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Superfluid-spiral state of quantum ferrimagnets in magnetic field

Super?uid-spiral state of quantum ferrimagnets in magnetic ?eld
M. Abolfatha , and A. Langarib,c
b

Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019-0225 Max-Planck-Institut f¨r Physik komplexer Systeme, N¨thnitzer Strasse 38, D-01187 Dresden, Germany u o c Institute for Advanced Studies in Basic Sciences, Zanjan 45195-159, Iran (February 1, 2008) We study the phase diagram of one-dimensional quantum ferrimagnets by using a numerical exact diagonalization of a ?nite size system along with a ?eld-theoretical non-linear σ model of the quantum ferrimagnets at zero temperature and its e?ective description in the presence of the external magnetic ?eld in terms of the quantum XY-model. The low- and the high-?eld phases correspond respectively to the classical N?el and the fully polarized ferromagnetic states where in e the intermediate magnetic ?eld (hc1 < h < hc2 ), it is an XXZ+h model with easy plane anisotropy, which possess the spiral (super?uid) states that carry the dissipationless spin-supercurrent. We derive the critical exponents, and then will study the stability of the XY spiral state against these spin-supercurrents and the hard axis ?uctuations. We will show a ?rst order phase transition from the easy plane spiral state to a saturated ferromagnetic state occurs at h = hc2 if the spinsupercurrent reaches to its critical value. PACS number: 76.50.+g, 75.50.Gg, 75.10.Jm

a

arXiv:cond-mat/0101194v1 12 Jan 2001

I. INTRODUCTION

Recently, antiferromagnetically coupled mixed-spin chains with an alternating array of two kinds of spins have attracted interest among researchers [1–5]. Integrable models of mixed-spin antiferromagnetic chains were constructed by de Vega and Woynarovich [1] and the simplest case of such chains with spins S = 1 and 1/2 were subsequently studied [2]. Since these integrable models are exactly solvable, they are very useful for studying (quantum) statistical mechanical properties. Although ferrimagnetic spin chains exhibit both ferromagnetic and antiferromagnetic features, they show some peculiar, and sometimes surprising, features uncharacteristic of either the ferromagnet or the antiferromagnet—an example being the existence of gapless excitations with very small correlation length. It is important to understand these features more clearly. In this paper we ?rst present a numerical study on the quantum ferrimagnetic spin chain in magnetic ?eld by an exact diagonalization of a ?nite size system. We show that the low- and the high-?eld phases correspond respectively to the classical N?el and the fully polarized ferromagnetic states where e in the intermediate magnetic ?eld (hc1 < h < hc2 ), it is an XXZ+h model with easy plane anisotropy [4,5], and the critical exponents are derived numerically. Then the physical properties of the quantum ferrimagnets in the presence of the uniform magnetic ?eld, as described by a phenomenological ?eld theory based on a continuum non-linear σ model (NLσM) formulation are examined further and will explore its novel features. When h ≥ hc1 , the quantum ferrimagnets can be described by an easyplane anisotropy state where the broken U (1) symmetry of spins corresponds to a topological spiral (super?uid) state. A metastable state with a non-zero topological 1

spin-supercurrent Js , is the super?uid state of the order parameter in XY-plane. Such possibility, may be led to the remarkable spin-dependent transport phenomenon in the ferrimagnetic chains, i.e., the spin-supercurrent is carried collectively (rather than by quasiparticles). Because the spin current is non-zero when the system is in equilibrium, it ?ows without dissipation. The spiral state can be characterized by a wave-vector Q, i.e., the soliton (saw tooth) lattice spacing a = 2π/Q is a length scale through which the in-plane phase ? of the order parameter changes by 2π. Hence, the spin-supercurrent is characterized by these wave-vectors, e.g., Js ∝ Q. In analogy with the dissipation mechanism of the supercurrent in a one-dimensional superconductors [6], at large Js , hard-axis ?uctuations of the order parameter becomes unstable and a ?rst order phase transition to a uniform in-plane state (? = cte) takes place, and the long range order of the soliton lattice destroys. By overcoming the nucleation energy barrier of the order parameter vortices, a ?rst order phase transition from easy plane spiral state to a saturated ferromagnetic state occurs at h = hc2 , depends on Js , and Q. This transition turns out to a continuous cross-over if the initial state of the system at h = hc1 holds the zero spin-supercurrent (a uniform easy plane with Q = 0). The possibility of the existance of the super?uid phase and crossing onto a saturated ferromagnetic state (the dependence of hc2 on Q, and Js ) are discussed in some detail. The zero-temperature quantum ferrimagnetic chain is de?ned by H = J <ij> Si · sj ? h i (Si + si ) where S = s. The low energy physics of the quantum ferrimagnets in the presence of the external magnetic ?eld can be obtained by a 1-dimensional NLσM. The external magnetic ?eld couples with n, the N?el order parameter e

Lh = iM0 A(n) · ?τ n + LNLσM+h + Ltop ? M0 h · n, (1) where LNLσM+h = 1 2g vs (?x n)2 + 1 [2ih + n × ?τ n]2 . (2) vs

ization Group (QRG) approach. The main part of this paper, the spiral (super?uid) state of the quantum ferrimagnets is presented in section III. The details of the collective modes in the intermediate magnetic phase is given as the appendix in Sec. IV.
II. NUMERICAL RESULTS

The ?rst term in Eq. (1) is the usual (dynamical) Berry’s phase of a quantum ferromagnet, with M0 ≡ |S ? s|/a0 the magnetization per unit cell (pair of sites). Here Ltop is the topological term, g = 4/(s + S), and vs = 4a0 JsS/(s + S) [5]. It is this term that results in the ferromagnetic branch of the spin waves and corresponds to the trajectory of spin over a closed orbit on the unit sphere in the presence of a unit magnetic monopole at the center. The contribution of the ?rst term in Lh is equivalent to the area enclosed by this trajectory and since either of the smaller or the larger enclosed areas on the unit sphere must lead to the same Berry’s phase, the magnetic moment per unit cell, i.e. 2M0 a0 , must be quantized with integral values [7]. Setting M0 = 0 in action (1) follows to the usual O(3) NLσM in the magnetic ?eld [8], applicable to the Heisenberg antiferromagnets. Similar to zero magnetic ?eld [5] we can ?nd the spin-wave modes. At h = 0 the ferrimagnetic spin-waves consist of both gapless (ferromagnetic) and gapped (antiferromagnetic) modes, and at T = 0 the low (high) energy physics of quantum ferrimagnets is e?ectively like that of a ferromagnet (antiferromagnet) which is formed by the chain of (dimerized) unit cells with magnetic moment M0 (= |S ? s|). Applying an external magnetic ?eld, h, develops a gapped ferromagnetic spin-waves; in which case the energy cost for the ferromagnetic transitions is proportional to the Zeeman splitting factor. Unlike to ferromagnetic mode, the e?ect of the external magnetic ?eld is to suppress the antiferromagnetic gap. Clearly, this reveals similarities between the ferrimagnets and the integer spin Heisenberg antiferromagnets in magnetic ?eld [9]. The ground state of the ferrimagnet corresponds to the staggered con?guration of spins, unless h ≥ hc1 = 2J|S ? s|. At this point the staggered state becomes unstable against the non-collinear spin-?op phase (the partially polarized state) of the spins when the spectrum becomes soft at k = 0. When the external magnetic ?eld exceeds hc2 , the system will be in a saturated ferromagnetic phase, with a quantized magnetization per unit cell. The hc2 ≡ 2J(S + s) is obtained by using the dispersion relation of the spin waves based on the fully polarized state of the ferrimagnets. It is the lower-bound of the external magnetic ?eld, in the sense that the spin waves of the fully polarized state become soft at k = 0. The outline of this paper is as follows : In next section we will present the numerical computations of Lanczos method to compare the behavior of the correlation functions on the plateau and between two plateaux. We also derive the critical exponents, and the e?ective Hamiltonian for the latter region by using a Quantum Renormal2

In order to have more accurate physical picture, we present the numerical results, by applying a Lanczos method to each Sz -sector of Hilbert space from Sz = N N 2 (S ? s) to Sz = 2 (S + s). Here N (= 20) is the number of sites that is used in the exact diagonalization. A curve for magnetization vs. magnetic ?eld for a chain of (1/2, 1)-ferrimagnet has been given in Ref. [10]. An extrapolation to N → ∞ on exact diagonalization calculation shows there are two plateaux of magnetization below hc1 = ?0 = 1.7589J and above hc2 = 3J, M associated with the magnetization m ≡ S+s = 1/3 and m = 1. Here ?0 is the energy gap between the Ferroand Antiferromagnetic spin wave modes of this model. In Fig. 1 the numerical results of the in-plane spin-spin correlations is presented for a point on the plateau (at m = 1/3) and some intermediate points between two plateaux, i.e., m = 2/5, 8/15, 2/3 and 4/5. This exhibits the in-plane spin-spin correlation functions within the intermediate magnetic ?eld region which falls o? as power law, and manifests the critical behavior. Thus the in-plane correlation is expected to have the asymptotic form S x (0)S x (r) ? r?η . For instance we have calculated this exponent for m=2/3 by using data of exact diagonalization of a chain with length N = 20. Since there are correlations between di?erent types of sublattices we ?nd an exponent for each case. For correlations on sublattice A (Fig.1-(a)) and m=2/3 we obtained η = 0.44 ± 0.01 and for sublattice B (Fig.1-(b)) we have η = 0.42 ± 0.04. A similar behavior is seen for the correlations of di?erent sublattices A and B (Fig.1(c)) where η = 0.47 ± 0.01. To have more qualitative picture of the transient region between two plateaux, we can use a quantum renormalization group. To perform this, we choose two adjacent spins (S = 1, s = 1/2) as the building block. The block Hamiltonian is then HB = JS·s?h(Sz +sz ). For low ?eld limit (e.g. h < hc1 ) the lowest lying states of HB are a spin-1/2 doublet. This comes out with an e?ective Hamiltonian of ferromagnetic Heisenberg chain with S = 1/2 in magnetic ?eld, corresponds to the m = 1/3 plateau. For high ?eld limit z h > 3J/2 where two states |ST = 1/2, ST = 1/2 , and z |ST = 3/2, ST = 3/2 are nearly degenerate, we arrive with a spin-1/2 antiferromagnetic XXZ + h Hamiltonian. The e?ective Hamiltonian is He? = 2J 3
N/2 ⊥ ⊥ z z (τn · τn+1 + ?τn τn+1 ) ? h′ n=1 n=1 N/2 z τn

(3)

10

0

m

< S (0) S (r) >

10

?1

1/3 (plateau) 2/5 8/15 2/3 4/5

(a)

(spin?1, spin?1)

tially polarized phase (between the two plateaux) goes to the isotropic XY ?xed point [12].
III. SUPERFLUID PHASE

x

0.2 0.15 10
?2

m=1/3 m=2/3

x

0.1 0.05 0 2 4 6 8 10 r m
?1

10

?3

1
0

10

By generalizing the NLσM+h we propose a phenomenological ?eld theory which can present the long wave length limit of the quantum ferrimagnets within h ≥ hc1 E= dx 1 1 ⊥ ρ (?x n⊥ )2 + ρz (?x nz )2 + βn2 ? h? nz , z 2 s 2 s (4)

10

<s (0)s (r)>

10

1/3 (plateau) 2/5 8/15 2/3 4/5

(b) (spin?1/2, spin?1/2)

x

x

10

?2

0.4 0.3
m=1/3 m=2/3

10

?3

0.2 0.1 0 0 2 4 6 8 10 r
m
1/3 2/5 8/15 2/3 4/5 (plateau)

10

?4

1
0

10

10

(c) (spin?1, spin?1/2)

10

?1

0.3 10
?2

0.2 0.1 0

m=1/3 m=2/3

10

?3

1

3

5

7

9 r 10

1

FIG. 1. The log-log plot of in-plane spin-spin correlation function is shown within the plateau (m = 1/3) and within the partially spin polarized state (m = 2/5, 8/15, 2/3, 4/5). The former shows the exponential decay while in the latter region the power law behavior of correlation function is seen. The correlation on sublattice A (spin-1, spin-1) is shown in (a), on sublattice B (spin-1/2, spin1/2) in (b) and between the two sublattices A (spin-1) and B(spin-1/2) in (c). In (c) spin-1 is ?xed and the position of spin-1/2 is running to show the correlations. In each part an inset represents the correlation for m = 1/3 and 2/3 in a normal scale. The former approaches to zero exponentially, and the latter falls o? algebraically.
3 where τ is spin-1/2 operator, ? = 1/3, h′ = 2 h ? 19 J. 6 It is known such model Hamiltonian in the presence of a magnetic ?eld (h) has a critical line which separates the partially polarized phase from the fully polarized one [11]. The magnetization of the ferrimagnetic chain (m) is related to the magnetization of XXZ + h model by m = 1 + mXXZ+h which leads to the same behavior as the numerical results but with some discrepancies for the critical ?elds (hc1 = 1.5J , hc2 = 3J). However one may continue the quantum RG procedure for the XXZ +h model and obtain that the RG ?ow for the par-

where we applied the spin coherent state formalism in Eq.(4). Here h? = h ? hc1 is the e?ective magnetic ?eld. ρ⊥ and ρz are in-plane and out of plane spin sti?ness. s s β > 0 gives an easy plane anisotropy. To study a super?uid phase, it is convenient to introduce the following variational solutions n(x) = (sin θ cos ?, sin θ sin ?, cos θ), where θ = θ(x) and ? = Qx. Here Q is a spiral state wave vector responsible for the super?uid phase slip, and a = 2π/Q is the soliton lattice spacing, i.e., the phase ? changes by 2π along the chain. Clearly nz = cos θ is the uniform solution (θ is a constant) and ? ? E = ρ⊥ Q2 (1 ? n2 )/2 + β n2 ? h? nz is the correspond?z ?z ? s ing energy per unit length. Note nz is the momentum density conjugate to ?eld variable ?, and the Hamiltonian (4) gives a linearly dispersing collective mode, associated with the U (1) symmetry breaking phase, i.e., the super?uid phase. It is easy to check how the classical solutions and the ?uctuating modes can be derived by ? n ? Eq.(4). For example dE/d? z = 0 leads to nz = h? /Kzz ? ? zz ≡ d2 E/d? 2 = 2β ? ρ⊥ Q2 is the energy gap ? nz where K s of the out-of-plane modes at zero wave vector. Then Js = (1/? )dE/dQ = n2 ρ⊥ Q/? is the gauge invariance h ? ?⊥ s h topological spin-supercurrent density carried along the x-direction. Furthermore, it is necessary to study the stability of the super?uid phase against the quantum ?uctuations. This governs with the in-plane ?uctuations K?? (k) = 2ρ⊥ n2 k 2 , and the out of plane ?uctuations s ?⊥ Kzz (k) = ρ⊥ n2 (k 2 + Q2 ) ? ρ⊥ n2 Q2 s ?z s ?⊥ z 2 2 2 +ρs n⊥ k + 2β(? ⊥ ? n2 ) + h? nz . ? n ?z ? (5)

| <S (1) s (r+1)> |

x

x

? As one can see Kzz = Kzz (k = 0) at h? = 0. The gapless linear super?uid mode can be obtained by hω = ? 2 K?? Kzz . The details of the collective modes calculation is presented in Appendix. Through out this formulation, the zero temperature phase diagram of this system can be obtained. At h? = 0 where nz = 0 the out of plane ? instability occurs at Qc = 2β/ρ⊥ . At this point the ens ergy gap of the zz-modes vanishes and the O(3) symmetry of the Hamiltonian is restored (the dispersion relation of the collective modes become imaginary at small k). It 3

follows the transition to a uniform solution must occurs at this point since π1 (S 2 ) = 0 where the linear solution can be considered as a map from the compacti?ed physical space to the equator of the order parameter space S 2 . Such instability in order parameter space is induced by the local ?uctuations of the order parameter, out of the equator toward the north pole of the order parameter space S 2 . When Js is close to its critical values (Jsc ), the local ?uctuations are very strong, and it is likely the order parameter pass the north pole, and the phase ? becomes singular, i.e., a vortex in ? is nucleated, and the phase winding is lost. The e?ect of h? is pushing the spins to be aligned along z-axis, e.g., nz = 1 at large enough h? . We ? notice if h > hc1 , system undergoes onto the saturated ferromagnetic phase by increasing the spin-supercurrent density. This happens at Q? = (2β ? h? )/ρ⊥ < Qc , s corresponding to Js = 0, before the outset of the easyplane uniform state. To study the nucleating mechanism of the vortices, we implement a mechanical analogy to the classical ?eld theory. This formalism has been developed by Langer, and Ambegaokar [6] for one dimensional superconductors. To avoid the complexity let us neglect ρz in energy functional (4) from now on. To s start this calculation, we make the following transformations nx (x) = f (x) cos ?(x), ny (x) = f (x) sin ?(x), and nz (x) = 1 ? f 2 (x) (where f ≡ n⊥ ), and E[f ] = dx L2 ρ⊥ s (?x f )2 + 2 2 f 1 ? f2 , (6)

2.0
Q=0 Q=.33 Q=0.66 Q=1.0 Q=1.33 Q=1.66

1.5

1.0

?Ueff

0.5

0.0

?0.5 0.0

0.2

0.4

0.6

0.8

1.0

f
FIG. 2. ?Ue? (f ), the e?ective potential of the spiral state vs. f is plotted for h? = J. The minimum energy solution at Q = Q? is f = 0 where the spiral state destroys.

+β(1 ? f 2 ) ? h?

where L is the momentum conjugate to ? and it is proportional to the spin-supercurrent density Js , because ?x ? = L/f 2 is the solution of the δE[f, ?]/δ? = 0. It is straightforward to show how δE[f ]/δf = 0 yields
f (x)

x ? x0 =
f (x0 )

df 2 Ee? ? Ue? (f )
f (x)

,

(7a)

?(x) ? ?(x0 ) = L
f (x0 )

dx , f2

(7b)

? Ue? (f ) =

ρ⊥ L 2 s + β(1 ? f 2 ) ? h? 2f 2

1 ? f 2.

(7c)

? Uniform solutions (f = f = cte) are one of the solu? tions of Eqs. (7). It leads to ? = Qx and L = Qf 2 . The energy potential associated with the uniform solutions are depicted in Fig. 2 where ?Ue? (f ) vs. f is plotted for h? = J for di?erent Q’s, or equivalently for di?erent spin-supercurrent Js . Here ρ⊥ = 2J/3, and s β = J. The e?ective potential, ?Ue? , has one minimum ? at f = 1 ? [h? /(2β ? ρ⊥ Q2 )]2 = 0 if Q ≤ Q? . s 4

This minimum energy solution is disappeared if Q > Q? (= (2β ? h? )ρs ), and is replaced by another min? imum at f = 0. At Q? a ?rst order phase tran? ? sition from f = 0 onto f = 0 takes place by nucleating of a vortex. One can follow the transition between these minimum energy solutions by changing the parameters, Q and h? , as illustrated in Fig. 2. The saddle point solutions can be derived by setting ? ? d2 Ue? /df 2 = 0 at the extremum of Ue? (f ). It follows (2β ? h? )/ρ⊥ (or equivalentlyJs = 0), the at Q? = s ? local minima at ?nite f is disappeared and nz = 1 be? comes a unique minima of Ue? . Moreover, the crossover from the uniform state [a broken U (1) symmetry state with zero spin-supercurrent] to a saturated ferromagnetic state occurs at h? = 2β = 2J (Q? = 0), consistent with the exact diagonalization results (hc2 = 3J). Obviously at h = hc1 and Q = Qc (= 2β/ρs ), the sign of curvature of ?Ue? changes, and therefore Qc can be considered as the saddle point of ?Ue? , at which the spiral state becomes unstable against the hard-axis ?uctuations and a transition to a uniform solution occurs. This is seen if the spin-supercurrent Js arrives to its critical value, Jsc . As was mentioned earlier, the mechanism for such transition is nucleating of a vortex in the order parameter space. For a given Q and h? , it is straight? forward to show E(Q) = ρ⊥ Q2 /2 ? h?2 /(2β ? ρ⊥ Q2 ). s s ? Recalling Js = (1/? )dE(Q)/dQ leads to Js = ρ⊥ f 2 Q/? h ? h s (Js = ρ⊥ L/? ) which is the spin-supercurrent density and h s nz = h? /(2β ? ρ⊥ Q2 ) is the classical solution of Eq.(7). ? s The dependence of L(= Q[1 ? h?2 /(2β ? ρ⊥ Q2 )2 ]) with s respect to Q for various h? is illustrated in Fig. 3. It follows that Js vanishes at Q? = (2β ? h? )/ρ⊥ where s the system crosses to the saturated ferromagnetic phase. ? The maximum spin-supercurrent Js (< Jsc ), which can pass through the system at ?nite wave vector Q (and

h? = 0), is a monotonically decreasing function of h? .

δHf d?(q) 2 = h = Kzz (q)nz (q), ? dt h ? δ 2 nz (?q)

(9b)

2.0
h=0 * h = 0.16 J * h = 0.32 J * h = 0.48 J * h = 0.64 J * h = 0.8 J
*

1.0

where nz is the momentum density associated with the ?eld variable ?. In the spin coherent representation, nz and n+ are given by ?i? ?/??, and ei? respectively. Then h it is easy to check the transformation nz → ?t ? can be obtained by canonical quantization, i.e., nz can be considered as the momentum density conjugate to ?eld variable ?. Making the derivative with respect to time we ?nd the equation of motion: nz = ?(2/? )K?? (q)?(q) = ¨ h ˙ ?(2/? )2 K?? (q)Kzz (q)nz , where ? ≡ d?/dt. This clearly h ˙ gives h ? ω = 2 K?? (q)Kzz (q). (10)

L

0.0 0.0

0.5

1.0

1.5

2.0

Q
FIG. 3. L vs. Q is plotted for di?erent h? . For any h? there ? is a maximum spin-supercurrent Js = Js (< Jsc ) which can ? pass through the system at wave vector Q′ . Js is a mono? tonically decreasing function of h . At Q = Q? = 0, the spin-supercurrent density vanishes (L = 0).

Eqs.(9a-9b) are similar to the coupled Josephson junction relations in superconductivity.

V. ACKNOWLEDGEMENT

In conclusion, we have predicted the intermediate magnetic ?eld region of ferrimagnets can support the dissipationless ?ow of the spin-supercurrents, which may be observed by the advance techniques of spintronics. The idea of the collective spin-supercurrent transport which has been presented in this paper is completely general, and can be applicable to any 1-dimensional spin system with easy plane anisotropy. The intermediate magnetic ?eld phase of the ferrimagnets is one example. Other example of this kind is the isospin-supercurrent transport of the quantum Hall bars in the bilayer electron systems. [13]
IV. APPENDIX

We are grateful to Homayoun Hamidian who was involved in the early stage of this study. Useful conversation with Allan MacDonald, Miguel Martin-Delgado and Kieran Mullen is acknowledged. Work at the University of Oklahoma was supported by the NSF under grant No. EPS-9720651 and a grant from the Oklahoma State Regents for Higher Education.

Here we present the detailed calculation of the collective modes in the intermediate magnetic ?eld phase. The e?ective Hamiltonian in terms of the in-plane and the out of plane ?uctuations is given by Hf [?, nz ] = 1 2 + 1 2 ?(?q)K?? (q)?(q)
q

nz (?q)Kzz (q)nz (q),
q

(8)

where K?? = δ 2 E/δ?2 and Kzz = δ 2 E/δn2 . The equaz tion of motion for the ?eld variables ?, and nz can be obtained by the Hamilton equations (p = ?δH/δq and ˙ q = δH/δp) ˙ dnz (q) 2 δHf = ? K?? (q)?(q), =? h ? dt h ? δ 2 ?(?q) (9a)

[1] H. J. de Vega and F. Woynarovich, J. Phys A 25, 4499 (1992); H. J. de Vega, L. Mezincescu, and R. I. Nepomechie, Phys. Rev. B 49, 13223 (1994). [2] A. K. Kolezhuk, H.-J. Mikeska, and S. Yamamoto, Phys. Rev. B 55, R3336 (1997); S. K. Pati, S. Ramasesha, and D. Sen, Phys. Rev. B 55, 8894 (1997); S. Brehmer, H.J. Mikeska, and S. Yamamoto, J. Phys.: Condens. Matter 9, 3921 (1997); S. K. Pati, S. Ramasesha, and D. Sen, J. Phys. Condens. Matter 9, 8707 (1997); G.-S. Tian, Phys. Rev. B 56, 5355 (1997); S. Yamamoto and T. Fukuii, Phys. Rev. B 57 R14008 (1998); N. B. Ivanov, Phys. Rev. B 57 R14024 (1998); K. Maisinger, U. Schollw¨ck, o S. Brehmer, H. J. Mikeska, and Shoji Yamamoto, Phys. Rev. B 58, R5908 (1998). [3] Shoji Yamamoto, S. Brehmer, H. J. Mikeska, Phys. Rev. B 57, 13610 (1998). [4] A. K. Kolezhuk, H.-J. Mikeska, K. Maisinger, and U. Schollw¨ck, Phys. Rev. B 59, 13565 (1999). o [5] M. Abolfath, H. Hamidian, A. Langari, preprint condmat/9901063.

5

[6] J. S. Langer, and Vinay Ambegaokar, Phys. Rev. 164, 498 (1967). [7] N. Read, and S. Sachdev, Phys. Rev. Lett. 75, 3509 (1995). [8] V. Nikos Nicopoulos, and A. M. Tsvelik, Phys. Rev. B 44, 9385 (1991). [9] T?ru Sakai, and Minoru Takahashi, Phys. Rev. B 43 o 13383 (1991).

[10] The magnetization curve of a 1d ferrimagnets can be obtained by setting γ = 1 in the SBA con?guration of twoleg ladder, presented in A. Langari and M. A. MartinDelgado, Phys. Rev. B 62, 11725 (2000), and normalizing the magnetic ?eld by 2. [11] C.N. Yang and C. P. Yang, Phys. Rev. 150, 321 (1966). [12] A. Langari, Phys. Rev. B58, 14467 (1998). [13] M. Abolfath, and Allan H. MacDonald (unpublished).

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