Nuclear Magnetic Relaxation in the Haldane-Gap Antiferromagnet Ni(C2H8N2)2NO2(ClO4) Shoji Yamamoto and Hiromitsu Hori Division of Physics, Hokkaido University, Sapporo 060-0810, Japan 4 (Received 13 November2003) 0 0 A new theory is proposed to interpret nuclear spin-lattice relaxation-time (T1) measurements 2 onthespin-1quasi-one-dimensionalHeisenbergantiferromagnet Ni(C2H8N2)2NO2(ClO4)(NENP). While Sagi and Affleck pioneeringly discussed this subject in terms of field-theoretical languages, n there is notheoretical attempt yet to explicitly simulate thenovelobservations of T−1 reported by a 1 J Fujiwara et al.. By means of modified spin waves, we solve the minimum of T1−1 as a function of an applied field, pendingfor thepast decade. 4 2 PACS numbers: 76.60.−k, 76.50.+g, 75.10.Jm ] h Predicting a striking contrast between integer- and temperature relaxation rate of 31P and 51V nuclei of c e half-odd-integer-spin one-dimensional Heisenberg anti- AgVP S , which is also an ideal spin-1 Haldane-gap 2 6 m ferromagnets, Haldane [1,2] sparked renewed interest antiferromagnet, and observed the diffusive dynamics at- ignaplo,wt-hdaimt eins,sioanamlaqgunaentticumexmcitaagtnioetnismga.pTimhemHedailadtaenlye wT1h−e1th∝erHth−e1s/p2in[3t7r]a.nsTphoerrteinisquaanhtoutmarsgpuinm-egnatpp[3ed8–a4n1-] t above the ground state, was not only calculated by var- tiferromagnets should be diffusive or ballistic at finite s . ious numerical tools [3–8] but indeed observed in spin-1 temperatures. Onthe otherhand,Gaveauet al. [42]and t a quasi-one-dimensionalHeisenberg antiferromagnets such Fujiwara et al. [43,44]measuredthe low-temperature re- m as CsNiCl [9] and Ni(C H N ) NO (ClO ) (NENP) laxation rate of 1H nuclei of NENP. As an applied field 3 2 8 2 2 2 4 - [10]. The valence-bond-solid model [11,12] significantly increases, the first excited state moves down and then d contributed toward understanding novel features of the crossestheground-stateenergylevel. IndeedT−1reaches 1 n Haldane massive phase such as fractional spins induced apeaknearthe criticalfieldH ∆ /gµ ateverytem- o c ≡ 0 B on the boundaries [13,14], a string order hidden in the perature, but it is not a monotonically increasing func- c [ ground state [15,16] and magnon excitations against the tionofH,atlowtemperaturesT <∆0/kB inparticular. hidden order [17–20]. The nonlinear-σ-model quantum Sagi and Affleck [45] formulated∼the nuclear magnetic 1 field theory [21,22] skillfully visualized the competition resonance in Haldane-gap antiferromagnets in terms of v between massive and massless phases, while a general- field-theoretical languages. However, few investigations 8 izedLieb-Schultz-Mattistheorem[23,24]gaveacriterion have followed their pioneering argument and there is no 7 4 for the gap formation in a magnetic field. attemptyettoexplicitlyfitatheoryforthe aboveobser- 1 Recent progress in the experimental studies also de- vations. 0 serves special mention. Not only the single-magnon dis- In such circumstances, we revisit the low-temperature 4 persion relation [25] but also the two-magnon contin- nuclear magnetic relaxation in Haldane-gap antiferro- 0 uum [3,26] was directly observed by inelastic-neutron- magnets with particular emphasis on its field depen- / t scatteringmeasurementsonNENP[27]andCsNiCl [28]. dence. Excluding any phenomenological assumption a 3 m An applied magnetic field may destroy the Haldane gap from our argument, we calculate the nuclear spin-lattice and bring back magnetism to the system. Such a field- relaxation rate by means of modified spin waves. A d- induced long-range order was indeed realized in a nickel newtheoryclaimsthatas an applied field increases, T1−1 n compound Ni(C H N ) N (PF ) [29]. Another family shouldinitiallydecreaselogarithmically andthenincrease 5 14 2 2 3 6 o of linear-chain nickelates of general formula R BaNiO exponentially, wellexplainingexperimentalobservations. 2 5 c (R = rare earth or Y) [30] exhibited a novel scenario of We employ the spin-1 one-dimensional Heisenberg : v one- to three-dimensional crossover [31,32]. When the Hamiltonian i nonmagneticY3+ ionsaresubstitutedbyothermagnetic X L L ar rstaartee-e,atrhtehreioanpspienarYs2aBatNhrieOe5-dwimitehnsaiodniaslorlodnegre-rdanggroeuonrd- H=JXl=1Sl·Sl+1−gµBHXl=1Slz. (1) der,whiletheone-dimensionalgappedexcitationspersist both above and below the N´eel temperature [33–35]. Inordertoilluminatetheessentialrelaxationmechanism Nuclear spin-lattice relaxationtime (T )has alsobeen inspin-gappedantiferromagnetsas analyticallyas possi- 1 measured on Haldane-gap antiferromagnets. The field ble, we do not take any anisotropy into consideration in (H) dependence of T−1 is of particular interest at both thisletter. Magneticanisotropyquantitativelyaffectsthe 1 high and low temperatures. The high-temperature re- gapamplitude but has no qualitative effect onthe whole laxation rate has been discussed in the context of trans- scenario. Scalingtemperatureandanappliedfieldbythe port properties. Takigawa et al. [36] measured the high- Haldanegap,wepresentauniversaltheory. Quantitative 1 refinement of the final product in the presence of single- Now we calculate the relaxation rate in terms of the ionandorthorhombicanisotropytermswillbeconsidered modified spin waves. Considering the electronic-nuclear elsewhere. Introducing the Holstein-Primakoff bosonic energy-conservation requirement, the Raman scattering operators predominates in spin-gapped antiferromagnets. The Ra- man relaxation rate is generally given by S+ = 2S a†a a , Sz =S a†a , S22+nn−=1 b†nqq2S−−b†nnbnn, n S22znn−=1 −S+−b†nnbnn, (2) T11 = ¯h4π(giµe−BE¯hiγ/NkB)2T Xi,j e−Ei/kBT P andretainingonlybilineartermsofthem,werewritethe j A Sz i 2δ(E E ¯hω ), (10) × h | l l l| i j − i− N Hamiltonian as (cid:12) P (cid:12) where A is(cid:12)the dipolar c(cid:12)oupling constants between the l = 2JS2N nuclear and electronic spins in the lth site, ω γ H is N N H − ≡ N N theLarmorfrequencyofthenucleiwithγN beingthegy- +(2JS+gµ H) a†a +(2JS gµ H) b†b romagneticratio,andthesummation istakenoverall B n n − B n n i nX=1 nX=1 the electronic eigenstates i with enePrgy Ei. By means | i N of the modified spin waves, eq. (10) is rewritten as +JS a†b† +a b +b†a† +b a , (3) n n n n n n+1 n n+1 nX=1(cid:16) (cid:17) 1 = 4π(gµB¯hγN)2 e−Ei/kBTδ(E E ¯hω ) where N L/2. In order to preserve the up-down sym- T1 ¯hN2 ie−Ei/kBT Xi,j j − i− N ≡ P mwaevtrey,diosrtrtihbeutsiuobnlaftutniccetisoynmsmcoentrsytr,awineinopgttimheizteotthael ssptaing-- ×(cid:12)hj| k,k′Ak′−k α†k′αk−βk†′βk coshθk′coshθk gered magnetization to be zero [46–49]: (cid:12)(cid:12)− Pαk′α†k−βk′(cid:2)β(cid:0)k† sinhθk′sinhθk(cid:1) |ii(cid:12)2, (11) a†nan+b†nbn =2NS. (4) where A(cid:0) = eiqlA .(cid:1) The Fourie(cid:3)r c(cid:12)(cid:12)omponents of Xn (cid:0) (cid:1) q l l the hyperfine cPoupling constant exhibit little momen- The constraint is enforced by introducing a Lagrange tumdependencewhenthenucleitakeunsymmetricalpo- multiplier and diagonalizing an effective Hamiltonian sitions in the crystal, which is the case with the pro- tons in NENP [44]. Hence we assume in the following H=H+2JλXn (cid:0)a†nan+b†nbn(cid:1) . (5) tehnacet Abeqtw≃eenAqt=h0e ≡elecAt.ronDicueantdo nthueclesaigrneifincearngyt dsciffaelers- e (h¯ω <10−5J), eq. (11) ends in N Via the Bogoliubov transformation ∼ 1 4(gµ ¯hγ A)2 π/2 n¯σ(n¯σ+1) √1N eik(2n−1/2)an = αkcoshθk−βk†sinhθk, T1 = Bπ¯hN Z−π/2 P√σv=2±k2k+vk¯hωN dk, (12) Xn (6) 1 √NXn e−ik(2n+1/2)bn = βkcoshθk−α†ksinhθk, ωwk−h=eπre/2 =as2suJm(Sin+gλ)m−ogdµerBaHte, wteemhapveeraatpuprreosximkBaTtedt≪he dispersion relations as with tanh2θ = Scosk/(S +λ), we reach the spin-wave k Hamiltonian Jω± ∆+vk2 gµ H, (13) k ≃ ± B = 2JS2N 2J(S+λ)N +J ωk with H − − Xk JS2 +J ω−α†α +ω+β†β , (7) ∆=2J λ(2S+λ), v = . (14) k k k k k k λ(2S+λ) Xk (cid:16) (cid:17) p p Equation (12) claims that an applied field produces two where distinct effects on T−1, one of which originates from the 1 ω± gµ H/J =2 (S+λ)2 S2cos2k ω . (8) Zeeman energy and appears in n¯σ, while the other of k ∓ B p − ≡ k which appears via the nuclear spikns giving the charac- Minimization of the free energy gives the optimum dis- teristic weight (v2k2+v¯hω )−1/2 to the electronic tran- N tthriebnutdieotnerfmunincteidontshraosugn¯h±k = (eJωk±/kBT −1)−1 and λ is sspitiinosn ersactaepn¯esσk(on¯bσks+erv1a)t.ioTnhienficerlidticeafflecstpionncthhaeinnsucwleitahr a linear dispersion at small momenta. The prefactor n¯−k +n¯+k +1 cosh2θk =(2S+1)N. (9) (v2k2 +v¯hωN)−1/2 is the consequence of the nature of Xk (cid:0) (cid:1) the delta function, 2 δ(x xi) The temperature dependence is mainly described by δ[f(x)]= − , (15) f′(x ) the term e−(∆0−gµBH)/kBT but further decorated due Xi | i | to the temperature-dependent energy spectrum. The where xi is a zero point of an arbitrary regular func- inelastic-neutron-scattering peak position of the lowest- tion f(x), and therefore generally arises from quadratic energy mode exhibits an upward behavior with increas- dispersionrelationsoftherelevantelectronicexcitations, ing temperature, for k T > ∆ /2 in particular [54–56], B 0 which are the case with ferromagnets [50–52] as well as where the slope of lnT−1∼to T−1 correspondingly in- 1 spin-gapped antiferromagnets. creases with increasing temperature. Letusfiteq. (12)fortheprotonspin-latticerelaxation- The nuclear magnetic relaxation in the Haldane-gap time measurements on NENP [44]. Although ∆ and v, antiferromagnet NENP has been interpreted in terms of given in eq. (14), depend on temperature in principle, a modified spin-wave theory. The field dependence of here we fix (v2k2 +v¯hω )−1/2 to its zero-temperature N T−1 was analyzed in detail and the minimum of T−1 as value in the integration (12), which is well justified for 1 1 a function of H, pending for the past decade, was solved. k T < ∆ gµ H and allows us to inquire further into eqB. (1∼2) an−alytBically. We compare the calculations with We considerthatsucha field dependence ofT1−1 is qual- itatively common to spin-gapped antiferromagnets. We the observations in Fig. 1. Assuming that g = 2 and encourage low-temperature T measurements on related J/k = 55K [44], we have set A equal to 0.024˚A−3, 1 B materials such as the ferromagnetic-antiferromagnetic whichsuggests the distance betweenthe interacting pro- bond-alternating compound (CH ) CHNH CuCl [57] ton and electron spins being about 3.5˚A and is consis- 3 2 3 3 and the two-leg ladder antiferromagnet SrCu O [58]. 2 3 tent very well with the structural analysis [53]. Under The authors are gratefulto Dr. N. Fujiwara and Prof. the present parametrization,the lowest excitation gap is T.Gotoforfruitful discussion. Thisworkwassupported given by ∆(T =0)/k ∆ /k 4.0K, which is some- B ≡ 0 B ≃ by the Ministry of Education, Culture, Sports, Science what smaller than that of NENP, 12.8K [44]. However, and Technology of Japan, and the Nissan Science Foun- thescaledfunction∆(T)/∆ wellreproducestheupward 0 dation. behavior of the Haldane-gap mode as a function of tem- perature [49]. Withincreasingfield,therelaxationratefirstdecreases moderately and then increases much more rapidly, at lowtemperatures inparticular. Althoughwe cannotcal- culate beyond the critical field H ∆ /gµ 9.5T c 0 B ≡ ≃ on the present formulation, our theory well reproduces [1] F. D.M. Haldane: Phys. Lett. 93A (1983) 464. the observations over a wide field range. For kBT < [2] F. D.M. Haldane: Phys. Rev.Lett. 50 (1983) 1153. ∆ gµBH, the distribution functions may be approx∼i- [3] S. R. White and D. A. Huse: Phys. Rev. B 48 (1993) ma−ted as n¯±k(n¯±k +1)≃e−ωk±/kBT and therefore eq. (12) 3844. can further be calculated as [4] O. Golinelli, Th. Jolicœur and R. Lacaze: Phys. Rev. B 50 (1994) 3037. 1 8(gµB¯hγNA)2e−∆0/kBTcosh gµBH K ¯hωN , [5] S. Yamamoto: Phys. Rev.Lett. 75 (1995) 3348. T1 ≃ π¯hv (cid:16) kBT (cid:17) 0(cid:16)2kBT(cid:17) [6] U. Schollw¨ock and Th. Jolicœur: Europhys. Lett. 30 (1995) 493. (16) [7] X. Wang, S. Qin and L. Yu: Phys. Rev. B 60 (1999) where K is the modified Bessel function of the second 14529. 0 kind. Provided h¯ω k T < ∆ gµ H, we further [8] S.TodoandK.Kato: Phys.Rev.Lett.87(2001)047203. N B B reach ≪ ∼ − [9] W. J. L. Buyers, R. M. Morra, R. L. Armstrong, M. J. Hogan,P.GerlachandK.Hirakawa: Phys.Rev.Lett.56 1 8(gµ ¯hγ A)2 gµ H B N e−∆0/kBTcosh B (1986) 371. T1 ≃ π¯hv (cid:16) kBT (cid:17) [10] J. P. Renard, M. Verdaguer, L. P. Regnault, W. A. C. ¯hω Erkelens, J. Rossat-Mignod and W. G. Stirling: Euro- N 0.80908 ln . (17) phys.Lett. 3 (1987) 945. ×(cid:20) − (cid:16)kBT(cid:17)(cid:21) [11] I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki: Phys. Thus, as H increases, T−1 should initially decrease log- Rev. Lett.59 (1987) 799. 1 arithmically and then increase exponentially. Figure 2 [12] I. Affleck, T. Kennedy,E. H. Lieb and H. Tasaki: Com- suggests that the initial decrease of T−1 turns from mun. Math. Phys.115 (1988) 477. T−1 lnH to T−1 H−1/2 with d1ecreasing tem- [13] S. R.White: Phys.Rev. Lett. 69 (1992) 2863. pe1ratu∝re−. The mom1entu∝m distribution functions n¯± are [14] S.Miyashitaand S.Yamamoto: Phys.Rev.B 48(1993) k 913. peaked at k = 0 and their peaks are sharpened as T [15] M. den Nijs and K. Rommelse: Phys. Rev. B 40 (1989) decreases. n¯± behaves as δ(k) in the low-temperature k 4709. limit. When we replace n¯±k by δ(k), eq. (12) gives a [16] S.YamamotoandS.Miyashita: Phys.Rev.B 48(1993) H−1/2-linear field dependence of T−1. 9528. 1 3 [17] S.Knabe: J. Stat. Phys.52 (1988) 627. C. Vettier: J. Appl.Phys. 63 (1988) 3538. [18] G. F´ath and J. S´olyom: J. Phys.: Condens. Matter 5 [55] Z. Tun,W. J. L. Buyers, R. L. Armstrong, K. Kirakawa (1993) 8983. and B. Briat: Phys.Rev.B 42 (1990) 4677. [19] S.Yamamoto: Phys. Lett. A 225 (1997) 157. [56] T.Sakaguchi,K.Kakurai,T.YokooandJ.Akimitsu: J. [20] S.Yamamoto: Int.J. Mod. Phys. B 12 (1998) 1795. Phys. Soc. Jpn. 65 (1996) 3025. [21] I.Affleck: Nucl. Phys.B 257 (1985) 397. [57] H.Manaka,I.YamadaandK.Yamagushi: J.Phys.Soc. [22] I.Affleck: Nucl. Phys.B 265 (1986) 409. Jpn. 66 (1997) 564. [23] M. Oshikawa, M. Yamanaka and I. Affleck: Phys. Rev. [58] M. Azuma, Z. Hiroi, M. Takano, K. Ishida and Y. Ki- Lett.78 (1997) 1984. taoka: Phys. Rev.Lett. 73 (1994) 3463. [24] K.Totsuka: Phys.Lett. A 228 (1997) 103. [25] M. Takahashi: Phys.Rev. Lett. 62 (1989) 2313. [26] S.YamamotoandS.Miyashita: Phys.Lett.A235(1997) 102 102 545. (a) (b) 3.01 T [27] S. Ma, C. Broholm, D. H. Reich, B. J. Sternlieb and R. 4.32 T W. Erwin: Phys.Rev.Lett. 69 (1992) 3571. 5.45 T [28] I. A. Zaliznyak, S.-H. Lee and S. V. Petrov: Phys. Rev. 101 Lett.87 (2001) 017202. [29] Z. Honda, H. Asakawa and K. Katsumata: Phys. Rev. 101 Lett.81 (1998) 2566. ] [30] J. Darriet and L. P. Regnault: Solid State Commun. 86 -1[s 100 (1993) 409. 1 [31] S.Maslov and A. Zheludev: Phys. Rev.B 57 (1998) 68. -T1 [32] Y. Takushima, A. Koga and N. Kawakami: Phys. Rev. 100 B 61 (2000) 15189. [33] A. Zheludev, J. M. Tranquada, T. Vogt and D. J. But- 10-1 trey: Phys.Rev.B 54 (1996) 6437. 4.2 K [34] A. Zheludev, J. M. Tranquada, T. Vogt and D. J. But- 8.0 K trey: Phys.Rev.B 54 (1996) 7210. 10.0 K [35] T. Yokoo, A. Zheludev, M. Nakamura and J. Akimitsu: 10-1 10-2 0.0 0.5 1.0 2.0 4.0 6.0 8.0 Phys.Rev.B 55 (1997) 11516. gm H/D D /k T B 0 0 B [36] M. Takigawa, T. Asano, Y. Ajiro, M. Mekata and Y. J. FIG.1. Dependences of the nuclear spin-lattice relaxation Uemura: Phys.Rev. Lett.76 (1996) 2173. rate on a field parallel to the chain (a) and temperature [37] F. Borsa and M. Mali: Phys. Rev.B 9 (1974) 2215. (b) in NENP (symbols) [44] are compared with the modified [38] S. Sachdev and K. Damle: Phys. Rev. Lett. 78 (1997) spin-wave calculations (lines), where the field and tempera- 943. [39] S.Fujimoto: J. Phys.Soc. Jpn. 68 (1999) 2810. tureare scaled by thelowest excitation gap ∆0. [40] S. Sachdev and K. Damle: J. Phys. Soc. Jpn. 69 (2000) 2712. [41] S.Fujimoto: J. Phys.Soc. Jpn. 69 (2000) 2714. 0.001 k T/D gm H/D k T/D gm H/D B 0 B 0 B 0 B 0 [42] P. Gaveau, J. P. Boucher, L. P. Regnault and J. P. Re- =0.10 =0.05 =0.10 =0.10 nard: Europhys.Lett. 12 (1990) 647. k n [43] N. Fujiwara, T. Goto, S. Maegawa and T. Kohmoto: Phys.Rev.B 45 (1992) 7837. [44] N. Fujiwara, T. Goto, S. Maegawa and T. Kohmoto: 0 Phys.Rev.B 47 (1993) 11860. [45] J. Sagi and Affleck: Phys. Rev.B 53 (1996) 9188. 0.2 kBT / D 0 gm BH / D 0 kBT / D 0 gm BH / D 0 =0.50 =0.05 =0.50 =0.10 [46] M. Takahashi: Phys.Rev. B 40 (1989) 2494. [47] J. E. Hirsch and S. Tang: Phys.Rev.B 40 (1989) 4769. nk [48] S.Tang, M. E. Lazzouniand J. E. Hirsch: Phys.Rev.B 40 (1989) 5000. 0.0 [49] S.Yamamotoand H.Hori: J. Phys.Soc.Jpn. 72(2003) 769. 0.3 kBT / D 0 gm BH / D 0 kBT / D 0 gm BH / D 0 [50] N.FujiwaraandM.Hagiwara: SolidStateCommun.113 = 1.00 = 0.05 = 1.00 = 0.10 (2000) 433. k n [51] S.Yamamoto: Phys. Rev.B 61 (2000) R842. [52] S.Yamamoto: J. Phys.Soc. Jpn.69 (2000) 2324. [53] A.Meyer,A.Gleizes, J.J. Girerd,M. VerdaguerandO. 0.0 Kahn: Inorg. Chem. 21 (1982) 1729. -0.4 -0.2 k / p 0.2 0.4 -0.4 -0.2 k / p 0.2 0.4 [54] J. P. Renard, M. Verdaguer, L. P. Regnault, W. A. C. FIG.2. The momentum distribution functions n¯+ (dotted k Erkelens,J.Rossat-Mignod,J.Ribas,W.G.Stirlingand lines) and n¯− (solid lines). k 4