8/3/2019 I. Harada, T. Kimura and T. Tonegawa- Disorder Line in a Quantum Spin Chain with Competing Interactions

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JOURNAL DE PHYSIQUE.Colloque C8, SupplBment au no 12, Tome 49, dbcembre 1988

DISORDER LINE IN A QUANTUM SPIN CHAIN WITH COMPETINGINTERACTIONS

I. Harada, T. Kimural and T. Tonegawa2Institu t fGr Theore tische Physik, Un ive rsit it Hannover, Appe lstrasse 2, 3000 Hannover 1 , F.R . G.

Abstract. - The spin-112 Heisenberg chain with antiferromagnetic first- and second-neighbor interactions is studied bythe cluster transfer-matrix method. It is found that this quantum model exhibits a disorder temperature above which thespin correlation function decays with an incommensurate oscillation while below which it decays antiferromagnetically.

It is well known that, in a classical spin chainwith competing interactions, the two-spin correlationfunction shows not only monotonic exponential decaybut exponential decay modified by an incommensurateoscillation [I , 2, 31. There is a definite temperaturecalled a disorder point a t which the nature of short-range order changes as described above.

In this paper, we report the results of our calcula-tions for the two-spin correlation function in the one-dimensional spin-112 magnet with antiferromagneticfirst- and second-neighbor interactions. We expressthe Hamiltonian describing the systepl as

with periodic boundary conditions s ~ + i s;, wheres; s the spin-112 operator a t the site i; Jt and J 2 arethe first- and the second-neighbor interaction constant,respectively, and are assumed to be positive (antifer-romagnetic); 7 s the parameter describing the Isinganisotropy (0 5 7 5 1) ; N is the number of spins inthe chain. A portion of our work on the ground stateproperties has been reported in separate papers [4, 51.

The method used in this paper is the clustertransfer-matrix method based on the Suzuki-Trottertheorem [6]. Details of the method for the present sys-tem have been given,in our previous paper [71. To makethis paper self-contained, however, we briefly summa-rize the results.According to the Suzuki-Trotter theorem [6 ] ,we ap-proximate the partition function as

(see Fig. la) ; m is the so-called Trotter number. It isnoted that, when either n or m is infinite, Z (n, m)yields the exact partition function. In this paper, weperform the numerical calculations for n = 4, 6, 8 and10 with m = 1, aiming to take into account effectsof the competition properly. We can find the two-dimensional Ising system whose partition function isequivalent to that given by equation (2). The Isingsystem for m = 1 and n = 10 is depicted in figure lb .

Peal-space direction( b )Fig. 1. - Schematic illustration of (a) the 10-spin clusterdecomposition and (b) the corresponding Ising system ona checkerboard-like lattice in the case of m = 1. At thebottom the transfer matrices along the real-space direction,Tn nd TB,are indicated.

Applying the transfer matrix method to this Isingsystem, we obtain2 (n, m) = TI^ {exp (-pH; (n ) /m) x

(n, m) = (XI)N/2(n-2) 9 (3)xexp lm )1 ' (2) where hl is the largest eigenvalue of the matrix T =

where ,B = l / k ~ T ;H (n) and H (n) are the wspin TnTs (see Fig. 1). w e note that in the case of m = 1,cluster Hamiltonian such tha t T*=Ts holds for any value of n. In a similar way, thecorrelation function (sf st+k) in the k * oo limit is

H = Ck (HIP (n)+H (n)) expressed in terms of the largest eigenvalueA1 and thesecond largest eigenvalue X2 of T :'NEC Corporation, Kawasaki, Kanagawa 211, Japan.2 ~ e p t .f Phys., Kobe Univ., Kobe 657, Japan. (sIsI+k)0:exP (-k/5) cos (qk) (4)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19888647

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8/3/2019 I. Harada, T. Kimura and T. Tonegawa- Disorder Line in a Quantum Spin Chain with Competing Interactions

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C8 - 1410 JOURNAL DE PHYSIQUEwhere the inverse correlation length 1/< and thewavenumber q are given by

I/< = In (XI/ IX21) /2 (n - 21, (5)q = an-' {fm (X2) /R e (X2)) 12 (n - 2) . (6)

Note that in our units, the lattice constant a = 1.Now, we present our numerical results of 1/ 1/2 there is no disorder temper*ture and the curve reaches its own value at t = 0. Onthe other hand, 1/