The effect of spatial modulation of the single-site anisotropy D on the
ground state of the S=1 Heisenberg chains is investigated. In the case of
period 2 modulation, it is found that the phase diagram contains the Haldane
phase, large-D phase, Neel phase of udud-type and u0d0-type. It is shown that
the hidden antiferromagnetic order in the Haldane phase compatible with the
spatial modulation of D-term get frozen resulting in the emergence of various
types of Neel orders. The investigation of the model with longer period
D-modulation also confirms this picture.Comment: 4 pages, 3 figure

Chen, Wei

Hida, Kazuo

ヒダ, カズオ

飛田, 和男

English

arXiv

The effect of spatial modulation of the single-site anisotropy D on the ground state of the phase diagram contains the Haldane phase, large-D phase, N´eel phase of udud-type and u0d0-type. It is shown that the hidden antiferromagnetic order in the Haldane phase compatible with the spatial modulation of D-term get frozen resulting in the emergence of various types of N´eel orders. The investigation of the model with longer period D-modulation also confirms this picture.Copyright(c)2005 The Physical Society of Japa

飛田 和男

Saitama University Cyber Repository of Academic Resources

Typeset with jpsj2.cls <ver.1.2> Full PaperEmergence of Long Period Antiferromagnetic Orders from Haldane Phase in S = 1Heisenberg Chains with D-ModulationKazuo Hida¤ and Wei Chen1Department of Physics, Saitama University, Saitama, Saitama, 338-85701Institute of Biomedical Engineering, University of Montreal, Montreal, Qc. H3C 3J7 Canada(Received February 19, 2007)The e®ect of spatial modulation of the single-site anisotropy D on the ground state of theS = 1 Heisenberg chains is investigated. In the case of period 2 modulation, it is found that thephase diagram contains the Haldane phase, large-D phase, N¶eel phase of udud-type and u0d0-type. It is shown that the hidden antiferromagnetic order in the Haldane phase compatiblewith the spatial modulation of D-term get frozen resulting in the emergence of various typesof N¶eel orders. The investigation of the model with longer period D-modulation also con¯rmsthis picture.KEYWORDS: S=1 Heisenberg chain, single-site anisotropy, spatial modulation, Haldane phase, large-Dphase, N¶eel phase, hidden antiferromagnetic order1. IntroductionAntiferromagnetic quantum spin chains have been con-tinually the subject of experimental and theoretical re-search in the past two decades. Among them, the Hal-dane ground state of the S = 1 antiferromagnetic Heisen-berg chain with a gap to the ¯rst excited state1) hasbeen extensively studied by many authors. This state ischaracterized by the hidden antiferromagnetic string or-der accompanied by the Z2 £ Z2 symmetry breakdown.The easy plane single-site anisotropy D(> 0) destroysthe Haldane ground state leading to the large-D statewithout speci¯c order. On the other hand, the easy axissingle-site anisotropy (D < 0) drives the Haldane stateinto the N¶eel state.2{4)In this context, it is an interesting issue to investigatehow the Haldane phase is modi¯ed if the easy-axis andeasy-plane D-terms coexist in a single chain. As a sim-plest example of such competition, we ¯rst investigatethe ground state phase diagram of the S = 1 antiferro-magnetic Heisenberg chain with alternating D-term,H =NXl=1JSlSl+1 +D+N=2Xl=1Sz22l¡1+ D¡N=2Xl=1Sz22l ; (J > 0) (1)whereD+ = D0+±D;D¡ = D0¡±D and Si is the spin-1operator on the i-th site. The parametersD0 and ±D rep-resent uniform and alternating components of uniaxialsingle-ion anisotropy, respectively. The periodic bound-ary condition is assumed unless speci¯cally mentioned.The most remarkable conclusion on the ground state ofthis model obtained in the present work is that the u0d0-type long period N¶eel state emerges for large ±D althoughthis type of order is not compatible with the short rangeorder in the Haldane phase at D+ = D¡ = 0. Here u, 0and d stand for the single-site states jSzi i with Szi = 1; 0¤E-mail: hida@phy.saitama-u.ac.jpand ¡1, respectively. This implies that the single-siteanisotropy can reverse the sign of spin-spin correlation inthe corresponding isotropic model. This phenomenon canbe understood as a selection of the speci¯c antiferromag-netic order compatible with the D-alternation among thecomponents of the Haldane state which is the superposi-tion of various states with the hidden antiferromagneticorder. We also con¯rm this idea by investigating the sim-ilar model with long period D-modulation.This paper is organized as follows. In the next section,the e®ective Hamiltonians in various limiting cases arederived and the overall feature of the phase diagram isexplained. The numerical exact diagonalization resultsand the quantitative phase diagram are presented in x3. The model with long period D-modulation is discussedin x4. In the ¯nal section, we summarize our results.2. E®ective HamiltoniansLet us ¯rst consider the various limiting cases of theHamiltonian (1) in which the ground states can be de-termined by the e®ective Hamiltonians derived perturba-tionally. In the simplest limiting case of D+; D¡ >> J ,all spins are con¯ned to the state j0i and the groundstate is obviously the large-D phase.For D+; D¡ << ¡J , all spins are allowed to be in thestates j§1i. The e®ective coupling between these spinsare given byH(1)e® = J¢NXl=1Szl Szl+1; (Szl = §1) (2)up to the lowest order in J . Therefore we expect theudud-type N¶eel phase.In the limit of D+ >> J;D¡ << ¡J , the spins on theodd-th sites are con¯ned into the state j0i and those onthe even-th sites can take j§1i states. The latter spinsare coupled via the Ising type antiferromagnetic coupling12 J. Phys. Soc. Jpn. Full Paper Kazuo Hida and Wei Chenwithin the third order perturbation in J as,H(2)e® = Je®N=2Xi=1Sz2iSz2(i+1); Je® =J32±D2; (Szl = §1)(3)although the spins S2i and S2(i+1) are ferromagneticallycorrelated in the absence of D-terms. The origin of theantiferromagnetic interaction is as follows. Let us denotethe spin state of 3 successive sites of the original Hamil-tonian (1) by¯¯¯Sz2i Sz2i+1 Sz2(i+1)E. In the absence of theexchange term J , the ground states are j§1 0 § 1i whichare 4-fold degenerate. Due to the spin °ip term in (1), thej1 0 1i state is mixed with the j1 1 0i and j0 1 1i stateswhile the j1 0 ¡1i state is mixed with the j1 ¡1 0i andj0 1 ¡1i states. Therefore the energy of the j1 0 1i stateis raised by the nearest neighbour pair of up spins inthe intermediate state while the energy of the j1 0 ¡1istate is lowered by the nearest neighbour pair of up anddown spins in the intermediate state. This means thatthe e®ective interaction between Sz2i and Sz2(i+1) is anti-ferromagnetic. It should be noted that the spin °ip termsin the e®ective Hamiltonian appear only in the 4-th or-der perturbation in J . Therefore we expect the u0d0-typeN¶eel phase in this regime.This can be also interpreted using the concept of thehidden antiferromagnetic string order in the followingway. In the absence of the D-terms, the ground state hasa hidden string order which implies that the spins withj§1i are arranged antiferromagnetically if the sites withj0i are skipped.2,3) The position of the sites with j§1iand j0i strongly °uctuate quantum mechanically and thisantiferromagnetic order remains hidden because it is im-possible to observe the correlation between only the siteswith j§1i experimentally. In the presence of strong D-terms, only the states consistent with the constraint setby the D-terms survive among all states with hidden or-der. For D+; D¡ << ¡J , no j0i sites are allowed, so thatthe hidden order is frozen into the explicit udud order.For D+ >> J and D¡ << J , the odd-th site must be inthe state j0i and the even-th sites j§1i. To be compatiblewith the string order, spins must be arranged with u0d0order which is no more hidden. Thus the strong D-termselects the spin states among the states with hidden orderto realize the explicit N¶eel order of various periodicity.Near the line D0 » ±D >> J , we can also derive thee®ective Hamiltonian. The spins on the odd-th sites arecon¯ned into the state j0i and those on the even-th sitescan take all j§1i and j0i states. The e®ective couplingbetween the latter spins are obtained within the secondorder perturbation in J and ¯rst order in D0 ¡ ±D as,H(3)e® = Je®N=2Xi=1n12hS+2iS¡2(i+1) + S¡2iS+2(i+1)i+ De®Sz22(i+1)o(4)whereJe® =2J2D0 + ±D; De® = D0 ¡ ±D + 2J2D0 + ±D: (5)For the model (4), it is known that the large-D phase isstable for De®=Je® > 0:35. For 0:35 > De®=Je® > ¡2:0,the ground state is on the critical line between the Hal-dane and XY phase. For De®=Je® < ¡2:0, the groundstate is on the critical line between the N¶eel phase andthe XY phase.2{5) In the latter two cases, consideringthe continuity to other parts of the phase diagram, wemay expect that the ground state is actually the Hal-dane phase or the N¶eel phase due to the Ising compo-nent of the e®ective exchange interaction which appearas higher order corrections in J . It should be noted thatthis N¶eel phase is the u0d0 phase in the original Hamilto-nian (1). The large-D-Haldane phase boundary is givenby De®=Je® = 0:35 which gives,D20 ¡ ±D2 = ¡1:3J2: (6)The u0d0-Haldane phase boundary is given byDe®=Je® = ¡2:0 which gives,D20 ¡ ±D2 = ¡6:0J2: (7)These lines are plotted by the broken and dotted linesin the ground state phase diagram in Fig. 1 along withthe numerical results obtained in the next section. Wethus expect that the Haldane phase shrinks but does notvanish even in the limit of large D0 ' ±D.On the other hand, for large negative D0, the groundstate is the almost perfect udud state for D0 < ±D andthe almost perfect u0d0 state for D0 > ±D. The groundstate energy for the former case is given byEudud = N(D0 ¡ J) (8)and in the latter phaseEu0d0 =N2(D0 ¡ ±D): (9)Therefore the transition between these two phases shouldtake place around ±D ' ¡D0+2J . Around this line, how-ever, the hidden antiferromagnetic order cannot freeze,because the correlation between the 2i-th spin and 2i+2-th spin can be ferromagnetic or antiferromagnetic de-pending on whether the 2i + 1-th spin is j0i or j§1i.Therefore the hidden order remains hidden. Accordinglya narrow Haldane phase should survive in the neighbour-hood of the line ±D ' ¡D0+2J for arbitrarily large ±D.3. Numerical Analysis3.1 Haldane-Large-D transition lineThis phase transition is a Gaussian transition. In or-der to determine the phase boundary with high accu-racy, we use the twisted boundary method of Kitazawaand Nomura.6,7) The Hamiltonian is numerically diag-onalized to calculate the two low lying energy levelswith the twisted boundary condition (SxN+1 = ¡Sx1 ;SyN+1 = ¡Sy1 ; SzN+1 = Sz1 ) for N = 8; 12 and 16 bythe Lanczos method.For small D0 and ±D, the ground state is the Haldanephase with a valence bond solid (VBS) structure.8) Underthe twisted boundary condition, the eigenvalue of thespin reversal T (Szi ! ¡Szi ; S§i ! ¡S¨i ) is equal to ¡1in this phase.6,7) As D0 increases with positive value,a phase transition takes place from the Haldane to theJ. Phys. Soc. Jpn. Full Paper Kazuo Hida and Wei Chen 30 2 4 605δDD0u0d0large−DHaldaneududFig. 1. Ground state phase diagram of the Hamiltonian (1). Theerror bars are within the size of the symbols. The solid linesare the guide for eye. The broken and dotted lines represent theapproximate phase boundary given by Eqs. (6) and (7), respec-tively. The dash-dotted line is the line ±D = ¡D0 + 2J .large-D phase for which T = 1. Because the quantumnumber T of Haldane and large-D phases are di®erent fortwisted boundary condition, the ground state energy ofthese two phases cross each other at the phase boundary.For example, if ±D is ¯xed, the two levels cross at the¯nite size critical point D0c(N). Extrapolation to thethermodynamic limit is carried out assuming the formulaD0c(N) = D0c(1) + aN¡2 + bN¡4.3.2 Haldane-udud and Haldane-u0d0 transition linesFrom symmetry consideration, these transitions areIsing type transitions. We therefore employ the phe-nomenological renormalization group (PRG) method todetermine the transition lines. The Hamiltonian is nu-merically diagonalized to calculate the lowest energy gap¢E(N;D0; ±D) with total Sz = 0 in the periodic bound-ary condition using the Lanczos algorithm. Both N¶eeltype states have Sz = 0 and are two fold degenerate inthe thermodynamic limit. For ¯nite N , this degeneracy islifted and the energy di®erence between them gives thesmallest gap ¢E(N;D0; ±D) which decreases exponen-tially with N . On the other hand, in the Haldane phase,the energy gap ¢E(N;D0; ±D) remains ¯nite in the ther-modynamic limit. Thus the product N¢E(N;D0; ±D)increases (decreases) with N in the Haldane(udud oru0d0) phase. The intersection of N¢E(N;D0; ±D) fortwo successive sizes N = N1 and N2 de¯nes the ¯-nite size critical point ±Dc(N1; N2) for ¯xed D0. Tak-ing into account the Ising universality class, these val-ues are extrapolated using the formula ±Dc(N1; N2) =±Dc(1)+2C1=(N1+N2)3.9,10) The same procedure canbe also carried out interchanging the roles of ±D and D0.For the Haldane-udud phase boundary, the system sizesN = 8; 10; 12; 14 and 16 are used. For the Haldane-u0d0phase boundary, only the sizes of multiples of 4 are com-patible with the u0d0 order. We therefore use the data0 0.1 0.2 0.311.21.4δD∆EN=8N=12N=16J/δDFig. 2. Energy gap ¢E on the line ±D = ¡D0 + 2J plottedagainst ±D. The vertical axis is multiplied by ±D to clarify the±D-dependence of ¢E The data for N = 12 and N = 16 over-wrap within the size of the symbols.forN = 4; 8; 12 and 16 to determine this phase boundary.The numerical phase diagram also shows that thereexists an intermediate phase between the u0d0 phase andthe udud phase which is continuously connected withthe Haldane phase. This is consistent with the analyticalargument in the preceding section. To further con¯rmthe presence of this phase, we calculate the energy gap¢E to the lowest excited state which has total Sz = 1along the line ±D = ¡D0+2J up to ±D = 40J as shownin Fig. 2 for N = 8; 12 and 16. It is clear that the energygap does not depend on the system size signi¯cantly andbehaves as ±D¡1 for large ±D. We have also checkedthat the parity T with respect to the spin inversion isalways ¡1 with twisted boundary condition along theline ±D = ¡D0 + 2J . This con¯rms the VBS nature ofthis phase.4. Long Period ModulationThe mechanism of the selection of u0d0-phase givenin x2 suggests that the similar mechanism also works fora variety of modulation patterns of D-terms producinga variety of long range N¶eel order. To con¯rm this idea,we also carried out the numerical calculation with theHamiltonian with period p modulation of the D-terms.Hp =NXl=1JSlSl+1 ¡ ±DN=pXl=1Sz2pl+ ±DN=pXl=1p¡1Xq=1Sz2pl+q; (10)where Sl are spin-1 operators. For p = 2, this reducesto the Hamiltonian (1) with D0 = 0. The Haldane-N¶eel-type transition takes place at ±Dc ' 2:69 for p = 4 (es-timated from N = 8 and 16), ±Dc ' 2:75 for p = 3(estimated from N = 6; 12 and 18) and ±Dc ' 3:29 forp = 2 (estimated from N = 4, 8, 12 and 16).For ±D > ±Dc, only the spins Spl are alive. The spin-spin correlation functions among these alive spins areplotted in Fig. 3 for ±D = 4. It is clear that the alivespins are aligned antiferromagnetically with period p ir-respective of the short range correlation expected in the4 J. Phys. Soc. Jpn. Full Paper Kazuo Hida and Wei Chen0 5 10 15−101p|i−j|<SzpSzpi>Fig. 3. Correlation functions between the alive spinsDSzpSzpiEforp = 1(N = 16, circles), p = 2(N = 16, squares), p = 3(N = 18,diamonds) and p = 4(N = 16, triangles). ±D = 4.absence of ±D-term.5. SummaryThe ground state phase diagram of the spin-1 Heisen-berg chains with alternating single-site anisotropy isdetermined by analyzing the numerical diagonalizationdata and the perturbation analysis of various limitingcases. In addition to the Haldane, large-D and conven-tional udud N¶eel phases, the N¶eel phase with u0d0 spincon¯guration is found. The emergence of these two typesof N¶eel phases is explained as a selection of spin con-¯guration compatible with the spatial modulation of D-terms among various spin con¯gurations with hidden an-tiferromagnetic string order in the Haldane phase.We ¯nd no direct transition between the udud andu0d0 phase and the intermediate gapful phase is alwayspresent. This gapful phase is continuously connectedwith the Haldane phase. Between the large-D phase andthe u0d0 phase, we also ¯nd the Haldane phase. This ro-bustness of the Haldane phase is also understood by theabove picture.The case of longer period D-modulation is also inves-tigated to con¯rm the above picture. It is natural to ex-pect various types of N¶eel order emerge from the Haldanestate in the presence of various types of spatial modula-tion of D.So far, it is impossible to observe the string order it-self experimentally, because the spatial spin structurecorresponding to the string order is °uctuating quan-tum mechanically. However, if the S = 1 chain materialwith spatially modulated single-site anisotropy D whosemagnitude is larger than the exchange coupling J is re-alized, the observation of the N¶eel order in such ma-terial can be regarded as the observation of the frozenstring order. Unfortunately, the anisotropy parameterD is smaller than the exchange coupling J for typicalquasi-one-dimensional magnetic materials which are sofar available. However, it is well known that the exchangecoupling is quite sensitive to the bond angle. Therefore itwould be still possible to synthesize the material with Jsmaller than jDj by ¯ne tuning. If the D-term is spatiallymodulated by choosing the magnetic ions and their en-vironment appropriately, the frozen string order shouldbe observable. The investigation in this direction wouldbe a challenging attempt for the experimentalists.The computation in this work has been done usingthe facilities of the Supercomputer Center, Institute forSolid State Physics, University of Tokyo, the InformationProcessing Center of Saitama University. The diagonal-ization program is based on the TITPACK ver.2 codedby H. Nishimori. This work is supported by a researchgrant from a Grant-in-Aid for Scienti¯c Research fromthe Ministry of Education, Science, Sports and Cultureof Japan.1) F. D. M. Haldane: Phys. Lett. 93A (1983); Phys. Rev. Lett.50 (1983) 1153.2) M. den Nijs and K. Rommelse: Phys. Rev. B40 (1989) 4709.3) H. Tasaki: Phys. Rev. Lett 66 (1991) 798.4) W.Chen, K.Hida and B.C. Sanctuary: Phys.Rev.B67 (2003)104401.5) A. Kitazawa, K. Hijii and K. Nomura : J. Phys. A: Math. Gen.36 (2003) L351.6) A. Kitazawa : J. Phys. A Math. Gen. 30 (1997) L285.7) A. Kitazawa and K. Nomura: J. Phys. Soc. Jpn. 66 (1997)3944.8) I. A²eck, T. Kennedy, E. H. Lieb and H. Tasaki : Commn.Math. Phys. 115 (1988) 477; Phys. Rev. Lett. 59 799 (1987).9) M. N. Barber: Phase Transitions and Critical Phenomena 8,ed. C. Domb and J. L. Lebowitz (Academic Press, London,New York, 1983) 146.10) T. Sakai and M. Takahashi: J. Phys. Soc. Jpn. 59 (1990) 2688.

Emergence of Long Period Antiferromagnetic Orders from Haldane Phase in S=1 Heisenberg Chains with D-Modulation

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