The plateau-non-plateau transition in the
antiferromagnetic-(ferromagnetic)$_{n}$ polymerized $S=1/2$ XXZ chains under
the magnetic field is investigated. The universality class of this transition
belongs to the Brezinskii-Kosterlitz-Thouless (BKT) type. The critical points
are determined by level spectroscopy analysis of the numerical diagonalization
data for $4 \leq p \leq 13$ where $p(\equiv n+1)$ is the size of a unit cell.
It is found that the critical strength of ferromagnetic coupling decreases with
$p$ for small $p$ but increases for larger enough $p$. It is also found that
the plateau for large $p$ is wide enough for moderate values of exchange
coupling so that it should be easily observed experimentally. This is in
contrast to the plateaus for $p = 3$ chains which are narrow for a wide range
of exchange coupling even away from the critical point

Ajiro Y.

Hida K.

Kitazawa A.

Nakano H.

Narumi Y.

Nomura K.

Okamoto K.

Oshikawa M.

Sakai T.

Shanks D.

Shiramura W.

Tonegawa T.

Totsuka K.

English

arXiv

http://jpsj.ipap.jp/link?JPSJ/69/3414/ | http://jpsj.ipap.jp/link?JPSJ/69/3414/arXiv:cond-mat/0005544v1The plateau-non-plateau transition in the antiferromagnetic-(ferromagnetic)(n) polymerized S = 1/2 XXZ chains under the magnetic field is investigated. The universality class of this transition belongs to the Brezinskii-Kosterlitz-Thouless (BKT) type. The critical points are determined by level spectroscopy analysis of the numerical diagonalization data for 4 less than or equal to p less than or equal to 13 where p(equivalent to n + 1) is the size of a unit cell. It is found that the critical strength of ferromagnetic coupling decreases with p for small p but increases for large enough p. It is also found that the plateau for large p is wide enough for moderate values of exchange coupling so that it should be easily observed experimentally. This is in contrast to the plateaus for p = 3 chains which are narrow for a wide range of exchange coupling even away from the critical point.textapplication/pdfjournal articl

Chen, Wei

飛田, 和男

ヒダ, カズオ

Sanctuary, Bryan, Clifford

Saitama University Cyber Repository of Academic Resources

arXiv:cond-mat/0005544v1  [cond-mat.stat-mech]  31 May 2000typeset using JPSJ.sty <ver.1.0b>Magnetization plateaus in antiferromagnetic-(ferromagnetic)npolymerized S = 1/2 XXZ chainsWei Chen∗, Kazuo Hida∗∗ and B. C. Sanctuary1Department of Physics, Saitama University, Urawa, Saitama, Japan 338-85701 Department of Chemistry, McGill University, Montreal, PQ, Canada H3A 2K6(Received August 12, 2008)The plateau-non-plateau transition in the antiferromagnetic-(ferromagnetic)n polymerized S =1/2 XXZ chains under the magnetic field is investigated. The universality class of this transitionbelongs to the Brezinskii-Kosterlitz-Thouless (BKT) type. The critical points are determined bylevel spectroscopy analysis of the numerical diagonalization data for 4 ≤ p ≤ 13 where p(≡ n+1)is the size of a unit cell. It is found that the critical strength of ferromagnetic coupling decreaseswith p for small p but increases for larger enough p. It is also found that the plateau for largep is wide enough for moderate values of exchange coupling so that it should be easily observedexperimentally. This is in contrast to the plateaus for p = 3 chains which are narrow for a widerange of exchange coupling even away from the critical point.KEYWORDS: magnetization plateau, level spectroscopy, exact diagonalization, BKT transition, twisted boundarycondition§1. IntroductionMagnetization plateaus in one dimensional Heisenbergchains are attracting much attention as the field inducedspin gap states. On the plateau, some spins are partlyquenched by the magnetic field while the remaining spinsform a spin gap state.2, 7, 3, 4, 1, 5, 6, 8, 9, 10, 11, 12, 13) It is evi-dent that such magnetization plateaus can be realized invarious non-homogeneous Heisenberg chains. Oshikawa,Yamanaka and Affleck1) proposed the necessary condi-tion for the magnetization plateaus as p(S − mz) = q ≡integer where p is the periodicity of the magnetic groundstate in the thermodynamic limit, S is the magnitude ofthe spin, and mz is the magnetization per site.In this context, the spin chains consisting of pe-riodic arrays of ferromagnetic and antiferromagneticbonds have been widely investigated. The S = 1/2ferromagnetic-antiferromagnetic alternating chain has aspin gap ground state14, 15, 16) for arbitrary values ofcoupling constants. This state can be regarded as theplateau state with zero magnetization (p = 2, S =1/2, q = 1 and mz = 0). In the isotropic case, thisenergy gap survives as the well-known Haldane gap evenif the strength of the ferromagnetic bonds j is infinite.14)There appears a magnetization plateau in the magneti-zation curve of the S = 1/2 ferromagnetic-ferromagnetic-antiferromagnetic trimerized chain at 1/3 of the satura-tion magnetization (p = 3, S = 1/2, q = 1 and mz =1/6).2) In this case, Kitazawa and Okamato9) have shownthat the plateau vanishes by the Brezinskii-Kosterlitz-Thouless (BKT) type transition for j ≥ jc ≃ 15.4 wherethe energy unit is the strength of the antiferromagnetic∗ E-mail: chen@chemistry.mcgill.ca, Present address: Depart-ment of Chemistry, McGill University, Montreal, PQ, Canada H3A2K6∗∗ E-mail: hida@phy.saitama-u.ac.jpbonds.In the present work, we investigate the plateau-non-plateau transition for general values of p using themethod of ref. 9. Intuitively, the quantum fluctuationsare expected to be suppressed as the number of the fer-romagnetic bonds increases. Therefore it is natural toexpect that the magnetization plateau, which is an es-sentially quantum phenomenon, becomes less favourableas the number of the ferromagnetic bonds increases. Sur-prisingly, our numerical calculation shows that this is in-correct. We discuss the physical origin of this unexpectedbehavior.This paper is organized as follows. In the next section,the model Hamiltonian is defined and the numerical re-sults of the magnetization curve and the phase diagramof the plateau-non-plateau transition is presented. Thephysical reason of the enhancement of the plateau statefor large p is also explained in this section. The finalsection is devoted to a summary and discussion.§2. Numerical Results2.1 Model HamiltonianThe Hamiltonian of the antiferromagnetic-(ferromagnetic)npolymerized S = 1/2 XXZ chain in the magnetic field isgiven byH = −jL∑l=1n−1∑i=0H(n+1)l+i(1) +L∑l=1H(n+1)l+n(∆)− gµBH(n+1)L∑l=1Szl . (2.1)whereHl(∆) = Sxl Sxl+1 + Syl Syl+1 + ∆Szl Szl+1, (2.2)12 W. Chen, K. Hida and B. C. SanctuaryThe ferromagnetic coupling constant, number of the fer-romagnetic bond and anisotropy parameter of the anti-ferromagnetic bond are represented by j, n and ∆, re-spectively. The ferromagnetic bonds are assumed to beisotropic. The magnetic field, the electronic g-factor andBohr magneton are denoted by H, g and µB, respectively.This model has spatial periodicity p ≡ n + 1. In the fol-lowing, we take gµB = 1 as unit.In the limiting case of j → ∞, this model tends tothe S = p/2 uniform antiferromagnetic XXZ chain, sothat no plateau is expected. Therefore the BKT typeplateau-non-plateau transition is expected to take placeat a finite critical value of j.2.2 Magnetization plateausThe schematic magnetization curve of the presentmodel is shown in Fig. 1. In the present paper, weconcentrate on the highest plateau with q = 1 andmz = 1/2 − 1/p. The total magnetization M ≡ mzon the plateau is given by M = Mp ≡ pL(1/2 − 1/p) =Ms(1 − 2/p) where Ms ≡ pL/2 is the saturation mag-netization. In Fig. 1 the critical fields Hc0 corresponds01.5301M/MsHc0 Hc1 Hc2 HsGapGapPlateau1–2/pFig. 1. Schematic magnetization curve of the present model.to the energy gap in the absence of the magnetic field,Hc1, to the energy gap between the lowest energy withMz = Mp and that with Mz = Mp − 1, Hc2, the energygap between the lowest energy with Mz = Mp + 1 andthat with Mz = Mp. Finally Hs is the saturation field.The lower part of the magnetization curve is representedby the dashed line where lower plateaus may appear.We use an exact diagonalization method to calcu-late the magnetization curve. As an example, the j-dependence of the 3 critical fields Hc0, Hc1 and Hc2 forp = 4 and ∆ = 1 are shown in Fig. 2 by the symbols •, 3and ◦, respectively, which are obtained by the Shanks’17)extrapolation of the results of the Lanczos exact diago-nalization for 4L = 8, 16 and 24 with periodic boundarycondition. The plateau seems to close at a finite valueof j. Because the non-plateau state corresponds to thegapless spin liquid and the plateau state to the spin gapstate, this transition is expected to be the BKT-typetransition. In the next subsection. we determine thecritical point of the plateau-non-plateau transition fromnumerical diagonalization data for general values of p.0 500.51Hjm=1/4Fig. 2. The critical fields for p = 4 and ∆ = 1. The critical fieldsHc0, Hc1 and Hc2 are represented by •, 3 and ◦.2.3 BKT critical pointIt is difficult to estimate the BKT critical pointprecisely from standard finite size analysis of the nu-merical calculation data. To circumvent this diffi-culty Nomura and Kitazawa19) proposed to use levelspectroscopy method18) with twisted boundary con-dition. This method has been successfully appliedto the ferromagnetic-ferromagnetic-antiferromagnetictrimerized Heisenberg chain by Kitazawa and Okamoto.9)In the present work, we employ this method for thepresent model. Here we do not explain how and whythis method works, because methodological details arewell described in ref. 9.The finite size critical point is determined from thecrossing point of ∆E0,2 with the lower of ∆Ec1/2,0 or∆Es1/2,0 defined by,∆E0,2 =12{E0(Mp + 2, 0, 1) + E0(Mp − 2, 0, 1)}− E0(Mp, 0, 1). (2.3)and∆Ec1/2,0 = ETBC(Mp, 1) − E(Mp, 0, 1) (2.4)∆Es1/2,0 = ETBC(Mp,−1) − E(Mp, 0, 1) (2.5)where E0(Mz, k, P ) is the lowest energy under a periodicboundary condition with magnetization Mz, wave num-ber k and parity P . The energy ETBC(Mz, P ) is thelowest energy with the twist boundary conditions withmagnetization Mz and parity P .To confirm the reliability of this method the followingaveragex =xc1/2,0(L) + 3xs1/2(L)4. (2.6)is close to 0.5 at the critical points9) where xc,s1/2,0(L)are the scaling exponents corresponding to Ec,s1/2,0 anddefined byxc,s1/2,0 =L2πvs∆Ec,s1/2,0. (2.7)Magnetization plateaus in polymerized S = 1/2 chains 3Here vs is the spin wave velocity given byvs = limL→∞L2π[EMp,k1(L) − EMp(L)], (2.8)where EMp,k1(L) is the energy of the excited state withwave number k1 =2πL and Mz = Mp. The results areshown in Fig. 3 for p = 4 which confirms that x = 0.5holds with good accuracy.Figure 4 shows the behavior of ∆E0,2 and ∆Es1/2,0 forL = 6 unit cells and p = 4. From the crossing point,we obtain jc(L) = 6.310 for L = 6 unit cells. The BKTtransition point for the infinte system can be obtainedby extrapolating from L = 2, 4 and 6 to N → ∞ asjc = 6.260 assuming the extrapolation formula jc(L) =jc +c1L2 +c2L4 .–1 0 10.40.50.6x∆Fig. 3. The extrapolated values of the averaged scaling dimensionx on the critical points for p = 4. The solid line is x = 0.56.3 6.20.0960.0950.094L=8j∆E ∆ E0,2∆ E1/2,0sFig. 4. The j-dependence of the energies ∆E0,2 and ∆Es1/2,0 rep-resented by ◦ and •, respectively, for p = 4,∆ = 1 and L = 6.The phase boundaries are shown in Fig. 5 for p = 3, 4and 5 by 2, • and ◦, respectively. Extrapolation is madefrom L = 2, 4 and 6 for p = 4 and 5. The results for p = 3are taken from ref. 9. From this figure, we see that thecritical points jc does not decrease monotonically withan increase of periodicity p. Therefore we further calcu-late the cases of longer periodicity up to p = 13. The pdependence of jc is shown in Fig. 6 for ∆ = 1.0, 0.5, 0.0and −0.5. The different symbols represent the criticalpoints for different system size L. The data for p = 2is also included for ∆ = −0.5. When ∆ ≥ 0 and p = 2then jc equals infinity.15, 16) For p = 4 and 5, the calcu-lations are carried out for L = 2, 4 and 6 and for p = 6and 7, L = 2 and 4. The extrapolation to L → ∞ ismade for these cases. Nevertheless, the L-dependence of–1 0 1051015∆j No PlateauPlateauFig. 5. The phase boundaries on the ∆− j-plane for p = 3, 4 and5 represented by 2 • and ◦ , respectively. The lines are fitted.4 8 12051015jcp∆=1.0∆=0.5No PlateauPlateau∆=0.0∆=−0.5L=2L=4L=6L=8Fig. 6. The p-dependence of the critical point jc for ∆ =1.0, 0.5, 0.0 and −0.5 from top to bottom. The different sym-bols distinguish jc’s for different system sizes L. The + symbolsare the values of jc extrapolated to L → ∞. The result for p = 3is taken from ref. 9. For p = 2, jc → ∞ for ∆ ≥ 0.15, 16)the critical point becomes weaker as the periodicity p be-comes larger as shown in Fig. 6. Therefore for p ≥ 8, wecan safely estimate approximate values of the thermody-namic critical point jc from that for L = 2. Surprisingly,the p-dependence of the critical point is strongly non-monotonic. The decrease of jc with p for small valuesof p is consistent with intuition that the increasing thenumber of the ferromagnetic bonds suppresses the quan-tum fluctuation. The magnetization plateau, which isthe quantum origin, becomes less favoured.With the further increase of p, however, jc increasesagain almost linearly with p. This phenomenon can beexplained by the following physical picture. On the mag-netization plateau Mz = Mp, the singlet pairs are lo-cated on the antiferromagnetic bonds and other spinsare alligned in the direction of the magnetic field. Thisis schematically shown in Fig. 7(a). If the magne-tization becomes Mz = Mp + 1, one singlet pair isbroken as shown in Fig. 7(b). The energy difference4 W. Chen, K. Hida and B. C. SanctuaryE0(Mp + 1) − E0(Mp) determines the upper edge ofthe plateau Hc2. Because the additional up spin is lo-calized around the antiferromagnetic bonds, the criti-cal field Hc2 tends to a constant value of the order ofthe antiferromagnetic coupling constant ∼ O(1) as p in-creases. On the other hand, if the magnetization be-comes Mz = Mp − 1, one of p− 2 spins in the ferromag-netic segments is inverted as shown in Fig. 7(c). In theabsence of coupling to the 2 boundary spins forming alocal singlet, the segment of the remaining p − 2 spins,which contain one inverted spin, form a state with totalspin S = p−22 and magnetization Sz = p−22 − 1. Sincethe ferromagnetic bonds are isotropic this state has thesame energy as the fully polarized state with S = p−22and Sz = p−22 . The energy change E0(Mp)−E0(Mp−1)is due to the interaction of the inverted spin with thesinglet pair on the antiferroamgnetic bonds at the bothends of the ferromagnetic segment. The probabilitythat the inverted spin stays on the boundary sites ofthe segment is proportional to 1p−2 . Hence, the energydiference E0(Mp) − E0(Mp − 1), which determines thelower edge of the plateau Hc1 is estimated as O(jp−2 ) forlarge enough p. Because the plateau-non-plateau tran-sition takes place around the point Hc1 ∼ Hc2, we findjc ∝ p − 2 for large enough p. This explains why jc in-creases linearly with p for large p. To substantiate thisargument, Hc1 and Hc2 are calculated numerically byexact diagonalization for L = 2 and ∆ = 1.0. Figure 8shows that Hc1 scales withjp−2 for p = 10, 11, 12 and 13from top to bottom although there is some deviation asthe critical point is approached. Figure 9 shows Hc2 isapproximately independent of p and tends to a constantvalue for large j. These results show that the increase ofthe critical point for large p is a consequence of delocal-ization of the down spin in a wide region of length p. InFig. 10, jcp−2 is plotted against p. It is evident thatjcp−2tends to a constant value for large enough p.§3. Summary and DiscussionThe magnetization plateaus at mz = 1/2 − 1/p inthe S = 1/2 antiferromagnetic-(ferromagnetic)n poly-merized XXZ chains are investigated by exact diagonal-ization of finite size systems. The plateau-non-plateauBKT transition points are determined precisely by thelevel spectroscopy method with twisted boundary condi-tion.19, 9)The p-dependence of the critical point is not a mono-tonic function. For small p, the critical point decreaseswith p in agreement with the simple-minded intuitionthat an increase of the number of the ferromagneticbonds suppresses the quantum fluctuation and the mag-netization plateau, which is the quantum origin, becomesless favoured. With a further increase of p, however, jcincreases again. This can be explained by the delocaliza-tion of down spin within the ferromagnetic segment withlength p − 2.The plateau for p = 3 is narrow for a wide range of1 2 p p+1 p+2 2paff(p–2)f f f(a)1 2 p p+1 p+2 2paff(p–2)f f f(b)1 2 p p+1 p+2 2paff(p–2)f f f(c)Fig. 7. The spin configurations (a) on the magnetization plateauMz = Mp, (b) at the magnetization Mz = Mp + 1 and (c) atthe magnetization Mz = Mp − 1. The spin pairs connected byovals form local singlets.0 0.5 10.10.2j/(p–2)HC1p=10p=11p=12p=13Fig. 8. The jp−2dependence of Hc1 for p = 10, 11, 12 and 13from top to bottom, Hc1 scales asjp−2.j ≤ jc.2) This is the reason why so far this simplestexample of the magnetization plateau has not even ob-served experimentally. Actually, no visible plateau is ob-served in the magnetization curve of 3CuCl2· 2dx whichhas j ∼ 4.5 << jc.2, 20) In contrast, the magnetizationplateaus for large p is fairly wide compared to that forp = 3 for moderate values of j. As an example, we showthe magnetization curve for p = 12, ∆ = 1 and j = 4.0obtained by exact diagonalization for L = 2 in Fig. 11.It is evident that this plateau is wide enough for experi-mental measurement, although the value of jc for p = 12is less than that for p = 3. Thus a systematic synthesisof materials with broad spatial periodicity is hoped toexperimentally confirm our predictions.Chen and Hida thank K. Okamoto for fruitful discus-sion. We also thank A. Kitazawa and K. Okamoto forMagnetization plateaus in polymerized S = 1/2 chains 50 100.51jHc2p=10p=11p=12p=13Fig. 9. The j-dependence of Hc2 for p = 10, 11, 12 and 13. Hc2is almost independent of p and the tends to a constant value forlarge j.4 8 120123pjc/(p–2)∆=1.0∆=0.5∆=0.0∆=−0.5L=2L=4L=6Fig. 10. The p-dependence of jc/(p − 2).0 0.1 0.2 0.30.51HMz /Msj=4.0Fig. 11. The magnetization curve for p = 12, ∆ = 1 and j = 4obtained by exact diagonalization with L = 2.supplying us the numerical data of ref. 9. The numeri-cal calculation is performed using the program packageTITPACK version 2 coded by H. Nishimori on HITACS820 and SR2201 at the Information Processing Cen-ter of Saitama University and the FACOM VPP500 atthe Supercomputer Center of Institute for Solid StatePhysics, University of Tokyo. This work is supported bya Grant-in-Aid for Scientific Research from the Ministryof Education, Science, Sports and Culture of Japan and aresearch grant from the Natural Science and EngineeringResearch Council of Canada (NSERC).[1] M. Oshikawa, M. Yamanaka and I. Affleck: Phys. Rev. Lett.78 (1997) 1984.[2] K. Hida: J. Phys. Soc. Jpn. 63 (1994) 2359.[3] K. Okamoto: Solid State Commun. 98 (1996) 245.[4] T. Tonegawa, T. Nakao and M. Kaburagi: J. Phys. Soc. Jpn.65 (1996) 3317.[5] K. Totsuka: Phys. Lett. A228 (1997) 103.[6] K. Totsuka: Phys. Rev. B 57 (1998) 3454.[7] T. Sakai and M. Takahashi: Phys. Rev. B 57 (1998) R3201.[8] H. Nakano and M. Takahashi: J. Phys. Jpn. 67 (1998) 1126.[9] A. Kitazawa and K. Okamoto: J. Phys. Condens. Matter 11(1999) 9765.[10] K. Okamoto and A. Kitazawa: J. Phys. A: Gen. 32 (1999)4601.[11] K. Okamoto and A. Kitazawa: cond-mat/9907481.[12] Y. Narumi, M. Hagiwara, R. Sato, K. Kindo, H. Nakano andM. Takahashi: Physica 246-247 (1998) 234.[13] W. Shiramura, K. Takatsu, B. Kurniawan, H. Tanaka, H.Uekusa, Y. Ohashi, K. Takizawa, H. Mitamura and T. Goto:J. Phys. Soc. Jpn. 67 (1998) 1548.[14] K. Hida: :Phys.Rev. B45 (1992) 2207.[15] K. Hida: J. Phys. Jpn. 62 (1993) 3357.[16] K. Hida: J. Phys. Jpn. 62 (1993) 1463.[17] D. Shanks: J. Math. Phys. 34 (1955) 1.[18] K. Nomura: J. Phys. A: Math. Gen. 28 (1995) 5451.[19] K. Nomura and A. Kitazawa: J. Phys. A.: Math. Gen. 31(1998) 7341.[20] Y. Ajiro, T. Asano, T. Inami, H. Aruga-Katori and T. Goto:J. Phys. Soc. Jpn. 63 (1994) 859.

Magnetization plateaus in antiferromagnetic-(ferromagnetic), polymerized S=1/2 XXZ chains

The plateau-non-plateau transition in the antiferromagnetic-(ferromagnetic)(n) polymerized S = 1/2 XXZ chains under the magnetic field is investigated. The universality class of this transition belongs to the Brezinskii-Kosterlitz-Thouless (BKT) type. The critical points are determined by level spectroscopy analysis of the numerical diagonalization data for 4 less than or equal to p less than or equal to 13 where p(equivalent to n + 1) is the size of a unit cell. It is found that the critical strength of ferromagnetic coupling decreases with p for small p but increases for large enough p. It is also found that the plateau for large p is wide enough for moderate values of exchange coupling so that it should be easily observed experimentally. This is in contrast to the plateaus for p = 3 chains which are narrow for a wide range of exchange coupling even away from the critical point

Chen Wei

飛田 和男

Sanctuary, Bryan Clifford

arXiv:cond-mat/0005544v1  [cond-mat.stat-mech]  31 May 2000typeset using JPSJ.sty <ver.1.0b>Magnetization plateaus in antiferromagnetic-(ferromagnetic)npolymerized S = 1/2 XXZ chainsWei Chen∗, Kazuo Hida∗∗ and B. C. Sanctuary1Department of Physics, Saitama University, Urawa, Saitama, Japan 338-85701 Department of Chemistry, McGill University, Montreal, PQ, Canada H3A 2K6(Received August 12, 2008)The plateau-non-plateau transition in the antiferromagnetic-(ferromagnetic)n polymerized S =1/2 XXZ chains under the magnetic field is investigated. The universality class of this transitionbelongs to the Brezinskii-Kosterlitz-Thouless (BKT) type. The critical points are determined bylevel spectroscopy analysis of the numerical diagonalization data for 4 ≤ p ≤ 13 where p(≡ n+1)is the size of a unit cell. It is found that the critical strength of ferromagnetic coupling decreaseswith p for small p but increases for larger enough p. It is also found that the plateau for largep is wide enough for moderate values of exchange coupling so that it should be easily observedexperimentally. This is in contrast to the plateaus for p = 3 chains which are narrow for a widerange of exchange coupling even away from the critical point.KEYWORDS: magnetization plateau, level spectroscopy, exact diagonalization, BKT transition, twisted boundarycondition§1. IntroductionMagnetization plateaus in one dimensional Heisenbergchains are attracting much attention as the field inducedspin gap states. On the plateau, some spins are partlyquenched by the magnetic field while the remaining spinsform a spin gap state.2, 7, 3, 4, 1, 5, 6, 8, 9, 10, 11, 12, 13) It is evi-dent that such magnetization plateaus can be realized invarious non-homogeneous Heisenberg chains. Oshikawa,Yamanaka and Aﬄeck1) proposed the necessary condi-tion for the magnetization plateaus as p(S −mz) = q ≡integer where p is the periodicity of the magnetic groundstate in the thermodynamic limit, S is the magnitude ofthe spin, and mz is the magnetization per site.In this context, the spin chains consisting of pe-riodic arrays of ferromagnetic and antiferromagneticbonds have been widely investigated. The S = 1/2ferromagnetic-antiferromagnetic alternating chain has aspin gap ground state14, 15, 16) for arbitrary values ofcoupling constants. This state can be regarded as theplateau state with zero magnetization (p = 2, S =1/2, q = 1 and mz = 0). In the isotropic case, thisenergy gap survives as the well-known Haldane gap evenif the strength of the ferromagnetic bonds j is infinite.14)There appears a magnetization plateau in the magneti-zation curve of the S = 1/2 ferromagnetic-ferromagnetic-antiferromagnetic trimerized chain at 1/3 of the satura-tion magnetization (p = 3, S = 1/2, q = 1 and mz =1/6).2) In this case, Kitazawa and Okamato9) have shownthat the plateau vanishes by the Brezinskii-Kosterlitz-Thouless (BKT) type transition for j ≥ jc ≃ 15.4 wherethe energy unit is the strength of the antiferromagnetic∗ E-mail: chen@chemistry.mcgill.ca, Present address: Depart-ment of Chemistry, McGill University, Montreal, PQ, Canada H3A2K6∗∗ E-mail: hida@phy.saitama-u.ac.jpbonds.In the present work, we investigate the plateau-non-plateau transition for general values of p using themethod of ref. 9. Intuitively, the quantum fluctuationsare expected to be suppressed as the number of the fer-romagnetic bonds increases. Therefore it is natural toexpect that the magnetization plateau, which is an es-sentially quantum phenomenon, becomes less favourableas the number of the ferromagnetic bonds increases. Sur-prisingly, our numerical calculation shows that this is in-correct. We discuss the physical origin of this unexpectedbehavior.This paper is organized as follows. In the next section,the model Hamiltonian is defined and the numerical re-sults of the magnetization curve and the phase diagramof the plateau-non-plateau transition is presented. Thephysical reason of the enhancement of the plateau statefor large p is also explained in this section. The finalsection is devoted to a summary and discussion.§2. Numerical Results2.1 Model HamiltonianThe Hamiltonian of the antiferromagnetic-(ferromagnetic)npolymerized S = 1/2 XXZ chain in the magnetic field isgiven byH = −jL∑l=1n−1∑i=0H(n+1)l+i(1) +L∑l=1H(n+1)l+n(∆)− gµBH(n+1)L∑l=1Szl . (2.1)whereHl(∆) = Sxl Sxl+1 + Syl Syl+1 +∆Szl Szl+1, (2.2)12 W. Chen, K. Hida and B. C. SanctuaryThe ferromagnetic coupling constant, number of the fer-romagnetic bond and anisotropy parameter of the anti-ferromagnetic bond are represented by j, n and ∆, re-spectively. The ferromagnetic bonds are assumed to beisotropic. The magnetic field, the electronic g-factor andBohr magneton are denoted byH, g and µB, respectively.This model has spatial periodicity p ≡ n+ 1. In the fol-lowing, we take gµB = 1 as unit.In the limiting case of j → ∞, this model tends tothe S = p/2 uniform antiferromagnetic XXZ chain, sothat no plateau is expected. Therefore the BKT typeplateau-non-plateau transition is expected to take placeat a finite critical value of j.2.2 Magnetization plateausThe schematic magnetization curve of the presentmodel is shown in Fig. 1. In the present paper, weconcentrate on the highest plateau with q = 1 andmz = 1/2 − 1/p. The total magnetization M ≡ mzon the plateau is given by M = Mp ≡ pL(1/2− 1/p) =Ms(1 − 2/p) where Ms ≡ pL/2 is the saturation mag-netization. In Fig. 1 the critical fields Hc0 corresponds01.5301M/MsHc0 Hc1 Hc2 HsGapGapPlateau1–2/pFig. 1. Schematic magnetization curve of the present model.to the energy gap in the absence of the magnetic field,Hc1, to the energy gap between the lowest energy withMz =Mp and that with Mz =Mp − 1, Hc2, the energygap between the lowest energy with Mz = Mp + 1 andthat with Mz = Mp. Finally Hs is the saturation field.The lower part of the magnetization curve is representedby the dashed line where lower plateaus may appear.We use an exact diagonalization method to calcu-late the magnetization curve. As an example, the j-dependence of the 3 critical fields Hc0, Hc1 and Hc2 forp = 4 and ∆ = 1 are shown in Fig. 2 by the symbols •, 3and ◦, respectively, which are obtained by the Shanks’17)extrapolation of the results of the Lanczos exact diago-nalization for 4L = 8, 16 and 24 with periodic boundarycondition. The plateau seems to close at a finite valueof j. Because the non-plateau state corresponds to thegapless spin liquid and the plateau state to the spin gapstate, this transition is expected to be the BKT-typetransition. In the next subsection. we determine thecritical point of the plateau-non-plateau transition fromnumerical diagonalization data for general values of p.0 500.51Hjm=1/4Fig. 2. The critical fields for p = 4 and ∆ = 1. The critical fieldsHc0, Hc1 and Hc2 are represented by •, 3 and ◦.2.3 BKT critical pointIt is difficult to estimate the BKT critical pointprecisely from standard finite size analysis of the nu-merical calculation data. To circumvent this diffi-culty Nomura and Kitazawa19) proposed to use levelspectroscopy method18) with twisted boundary con-dition. This method has been successfully appliedto the ferromagnetic-ferromagnetic-antiferromagnetictrimerized Heisenberg chain by Kitazawa and Okamoto.9)In the present work, we employ this method for thepresent model. Here we do not explain how and whythis method works, because methodological details arewell described in ref. 9.The finite size critical point is determined from thecrossing point of ∆E0,2 with the lower of ∆Ec1/2,0 or∆Es1/2,0 defined by,∆E0,2 =12{E0(Mp + 2, 0, 1) + E0(Mp − 2, 0, 1)}− E0(Mp, 0, 1). (2.3)and∆Ec1/2,0 = ETBC(Mp, 1)− E(Mp, 0, 1) (2.4)∆Es1/2,0 = ETBC(Mp,−1)− E(Mp, 0, 1) (2.5)where E0(Mz, k, P ) is the lowest energy under a periodicboundary condition with magnetization Mz, wave num-ber k and parity P . The energy ETBC(Mz, P ) is thelowest energy with the twist boundary conditions withmagnetization Mz and parity P .To confirm the reliability of this method the followingaveragex =xc1/2,0(L) + 3xs1/2(L)4. (2.6)is close to 0.5 at the critical points9) where xc,s1/2,0(L)are the scaling exponents corresponding to Ec,s1/2,0 anddefined byxc,s1/2,0 =L2pivs∆Ec,s1/2,0. (2.7)Magnetization plateaus in polymerized S = 1/2 chains 3Here vs is the spin wave velocity given byvs = limL→∞L2pi[EMp,k1(L)− EMp(L)], (2.8)where EMp,k1(L) is the energy of the excited state withwave number k1 =2piL and Mz = Mp. The results areshown in Fig. 3 for p = 4 which confirms that x = 0.5holds with good accuracy.Figure 4 shows the behavior of ∆E0,2 and ∆Es1/2,0 forL = 6 unit cells and p = 4. From the crossing point,we obtain jc(L) = 6.310 for L = 6 unit cells. The BKTtransition point for the infinte system can be obtainedby extrapolating from L = 2, 4 and 6 to N → ∞ asjc = 6.260 assuming the extrapolation formula jc(L) =jc +c1L2 +c2L4 .–1 0 10.40.50.6x∆Fig. 3. The extrapolated values of the averaged scaling dimensionx on the critical points for p = 4. The solid line is x = 0.56.3 6.20.0960.0950.094L=8j∆E ∆ E0,2∆ E1/2,0sFig. 4. The j-dependence of the energies ∆E0,2 and ∆Es1/2,0 rep-resented by ◦ and •, respectively, for p = 4,∆ = 1 and L = 6.The phase boundaries are shown in Fig. 5 for p = 3, 4and 5 by 2, • and ◦, respectively. Extrapolation is madefrom L = 2, 4 and 6 for p = 4 and 5. The results for p = 3are taken from ref. 9. From this figure, we see that thecritical points jc does not decrease monotonically withan increase of periodicity p. Therefore we further calcu-late the cases of longer periodicity up to p = 13. The pdependence of jc is shown in Fig. 6 for ∆ = 1.0, 0.5, 0.0and −0.5. The different symbols represent the criticalpoints for different system size L. The data for p = 2is also included for ∆ = −0.5. When ∆ ≥ 0 and p = 2then jc equals infinity.15, 16) For p = 4 and 5, the calcu-lations are carried out for L = 2, 4 and 6 and for p = 6and 7, L = 2 and 4. The extrapolation to L → ∞ ismade for these cases. Nevertheless, the L-dependence of–1 0 1051015∆j No PlateauPlateauFig. 5. The phase boundaries on the ∆− j-plane for p = 3, 4 and5 represented by 2 • and ◦ , respectively. The lines are fitted.4 8 12051015jcp∆=1.0∆=0.5No PlateauPlateau∆=0.0∆=−0.5L=2L=4L=6L=8Fig. 6. The p-dependence of the critical point jc for ∆ =1.0, 0.5, 0.0 and −0.5 from top to bottom. The different sym-bols distinguish jc’s for different system sizes L. The + symbolsare the values of jc extrapolated to L →∞. The result for p = 3is taken from ref. 9. For p = 2, jc →∞ for ∆ ≥ 0.15, 16)the critical point becomes weaker as the periodicity p be-comes larger as shown in Fig. 6. Therefore for p ≥ 8, wecan safely estimate approximate values of the thermody-namic critical point jc from that for L = 2. Surprisingly,the p-dependence of the critical point is strongly non-monotonic. The decrease of jc with p for small valuesof p is consistent with intuition that the increasing thenumber of the ferromagnetic bonds suppresses the quan-tum fluctuation. The magnetization plateau, which isthe quantum origin, becomes less favoured.With the further increase of p, however, jc increasesagain almost linearly with p. This phenomenon can beexplained by the following physical picture. On the mag-netization plateau Mz = Mp, the singlet pairs are lo-cated on the antiferromagnetic bonds and other spinsare alligned in the direction of the magnetic field. Thisis schematically shown in Fig. 7(a). If the magne-tization becomes Mz = Mp + 1, one singlet pair isbroken as shown in Fig. 7(b). The energy difference4 W. Chen, K. Hida and B. C. SanctuaryE0(Mp + 1) − E0(Mp) determines the upper edge ofthe plateau Hc2. Because the additional up spin is lo-calized around the antiferromagnetic bonds, the criti-cal field Hc2 tends to a constant value of the order ofthe antiferromagnetic coupling constant ∼ O(1) as p in-creases. On the other hand, if the magnetization be-comes Mz =Mp − 1, one of p− 2 spins in the ferromag-netic segments is inverted as shown in Fig. 7(c). In theabsence of coupling to the 2 boundary spins forming alocal singlet, the segment of the remaining p − 2 spins,which contain one inverted spin, form a state with totalspin S = p−22 and magnetization Sz = p−22 − 1. Sincethe ferromagnetic bonds are isotropic this state has thesame energy as the fully polarized state with S = p−22and Sz = p−22 . The energy change E0(Mp)−E0(Mp−1)is due to the interaction of the inverted spin with thesinglet pair on the antiferroamgnetic bonds at the bothends of the ferromagnetic segment. The probabilitythat the inverted spin stays on the boundary sites ofthe segment is proportional to 1p−2 . Hence, the energydiference E0(Mp) − E0(Mp − 1), which determines thelower edge of the plateau Hc1 is estimated as O(jp−2 ) forlarge enough p. Because the plateau-non-plateau tran-sition takes place around the point Hc1 ∼ Hc2, we findjc ∝ p − 2 for large enough p. This explains why jc in-creases linearly with p for large p. To substantiate thisargument, Hc1 and Hc2 are calculated numerically byexact diagonalization for L = 2 and ∆ = 1.0. Figure 8shows that Hc1 scales withjp−2 for p = 10, 11, 12 and 13from top to bottom although there is some deviation asthe critical point is approached. Figure 9 shows Hc2 isapproximately independent of p and tends to a constantvalue for large j. These results show that the increase ofthe critical point for large p is a consequence of delocal-ization of the down spin in a wide region of length p. InFig. 10, jcp−2 is plotted against p. It is evident thatjcp−2tends to a constant value for large enough p.§3. Summary and DiscussionThe magnetization plateaus at mz = 1/2 − 1/p inthe S = 1/2 antiferromagnetic-(ferromagnetic)n poly-merized XXZ chains are investigated by exact diagonal-ization of finite size systems. The plateau-non-plateauBKT transition points are determined precisely by thelevel spectroscopy method with twisted boundary condi-tion.19, 9)The p-dependence of the critical point is not a mono-tonic function. For small p, the critical point decreaseswith p in agreement with the simple-minded intuitionthat an increase of the number of the ferromagneticbonds suppresses the quantum fluctuation and the mag-netization plateau, which is the quantum origin, becomesless favoured. With a further increase of p, however, jcincreases again. This can be explained by the delocaliza-tion of down spin within the ferromagnetic segment withlength p− 2.The plateau for p = 3 is narrow for a wide range of1 2 p p+1 p+2 2paff(p–2)f f f(a)1 2 p p+1 p+2 2paff(p–2)f f f(b)1 2 p p+1 p+2 2paff(p–2)f f f(c)Fig. 7. The spin configurations (a) on the magnetization plateauMz = Mp, (b) at the magnetization Mz = Mp + 1 and (c) atthe magnetization Mz = Mp − 1. The spin pairs connected byovals form local singlets.0 0.5 10.10.2j/(p–2)HC1p=10p=11p=12p=13Fig. 8. The jp−2dependence of Hc1 for p = 10, 11, 12 and 13from top to bottom, Hc1 scales asjp−2.j ≤ jc.2) This is the reason why so far this simplestexample of the magnetization plateau has not even ob-served experimentally. Actually, no visible plateau is ob-served in the magnetization curve of 3CuCl2· 2dx whichhas j ∼ 4.5 << jc.2, 20) In contrast, the magnetizationplateaus for large p is fairly wide compared to that forp = 3 for moderate values of j. As an example, we showthe magnetization curve for p = 12,∆ = 1 and j = 4.0obtained by exact diagonalization for L = 2 in Fig. 11.It is evident that this plateau is wide enough for experi-mental measurement, although the value of jc for p = 12is less than that for p = 3. Thus a systematic synthesisof materials with broad spatial periodicity is hoped toexperimentally confirm our predictions.Chen and Hida thank K. Okamoto for fruitful discus-sion. We also thank A. Kitazawa and K. Okamoto forMagnetization plateaus in polymerized S = 1/2 chains 50 100.51jHc2p=10p=11p=12p=13Fig. 9. The j-dependence of Hc2 for p = 10, 11, 12 and 13. Hc2is almost independent of p and the tends to a constant value forlarge j.4 8 120123pjc/(p–2)∆=1.0∆=0.5∆=0.0∆=−0.5L=2L=4L=6Fig. 10. The p-dependence of jc/(p − 2).0 0.1 0.2 0.30.51HMz /Msj=4.0Fig. 11. The magnetization curve for p = 12,∆ = 1 and j = 4obtained by exact diagonalization with L = 2.supplying us the numerical data of ref. 9. The numeri-cal calculation is performed using the program packageTITPACK version 2 coded by H. Nishimori on HITACS820 and SR2201 at the Information Processing Cen-ter of Saitama University and the FACOM VPP500 atthe Supercomputer Center of Institute for Solid StatePhysics, University of Tokyo. This work is supported bya Grant-in-Aid for Scientific Research from the Ministryof Education, Science, Sports and Culture of Japan and aresearch grant from the Natural Science and EngineeringResearch Council of Canada (NSERC).[1] M. Oshikawa, M. Yamanaka and I. Aﬄeck: Phys. Rev. Lett.78 (1997) 1984.[2] K. Hida: J. Phys. Soc. Jpn. 63 (1994) 2359.[3] K. Okamoto: Solid State Commun. 98 (1996) 245.[4] T. Tonegawa, T. Nakao and M. Kaburagi: J. Phys. Soc. Jpn.65 (1996) 3317.[5] K. Totsuka: Phys. Lett. A228 (1997) 103.[6] K. Totsuka: Phys. Rev. B 57 (1998) 3454.[7] T. Sakai and M. Takahashi: Phys. Rev. B 57 (1998) R3201.[8] H. Nakano and M. Takahashi: J. Phys. Jpn. 67 (1998) 1126.[9] A. 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