yyy () /2.Tocomputeyy (),weas-sumethatthemagneticatomsformathreedimensionallatticeandauniformferromagneticcouplingexistsbe-tweentheadjacentchainstostabilizethelong-rangeor-der. Then,theHamiltonianis interpreted as an eectiveone-dimensional model based on the interchain mean eldpictureandiswrittenas,H = J2

j[eiQ0cS+jSj+1 +eiQ0cSjS+j+1]J

jSzjSzj+1 +K

j(Szj)2 H0

jSj, (1)whereSjrepresentsaspinlocatedatthej-thsitealongthehelicalaxis(z-axis)andSj= Sxj iSyj.Themono-axial DMvector is D=DezandJ =[J+iD[ =J2+D2. Thelatticeconstant is c. Weinclude theeasy-planeanisotropywithstrengthK. ForH0=0,FIG.2: (a)HelimagnondispersionsforH0/H0c= 0,0.7,and1.. WetookD/J=0.5andK/J=2. Blackdotsindicatethelocationoftheresonanceenergies. (b)FielddependenceoftheresonanceenergyasafunctionofH0/H0c.theplanar helicalstructure isstableundertheconditionK/J>1 _1 + (D/J)2whichisassumedtobesat-ised. For0