PHYSICAL REVIEW 8 VOLUME 43, NUMBER 16 1 JUNE 1991 Continued-fraction formalism applied to the spin- —' XYZ model Edson Sardella* UniUersidade Estadual Paulista Julio de Mesquita Filho, Campus de Ilha Solteiva, I/ha Solteiva —Sao Paulo, Cep 15378, Brazil (Received 1 October 1990; revised manuscript received 16 December 1990) In this paper, we evaluate the correlation functions of the spin- ~ XYZ model for some particular cases by using the Mori continued-fraction formalism. The results are exactly the same as those well-known ones. This removes any doubt about the convergence of the continued fraction recently raised by some authors. By introducing a projection-operator technique, it has been demonstrated by Mori' that the Laplace transform of the correlation function (the relaxation function) could be written as an infinite continued fraction. This contin- ued fraction can be obtained by solving a set of coupled Volterra equations for a hierarchy of memory functions. In general, it cannot be evaluated exactly for most of the physical systems. Consequently, some approximation is required in order to obtain a closed-form expression for the relaxation function. A very often used one is the N- pole approximation in which the Nth-order memory function is taken as a constant. This formalism has been revealed itself as a powerful computational method of calculating the relaxation function, and it has been ap- plied successfully in many problems of theory of relaxa- tion (for instance, see Refs. 2 and 3). Very recently, Oitmaa, Linbasky and Aydin have eval- uated numerically the relaxation function of the spin- —, ' XYZ model (spin- —, anisotropic Heisenberg chain). In contrast to previous works, their results show that the continued-fraction approach, together with the N-pole approximation, suffers from a serious lack of conver- gence. In this paper some analytical calculations will be presented which are in total disagreement with this con- clusion. In other words, we will prove that the continued fraction converges and provide results which coincide with previous well-established ones. We then point out what might be the mistake in those numerical works. The system to be investigated is the spin- —, anisotropic Heisenberg chain described by the Hamiltonian " H: 2 g (Jxulcrt+)+ JyoIol+r+Jzo'tot+&) ' (1)'' I = lim 2 Tr[ .o(ot)o „(0)], &—+ oo (2) where N is the number of spins. The spatial Fourier transform of C (n, t ) is defined by C (k, t)= g e '""C (n, t) . (3) For the special case J,=0, these correlation functions can be evaluated exactly. ' Let us analyze separately two branches of this particular case. (a) Calculation of C'(n, t) for J =J =J. The expan- sions of the C (k, t ) in powers of r is given by the mo- ments according to C (k, r)= g, M»(k)r»,( —1)'— (2l )! (4) where I M z&(k ) =M&&(0)+2 g M&I(n )cos(nk ), n =1 (5) where M (2n1) are the moments of the power-series ex- pansion of C (n, t ). In fact, the sum in Eq. (5) should go up to n = ~. Nevertheless, Mz&(n )=0 if n ~ 1+1, ' so that we can stop at n =l. These moments were found in Ref. 5(a) up to tenth order. For our purpose, it will be sufficient to retain terms only up to sixth order. We have where o (n=x, y, z) are the Pauli operators. The corre- lation functions for this system at infinite temperature are defined by C (n, t)=(cro(t)o. „(0))T M o(k)=1, M z(k)=2(J„+J ) —4J J~cosk, M '(k) =4[2(J4+J~)+5J2J +(J2+J )J, ]—8J J [3(J„+J )+J, ]cask+ 12J J cos2k, M 6(k)=4I8(J„+J )+42J J (J„+J )+[18(J +J )+19J„Jy]J, +4(J +Jy )J, j —4J J [32(J„+Jy )+86J J +35(J„+J )J, +8J, ]cosk +60J„J [2(J„+J )+J, ]cos(2k) —40J J cos(3k) . (6a) (6b) (6c) (6d) 43 13 653 1991 The American Physical Society 13 654 BRIEF REPORTS 43 For the special case J, =0, J =J =J, tedious but straightforward calculation leads us to M~(k)=4(2J +J J ), M6(k)=4(8J +18J J +4J J„), (1 lc) (1 ld) I (k, s)= 6", (k )s+ 6~(k )s+ s+ '. (8) where the Mori parameters are given by M 0(k ) = 1, M ~(k ) =8a =2 a M'(k)=96a =3X2'a M;(k) =1280a'=5 X2'a', where a ==J sin(k/2). Let r (k, s ) denote the Laplace transform of C (k, t ). According to Mori, ' the continued-fraction representa- tion of this function is My(k) =1, My~(k) =2J2, M;(k ) =4(2J„'+J'J') M;(k ) =4(8J.'+18J,'J+4J„'J,') . (12a) (12b) (12c) (12d) Note that as we make the change specified above, all the k-dependent terms in the moments are proportional to J, . Because we are considering J, =O, M &i(k) be- come k independent. It has been pointed out in Ref. S(a) that, for J, =0, Mz&( n) is nonzero only if n =0. There- fore, Eq. (3) implies that C (k, t)=C (O, t). Thus, substi- tuting Eqs. (11) and (12) into Eq. (9), the Mori parameters take the form M ~(k) 5, (k)= M 0(k) M4(k) M ~(k) 6~(k) = M 2(k) M 0(k) (9) 6",(k ) =2J 6,(k)= 62(k)=2(J„+J ), 2J (SJ +J ) (J+J ) (13a) (13b) (13c) M 6(k) —[M 4(k )] /M ~(k ) 63(k ) = M 4(k) —[M 2(k )] /M 0(k ) which are particular cases of a more general expression 5i(k)=b, iht 2/(D, I, ), where 6,=60=1, and for 1~1, Co Ci C2 (10) CI CI+1 ' . . C21 with C2i+, =0 and C~, =M ~i(k ). Now, it follows from Eqs. (7) and {9) that 6i(k)=8a, 62(k)=53(k)= =4a . Introducing these expres- sions into Eq. (8), we obtain I (k', s) =1/[s+2f(k, s)], where f ( k, s ) =4a /[s +f(s, k ) ]. This immediately gives f(s, k)= —s/2+(s /4+4a )' and I '(k, s)= 1/(s +16a )'y . Hence C'(k, t) is just the Laplace transform of the Bessel function of order zero, Jo(4at)=J0[4Jt sin(k/2)]. This is exactly the same re- sult which was found in Ref. 5(a) by using a dift'erent ap- proach. For completeness, we just quote the result C'( n, t ) = [J„(2Jt )], which may be easily obtained by taking inverse Fourier transform of C '(k, t ). (b) Calculation of C (O, t) and C (O, t). The moments of the expansion of C ( k, t ) (cz =x,y ) may be found from Eq. (6) by making the changes J ~J, J ~J„and J,—+J . We then obtain, with J, =O, 5y(k ) =2J, , 5y(k)=2(J„+J ), 2Jy2(SJy2+ J2) 5 (k)= (J„'+Jy') (14a) (14b) (14c) For the Ising case J„WO and J =0, Eqs. (13a) and (14) are reduced to 6$(0) =0, 5,(0) =5~(0)=2J, and 6y3(0)=0. Consequently, with the help of Eq. (8), the re- laxation functions become I (O, s ) = 1/s and I y(O, s)=(s +2J )/s(s +4J, ), which are the Laplace transform of C (O, t)=1 and Cy(O, t)=cos J„t, respec- tively. This is again in agreement with the results of Ref. 5(b). As a final case, let us take the isotropic situation J =J =J for which C'(0, t ) = Cy(0, t ). Then 6, (0)=2J, 52(0)=4J, 63(0)=6J, etc. , and r (O, s)= 1 2J 4J2 6Js+ S+ (15) I {0s)=- 1/&ZJ 1 +V'2J s + '. &ZJ (16) with o.=x,y. This expression can be rearranged in a more con- venient form as follows: Mo(k)=1, M ~(k)=2J (1 la) (11b) to According to Wall, this continued fraction converges 43 BRIEF REPORTS 13 655 (I/v'2J )exp( —s /4J ) J & J—e " ~ du, which in turn is precisely the Laplace transform of &2, 2 e ' . Once more, this is a merely confirmation of well- known results. To the best of my knowledge, I could not find any closed form of the relaxation function for which the con- tinued fraction converges in the anisotropic case J WJ . Even so, I consider that the previous calculation is sufhcient to believe that the continued-fraction approach produces the correct answer for the correlation functions of the spin- —, ' XYZ model in any situation. Let us now make some comments about the conver- gence of the continued fraction. The relaxation function of Eq. (8) can be generated from a hierarchy of relaxation functions coupled one each other by with 50(k ) = 1 and 1 0(k, z ) =1 (k,z ). We can also write this as I I+,(k, s ) = —s+5& (k )/1 I (k, s ). Having knowledge of the exact result for 1 0(k, z), in Ref. 4 they evaluated (numerically) I I (k, s ) up to l =5. Surprisingly, they did not find any function g(k, s) for which I I (k, s) approaches with increasing values of l. For instance, in case (a) investigated above, I i(k, s) =s/2 +(s /4+4a )' for all I ) 2. It must be emphasized that Eq. (8) converges for all Re(s) )0 (the proof of this fact can be found in Ref. 7). It seems that in Ref. 4 they did not attempt to solve this problem. This might be the origin of all their difhculties in not finding the conver- gence of the continued fraction. It is hard to alarm whether this is in fact the problem; nevertheless, our analysis certainly indicates that those numerical works must be revised. I would like to thank T. J. Newman for useful discus- sions. This work was supported by CNPq-Conselho Na- cional de De senvolvimento Cienti'fico e Tecnologico- Brazil under Contract No. 200471/88. 0. *Present address: Department of Theoretical Physics, Universi- ty of Manchester, Manchester M13 9PL, United Kingdom. H. Mori, Prog. Theor. Phys. 34, 399 (1965)~ M. W. Evans, P. Grigolini, and P. Parravicini, Advances in Chemical Physics (Wiley, New York, 1985), Vol. 62. S. W. Lovesey and R. A. Meserve, J. Phys. C 6, 79 (1973). 4J. Oitmaa, I. Linbasky, and M. Aydin, Phys. Rev. B 40, 5201 (1989) and references therein. ~(a) J. M. R. Roldan, B. M. McCoy, and J. H. H. Perk, Physica 136A, 255 (1986); (b) H. W. Capel and J. H. H. Perk, ibid. 87A, 211 (1977). We will follow closely the notation of Ref. 5(a). 7M. Dupuis, Prog. Theor. Phys. 37, 502 (1967). To convince ourselves that 6&{k ) =4a for all l )2, one could say that we should go a few more stages beyond l = 3. In fact, from Eqs. (2.19) and (2.20) of Ref. 5(a) with J,=0 and J„=J =J, we find M;(k ) = 35 X 2 a y M, o(k ) =315X 2' a ' . These moments and those of Eq. (7) as substituted into Eq. (10) produce 6& =2'a, 62=2'a, 63=2 "a ', A4 =2 a, and 55 =2 a . This sequence suggests to us that Al = 2I't+z)a I(I+') for all l ) 1. Upon using 61(k)=616~ 2/(AI &), one finally obtains 51(k ) =4a for all l) 2. 9H. S. Wall, Continued-Fractions (Van-Nostrand, New York, 1948), p. 356.