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Massive triplet excitations in a magnetized anisotropic Haldane spin chain. A.Zheludev,1Z.Honda,2C.L.Broholm,3,4K.Katsumata,5S.M.Shapiro,6A.Kolezhuk,7,8S.Park,4,9andY.Qiu4,9 1Condensed Matter Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6393, USA. 2Faculty of Engineering, Saitama University, Urawa, Saitama 338-8570, Japan. 3Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA. 4NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. 3 5The RIKEN Harima Institute, Mikazuki, Sayo, Hyogo 679-5148, Japan. 0 6Physics Department, Brookhaven National Laboratory, Upton, NY 11973-5000, USA. 0 7Institut fu¨r Theoretische Physik, Universität Hannover, Appelstraße 2, 30167, Hannover, Germany. 2 8 Institute of Magnetism, National Academy of Sciences & Ministry of Education of Ukraine, 36(B) Vernadskii avenue, 03142 Kiev, Ukraine. n 9Department of Materials and Nuclear Engineering, a University of Maryland, College Park, MD 20743, USA. J (Dated: February 2, 2008) 4 2 Inelastic neutron scattering experiments on the Haldane-gap quantum antiferromagnet Ni(C5D14N2)2N3(PF6) are performed at mK temperatures in magnetic fields of almost twice the l] critical field Hc applied perpendicular to the spin cahins. Above Hc a re-opening of the spin gap e is clearly observed. In the high-field Néel-ordered state the spectrum is dominated by three dis- - tinct long-lived excitation branches. Several field-theoretical models are tested in a quantitative r t comparison with the experimental data. s . t a m One-dimensional (1D) integer-spin antiferromagnets thespectrumretainsaconsiderablequasielastic(gapless) (AFs) are famous for having a disordered “spin liquid” component at higher fields. The theoretically predicted - d ground state and an energy gap ∆ exp( πS) in the reopeningofthegapwasthusnotobserved. Atthetime, ∼ − n excitation spectrum [1]. Elementary excitations are a thisbehaviorwasnotfullyunderstood,thoughseveralin- o triplet of massive (gapped) long-lived “magnons”. An triguing explanations were put forward. One attributed c external magnetic field modifies the magnon energies by the phenomenonto the 1Ddiffusionofthermallyexcited [ virtue of Zeeman effect [2, 3]. At a certain critical field classical solitons [11], while another drew parallels with 2 H the gapin one ofthe branchesapproacheszero[3, 4]. the incommensurate Luttinger liquid state in the AS ge- c v The result is a condensation of magnons[3, 5] and the ometry. To better understand this issue, we carried out 4 2 emergence of a qualitatively new ground state. What a new seriesof measurementsat considerablylowertem- 4 this new ground state actually is, depends on the sym- peratures and in higher magnetic fields, overcoming any 1 metry of the problem. Theory predicts that in an axi- finite-T effects to directly probe the ground state prop- 0 ally symmetric (AS) Heisenberg orXY-like scenariosthe erties. The spin dynamics in this regime was found to 3 high-field phase is a gapless “Luttinger spin liquid” with be qualitativelydifferentfromthatpreviouslyseenatel- 0 quasi-long-range order and a diffuse continuum of exci- evated temperatures. The new data allow a quantitative / t a tations (no sharp magnons) [6]. The high-field phase in comparison with several quantum field-theoretical mod- m the axially asymmetric (AA) case is expected to be to- els, while emphasizing dramatic differences between the - tally different. Here the ground state should have true high-field phase and a classical magnet. d long-rangeNéelorder(“spin solid”). A simple bosonde- Five newly-grown deuterated NDMAP single crystals n scription[3] predicts a re-opening of the gapat H >H , wereco-alignedbyneutrondiffractiontoproduce asam- o c andimplies the restorationofa single-particleexcitation ple of total mass 1.4 g and a mosaic spread of 3◦. c : spectrum. Is this state then similar to a classical easy- NDMAP crystallizes in the orthorhombic space group v plane AF in a field, that also features long-range order Pnmn. The S =1 AF spin chains, formed by Ni2+ ions i X and sharp gapped spin waves? bridged by azido-groups, run along the crystallographic r Onlyrecentlydidexperiments,whicharethekeytoun- caxis,and(a,b)isthe magneticeasyplane. Inzerofield a derstanding the high-field behavior, become technically the Haldane gap energies were previously determined to feasible. Thiswasinpartduetothediscoveryofthevery be∆ =0.42meV,∆ =0.52meV,and∆ =1.89meV x y z usefulmodelmaterialNi(C D N ) N (PF )(NDMAP) [9]. Inourexperimentsthesamplewasmountedwiththe 5 14 2 2 3 6 [7], where various techniques [7, 8, 9] confirmed a quan- aaxisvertical,andthedatawerecollectedinthe(0,k,l) tum phase transition at an easily accessible critical field reciprocal-space plane. The sample environment was a of H 6 T. Inelastic neutron studies were carried out cryomagnet with a dilution refrigerator. The measure- c ≈ in the AA geometry in the thermally-disordered phase: ments were performed at T = 30 mK in magnetic fields at H > H , but at temperatures high enough to de- of up to 11 T applied along the crystallographica axis. c stroy long-range Néel order [10]. Somewhat unexpect- The first series of experiments was performed at the edly, it was found that as the Haldane gap closes at H , SPINS 3-axis cold-neutron spectrometer installed at the c 2 FIG. 2: Measured field dependence of the gap energies in NDMAP at T = 30 mK and H applied along the crystallographic a axis (open symbols). Dashed and dash-dot lines arepredictionsofthetheoreticalmodelsproposedinRefs.[3] and[4],respectively. Thesolid linesareabestfittothedata using an alternative model described in this work. the gap in the lower mode vanishes altogether,to within FIG. 1: A series of constant-q scans measured in NDMAP experimentalerror,asillustratedinFig.1b. Atthesame k at T = 30 mK for different values of magnetic field applied time, long-range AF order sets in [8]. The main new along the a axis (symbols). The lines are fits to a simple resultofthepresentstudyistheobservationthatatT = single-mode cross section function as described in the text. 30 mK at H > H the gap in the lower mode re-opens c and increases with field (Fig. 1c,d). In the ordered state thespectrumatq =π thuscontainsthree distinct sharp k NIST Center for High Resolution Neutron Scattering excitation branches, just as in the low-field disordered (CHRNS).Themainpurposewastomeasurethefieldde- phase. pendence ofscatteringatthe 1DAF zone-centerl =0.5. Thenew30mKdataclearlyshowthatthequasielastic Neutrons with a 3.7 meV fixed-final energy were used scattering previously observed at H > H and T = 2 K c with a horizontally focusing analyzer and a BeO filter is absent, and must therefore be a finite-T effect. In the after the sample. Energy scans were performed on the constant-q scans collected at T =30 mK all three peaks (0,k,0.5)reciprocal-spacerod. The wave vector transfer have resolution-limited widths at all fields. In fact, very perpendicular to the chains was continuously adjusted good fits to the data (solid lines in Fig. 1) can be ob- to maintain the sample c axis directed towards the ana- tained using a simple model cross section that involves lyzerforoptimalwavevectorresolutionalongthechains. three excitations with a zero intrinsic energy width [9]. Thebackground(featurelessandtypically4counts/min) The cross section was numerically convoluted with the was measured away from the 1D AF zone-center, at spectrometer resolutionfunction, and the adjustable pa- (0,k,0.35) and (0,k,0.65). rameters at each field were the three gap energies and Typicalbackground-subtracteddata sets are shownin intensity prefactors for each mode. In Fig. 1 the partial Fig. 1. A similar scan previously measured in zero field contributions of the three branches are shown as shaded (Fig. 3 in Ref. [9]) clearly shows two peaks at roughly areas. The field dependence of the gap energies deduced 0.47 mV and 1.9 meV energy transfer, respectively. The from these fits is shown in Fig. 2 in open symbols. data plotted in Fig. 1a corresponds to H = 3 T, still To study the wave vector dependence of the dynamic well below H 6 T. At this field the lower-energypeak structurefactoradditionalmeasurementswereperformed c ≈ is visibly split in two components. At H = 6 T H using the Disk Chopper Spectrometer (DCS) at NIST c ≈ 3 representation of excitations in isotropic Haldane spin chains. The present data show that this model does not apply in highly anisotropic case of NDMAP: the spectrum is truly gapped and has no intrinsic incommensurate features. The quasielastic scattering at T = 2 K, H > H is thus to be attributed to a diffusion of ther- c mallyexcitedtopologicalsolitons[10],anditsanomalous q-widthisduetotheT-dependentmeandistancebetween solitons [11]. For the following discussion of the observed low-T properties it is crucial to note that at T = 30 mK the spin chains are antiferromagneticallyordered at H >H c [8], with a static staggered magnetization as large as m 1µ per site at H = 11 T. Clearly, inter-chain B ∼ coupling is needed to stabilize order at a non-zero temperature. However, in the AA geometry, even an iso- lated chain orders at H > H at T = 0, the system be- c ing equivalent to the (1+1)-dimensional Ising-model [3]. Since inter-chain interactions in NDMAP are very weak [9], we can assume that a purely 1D problem is realized: long-rangeAFcorrelationsareintrinsictothe1Dchains, andthesoleroleofresidual3Dinteractionsistomaintain their stability at a finite temperature. The conventional approach to describing spin excitations in ordered systems is the quasiclassical spin wave theory(SWT).Inthismodelthemagnonsareprecessions of staggeredmagnetization m around its equilibrium di- rection. As a consequence, in SWT there are only two sharpexcitationbranches,polarizedperpendiculartothe FIG.3: InelasticspectrameasuredinNDMAPusingtheDisk m. In our case three sharp magnons are seen in the or- Chopperspectrometerforseveralappliedfields. Therangeof deredstate(H >H ). Atleastoneofthethreebranches the false-color scale in the lower panel is 0 to 15 arb. u. c musthavethecharacterofa“longitudinal”magnonthat Contour lines are drawn with 3 arb. u. steps in all panels. Arrows indicate the gap energies for the different excitation is not a precession mode, but is polarized along the or- branches. dered moment. Thus, at H > Hc quantum-mechanical effects remain crucial, and the SWT is inapplicable. In- stead,thethreeobservedexcitationbranchescanbevisu- CHRNS. The data were collected using a fixed incident alized as soliton-antisolitonbreathers: the three massive neutron energy of 4.5 meV. The sample was mounted bound states formed by the two types of topological de- withthe(b,c)planehorizontal,andthechainaxisalmost fects allowedin ananisotropicsemisiclassical1Dmagnet perpendiculartotheincidentbeam. Thebackgroundwas [11]. In more detail our experiments can be understood measured separately, with the sample removed from the in the frameworks of severalfield-theoretical models. cryostat. The background-subtracted data collected at The approach due to Affleck [3] is based on coarse- H = 0, H = 6 T and H = 10 T are visualized in the graining the (1+1)-dimensional O(3) non-linear sigma false-color and contour plots in Fig. 3, and correspond model (NLSM), to which, in turn, the S = 1 Heisen- to a typical counting time of 20 hours. They are to be berg chain canbe mapped [1]. The resulting Lagrangian compared to similar 3-axis data measured previously at isthatofanunconstrainedrealvectorfieldϕ(r)withthe T =2KandshowninFig.2ofRef.[10]. Thenewlow-T ϕ4-type interaction. Anisotropy is added by postulating experimentshowsthatthe excitationsatH >Hc havea separate masses ∆α for the different components of this simple relativistic (hyperbolic) dispersion relation, with vectorfield. ForH >Hc thegroundstatehasanon-zero a spin wave velocity equal to that at H < H (see solid staggered magnetization L = ϕ and uniform magneti- lines in Fig. 3). The “inverted” hyperboliccdispersion zation M H ϕ . The ϕh4-mi odel captures the ba- ∝ h × i curveswith“negativegaps”showninsolidlinesinFig.3 sic physics involved, but suffers from several drawbacks. of Ref. [10] are clearly inconsistent with the new data. In particular, the predicted value of the critical field for This latter dispersionform was proposedas one possible H e isgµ H(α) =∆ . Thisisinconsistentwithestab- α B c α k interpretation of the anomalous q-width of quasielastic lished experimental [8, 12, 13, 14, 15] and numerical[16] scattering at T = 2 K [10] and is based on a Fermion results, as well as with simple arguments based on the 4 perturbation theory [16, 17]. Another potential weak- helps clarify the nature of the previously investigated ness is that at the mean field (MF) level M(H) and the finite-T spindynamics. Thenextexperimentalchallenge willbetoinvestigatetheASgeometry,searchingforman- lowest gap ∆(H) come out to be ∝ qH −Hc(α), while ifestations of the Luttinger spin liquid regime. a roughly linear behavior is seen experimentally and numerically. ForNDMAP,thederivationsofRef.3mustbe generalizedto allowforanarbitraryfielddirection,since We would like to thank H.-J. Mikeska, F. Eßler, the anisotropy axis of the Ni2+ ions forms an angle of A. Tsvelik, and I. Zaliznyakfor enlightening discussions. about16◦ withthecrystallographiccaxis. Theresulting Work at ORNL and BNL was carried out under DOE predictions are shown in dashed lines in Fig. 2. Contracts No. DE-AC05-00OR22725 and DE-AC02- Another theory, due to Tsvelik [4], stems from the in- 98CH10886, respectively. Work at JHU was supported tegrable SU(3)model of a S =1 chain. It involvesthree by the NSF through DMR-0074571. Experiments at Majorana fields with masses ∆ . The predicted critical NISTweresupportedbytheNSFthroughDMR-0086210 α fields gµ H(α) = ∆ ∆ coincide with the perturba- andDMR-9986442. Thehigh-fieldmagnetwasfundedby B c β γ tive formulas of [16p, 17]. Again, incorporating an arbi- NSF through DMR-9704257. Work at RIKEN was sup- traryfielddirection,andusingtheknownparametersfor ported in part by a Grant-in-Aid for Scientific Research NDMAP, we can directly compare the predicted gap en- from the Japan Sosciety for the Promotion of Science. ergies(dash-dotlinesinFig.2)tothosemeasuredinthis Oneofus(AK)wassupportedbythegrantI/75895from work. Unlike the ϕ4 model, Tsvelik’s approach predicts Volkswagen-Stiftung. the linear behavior of M(H) and ∆(H) near H . The c theory,however,failstoreproducethetwouppergapsat H >H . c In the context of the present experiments we would [1] F. D. M. Haldane, Phys. Lett. 93A, 464 (1983); Phys. like to introduce a different approach that is similar to Rev. Lett.50, 1153(1983). the model proposed for dimerized S = 1/2 chains in [2] K. Katsumata, H. Hori, T. Takeuchi, M. Date, A. Yam- Ref. 18. It is a Ginzburg-Landau-type theory written in agishi, J. P. Renard,Phys. Rev Lett. 63, 86 (1989). terms of a complex triplet field Φ(r) = A(r) +iB(r). [3] I. Affleck, Phys. Rev. B 41, 6697 (1990); Phys. Rev. B The uniform and staggered magnetization are written 43, 3215 (1991). as M (A B) and L A(1 A2 B2)1/2. In- [4] A. M. Tsvelik, Phys.Rev.B 42, 10499 (1990). teractio∝n term×s (Φ∗ Φ)2∝and (Φ−∗ Φ−)2 are natu- [5] M. Takahashi and T. Sakai, J. Phys. Soc. Jpn. 60, 760 · × (1991);M. Yajima and M. Takahashi, J. Phys. Soc Jpn. rally present in the model. Unlike the models discussed 63, 3634 (1994). above, anisotropy enters the Lagrangian through two [6] T.S.S.Sachdev,T.Senthil,andR.Shankar,Phys.Rev. sets of masses, separately in the A and B “channels”: B 50, 258 (1994). (m A2 + m B2). By integrating out the B-field [7] Z. Honda, H. Asakawa, and K. Katsumata, Phys. Rev. Ponαe obαtainαs an eαffeαctive ϕ4-type theory similar to that Lett. 81, 2566 (1998). g of Affleck. However, the Zeeman term H (ϕ ∂ ϕ) [8] Y. Chen, Z. Honda, A. Zheludev, C. Broholm, K. Kat- · × t sumata, and S. M. Shapiro, Phys. Rev. Lett. 86, 1618 of Ref. 3 becomes replaced by the anisotropy-dependent (2001). expression (m )−1ε H A (∂ A ). Remarkably, Pαβγ γ αβγ α β t γ [9] A. Zheludev, Y. Chen, C. Broholm, Z. Honda, and this model includefs Affleck’s theory as the special case K. Katsumata, Phys.Rev.B 63, 104410 (2001). mγ = const, and at the same time it reproduces the re- [10] A. Zheludev, Z. Honda, Y. Chen, C. Broholm, and sults of the Tsvelik’s theory for ∆(H) and M(H) below K. Katsumata, Phys.Rev.Lett. 88, 077206 (2002). f H in another special case m = m . There are no par- [11] H.-J. Mikeska and M. Steiner, Adv. Phys. 40, 191 c γ γ ticular reasons why either of these special cases should (1991). f [12] L.P.Regnault,I.Zaliznyak,J.P.Renard,andC.Vettier, correspond to the actual Heisenberg S = 1 chain with Phys. Rev.B 50, 9174 (1994). single-ion anisotropy realized in NDMAP. The masses [13] Z. Honda, K. Katsumata, H. A. Katori, K. Yamada, mα and mα at the present stage may be treated as ad- T.Ohishi,T.Manabe,andM.Yamashita,J.Phys.: Con- jgustable parameters to reproduce the measured gap en- dens. Matter 9, L83 (1997). ergies ∆ = √m m at H = 0 and the measured field [14] Z. Honda, K. Katsumata, M. Hagiwara, and M. Toku- α α α dependencies. Very good fits to our experimental data naga, Phys. Rev.B 60, 9272 (1999). g are obtained with m /m = 0.87, m /m = 0.83, and [15] A.ZheludevandZ.HondaandK.KatsumataandR.Fey- x x y y erherm and K. Prokes, Europhys.Lett., 55, 868 (2001). m /m =0.35 (solid lines in Fig. 2). A detailed descrip- z z f f [16] O. Golinelli, Th. Jolicoeur and R. Lacase, Phys. Rev. tfion of the application of this model to NDMAP will be B 45, 9798 (1992); J. Phys.: Condens. Matter 5, 7847 reported elsewhere. (1993). Insummary,the presentlow-T study isa directobser- [17] L.-P. Regnault, I. A. Zaliznyak, and S. V. Meshkov, J. vation of several fundamental quantum-mechanical fea- Phys: Condens. Matter 5, L677 (1993). tures predicted for anisotropic Haldane spin chains, and [18] A. K. Kolezhuk,Phys. Rev.B 53, 318 (1996).