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Critical behavior of the quantum spin-12 anisotropic Heisenberg model
Physica A 259 (1998) 138–144
Critical behavior of the quantum spin- 12 anisotropic Heisenberg model J. Ricardo de Sousa ∗ Instituto de Ciˆencias Exatas, Departamento de Fsica, Universidade do Amazonas, 3000-Japiim, 69077-000, Manaus-AM, Brazil Received 12 November 1997; revised 30 December 1997
Abstract A two-step renormalization group approach – a decimation followed by an e ective eld renormalization group (EFRG) – is proposed in this work to study the critical behavior of the quantum spin- 12 anisotropic Heisenberg model. The new method is illustrated by employing approximations in which clusters with one, two and three spins are used. The values of the critical parameter and critical exponent, in two- and three-dimensional lattices, for the Ising and isotropic Heisenberg limits are calculated and compared with other renormalization group c 1998 Elsevier Science B.V. All rights reserved. approaches and exact (or series) results. PACS: 64.60.Ak; 64.60.Fr; 68.35.Rh
1. Introduction The critical properties of quantum spin systems is one of the most fascinating subjects in solid state physics. The present author and Fittipaldi [1,2] proposed a new renormalization group (RG) method to study the quantum spin- 12 anisotropic Heisenberg (QAH) model. This new method, denoted as the e ective eld renormalization group (EFRG), was rst applied with success in the Ising model [ 4 –8]. The new approach (EFRG), which follows the same strategy as the mean eld renormalization group (MFRG) [9], is based on an alternative way for constructing e ective eld equations of states by using the Callen–Suzuki approximation [10] as a starting point. The method treats the e ects of the surroundings spins of each clusters through a convenient di erential expansion technique by which, in contrast with the usual mean eld approximation procedure, all the relevant self-spin correlations are taken exactly into ∗
E-mail: jrs@ sica.fua.br.
c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$19.00 Copyright PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 0 8 7 - 9
J. Ricardo de Sousa / Physica A 259 (1998) 138–144
139
account. On the implementation of the RG idea is achieved by considering, in a systematic and sucessive manner, larger clusters. The interactions within the clusters are treated exactly and the e ect of the remaining spins of the lattice is replaced by a symmetry breaking eld. In the standard e ective eld approximation this eld is identi ed with the order parameter of the system. In this scheme, both the magnetization of the clusters and respective symmetry breaking elds scale in the same way. The phase transition of the QAH has been studied in the literature by various RG approaches [1,2,11–16] and e ective methods [ 17 –19]. In these studies the exact result [21] in the isotropic limit and in two-dimensional lattices is obtained in the RG approaches. The exact result has not been obtained in the literature by e ective methods, because of the e ect of small size of the clusters utilized in this approaches. The exact result has not been obtained in the literature by e ective methods, because of the e ect of small size of the clusters utilized in this approaches. The increase of clusters size obtain more accurate values for the critical temperature and critical exponents. On the other hand, combination of RG methods has been applied in Ising [22,23] and Heisenberg [3] systems, where MFRG and the decimation method are treated in two-step renormalizations. The decimation method [24,25] contains two operations: bond moving and site decimation. The decimation transformation, however, cannot be performed exactly for quantum systems due to the noncommutativity of operators. An approximate decimation has been proposed by Suzuki and Takano [11] to circumvent the diculty of the noncommutativity of the spin- 12 anisotropic Heisenberg model. The purpose of this work, is to apply these new RG methods (EFRG) with decimation in the QAH. The outline of the remainder of this paper is as follows. In Section 2 we introduce the model and formalism by devoloping its formulation for the simple cases of clusters of one, two and three spins. In Section 3 we present numerical results and a discussion for critical parameter and critical exponent for the Ising and isotropic Heisenberg models. The results are compared with those obtained from RG approaches and exact (or series expansion) results. Finally we conclude in Section 4.
2. Model and formalism The e ective Hamiltonian of the QAH is given by H =K
X
[(1 − )(Six Sjx + Siy Sjy ) + Siz Sjz ] ;
(1)
hi;ji
where K = J=KB T (J ¿ 0 is the exchange coupling), Si ( = x; y; z) are components of a spin- 12 operator at site i; hi; ji denotes sum over all nearest-neighbor pairs, and the anisotropy parameter is restricted to the interval [0,1]. The Ising and isotropic Heisenberg models, correspond the = 1 and = 0 limits in Eq. (1), respectively. The critical temperature Tc , for = 0 in two-dimensional lattices has been proved to vanish [21].
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J. Ricardo de Sousa / Physica A 259 (1998) 138–144
The rst step of the new approach (EFRG with decimation) is aplly the MigdalKadano renormalization group (MKRG) approach (or decimation) [24,25]. For a renormalization with scaling factor L1 = 2, Suzuki and Takano [11] found the following recursion relation for the renormalized parameters K 0 and W 0 = K 0 (1 − 0 ) of the model (1) in d-dimensional lattice [exp(2K) + exp(−K)A− (k; )]2 ; (2) K 0 = f(k) = 2d−3 ln 4 exp(−K)A+ (K; ) W 0 = K 0 (1 − 0 ) =
1 4
ln{exp(−K)A+ (K; )}
(3)
with A± (K; ) = cosh[B()K] ±
sinh[B()K] ; B()
(4)
q where B() = 1 + 8(1 − )2 . The xed point K 0 = K = Kc and critical exponent Yt = ln(L1 )= ln()( = (@K 0 =@K)Kc ) are obtained in this approach as a function of and d. For d = 2 and = 0, the Eq. (2) found exact result, i.e., Kc = ∞ and Yt = 0. In the isotropic limit ( = 0) and in three-dimensional lattice (d = 3) from the Eq. (2) one obtains Kc = 0:344 and Yt = 0:714, result that are not good when compared with series results [26] Kc = 0:301 and Yt = 1:389. Now we apply the second step of this new RG method, that consist in the utilization of the EFRG approach. In this approach we utilize a simple example of renormalization in clusters of sizes N 00 = 1 and N 0 = 2 spins. Written the Hamiltonian (1) for these clusters in the axial approximation [1,2,17] we obtain the order parameters m1 = hS1z i (for N 00 = 1) and m2 = h1=2(S1z + S2z )i (for N 0 = 2) by using the Callen-Suzuki approximate relation derived in the Ref. [10]. The equations obtained for the order parameters are approximate and mathematically untractable. Using a decoupling procedure which ignores all high-order spin correlation (see Ref. [10] for more details) and, adopting hS1z i = b00 (for i 6= 1) and b0 (for i 6= 1; 2) for the state equations of the clusters with one and two spins, respectively we obtain m1 (K 00 ; b00 ) and m2 (K 0 ; b0 ). It should be noted here that this approximation is quite superior to the standard mean eld theory, since it neglects correlation between di erent spins but takes relation as (Siz )2 = 1(Siz = ±1) for all i, exactly into account through of the identity exp(aSiz ) = cosh(a) + Siz sinh(a). The usual mean eld approximation, on the other hand, neglects all correlations. The b0 and b00 are the parameters of the symmetric breaking e ective elds. Expanding the state equations, m1 (K 00 ; b00 ) and m2 (K 0 ; b0 ; 0 ), up to rst order in the parameters b0 and b00 and with the imposing of the same scaling relations betwen the order parameters and between symmetry breaking elds for the two clusters in study, respectively (i.e., m1 = m2 and b00 = b0 ), one nds the following recursion between the new (K 00 ) and the old (K 0 ; 0 ) parameters [1,2] B1 (K 00 ) = B2 (K 0 ; 0 )
(5)
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141
with B1 (K 00 ) = 2d x00 ( x00 )2d−1 tanh(x)|x=0 ;
(6)
B2 (K 0 ; 0 ) = 2(2d − 1) x0 ( x0 )2d−2 ( y0 )2d−1 g(x; y)|x;y=0 ;
(7)
where g(x; y) is de ned by g(x; y) =
sinh(x + y) ; cosh(x + y) + exp(−2K 0 ) cosh W (x; y)
(8)
where x00 = cosh(K 00 Dx ); x00q= sinh(K 00 Dx ); 0 = cosh(K 0 D ); 0 = sinh(K 0 D ) (with = x; y); W (x; y) = (x − y)2 + 4K 02 (1 − 0 )2 and D = @=@ is the differential operator. From the Eq. (7), we can obtain the xed point and the critical exponent in the EFRG approach as a function of and d. From the Refs. [1,2], the Eq. (7) was numerically solved. For d = 2 and = 0, the Mermin-Wagner theorem [21] is satis ed, and for the three-dimensional case we have Kc = 0:271 and Yt = 0:610. Replacing K 0 and 0 from Eqs. (2) and (3) in Eq. (5) gives a nal single approximate recursion relation for the renormalized coupling K 00 in terms of the original parameters (K; ). The e ective nal scaling parameter is L = L1 L2 , where L1 = 2 and L2 = 21=d . Linearizing the recursion relation around the xed points (K 00 = K 0 = Kc ) gives the thermal critical exponent as a function of and d through the relation 1 (@B2 =@K)Kc ln : (9) Yt = ln(L) (@B1 =@K 00 )Kc 3. Numerical results Solving Eqs. (5) and (9), we obtain the critical coupling and critical exponent for the quantum spin- 12 anisotropic Heisenberg model. In two-dimensional lattice, we have the exact result [21] in the isotropic Heisenberg limits ( = 0), and Kc = 0:526 (exact is 0.441) and Yt = 0:917 (exact is 1.000) for the Ising limit ( = 1). The present result for the Ising model in 2d lattice has better values for the critical parameters in comparison with the combination of the MFRG and decimation methods that are Kc = 0:538 and Yt = 0:760 [22]. In the 3d case, the Ising limit also present better values for Kc = 0:251 (series expansion is 0.222) and Yt = 0:926 (series expansion is 1.587) when we compare with MFRG with decimation, that present Kc = 0:252 and Yt = 0:892. The values of critical parameter and critical exponent, can be also compared with RG method more sophisticated, for example, the 2d Ising model present in this work Yt = 0:917, while the hierarchical lattice RG (HLRG) in clusters greater [27] present Yt = 0:919. In Table 1, the results of Kc and Yt for the 3d isotropic Heisenberg model are shown by the present method (EFRG+MK) and is compared with other RG methods. From Table 1, we have that our results are in excellent concordance with series expansion,
142
J. Ricardo de Sousa / Physica A 259 (1998) 138–144 Table 1 Comparison of critical properties determined by various RG approaches for the isotropic Heisenberg model on a simple cubic lattice Methods
Kc
Yt
MFRG [12] MK [11] EFRG [1,2] HLRG [13,14] MFRG+MK [3] EFRG+MK (present work) Series expansion [26]
0.275 0.344 0.271 0.208 0.312 0.309 0.301
0.450 0.714 0.610 0.855 0.758 0.833 1.389
Table 2 Value of Kc for the isotropic Heisenberg model on a simple cubic lattice obtained by various e ective methods Methods
Values of Kc
MFA [28] OTSC [28] TSC [28] KO [28] EFT [17] EFTM [19] Present work Series expansion [21]
0.167 0.179 0.240 0.273 0.204 0.232 0.309 0.301
and also with the RG method more sophisticated (HLRG). The present result for the critical parameter Kc = 0:309, in the three-dimensional isotropic Heisenberg model, may be compared also with several methods, such as Weiss molecular eld (MFA), Oguchi two-spin cluster (OTSC). Oguchi three spin cluster (TSC), Kramers-Opechowski (KO) (see Ref. [28]), e ective theory (EFT) [17] and EFT modi es (EFTM) [19]. In Table 2 we have the values of Kc obtained by these e ective methods. Recently, the model above has been studied by a variational cumulant expansion (VCE) [20] that gives Kc = 0:414. We observe, that the new RG method (EFRG with decimation) also present better result than the e ective methods. 4. Conclusions I have used the EFRG with decimation to study the critical behavior of the ddimensional quantum spin- 12 anisotropic Heisenberg (QAH) model. The QAH system proved dicult to use in solving a many-body problem involving noncommuting operators. With the EFRG approach an axial approximation [1,2,17] for the Hamiltonian
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(1) in a cluster with one two spins was utilized, and the direct diagonalization can be carried out. The decimation scheme for the scale factor L1 = 2 is here utilized of form approximate [11]. This new RG method presented in this work is simple, consistent and gives good results, even for small size clusters, for critical parameter Kc and critical exponent Yt of quantum system. The values of Kc and Yt for the Ising and isotropic Heisenberg models obtained in this work, however, are in excellent agreement with the series expansion predictions. I also have applied the MFRG method with decimation [3] to study the QAH model (1), but the results for Kc and Yt are not better than the values obtained in this work by EFRG with decimation. From calculations of the 3d isotropic Heisenberg model, one can see that the EFRG combined with decimation is superior to the other e ective methods, and also are compared quantitatively with the more accurate RG methods (HLRG). The results obtained here are encouraging for further investigations in other complex systems, such as the spin-1 Heisenberg model, the transverse Ising model, etc. This extension is in progress and will be published elsewhere. Acknowledgements The author likes to thank Professor F.S. de Aguiar for a critical reading of the manuscript. This work was nancially supported by CNPq (Brazilian Agency). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
J. Ricardo de Sousa, I.P. Fittipaldi, J. Appl. Phys. 75 (1994) 5835. J. Ricardo de Sousa, I.P. Fittipaldi, unpublished. J. Ricardo de Sousa, Phys. Lett. A 216 (1996) 312. Z.Y. Li, C.Z. Yang, Phys. Rev. B 37 (1988) 5744. I.P. Fittipaldi, D.F. de Albuquerque, J. Magn. Magn. Mater. 104–107 (1992) 236. I.P. Fittipaldi, J. Magn. Magn. Mater. 131 (1994) 43. Q. Jiang, Z.Y. Li, Phys. Rev. B 40 (1989) 11 264. D.F. de Albuquerque, I.P. Fittipaldi, J. Appl. Phys. 75 (1994) 5832. J.O. Indekeu, A. Maritan, A.C. Stella, J. Phys. A 15 (1982) L291. F.C. Sa Barreto, I.P. Fittipaldi, Physica A 129 (1985) 360. M. Suzuki, T. Takano, Phys. Lett. A 69 (1979) 426. J.A. Plascak, J. Phys. A 17 (1984) L597. A.O. Caride, C. Tsallis, S. Zanette, Phys. Rev. Lett. 51 (1983) 145. A.M. Mariz et al., J. Phys. Lett. A 108 (1985) 95. F. Lee, H.H. Chen, H.C. Tsen, Phys. Rev. B 38 (1988) 508. A.M. Polyakov, Phys. Lett. B 57 (1975) 79; A.M. Polyakov, Phys. Lett. B 59 (1975) 81. T. Idogaki, N. Uryu, Physica A 181 (1992) 173. J. Mielmicki, G. Wiatrowski, T. Balcerzak, J. Magn. Magn. Mater. 71 (1988) 186. K.G. Chakraborty, Phys. Lett. A 177 (1993) 263. R.J. Liu, T.L. Chen, Phys. Rev. B 50 (1994) 9169. N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17 (1966) 1133. O.F. de Alcˆantara Bon m, F.C. Sa Barreto, M.A. de Moura, J. Phys. C 17 (1984) L599. L.R. Evangelista, V.K. Saxena, J. Phys. A 18 (1985) L389. A.A. Migdal, Sov. Phys. JETP 42 (1976) 743.
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[25] L.P. Kadano , Ann. Phys. 100 (1976) 359. [26] G.S. Rushbrooke et al., Phase Transition and Critical Phenomena, vol. 3, C. Domb, M.S. Green (Eds.), Academic Press, London, 1974. [27] H.O. Martin, C. Tsallis, J. Phys. C 14 (1981) 5645. [28] B. Strieb, H.B. Callen, Phys. Rev. 130 (1963) 1798.
Critical behavior of the quantum spin- 12 anisotropic Heisenberg model J. Ricardo de Sousa ∗ Instituto de Ciˆencias Exatas, Departamento de Fsica, Universidade do Amazonas, 3000-Japiim, 69077-000, Manaus-AM, Brazil Received 12 November 1997; revised 30 December 1997
Abstract A two-step renormalization group approach – a decimation followed by an e ective eld renormalization group (EFRG) – is proposed in this work to study the critical behavior of the quantum spin- 12 anisotropic Heisenberg model. The new method is illustrated by employing approximations in which clusters with one, two and three spins are used. The values of the critical parameter and critical exponent, in two- and three-dimensional lattices, for the Ising and isotropic Heisenberg limits are calculated and compared with other renormalization group c 1998 Elsevier Science B.V. All rights reserved. approaches and exact (or series) results. PACS: 64.60.Ak; 64.60.Fr; 68.35.Rh
1. Introduction The critical properties of quantum spin systems is one of the most fascinating subjects in solid state physics. The present author and Fittipaldi [1,2] proposed a new renormalization group (RG) method to study the quantum spin- 12 anisotropic Heisenberg (QAH) model. This new method, denoted as the e ective eld renormalization group (EFRG), was rst applied with success in the Ising model [ 4 –8]. The new approach (EFRG), which follows the same strategy as the mean eld renormalization group (MFRG) [9], is based on an alternative way for constructing e ective eld equations of states by using the Callen–Suzuki approximation [10] as a starting point. The method treats the e ects of the surroundings spins of each clusters through a convenient di erential expansion technique by which, in contrast with the usual mean eld approximation procedure, all the relevant self-spin correlations are taken exactly into ∗
E-mail: jrs@ sica.fua.br.
c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$19.00 Copyright PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 0 8 7 - 9
J. Ricardo de Sousa / Physica A 259 (1998) 138–144
139
account. On the implementation of the RG idea is achieved by considering, in a systematic and sucessive manner, larger clusters. The interactions within the clusters are treated exactly and the e ect of the remaining spins of the lattice is replaced by a symmetry breaking eld. In the standard e ective eld approximation this eld is identi ed with the order parameter of the system. In this scheme, both the magnetization of the clusters and respective symmetry breaking elds scale in the same way. The phase transition of the QAH has been studied in the literature by various RG approaches [1,2,11–16] and e ective methods [ 17 –19]. In these studies the exact result [21] in the isotropic limit and in two-dimensional lattices is obtained in the RG approaches. The exact result has not been obtained in the literature by e ective methods, because of the e ect of small size of the clusters utilized in this approaches. The exact result has not been obtained in the literature by e ective methods, because of the e ect of small size of the clusters utilized in this approaches. The increase of clusters size obtain more accurate values for the critical temperature and critical exponents. On the other hand, combination of RG methods has been applied in Ising [22,23] and Heisenberg [3] systems, where MFRG and the decimation method are treated in two-step renormalizations. The decimation method [24,25] contains two operations: bond moving and site decimation. The decimation transformation, however, cannot be performed exactly for quantum systems due to the noncommutativity of operators. An approximate decimation has been proposed by Suzuki and Takano [11] to circumvent the diculty of the noncommutativity of the spin- 12 anisotropic Heisenberg model. The purpose of this work, is to apply these new RG methods (EFRG) with decimation in the QAH. The outline of the remainder of this paper is as follows. In Section 2 we introduce the model and formalism by devoloping its formulation for the simple cases of clusters of one, two and three spins. In Section 3 we present numerical results and a discussion for critical parameter and critical exponent for the Ising and isotropic Heisenberg models. The results are compared with those obtained from RG approaches and exact (or series expansion) results. Finally we conclude in Section 4.
2. Model and formalism The e ective Hamiltonian of the QAH is given by H =K
X
[(1 − )(Six Sjx + Siy Sjy ) + Siz Sjz ] ;
(1)
hi;ji
where K = J=KB T (J ¿ 0 is the exchange coupling), Si ( = x; y; z) are components of a spin- 12 operator at site i; hi; ji denotes sum over all nearest-neighbor pairs, and the anisotropy parameter is restricted to the interval [0,1]. The Ising and isotropic Heisenberg models, correspond the = 1 and = 0 limits in Eq. (1), respectively. The critical temperature Tc , for = 0 in two-dimensional lattices has been proved to vanish [21].
140
J. Ricardo de Sousa / Physica A 259 (1998) 138–144
The rst step of the new approach (EFRG with decimation) is aplly the MigdalKadano renormalization group (MKRG) approach (or decimation) [24,25]. For a renormalization with scaling factor L1 = 2, Suzuki and Takano [11] found the following recursion relation for the renormalized parameters K 0 and W 0 = K 0 (1 − 0 ) of the model (1) in d-dimensional lattice [exp(2K) + exp(−K)A− (k; )]2 ; (2) K 0 = f(k) = 2d−3 ln 4 exp(−K)A+ (K; ) W 0 = K 0 (1 − 0 ) =
1 4
ln{exp(−K)A+ (K; )}
(3)
with A± (K; ) = cosh[B()K] ±
sinh[B()K] ; B()
(4)
q where B() = 1 + 8(1 − )2 . The xed point K 0 = K = Kc and critical exponent Yt = ln(L1 )= ln()( = (@K 0 =@K)Kc ) are obtained in this approach as a function of and d. For d = 2 and = 0, the Eq. (2) found exact result, i.e., Kc = ∞ and Yt = 0. In the isotropic limit ( = 0) and in three-dimensional lattice (d = 3) from the Eq. (2) one obtains Kc = 0:344 and Yt = 0:714, result that are not good when compared with series results [26] Kc = 0:301 and Yt = 1:389. Now we apply the second step of this new RG method, that consist in the utilization of the EFRG approach. In this approach we utilize a simple example of renormalization in clusters of sizes N 00 = 1 and N 0 = 2 spins. Written the Hamiltonian (1) for these clusters in the axial approximation [1,2,17] we obtain the order parameters m1 = hS1z i (for N 00 = 1) and m2 = h1=2(S1z + S2z )i (for N 0 = 2) by using the Callen-Suzuki approximate relation derived in the Ref. [10]. The equations obtained for the order parameters are approximate and mathematically untractable. Using a decoupling procedure which ignores all high-order spin correlation (see Ref. [10] for more details) and, adopting hS1z i = b00 (for i 6= 1) and b0 (for i 6= 1; 2) for the state equations of the clusters with one and two spins, respectively we obtain m1 (K 00 ; b00 ) and m2 (K 0 ; b0 ). It should be noted here that this approximation is quite superior to the standard mean eld theory, since it neglects correlation between di erent spins but takes relation as (Siz )2 = 1(Siz = ±1) for all i, exactly into account through of the identity exp(aSiz ) = cosh(a) + Siz sinh(a). The usual mean eld approximation, on the other hand, neglects all correlations. The b0 and b00 are the parameters of the symmetric breaking e ective elds. Expanding the state equations, m1 (K 00 ; b00 ) and m2 (K 0 ; b0 ; 0 ), up to rst order in the parameters b0 and b00 and with the imposing of the same scaling relations betwen the order parameters and between symmetry breaking elds for the two clusters in study, respectively (i.e., m1 = m2 and b00 = b0 ), one nds the following recursion between the new (K 00 ) and the old (K 0 ; 0 ) parameters [1,2] B1 (K 00 ) = B2 (K 0 ; 0 )
(5)
J. Ricardo de Sousa / Physica A 259 (1998) 138–144
141
with B1 (K 00 ) = 2d x00 ( x00 )2d−1 tanh(x)|x=0 ;
(6)
B2 (K 0 ; 0 ) = 2(2d − 1) x0 ( x0 )2d−2 ( y0 )2d−1 g(x; y)|x;y=0 ;
(7)
where g(x; y) is de ned by g(x; y) =
sinh(x + y) ; cosh(x + y) + exp(−2K 0 ) cosh W (x; y)
(8)
where x00 = cosh(K 00 Dx ); x00q= sinh(K 00 Dx ); 0 = cosh(K 0 D ); 0 = sinh(K 0 D ) (with = x; y); W (x; y) = (x − y)2 + 4K 02 (1 − 0 )2 and D = @=@ is the differential operator. From the Eq. (7), we can obtain the xed point and the critical exponent in the EFRG approach as a function of and d. From the Refs. [1,2], the Eq. (7) was numerically solved. For d = 2 and = 0, the Mermin-Wagner theorem [21] is satis ed, and for the three-dimensional case we have Kc = 0:271 and Yt = 0:610. Replacing K 0 and 0 from Eqs. (2) and (3) in Eq. (5) gives a nal single approximate recursion relation for the renormalized coupling K 00 in terms of the original parameters (K; ). The e ective nal scaling parameter is L = L1 L2 , where L1 = 2 and L2 = 21=d . Linearizing the recursion relation around the xed points (K 00 = K 0 = Kc ) gives the thermal critical exponent as a function of and d through the relation 1 (@B2 =@K)Kc ln : (9) Yt = ln(L) (@B1 =@K 00 )Kc 3. Numerical results Solving Eqs. (5) and (9), we obtain the critical coupling and critical exponent for the quantum spin- 12 anisotropic Heisenberg model. In two-dimensional lattice, we have the exact result [21] in the isotropic Heisenberg limits ( = 0), and Kc = 0:526 (exact is 0.441) and Yt = 0:917 (exact is 1.000) for the Ising limit ( = 1). The present result for the Ising model in 2d lattice has better values for the critical parameters in comparison with the combination of the MFRG and decimation methods that are Kc = 0:538 and Yt = 0:760 [22]. In the 3d case, the Ising limit also present better values for Kc = 0:251 (series expansion is 0.222) and Yt = 0:926 (series expansion is 1.587) when we compare with MFRG with decimation, that present Kc = 0:252 and Yt = 0:892. The values of critical parameter and critical exponent, can be also compared with RG method more sophisticated, for example, the 2d Ising model present in this work Yt = 0:917, while the hierarchical lattice RG (HLRG) in clusters greater [27] present Yt = 0:919. In Table 1, the results of Kc and Yt for the 3d isotropic Heisenberg model are shown by the present method (EFRG+MK) and is compared with other RG methods. From Table 1, we have that our results are in excellent concordance with series expansion,
142
J. Ricardo de Sousa / Physica A 259 (1998) 138–144 Table 1 Comparison of critical properties determined by various RG approaches for the isotropic Heisenberg model on a simple cubic lattice Methods
Kc
Yt
MFRG [12] MK [11] EFRG [1,2] HLRG [13,14] MFRG+MK [3] EFRG+MK (present work) Series expansion [26]
0.275 0.344 0.271 0.208 0.312 0.309 0.301
0.450 0.714 0.610 0.855 0.758 0.833 1.389
Table 2 Value of Kc for the isotropic Heisenberg model on a simple cubic lattice obtained by various e ective methods Methods
Values of Kc
MFA [28] OTSC [28] TSC [28] KO [28] EFT [17] EFTM [19] Present work Series expansion [21]
0.167 0.179 0.240 0.273 0.204 0.232 0.309 0.301
and also with the RG method more sophisticated (HLRG). The present result for the critical parameter Kc = 0:309, in the three-dimensional isotropic Heisenberg model, may be compared also with several methods, such as Weiss molecular eld (MFA), Oguchi two-spin cluster (OTSC). Oguchi three spin cluster (TSC), Kramers-Opechowski (KO) (see Ref. [28]), e ective theory (EFT) [17] and EFT modi es (EFTM) [19]. In Table 2 we have the values of Kc obtained by these e ective methods. Recently, the model above has been studied by a variational cumulant expansion (VCE) [20] that gives Kc = 0:414. We observe, that the new RG method (EFRG with decimation) also present better result than the e ective methods. 4. Conclusions I have used the EFRG with decimation to study the critical behavior of the ddimensional quantum spin- 12 anisotropic Heisenberg (QAH) model. The QAH system proved dicult to use in solving a many-body problem involving noncommuting operators. With the EFRG approach an axial approximation [1,2,17] for the Hamiltonian
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(1) in a cluster with one two spins was utilized, and the direct diagonalization can be carried out. The decimation scheme for the scale factor L1 = 2 is here utilized of form approximate [11]. This new RG method presented in this work is simple, consistent and gives good results, even for small size clusters, for critical parameter Kc and critical exponent Yt of quantum system. The values of Kc and Yt for the Ising and isotropic Heisenberg models obtained in this work, however, are in excellent agreement with the series expansion predictions. I also have applied the MFRG method with decimation [3] to study the QAH model (1), but the results for Kc and Yt are not better than the values obtained in this work by EFRG with decimation. From calculations of the 3d isotropic Heisenberg model, one can see that the EFRG combined with decimation is superior to the other e ective methods, and also are compared quantitatively with the more accurate RG methods (HLRG). The results obtained here are encouraging for further investigations in other complex systems, such as the spin-1 Heisenberg model, the transverse Ising model, etc. This extension is in progress and will be published elsewhere. Acknowledgements The author likes to thank Professor F.S. de Aguiar for a critical reading of the manuscript. This work was nancially supported by CNPq (Brazilian Agency). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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