Vector chirality and inhomogeneous magnetization in frustrated spin tubes in high magnetic fields Masahiro Sato and Tˆoru Sakai Synchrotron Radiation Research Center, Japan Atomic Energy Agency, Sayo, Hyogo 679-5148, Japan and CREST JST, Japan (Dated: February 6, 2008) 7 Thelow-energyphysicsofthree-legfrustratedantiferromagneticspin-Stubesinthevicinityofthe 0 upper critical field are studied. Utilizing the effective field theory based on the spin-wave approx- 0 imation, we argue that in the intermediate-interchain-coupling regime, the ground state exhibits a 2 vector chiral order or an inhomogeneous magnetization for the interchain (rung) direction and the n low-energyexcitationsaredescribedbyaone-componentTomonaga-Luttingerliquid(TLL).Inboth a chiralandinhomogeneousphases,theZ2 paritysymmetryalongtherungdirectionisspontaneously J broken. Itisalsopredictedthatatwo-componentTLLappearsandallthesymmetriesarerestored 1 in the strong-rung-couplingcase. 1 PACSnumbers: 75.10.Jm,75.10.Pq,75.30.Kz,75.40.Cx,75.50.Ee ] l e - I. INTRODUCTION (l = 1,2,3), J > 0 (J > 0) is the intrachain (in- tr terchain) coupling, and⊥the PBC S~4,j = S~1,j is im- s . Frustrated spin systems1 have been continuously ex- posed. Focusing on the vicinity of the upper critical t a plored for more than five decades. Frustration is consid- field and applying an effective field theory approach, m eredasanimportantkeywordto generateexotic,uncon- we show the possibility of two interesting long-range- - ventional magnetic orders, disorders and excitations in- ordered states: for a certain high-magnetic-field area, a d cluding even spin-liquid states. Actually, frustrated sys- vector chirality z = (S~ S~ )z or an inhomo- n hVl,ji h l,j × l+1,j i tems have provided several peculiar concepts and phe- geneous magnetization along the rung direction occurs o nomena so far: resonating-valence-bond picture, non- in a one-component Tomonaga-Luttinger-liquid (TLL) c [ collinear orders, symmetry-unrelated degeneracy, order- state. In the chiral phase, the Z2 rung-parity symme- 3 by-disorder mechanism, etc. try Slα,j ↔ Slα+1,j, by which Vlα,j changes its sign, is In recent years, frustrated magnets containing four- spontaneously broken, while the inhomogeneous mag- v spin exchanges as well as standard two-spin ones have netization in another phase breaks the one-site trans- 9 been intensively studied.2 In such magnets, fascinating lational symmetry for the rung as well as the rung- 4 5 magnetic orders (nematic, chiral, dimer orders, etc.), parity one. We also predict that a two-component TLL 1 whichorderparameteris definedby productsofspinop- emerges and all the symmetries are preserved in the 1 erators,areshowntobepresent. Inasense,thesenewor- strong-rung-couplingregime. Recentlyaspintubemate- 6 ders are a naturalconsequence of the four-spinexchange rial [(CuCl tachH) Cl]Cl (Ref. 3) has been synthesized 2 3 2 0 becauseforsuchaninteraction,itispossibletoperforma anditsmagneticpropertiescouldbedescribedbyathree- / t mean-field approximation, SαSβSγSδ SαSβ SγSδ+ leg frustrated spin-tube model.4,5,6 This also promotes a i j k l →h i ji k l m SαSβ SγSδ SαSβ SγSδ . Furthermore, it is well the motivation of studying the spin tube (1). i jh k li − h i jih k li - knownthateffectsoffour-spinexchangesarefairlysmall Existingresultsofthemodel(1)aresummarizedhere. d in a large number of real magnets. Thus, to discover IntheS = 1 case,thezero-fieldgroundstatesaregapped n 2 intriguing magnetic orders within spin systems contain- and doubly degenerate with spontaneously breaking the o ing only two-spin exchanges could more stimulate many one-site translationalsymmetry along the chain, at least c : experimentalistsandwouldbe theoreticallyamorechal- when J & 0.5J.7 In addition, a semi-quantitative v lenging issue. ground-s⊥tatephasediagramintheJ -H plane(J >0), i X In one dimension, as representatives of geometrically which only shows gapless and gap⊥ful regimes, ⊥is con- r frustrated spin systems with only two-spin exchanges, structed in Ref. 8; there exists an intermediate magne- a onecanconsiderzigzagspinchainsandthree-legantifer- tization plateau with M = Sz = 1/6. In the case of romagnetic (AF) spin tubes, i.e., ladders with a periodic S = integer and H = 0, thhe ls,yjistem is predicted to be boundary condition (PBC) along the interchain (rung) always gapful and to conserve all symmetries.9 direction. In this paper, we study the latter model in a Beforeanalyzingthequantumspintube(1),todiscuss magnetic field. The Hamiltonian is written as its classical version is instructive. The classical ground 3 state is an umbrella structure as in Fig. 1. In this state, = JS~ S~ +J S~ S~ HSz ,(1) symmetries of the U(1) spin rotation around the spin H Xl=1Xj h l,j · l,j+1 ⊥ l,j · l+1,j − l,ji z axis, one-site translations, and parity transformations along both the chain and the rung directions are allbro- where S~ is spin-S operator on site j of the lth chain ken. Consequently, the system exhibits a finite vector l,j 2 spin moment U(1) degeneracy εK εK around the Sz axis band 4SJ interband width Κ=0 width j+1 3 3SJ⊥ H H K=+−2π/3 lowest gap 0 k 0 k 3 −π π −π π 0 0 saturation state magnon-condensed state j u u H > Hc H < Hc l=1 2 l=1 2 FIG. 2: Magnon bandsin Eq. (2). umbrella structure j-th rung FIG. 1: Classical ground state of the spin tube(1). we obtain the bosonic spin-wave Hamiltonian. As ex- pected,thebilinearpartof˜b reproducesthefree-spin- K,j chirality z = √3(1 H2 ). From this re- wave dispersion ǫK(k). In order to study the low-energy hVl,ji 2 − S2(4J+3J⊥)2 andlong-distance propertiesof the spin tube, we further sult, the vector chiral order is expected to exist even in introduce continuous boson fields Ψ as follows: the quantum version. However,since generally quantum q flduescttruoaytiaonnyisorqdueirtiensgt,roitnigsinnoonntreivdiiamlewnhsieotnhearnodrtnenodtsthtoe ˜b0,j →(−1)j√a0Ψ0(x), ˜b±23π,j →(−1)j√a0Ψ±(x), (5) chiralorderremainsand brokensymmetries are restored wherea is the latticespacing,andx=ja . Usingthese 0 0 in the model (1). andtakingintoaccountthemagnoninteractiontermsup tothelowestorderofthe1/S expansion,wearriveinthe following effective Hamiltonian, II. EFFECTIVE THEORY 1 = dx ∂ Ψ ∂ Ψ µ ρ Hereweconstructtheeffectivetheoryforthequantum Heff Z (cid:20)2m x †q x q− q q(cid:21) spin tube (1) in a high magnetic field. Let us begin with q=X0,+, q − the fully polarized state with M = S. For the state, +g ρ2+g (ρ +ρ )2+f ρ (ρ +ρ )+f ρ ρ 0 0 1 + 0 0 + 1 + ethxeacetnlyercgaylcduilsapteerdsiaosn of one magnon with ∆Sz = −1 is +λ0(Ψ20Ψ†+Ψ†−+h.−c.) − − ǫK(k)=H 2S(J +J )+2SJcosk+2SJ cosK,(2) +λ1(Ψ2+Ψ†0Ψ†−+Ψ2−Ψ†0Ψ†++h.c.)+··· , (6) − ⊥ ⊥ where K (= 0, 2π) is the wave number for the rung where ρq = Ψ†qΨq is the magnon-density field. (This ± 3 Hamiltonian can also be derived via the path-integral and that for the chain, k, is in k < π. The lowest | | approach.10) The first two terms correspond to the free- bandsǫ 2π arealwaysdegenerateduetotherung-parity symmet±ry3, transformation of which induces K K. spin-wave part, and if the chemical potential µq is posi- → − tive,themagnonΨ iscondensed.11Wesetµ =4SJ H As we explaininFig. 2, whenH becomes lowerthanthe and µ = µ = qS(4J + 3J ) H so th0at Hu −and u3SppJer),(lmowagern)oncrsitoifcatlhevallouweesHtcuba=nd4sSbJeg+in3StoJ⊥con(Hdeclns=e Hc′ are±fixed. Other parame⊥ter−s in Eq. (6) arec eval- (are⊥fully condensed). Moreover, as H < H = 4SJ, uated as 1/mq = 2SJa20 (mq = m), g0 = 2Ja0/3, c′ g = (4J + 3J )a /6, f = 8Ja /3, f = 4Ja /3, magnons in the remaining band ǫ are also condensed. 1 0 0 0 1 0 0 λ =(8J 3J )a⊥/6,andλ =(16J 3J )a /12. These Supposing that multimagnon bound states are absent va0lues wo−uld⊥be 0somewhat1changed−due⊥to0high-energy or their excitation energies are higher than those of one- modes, the curvature of the dispersion, higher-order in- magnon states (this is highly expected in antiferromag- teractions, and the hard-core property of magnons ne- netic systems and we have numerically verified it near glected in the spin-wave theory. the saturation), we may describe the low-energy physics around H Hu using one-magnon excitations. A suit- ∼ c able method for such a description is spin-wave theory III. LOWEST-BAND-MAGNON CONDENSED (1/S expansion). It makes spins bosonize as STATE Slz,j =S−nl,j, Sl−,j =b†l,j 2S−nl,j, (3) p Based on the effective theory (6), we investigate the where b is the magnon annihilation operator, and l,j spin tube near saturation. In this section, we consider nl,j = b†l,jbl,j denotes the magnon number. Substitut- thelowest-magnon-condensedcase,whereµ>0,µ0 <0, ing Eq. (3) in the model (1) and introducing the Fourier and max[Hl,H ] < H < Hu. For this case, the low- c c′ c transformation of b for the rung as energyphysicsmustbegovernedbytwocondensedfields l,j Ψ . The effective theory is derived by integrating out 1 b = eiKl˜b , (4) th±e massive magnon Ψ via the cumulant expansion in l,j K,j 0 √3 X terms of the free-spin-wave part of Ψ in the partition K=0, 2π/3 0 ± 3 function. The main effect of the Ψ sector is that an It is clear that as f˜ > 0, the potential is minimized by 0 1 attractiveinteractionbetweenρ andρ originatesfrom imposing ρ =ρ . Moreover,it is found that + + the second cumulant of the λ term. −As a result, the 6 − 0 coupling constant f is changed as 3 1 f f˜ =f C λ20a−02 , (7) ρ+−ρ− ∝˜b†23π,j˜b23π,j −˜b†−23π,j˜b−23π,j ∼Xl=1Vlz,j. (10) 1 1 1 → − m|µ0|3 We thus conclude that for f˜1 > 0, a finite long-range p vector chiral order z exists, and the rung-parity where C is a positive dimensionless constant of O(1). hVl,ji symmetry is spontaneously broken. For J J (i.e., Here, we have approximated the Matsubara Green’s µ /J 1) or J J [i.e., λ O(J⊥ )≪], f˜ < 0 faunndctiJoan0τhTτaΨre0(xsm,τa)lΨle†0r(0(,l0a)rigear)s 1th/aa0n (tzheero)cowrrheelnati|oxn| |gweh0ne|enraHll≪ybheoclodms,1e2s,1⊥c7low≫sehriletofoHr uJ,⊥f0˜∼∼iOnc(−Jre)as(ei.⊥se.a,nλd01t∼en0d)s, length (mµ ) 1/2 [τ : imaginary time], and assumed c 1 | 0| − to be positive. Consequently, the chiral phase is present that (mµ ) 1/2 is at most O(a ).12 For the resultant | 0| − 0 inanintermediate-rung-couplingregime. Supposingthat Hamiltonian [Ψ ], the Haldane’s harmonic-fluid ap- proach (i.e., bHoe′sffoniz±ation) (Refs. 13,14,15) could be ap- ρ+ >ρ holdsin the chiralphase,we canspeculate that the Ψ −mode constructs a massive spectrum, whereas plicable. Using the bosonization formulas ρ (x) {∂ρρx¯±φ±+,/wπ∂}ex1φo/±2b/Ptπai}∞nn=P−a∞n∞b=oe−si∞2onn(eiφiz2±en−d(φπH±ρ¯±−axmπ)ρ¯ei±l−txio)θn±ai,anndwoΨhfe†±rteh∼±eρ¯{p±ρ¯h±as≈+=e eTvthxihoeiseutΨsepn−+srcteuespedoanyfrcteitnhpoerRfoctevhhfi.ider8eaTs.lLoaILfrdTHiesLrLaalnssHodtausttu,ehp.et1ph0ToeNrLTtLaeLdmLiseblypyp,artrethadhemiecptecetrodeer--. fihe±ldis (φ ,θ ). Introducing the new fields φs,a = (φ+ would be close to the univers∼al vaclue 1. The correlation ± ± ± φ )/√2 and θs,a =(θ+ θ )/√2 further, we can repre- function of the chirality might exhibit a power decay: se−nt the phase-field Ham±ilto−nian as z z z 2 const/j2+ at j .10 hVl,jVl,0i≈hVl,ji − ··· →∞ Letusnowdiscussthecaseoff˜ <0,whereρ¯ =ρ¯ is H[φ,θ] = Z dxqX=s,a2vπq (cid:2)Kq(∂xθq)2+Kq−1(∂xφq)2(cid:3) irnesEtoqr.ed(8a)nydietlhdesbaosToLnLiz,awtiohnichisias1vsatirlaobnlgel.yTshtaeb+φilsizseedc−tboyr symmetries, while the low-energy physics of the φ sec- +g cos(2√2φ )+g cos(3√2θ )+ , (8) a φ a θ a ··· tor depends on whether cos(2√2φa) and cos(3√2θa) are relevant or not: the scaling dimensions of these two are where we have assumed ρ¯ = ρ¯ = ρ¯ (see below) and + 2K and 9/(2K ), respectively. The Hamiltonian dropped terms with spatially osc−illating factors ei2nπρ¯x. a a He′ff leadstoK ( ρ¯/f˜)1/2. Therefore,whenf˜ 0andρ¯ aTnhde gthφeanthdirgdθ ctuermmuslafnotr,erxeasmpepclteivoerliyg.inaUtenfforrotmunρa+teρly−, islargeenoau∝gh,−Ka is1alwaysmuchlargerthan1 ∼1. Atthis case,cos(3√2θ )and cos(2√2φ ) arerespectivelyhighly the values of g cannot be evaluated quantitatively a a φ,θ relevant and irrelevant, and then the φ sector obtains within the present approach. In the phase-field pic- a ture, a spin rotation around the Sz axis S+ eiγS+, a massive spectrum. If gθ > 0 (< 0), the phase field θa the one-site translation along the chain Slα,j → Sα l,j, is pinned on lines θa = √2(2n+1)π/6 (√2nπ/3) in the that along the rung Sα Sα , and thl,ej s→ite-pla,jr+it1y θ+-θ plane. Among these lines, only six lines intersect transformation along thl,ej c→hainl+S1α,j Sα are, respec- the p−hysically relevant “Brillouin” zone, π < θ+ π (tφivel(yx,+exapr)esseπdρ¯aas ,θθ± (→x+θa±)+l,jγπ→,),(φθ±l,−(xj),θθ±(x2))π/→3, astnadte−sπpo≤sseθs−s t<heπs.ixTfohlids dreesguelnteirmacpyl.ieTsothi−natvetshteigagtreo≤utnhde an±d (φ (x0),−θ (x±))0 ±( φ (0 x−),θ (±x→)). ±F±urther- physical meaning of locking θa and the ground-state de- more, t±he run±g-parit→y tra−nsf±orm−ation±S−α Sα may generacy,letus focusonthemagnetizationpersite. The 1,j ↔ 3,j bosonization represents it as be realizedbyρ¯ =ρ¯ and(φ ,θ ) (φ ,θ ). Owing + tothesesymmetries,i−nallverte±xo±per→ators∓wit∓houtoscil- 2 4 alarteinagllofawcetdortso,oenxliystcoins[2Enq(.φ(+8)−.φT−h)e]amnodstcorse[l3env(aθn+t−nθ=−)1] hSlz,ji≈M − 3ρ¯a0Dcos(cid:16)√2θa+ 3πl(cid:17)E+··· . (11) terms indeed appear in Eq. (8). One can see that the second term in Eq. (11) causes The bosonization approach for He′ff evaluates the ve- a down-down-up magnetization structure in the case of locityv asv ( f˜ρ¯/m)1/2. Therefore,iff˜ >0,then g >0,while forg <0anup-up-downstructureoccurs: a a 1 1 θ θ v becomesima≈gin−aryanditmeansthatthebosonization for instance, if θ is locked to zero for g < 0, Sz = a a θ h 1,ji isinvalid. Tounderstandthephysicalmeaningofthisin- Sz = M +δ and Sz = M 2δ [δ cos(√2θ ) ]. stability,16 we should consider the magnon-density part Wh e2,jtihus conclude thaht3a,njiinhomo−geneous∝mhagnetizaatioin in He′ff and then define the following Ginzburg-Landau for the rung is induced by pinning θa. Obviously, the (GL) potential: parity and translational symmetries for the rung direc- tion are spontaneously broken in this state. Three of = g (ρ +ρ )2+f˜ρ ρ µ(ρ +ρ ). (9) the sixfold degenerated states are indeed explained by 1 + 1 + + F − −− − 4 this inhomogeneousdistribution. The meaningofthe re- H (A) or (B) ? BKT 4SJ+3SJ⊥ maining twofold degeneracy is unknown.18 Remarkably, 1st order transition saturation transition (B) two-component TLL the inhomogeneously magnetized phase is not at all ex- M=1/2 pected from the classical tube system (see Fig. 1). We 4SJ (A) TLL with inhomogeneous note that this inhomogeneousdistributionmightslightly TLL with magnetization plateau be modified if cos(3n√2θ ) with n 2 are also rele- vector chiral M=1/6 a vant.19Fromthepredictionsofthechi≥ralorderforf˜1 >0 TLL order andtheinhomogeneousphaseundertheconditionf˜ <0 critical region 1 and f˜ 0, the boundary f˜ = 0 is expected to be a 0.28J first-|or1d|e∼r transition. 1 0 translational symmetry broken When f˜/ρ¯increases so that K < 9/4, cos(3√2θ ) 0 J⊥ M=0 1 a a − becomes irrelevant and the low-energy physics of the φ a sector is described by a Gaussian model. This transi- FIG. 3: Schematic ground-state phase diagram of the S = 1 2 tion must be of a Beresinskii-Kosterlitz-Thouless(BKT) spintube(1). Theareaawayfromthesaturationisdiscussed type.20 After the transition, the system is in a two- elsewhere (Ref. 24). Seethe Endnotes12 and 17. component TLL phase with all symmetries enjoying. If f˜/ρ¯ is further increased due to the growth of J or 1 t−he decrease of ρ¯, cos(2√2φ ) seems to become ⊥rele- a the boundary between these two regime, one might ob- vant. However, the exact results for the integrable Bose serve a weak singularity such as a magnetization cusp. gas21 imply that in a one-dimensional Bose system with a short-range repulsive interaction, the TLL parameter is not usually smaller than 1 even when the interaction becomes extremely strong. The two-component TLL is hence expected to continue even when J J or ρ¯ is small (see the Endnote 17). The predicti⊥on≫of the two- V. SUMMARY AND DISCUSSIONS componentTLL in the strong-rung-couplingregime is in agreement with a previous study applying the strong- We have studied the three-leg frustratedspin tube (1) rung-coupling approach to the S = 1 tube.22 2 near the upper critical field. It has been predicted that the vector chiral order or the inhomogeneously magne- tized order emerges in the magnetic-field-driven TLL IV. THREE-BAND-MAGNON CONDENSED phase in the intermediate-rung-coupling regime. It is STATE remarkable that in these two phases, the TLL critical- ity (massless modes) and the spontaneous breakdown Here, we consider the case where all three kinds of of discrete parity or translational symmetries for the magnons Ψ are condensed. This situation could be rung direction coexist. We have also shown that when +, ,0 realizedunder−the conditionofµ>0,µ >0,Hl <H < the rung coupling becomes strong enough, the inhomo- 0 c H , and J < 4J/3. This means that the three-band- geneous phase vanishes and instead the two-component c′ magnon co⊥ndensed state is allowed to exist only in the TLL occurs with preserving all the symmetries. weak-rung-couplingregime. LikeEq.(9),letusintroduce Combiningourresultsandtheexistentones,7,8 wecan the GL potential for the present case as follows: drawthe ground-statephasediagramfortheS = 1 tube 2 as in Fig. 3. The global phase structure near the sat- G = g0ρ20+g1(ρ++ρ−)2+f0ρ0(ρ++ρ−) uration would common to all the cases with arbitrary +f ρ ρ µ ρ µ(ρ +ρ ). (12) S, as far as S . O(1). Although in general the spin- 1 + 0 0 + −− − − wave approach used in this paper is not very reliable for To find the stable magnon-density profile (ρ ,ρ ,ρ ), small-S cases, we believe that it is valid if we consider 0 + the Hessian matrix H = [ ∂2 ] is useful. At the−lo- the regionwhere M is sufficiently closeto the saturation i,j ∂ρi∂Gρj value: insucharegion,multimagnonscatteringprocesses cal minimum point (ρ ,ρ¯,ρ¯) satisfying ∂ /∂ρ = 0, the 0 G j are expected to be negligible. When J is changed from eigenvalues of H are 4J/3, C , and C ( 4J/3 < i,j − 1 2 − +0 to + with M fixed near the sa⊥turation, the fol- C1 <0andC2 >0). The correspondingeigenvectorsare lowing sc∞enario is expected: TLL plus chirality [first- w(δhρe0r,eδρC+3,δ>ρ−0). ∝Th(e0,n1e,g−a1ti)v,e(e−igCe3n,v1a,l1u)e, −an4dJ/(3Ca3n,1d,1it)s, otirodner tr[BanKsTitiotrna]→nsitTioLnL] plutwsoin-choommpoogneennetouTsLmLa.g→netiza- eigenvector indicate that the ground state takes ρ+ → → − ρ = 0. Moreover, a positive eigenvalue C implies the Wefinallynotethatthepredictedfirst-orderandBKT 2 ex−is6tence of the TLL. We therefore predict that the chi- transitions could not be detected by observing the mag- ralorder(ρ =ρ )andaone-componentTLLstatestill netizationM becauseH couplesto∂ φ andρ +ρ ,but + x s + remainwhen6the−systemmovesfromthe lowest-magnon- it does not directly interact φ and ρ ρ . A sp−ecific- a + condensed regime to all-magnon-condensed one.23,24 At heat measurement would be efficient in−the−detection. 5 Acknowledgments Education, Culture, Sports, Science and Technology of Japan. This work is supported by a Grant-in-Aid for Scien- tific Research (B) (No. 17340100) from the Ministry of 1 Forexample,see Frustrated Spin Systems, editedbyH.T. ford University Press, New York,2004). Diep, (World Scientific, Singapore, 2005). 15 M.A.Cazalilla,andA.F.Ho,Phys.Rev.Lett.91,150403 2 For examplesee T.Hikihara, T. Momoi and X.Hu,Phys. (2003). Rev. Lett. 90, 087204 (2003); T. Momoi, T. Hikihara, M. 16 When in Ref. 9 we discussed the possibility of the two- Nakamura, and X. Hu, Phys. Rev. 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