HEISENBERG SPIN-ONE CHAIN IN STAGGERED MAGNETIC FIELD: A DENSITY MATRIX RENORMALIZATION GROUP STUDY

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1 Available at: ictp.trieste. it/"pub_ ff IC/99/35 United Natins Educatinal Scientific and Cultural Organizatin and Internatinal Atmic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS HEISENBERG SPIN-ONE CHAIN IN STAGGERED MAGNETIC FIELD: A DENSITY MATRIX RENORMALIZATION GROUP STUDY Jizhng Lu, Xi Dai, Shajin Qin, Zhabin Su Institute f Theretical Physics, P.O. Bx 2735, Beijing , Peple's Republic f China and Lu Yu Institute f Theretical Physics, P.O. Bx 2735, Beijing , Peple's Republic f China and The Abdus Salam Internatinal Centre fr Theretical Physics, Trieste, Italy. Abstract Using the density matrix renrmalizatin grup technique, we calculate numerically the lw energy excitatin spectrum and magnetizatin curve f the spin-1 antiferrmagnetic chain in a staggered magnetic field, which is expected t describe the physics f R^BaNiO^R ^ Y) family belw the Neel temperature f the magnetic rare-earth (R) sublattice. These results are valid in the entire range f the staggered field, and agree with thse given by the nn-linear a mdel study fr small fields, but differ frm the latter fr large fields. They are cnsistent with the available experimental data. The crrelatin functins fr this mdel are als calculated. The transverse crrelatins display the anticipated expnential decay with shrter crrelatin length, while the lngitudinal crrelatins shw explicitly the induced staggered magnetizatin. MIRAMARE - TRIESTE April 1999

2 The quasi-ne-dimensinal magnets have been the fcus f analytic, numerical and experimental studies since Haldane pinted ut the difference between the integer spin Heisenberg antiferrmagnetic (AF) chains and the half-integer chains in M By mapping the Heisenberg spin chains nt the 0(3) nnlinear tr-mdel, ^ he cnjectured that the lw-energy excitatin spectrum displays a finite gap fr the integer spin systems while it is gapless fr half-integer spin chains. This cnjecture has been verified by later experiments n quasi-ne-dimensinal spin-1 materials such as NENP and Y^BaNiOr-, which shw clear evidence f the Haldane gap. ^ Nwadays, the pure ne-dimensinal Haldane systems are fairly well understd, and a reliable estimate fr the Haldane gap A = (2) J fr spin-1 chains has been btained by bth density matrix renrmalizatin grup (DMRG) calculatin W and finite size exact diagnalizatin. ^ Mre recent develpments n the Haldane systems cncern varius effects f external perturbatins: dping with magnetic r nn-magnetic impurities ^ and applying external magnetic field. The impurity dping may intrduce bund states within the Haldane gap, ^ while applying unifrm external magnetic field splits the degenerate Haldane triplet state int transverse and lngitudinal mdes. ^ The lngitudinal mde becmes sftened upn increase f the magnetic field, and at a critical field H c the system enters a new phase with lng-range AF rder. Of curse, a staggered applied magnetic field is even mre interesting which wuld induce nn-vanishing staggered magnetizatin and affect the Haldane gap excitatin spectrum, but such a staggered field cannt be materialized by an external surce. Mst recently, a series f experiments perfrmed n the family f quasi-ne-dimensinal materials with a general frmula R2BaNiOs, I 9 ~ 16 1 where R is ne f the magnetic rare-earth elements substituting fully r partially Y (fr brevity we dente this replacement by R ^ Y), have made it pssible t study the effect f the staggered magnetic field n the Haldane systems in detail. All members f this family cntain spin-1 Ni' 2+ linear chains and the in-chain AF exchange cupling is rather strng. (The detailed structure f this family f cmpunds is described in Ref. [11]). The reference cmpund Y 2 BaNi0 5 is fund t be highly ne-dimensinal with negligible interchain interactins, and n magnetic rder has been bserved s far even at very lw temperatures. ^ Hence it is believed t be an almst ideal example f the Haldane-gap system. Other members have magnetic R 3+ ins in additin t the spin-1 Ni 2+ ins. These ins are psitined between tw neighbring Ni chains, weakly cupled t the Ni' 2+ ins and the cupling between themselves is als very weak. Nevertheless, these magnetic ins are AF rdered belw certain Neel temperature TV- These ins d nt affect the Ni chains substantially abve TJV, keeping their Haldane features untuched, but the 3D AF rdered R 3+ sublattice belw TN has dramatical effects n these chains, impsing a staggered magnetic field. The neutrn scattering experiments n pwder samples and small size single-crystals f Nd^BaNiO^, and Pr2BaNiO 5 ' ' shw an increase f the energy gap belw the Neel temperature. We assume that these chains can be still cnsidered ne-dimensinal, being put in a staggered field created by 3D rdered R 3+ ins at lw temperatures. The Hamiltnian can be then written as : JJ=J^[S r S I+1 + h(-lfs I z ], (1) i where J is the exchange cnstant (t be taken as energy unit, i.e., J =1 ). The dimensinless staggered field h = gs/dsh^/j with H^ as the physical staggered field, which, in turn, is prprtinal t the R sublattice magnetizatin MR H^ = am R. (2)

3 g=2 is the theretically predicted gyrmagnetic rati f the Ni in. The directin f the staggered field has been chsen as the z axis. This Hamiltnian has been cnsidered using the mean-field thery I 17 ' 16! as well as by mapping nt the 0(3) nnlinear a mdel (NLSM). I 20 l In this Cmmunicatin we use the DMRG technique t calculate the lw energy excitatin spectrum and magnetizatin curve f the Hamiltnian Eq.(l). The btained field dependence f the gap and the staggered magnetizatin is cnsistent with the experimental results. These results are valid in the entire range f the staggered field and recver thse given by nn-linear a mdel fr small fields but differ frm the latter fr large fields. Mrever, we calculate the spin-spin crrelatin functins fr this mdel. The transverse crrelatins display an expnential decay as anticipated fr the spin-1 AF chain, with a shrter crrelatin length, while the lngitudinal crrelatins shw explicitly the induced staggered magnetizatin. We fllw the standard DMRG algrithm I 4 ' 18 ' 19 ] t calculate the lw-energy excitatins f the Hamiltnian (1), adpting the peridic bundary cnditins (PBC). We use the infinite-chain algrithm up t chain length 7V=60 and keep as many as 400 ptimized states during each sweep. The largest truncatin errrs are f the rder f 10~ 8 fr smaller h, while fr bigger h, these errrs are as small as 10~ 13, which means ur results are even mre reliable fr bigger h. The numerical results fr the change f the lwest excitatin energies (the Haldane gap) f Hamiltnian (1) A A are presented in Fig. 1 as functins f the dimensinless staggered magnetic field h. In the absence f this field the lngitudinal (AL) and transverse (A T ) mdes are degenerate, frming the Haldane triplet. Fr nn-zer h, these mdes will split with respect t each ther. Bth f them will increase with the staggered magnetic field, while the lngitudinal gap increases faster than the transverse ne. Fr small staggered fields, the increase f the lngitudinal gap will be nearly three times faster then the transverse nes, while fr larger staggered fields, this rati will decrease, and is is apprximately tw fr the largest staggered field we cnsidered < 2.5 < A L A T Lu et al, Fig. 1,PRB - A a* > h 0 0 Q <) FIG. 1. The DMRG results fr the transverse (slid circle) and lngitudinal (empty circle) energy gaps as functins f the staggered magnetic field fr spin-1 chain. The staggered magnetic mment f the system with chain length N is defined as

4 the lngest chain in ur calculatin being N=60. Then the staggered magnetic mment fr infinite lng chain M n can be btained by extraplating M n = limjv^ M jr (N). Obviusly, this quantity is a functin f the staggered field, and ur numerical results are shwn in Fig. 2. Cnsidering Eq. (2), this figure is nthing but the relatin between the magnetizatin f the Ni sublattice and the R sublattice, and it is qualitatively in agreement with the experimental data in Fig. 2 f Ref. [16]. Our numerical results shw that in small staggered fields, the magnetizatin change linearly with the increase f the field, s we can easily extract the "zer field" staggered magnetic susceptibility x (^s- ) (0) = 18.50/J. This value fully agrees with the results btained frm the transfer-matrix renrmalizatin grup ^21 \ Quantum Mnte Carl I 22 1 and the NLSM calculatins. I 20! We fit ur results using the fllwing functin: ^ = a arctan(6 h) + (1 - a) tanh(c h d ) (4) with a=0.412, &=38.106, c=1.195, d= The fitting line is als shwn in Fig. 2. T cmpare with the NLSM results I 20! in detail, in Fig. 2, we use their analytic relatin = M 7r (l M M ) with their value x^h ) = 18.7/J as a reference. We see clearly that the analytic expressin is very gd fr small staggered fields, while fr larger staggered fields it deviates frm ur numerical results significantly. We have als calculated the magnetic mment fr large h which is nt shwn in Fig. 2. Our result indicates that the mment shuld saturate in large enugh staggered magnetic field fr bth istrpic and (single-in) anistrpic cases. The unsaturated mment at zer-temperature bserved s far in varius experiments tells us that the induced staggered magnetic field n Ni chains is, prbably, nt large enugh yet. This issue was als discussed earlier. I 16 1 Lu et al, Fig. 2, PRB (3) /i(cxm R ) DMRG result ur fitting line NLSM result FIG. 2. The staggered magnetizatin curve fr spin-1 chain. The staggered magnetic field is prprtinal t the magnetizatin f the R sublattice. The numerical results (slid circle) are fitted by a functin with fur parameters (slid line) (see Eq. (4 )), with =0.412, 6=38.106, c=1.195, d= 0.621, respectively; the analytic results given by nn-linear a mdel are als presented (dtted line). Frm the abve results, we btain the values f the transverse as well as the lngitudinal gap as functins f the magnetic mment f the Ni sites, which can be cmpared directly with the analytic NLSM result

5 (see Fig. 3). These results are als cnsistent with the experimental data. I 13 ' 20! As fr the cmparisn with the NLSM treatment we see again that in small staggered fields, the analytic and numerical results are in gd agreement with each ther, while fr larger staggered fields, the analytic results deviate significantly frm the numerical simulatins. Bth lngitudinal and transverse gaps increase faster in simulatins than the NLSM predicts. Since fr ur DMRG calculatins, larger is the staggered field, mre reliable are the results, s the disagreement f magnetizatin mment and the gaps between these tw appraches raises a questin whether the NLSM mapping is valid r nt fr large staggered magnetic fields. Lu et al, Fig. 3, PRB AT / AQ by DMRG All Al by DMRG A T / Al by NLSM Al / Al by NLSM FIG. 3. The numerical results fr the transverse gaps (slid circle) and the lngitudinal gaps (empty circle) versus the magnetic mment n Ni sites; the NLSM results are shwn ( slid line fr transverse gaps and dtted line fr lngitudinal gaps) in the same figure. Besides calculating the lw-energy spectrum f the ne-dimensinal systems, the DMRG methd als prvides a direct and simple way t calculate the spin-spin crrelatin functins which can shed sme further light n the nature f the system under study. Fr a pure Haldane system, the crrelatin decays expnentially, fllwing e «(5) where =6.03 is the crrelatin length btained by the numerical study, M and A is a cnstant. When a staggered field is applied, the AF lng-range rder will be induced alng the z-directin, s (SQS[) will nt decay any mre. Hwever, (SQS[) - (SQ)(S 1 Z ) will still decay expnentially. Of curse, the transverse crrelatins (SfiSf) decay expnentially as befre, but with mdified expnents. In Fig. 4(a) and Fig. 4(b), the functins C xx (l) = ln(vi$s$s?) ) and C zz (l) = ln(vl\(s^s[) - (S^)(S[)$ are shwn fr different staggered fields. We see that bth functins decay expnentially fllwing Eq. (5) and the crrelatin lengths xx and zz decrease with increasing staggered field. As h increases, the reduced lngitudinal crrelatins decay much faster. We have extracted the crrelatin lengths and extraplated them t infinite chain length by cnsidering the chain length dependence f ^xx and zz. We find that ^~ x ~ AT and t^~} ~ A/,, and bth results can be fitted by -1 = * A (Fig. 5), which cincides exactly with = 6.03 btained fr the istrpic spin chain (h = 0, A = 0.41). W This is an independent check f the self-cnsistency in ur calculatins.

6 Lu et al, Fig. 4, PRB 30 FIG. 4. Spin-spin crrelatin functins fr different values f staggered magnetic field, where / is the distance between the tw spins. Bth crrelatins (S Sf) and (SQS") (S')(Sf) decay expnentially, but the latter decays faster.

7 Lu et al., Fig. 5 FIG. 5. The inverse crrelatin lengths g, x^ an d S, zz fr infinite chain length vs. the transverse and lngitudinal gaps, respectively. The slid line is the fitting -1 = * A. In cnclusin, by cnsidering a mdel hamiltnian which describes the physics f a family f mixed-spin materials in the temperature range belw TN f the magnetic rare-earth sublattice, we have calculated numerically the energy gap and the staggered magnetic mment as functins f the staggered magnetic field created by the AF lng range rder. The btained results are cnsistent with the experimental data qualitatively. Our numerical results are als cmpared with the analytic cnsideratins based n the nnlinear a mdel. The cmparisn shws that the NLSM results are gd fr small staggered fields, while they deviate frm the numerical simulatins fr larger staggered fields. After submitting this paper we saw a new reprt n plarized neutrn study f lngitudinal Haldane-gap excitatins in Nd2BaNiO5. ' 2S ' which shw smewhat different behavir than expected frm the thery. The reasn f this discrepancy has t be understd. Acknwledgments Our wrk was supprted by the Natinal Natural Science Fundatin f China (NFSC). The cmputatins were perfrmed n machines in the State Key Labratry f Science and Engineering f China (LSEC) and the Chinese Center f Advanced Science and Technlgy (CCAST). One f the authrs (J. Lu) wuld like t thank Prf. T. K. Ng and Dr. Ta Xiang fr helpful discussins. References [1] F. D. M. Haldane, Phys. Lett. 93A, 464 (1983); Phys. Rev. Lett. 50, 1153 (1983). [2] Fr a review see Ian Affleck, in "Fields, Strings and Critical Phenmena", edited by E. Brezin and J. Zinn- Justin, Nrth-Hlland, Amsterdam, 1989, p [3] J. P. Renard et al., Eurphys. Lett. 3, 945 (1987); J. Darriet and L. P. Regnault, Slid State Cmmun. 86, 409 (1993); J. F. DiTusa et al., Physica B , 181. (1994). [4] S. R. White and D. A. Huse, Phys. Rev. B48, 3844 (1993). [5] O. Glinelli, Th. Jliceur, and R. Lacaze, Phys. Rev. B50, 3037 (1994).

8 [6] J. F. DiTusa et al, Phys. Rev. Lett. 73, 1857 (1994); K. Kjima et al, Phys. Rev. Lett. 74, 3471 (1995). [7] E. S. S0rensen and I. Affleck, Phys. Rev. B51, (1995); Wei Wang, Shajin Qin, Zhng-Yi Lu, Zha-Bin Su and Lu Yu, Phys. Rev. B53, 40 (1996); Xiaqun Wang and Steffen Mallwitz, Phys. Rev. B53, 492 (1996); Jizhng Lu, Shajin Qin, Zhabin Su and Lu Yu, Phys. Rev. B58, (1998). [8] E. S. S0rensen and I. Affleck, Phys. Rev. Lett. 71, 1633 (1993). Lett. 72, [9] Guangyng Xu et al, Phys. Rev. B54, R6827 (1996). [10] V. Sachan, D. J. Buttrey, J. M. Tranquada and G. Shirane, Phys. Rev. B49, 9658 (1994). [11] A. Zheludev, J.M. Tranquada, T. Vgt and D. J. Buttrey, Eurphys. Lett. 35, 385 (1996); Phys. Rev. B54, 6437 (1996). [12] A. Zheludev, J. M. Tranquada, T. Vgt and D. J. Buttrey, Phys. Rev. B54, 7210 (1996). [13] A. Zheludev, J. P. Hill and D. J. Buttrey, Phys. Rev. B54, 7216 (1996). [14] T. Yk, A. Zheludev, M. Nakamura and J. Akimitsu, Phys. Rev. B55, (1997). [15] T. Yk, S. Raymnd, A. Zheludev, S. Maslv, I. Zaliznyak, J. Akimitsu and R. Erwin, preprint: cndmat/ [16] A. Zheludev, E. Ressuche, S. Maslv, T. Yk, S. Raymnd and J. Akimitsu, Phys. Rev. Lett. 80, 3630 (1998). [17] S. Maslv and A. Zheludev, Phys. Rev. B57, 68 (1998). [18] S. R. White and R. M. Nack, Phys. Rev. Lett. 68, 3487 (1992); S. R. White, Phys. Rev. Lett. 69, 2863 (1992); S. R. White, Phys. Rev. B48, (1993). [19] Shajin Qin, Xiaqun Wang and Lu Yu, Phys. Rev. 56, R14251 (1997). [20] S. Maslv and A. Zheludev, Phys. Rev. Lett. 80, 5786 (1998). [21] Ta Xiang, Phys. Rev. B 58, 9142 (1998). [22] Y. J. Kim, M. Greven, U.-J. Wiese and R. J. Birgeneau, Eur. Phys. J. B 4, 291 (1998). [23] S. Raymnd et al., Preprint cnd-mat/

9 Lu et al., Fig. 5

10 Lu et al, Fig. 4, PRB O* A = 0.01 A 0 = 0.03 ^ \ \., /2 = 0.06 h = 0.\0 0 h = 0.20 A h = A 0. A <> A 0 \ «15 20 n 25 30

11 Lu et al, Fig. 3, PRB Af / Ag by DMRG A / Al by DMRG A /A by NLSM il by NLSM (N

12 Lu e? a/., Fig. 2,PRB cm R DMRG result ur fitting line NLSM result

13 < A L - A T - Lu et al., Fig. 1,PRB A A ( h