The $S=1/2$ frustrated Heisenberg chains with bond alternation is known to
exhibit a magnetization plateau at half of the saturation magnetization $\Ms$
accompanied by the spontanuous translational symmetry breakdown. The effect of
randomness on the magnetization process of this model is investigated. First we
consider the mixture of the two kinds of chains both of which possess the
$\Ms/2$-plateau in the common interval of the magnetization field. The plateau
at $\Ms/2$ is found to vanish immediately if the randomness is switched on in
agreement with Totsuka's prediction. The small plateau also appears near the
saturation field due to the localization of inverted spins around the minority
bond. On the other hand, if the stronger bond is replaced by the ferromagnetic
bonds randomly, the randomness induced fractional plateau appears as in the
nonfrustrated case. The plateau at $\Ms/2$ does not simply vanish but shifts
and splits into two smaller plateaux. The magnetization on this plateau varies
nonlinearly with $1-p$. The physical origin of this behavior is explained based
on the strong coupling picture.Comment: 5 pages, 9 figures, references corrected and adde

Hida, Kazuo

ヒダ, カズオ

飛田, 和男

English

arXiv

The S = 1/2 frustrated Heisenberg chains with bond alternation is known to exhibit a magnetization plateau at half of the saturation magnetization Ms accompanied by the spontaneous translational symmetry breakdown. The effect of randomness on the magnetization process of this model is investigated. First we consider the mixture of the two kinds of chains both of which possess the Ms/2-plateau in the common interval of the magnetization field. The plateau at Ms/2 is found to vanish immediately if the randomness is switched on in agreement with Totsuka’s prediction. The small plateau also appears near the saturation field due to the localization of inverted spins around the minority bond. On the other hand, if the stronger bond is replaced by the ferromagnetic bonds randomly, the randomness induced fractional plateau appears as in the nonfrustrated case. The plateau at Ms/2 does not simply vanish but shifts and splits into two smaller plateaux. The magnetization on this plateau varies nonlinearly with 1 − p. The physical origin of this behavior is explained based on the strong coupling picture.Copyright(c)2003 The Physical Society of Japa

飛田 和男

Saitama University Cyber Repository of Academic Resources

Typeset with jpsj2.cls <ver.1.2> Full PaperMagnetization Plateaux in Random Frustrated S=1/2 Heisenberg ChainsKazuo Hida ¤Department of Physics, Faculty of Science,Saitama University, Saitama 338-8570(Received February 19, 2007)The S = 1=2 frustrated Heisenberg chains with bond alternation is known to exhibit a mag-netization plateau at half of the saturation magnetizationMs accompanied by the spontanuoustranslational symmetry breakdown. The e®ect of randomness on the magnetization process ofthis model is investigated. First we consider the mixture of the two kinds of chains both ofwhich possess theMs=2-plateau in the common interval of the magnetization ¯eld. The plateauat Ms=2 is found to vanish immediately if the randomness is switched on in agreement withTotsuka's prediction. The small plateau also appears near the saturation ¯eld due to the local-ization of inverted spins around the minority bond. On the other hand, if the stronger bondis replaced by the ferromagnetic bonds randomly, the randomness induced fractional plateauappears as in the nonfrustrated case. The plateau at Ms=2 does not simply vanish but shiftsand splits into two smaller plateaux. The magnetization on this plateau varies nonlinearly with1¡ p. The physical origin of this behavior is explained based on the strong coupling picture.KEYWORDS: random Heisenberg chain, bond alternation, frustration, DMRG, magnetization plateau1. IntroductionIn the recent studies of quantum many body problem,magnetization plateaux in one dimensional Heisenbergchains are attracting much attention as the ¯eld inducedspin gap states. On the plateau, some spins are partlyquenched by the magnetic ¯eld while the remaining spinsform a spin gap state.1{4)Oshikawa, Yamanaka and A²eck5) proposed the nec-essary condition for the magnetization plateaus asQGS(S ¡M) = Z ´ integer (1)where QGS is the spatial periodicity of the magneticground state, S is the magnitude of the spin, and M isthe magnetization per site. It should be noted thatQGS isnot always equal to the spatial periodicity of the Hamil-tonian QHam. If the translational symmetry is spontan-uously broken, QGS = zQHam where z is an integer.Recently, the e®ect of randomness on the magneti-zation plateaux of one-dimensional quantum spin sys-tems has been studied theoretically.6{8) Among them,the present author found the randomness inducedplateau in the mixture of S = 1=2 antiferromagnetic-antiferromagnetic alternating Heisenberg chain andS = 1=2 ferromagnetic-antiferromagnetic alternat-ing Heisenberg chain8) which models the compound(CH3)2CHNH3Cu(ClxBr1¡x)3 studied by Manaka andcoworkers.9) Similar observation has been made by Cabraand coworkers6) for random q-merized chains. It shouldbe noted that these plateaux are realized at arbitraryvalues of magnetization depending on the concentrationratio of random bonds. Of course, even in the periodicsystem, one can realize the plateau at any arbitrary mag-netization value by choosing a periodic structure with ap-propriate periodicity. However, in the random systems,the position of the plateau can be continuously varied bychanging the bond concentration which is an experimen-tally tunable single parameter. From this point of view,the randomness plays an essential role in realizing theseplateaux.On the other hand, Totsuka7) discussed the e®ect ofrandomness on the magnetization plateaux which are al-ready present in periodic systems. He concluded thatthe plateau induced by the imposed periodicity (QGS =QHam) is stable against weak randomness whereas theplateau induced by the spontanuous translational sym-metry breakdown (STSB) (QGS 6= QHam) is unstableagainst randomness due to Imry-Ma e®ect.10)Therefore we know that randomness induces the mag-netization plateau on one hand and on the other handit destroys the plateau. It is the purpose of the presentwork to investigate the interplay of these two apparentlycontradicting aspects of randomness e®ect on the mag-netization plateau phenomenon using the density matrixrenormalization group (DMRG) method.11)This paper is organized as follows. In the next sec-tion, the model Hamiltonian is de¯ned. Frustrated ran-dom chains with bond alternation are investigated in sec-tion 3.1 and frustrated chains with bond alternation andrandom sign strong bonds are studied in section 3.2 us-ing the DMRG method. The physical interpretation ofthe numerical results are also given. The ¯nal section isdevoted to a summary and discussion.2. Model HamiltonianAs a simplest model which exhibits a plateau accom-panied by STSB, we consider the S = 1=2 Heisenbergchain with next-nearest-neighbour interaction depictedin Fig. 1. The Hamiltonian is given by,H =N¡1Xi=1JiSiSi+1 +N¡2Xi=1J±SiSi+2 (2)where the bond alternation and randomness is intro-duced in the distribution of Ji. In the present work, we12 J. Phys. Soc. Jpn. Full Paper Kazuo HidaJδ Jδ JδJδJδ Jδ JδJδJi−2 Ji−1 Ji Ji+1 Ji+2 Ji+3 Ji+4Ji−4 Ji−3Fig. 1. Heisenberg chain with next-nearest-neighbour interac-tion. The nearest neighbour interaction is spatially modulated.0 1 200.20.40.6MH(a) α=0.6 δ=0.20 1 200.20.40.6MH(b) α=0.5 δ=0.2Fig. 2. The magnetization curves of the pure frustrated randomHeisenberg chains with ± = 0:2, (a) ® = 0:5 and (b) ® = 0:6.The system size is N = 82.consider the following two types of randomness:(1) Random bond alternationJi =½J(1 + (¡1)i®1) : probability pJ(1 + (¡1)i®2) : probability 1¡ p (3)(2) Random sign strong bondsJ2i¡1 = J > 0J2i =½JA(> J) : probability pJF(< 0) : probability 1¡ p (4)In the following, we set J = 1 to de¯ne the energy unit.In the pure case (®1 = ®2 ´ ®), this model has aplateau at the magnetization M =Ms=2 for appropriatevalues of ® and ±2{4) where Ms is the saturation mag-netization per site (Ms ´ 1=2 in the present model). Toexplain this plateau, we have to set QGS = 2QHam inEq. (1). Therefore this plateau is accompanied by STSB.Figures 2(a) and (b) show typical examples of such caseswith ± = 0:2 and (a) ® = 0:5 and (b) ® = 0:6 calculatedby the DMRG method.3. Numerical Results3.1 Case 1: Frustrated Heisenberg chain with randombond alternationFrom Fig. 2, it is clearly seen that the magnetic ¯eldintervals of the plateau region of cases (a) and (b) have ¯-nite overlap. In this section, we investigate what happensif these two chains are mixed. The result is shown in Fig.3. Even in the magnetic ¯eld interval in which both (a)and (b) have plateau, no plateau appears in the mixedchain. This veri¯es the fragility of this kind of plateaudue to Imry-Ma e®ect10) as predicted by Totsuka.7)It should be also remarked that the plateau with M =0 (spin gap) remains stable. This plateau is due to theimposed bond alternation (QGS = QHam) so that thisresult is also in accord with Totsuka's prediction.7)The structure near the saturation is also induced byrandomness. It should be remarked that the saturation¯eld Hsat in the mixed chain is larger than those ofboth pure chains. The saturation ¯eld is determined asthe energy di®erence between the fully polarized state(Mtot = N=2) and the state with a single inverted spin(Mtot = N=2 ¡ 1). Here Mtot denotes the total magne-tization. In the presence of the impurity bond, the in-verted spin can be localized around the impurity bond.In Figure 4, the saturation ¯elds of the chains with a sin-gle impurity bond with strength Jimp in the host chainwith bond alternation ® = ®host is presented for N = 82chain. The ¯lled circles represent the saturation ¯eld ofthe chain with ®host = 0:5 and ± = 0:2 in which a sin-gle strong bond is replaced by the impurity bond withJimp = 1:6 corresponding to the strength of the strongbond of the chain with ® = 0:6. The open circles rep-resent the saturation ¯eld of the chain with ®host = 0:6and ± = 0:2 in which a single weak bond is replaced bythe impurity bond with Jimp = 0:5 corresponding to thestrength of the weak bond of the chain with ® = 0:5.The saturation ¯eld of the pure chain is easily calculatedanalytically asHsat =8>>>>><>>>>>:2 ± · 1¡ ®241 +2±1¡ ®2 +1¡ ®28±1¡ ®24®¸ ± ¸ 1¡ ®241 + 2± + ® ± ¸ 1¡ ®24®(5)as plotted in Fig. 4 by the solid and dotted lines for® = 0:5 and 0.6 with ± = 0:2, respectively. It is seen thatthe saturation ¯eld for the chain with an impurity is sub-stantially larger than the pure chains for small ± wherethe localized states are formed. Corresponding probabil-ity density of the inverted spin j x(i) j2 in the groundstate with Mtot = N=2¡1 is shown in Fig. 5 for ± = 0:2.This explicitly shows that the inverted spin is localizedJ. Phys. Soc. Jpn. Full Paper Kazuo Hida 30 1 200.20.40.6MHα1=0.5 α2=0.6 δ=0.2p=0.1p=0.5p=0.9Fig. 3. The magnetization curves of the random frustratedHeisenberg chains with ± = 0:2, ®1 = 0:5 and ®2 = 0:6 forvarious values of p. The system size is N = 82 and average istaken over 40 samples.0 122.533.5 Jimp=1.6 on strong bond αhost=0.5Jimp=0.5 on weak bond   αhost=0.6α=0.5α=0.6} uniformδHsatFig. 4. The saturation ¯elds Hsat of the frustrated Heisenbergchains with an impurity bond with N = 82. The ¯lled (open) cir-cles represent the saturation ¯eld of the chain with ® = 0:5(0:6)with a single strong (weak) bond replaced by the bond withJimp = 1:6 (0.5). The solid and dotted lines are the saturation¯elds of the pure chains with ® = 0:5 and 0.6, respectively.0 20 40 60 8000.51|x(i)|2iJimp=1.6 on strong bond αhost=0.5Jimp=0.5 on weak bond   αhost=0.6δ=0.2Fig. 5. The probability density j x(i) j2 of the inverted spin in theground state with Mtot = N=2¡ 1 of the frustrated Heisenbergchains with an impurity bond. The chain length is N = 82.The ¯lled (open) circles represent j x(i) j2 of the chain with® = 0:5(0:6) with a single strong (weak) bond replaced by thebond with Jimp = 1:6 (0.5) at i = 41(40) with ± = 0:2.around the impurity bond.For small but ¯nite density of impurity bonds, themagnetization of the most part of the chain is saturatedat the saturation ¯eld of the pure chain. However, theportion of the chain proportional to the density of theimpurity bond remains unpolarized until the saturation0 1 200.20.40.6MHp=1p=0.8p=0.6p=0.4p=0.2p=0JA=1.5 JF=−1.5 J=0.5 δ=0.2Fig. 6. The magnetization curves of the frustrated Heisenbergchains with bond alternation and random sign strong bonds withJA = 1:5; JF = ¡1:5; J = 1:0 and ± = 0:2 for various values of p.The system size is N = 82 and average is taken over 40 samples.The small low ¯eld magnetization for p = 1 is due to the ¯nitesize e®ect.¯eld of the chain with an impurity is reached. Thereforethere appears a small plateau at M = Msp for small pand at M =Ms(1¡p) for small 1¡p as observed in Fig.3. This can be also regarded as a randomness inducedplateau.3.2 Case 2 : Frustrated chains with bond alternationand random sign strong bondsIn the absence of frustration ± = 0, this model showsthe randomness induced plateau at M = Ms(1 ¡ p)as pointed out by the present author.8) The same phe-nomenon takes place in the presence of ¯nite ±.The fate of the plateau at M = Ms=2 is di®erentfrom that of case 1. The plateau no more appears atM = Ms=2 but it does not simply vanish. The plateausplits into two plateaux with magnetizations Mp1 andMp2 as depicted in Fig. 6. Actually, the lower one (Mp2)of these two plateaux is clearly visible only for relativelylarge values of 1¡ p in Fig. 6. The precise values of Mp1and Mp2 are plotted against p in Fig. 7. It should benoted that the magnetization on both plateaux dependsnonlinearly on 1¡ p.The nonlinear dependence of Mp1 and Mp2 on p canbe understood based on the strong coupling picture forthe plateau state proposed by Totsuka4,7) in the follow-ing way. On the plateau, all the spin pairs connected byferromagnetic bonds are polarized. This gives the contri-butionMs(1¡p) to the magnetization which is the valuefor the randomness induced plateau. By these bonds,the chain is cut into segments of ¯nite length. Consider-ing that the neighbouring triplet-triplet or singlet-singletpairs costs energy,4,7) the segments consisting of 2l + 1strongly coupled antiferromagnetic dimers have stablymagnetized state with l triplets pairs. The average num-ber of such segment is (N=2)(1¡ p)2p2l+1. So that thesesegments contribute to the magnetization by an amount4 J. Phys. Soc. Jpn. Full Paper Kazuo Hida0 0.5 100.20.4Mp1−pFig. 7. The 1¡p-dependence of the magnetization on the plateauwhich reduces to the M = Ms=2 plateau in the pure p = 1 limitfor the frustrated Heisenberg chains with bond alternation andrandom sign strong bonds. The ¯lled and open circles representMp1 and Mp2, respectively. The solid and dotted cureves repre-sent the formulae (9) and (10), respectively.N21Xl=1l(1¡ p)2p2l+1 (6)On the other hand, for the segments consisting of 2lstrongly coupled dimers, it is enevitable to have at leastone neighbouring triplet-triplet or singlet-singlet pairs.Therefore these segments have stably magnetized statewith l¡1 or l triplets pairs. The average number of suchsegment is (N=2)(1 ¡ p)2p2l. Therefore they contributeto the magnetization by amountsN21Xl=1(l ¡ 1)(1¡ p)2p2l; (7)orN21Xl=1l(1¡ p)2p2l (8)These summations are easily carried out and the stableplateaux are expected atMp1 =Ms(1¡ p+ p21 + p) (9)andMp2 =Ms(1¡ p+ p31 + p) (10)These are also plotted in Fig. 7 by solid and dottedcurves. The coincidence between the numerical resultsand above formulae is fairly good. The slight deviationis attributed to the ¯nite size e®ect.The nonlinear p-dependence of the plateau magnetiza-tion is in contrast to the case of the plateau in randomq-merized chain in which ± = 0 and Ji are randomlydistributed according to the distribution.Ji =½J(1 + ®i) : probability 1¡ pJ 0 : probability p (11)where ®i ´ ®; (¡®) if i = qn; (i 6= qn ). The XXmodel with this exchange distribution is investigatedby Cabra and coworkers.6) Here we present the mag-netization curve of the corresponding antiferromagneticHeisenberg model using the DMRG method in Fig. 8 forq = 4. In this case also, the plateau at M =Ms=2 in the0 1 200.20.40.6α=0.5 J’=0.2MHp=0p=0.2p=0.4p=0.6p=0.8p=1Fig. 8. The magnetization curves of the random q-merizedHeisenberg chains with q = 4; ® = 0:5; J 0 = 0:2 and J = 1:0for various values of p. The system size is N = 100 and averageis taken over 20 samples.0 0.5 100.20.4MppFig. 9. The p-dependence of the magnetization on the plateaurandom q-merized Heisenberg chains (open circles) with q = 4.The solid line is Eq. (12).pure chain shifts due to randomness. However, this shiftis proportional to p as,Mp =Ms2(1 + p); (12)as predicted by Cabra et al.6) and shown in Fig. 9 by thedotted curve.As discussed by Totsuka,4) the plateau due to STSBat M = Ms=2 in the model (2) without randomness isformed by the many body e®ect. This plateau state canbe regarded as a kind of Mott insulator of triplet pairs.This feature is also re°ected in the nonlinear dependenceof the plateau magnetization on p. On the other hand, theplateau state atM =Ms=2 of model (11) with p = 0 canbe understood by the local picture. Therefore the plateaumagnetization just shifts linearly with the number of theimpurity bonds.4. Summary and DiscussionThe magnetization plateaux in random quantum spinchains are investigated by the DMRG method. In therandom mixture of two frustrated and dimerized chainswith di®erent values of dimerization parameter, it is veri-¯ed that the plateau accompanied by STSB is destroyedby randomness even in the interval where the uniformchains have a plateau in common. It is also veri¯ed thatthe plateau due to imposed spatial periodicity is robust.J. Phys. Soc. Jpn. Full Paper Kazuo Hida 5These are consistent with the analytical prediction byTotsuka.7) The small randomness induced plateau is alsopredicted near the saturation ¯eld. This plateau is dueto the localization of inverted spins.For the frustrated chains with bond alternation andrandom sign strong bonds, it is found that the nontrivialplateau due to STSB splits into two smaller plateaux.The physical picture of these plateau states are clari¯ed.In contrast to the plateaux without STSB, the magneti-zation on the plateaux varies nonlinearly with p.So far, the experimental study of the interplay betweenthe randomness, frustration and magnetization plateauphenomenon has not yet been carried out. Consideringthe variety of phenomena predicted analytically and nu-merically and recent progress of high magnetic ¯eld tech-nique, fruitlful physics is expected in this ¯eld in the nearfuture.The computation in this work has been done usingthe facilities of the Supercomputer Center, Institute forSolid State Physics, University of Tokyo and the Informa-tion Processing Center, Saitama University. This work issupported by a Grant-in-Aid for Scienti¯c Research fromthe Ministry of Education, Culture, Sports, Science andTechnology, Japan.1) K. Hida: J. Phys. Soc. Jpn. 63 (1994) 2359.2) T. Tonegawa, T. Nishida and M. Kaburagi: Physica B 246-247(1998) 368.3) T. Tonegawa, T. Hikihara, K. Okamoto and M. Kaburagi:Physica B 294-295 (2001) 39.4) K. Totsuka: Phys. Rev. B57 (1998) 3454.5) M. Oshikawa, M. Yamanaka and I. A²eck: Phys. Rev. Lett.78 (1997) 1984.6) D. C. Cabra, A. De Martino, M. D. Grynberg, S. Peysson andP. Pujul: Phys. Rev. Lett. 85 (2000) 4791.7) K. Totsuka: Phys. Rev. B64 (2001) 134420.8) K. Hida: J. Phys. Soc. Jpn. in press, cond-mat/0111521; Prog.Theor. Phys. Suppl. 145 (2002) 320.9) H. Manaka, I. Yamada and H. Aruga Katori: Phys. Rev. B63(2001)104408 and references therein.10) Y. Imry and S.-K. Ma : Phys. Rev. Lett. 21 (1975) 1399.11) S. R. White: Phys. Rev. Lett. 69(1992) 2863; Phys. Rev.B48(1993) 10345.

Magnetization plateaux in random frustrated S=1/2 Heisenberg chains

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Magnetization Plateaux in Random Frustrated S=1/2 Heisenberg Chains

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