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Transverse Ising antiferromagnetic in a longitudinal magnetic field: study of the ground state
Physics Letters A 330 (2004) 322–325 www.elsevier.com/locate/pla
Transverse Ising antiferromagnetic in a longitudinal magnetic field: study of the ground state Minos A. Neto, J. Ricardo de Sousa ∗ Instituto de Ciências Exatas, Departamento de Física, Universidade Federal do Amazonas, 69077-000 Manaus AM, Brazil Received 3 March 2004; received in revised form 14 July 2004; accepted 14 July 2004 Available online 7 August 2004 Communicated by C.R. Doering
Abstract The two-dimensional antiferromagnetic Ising model in a transverse magnetic field (Ω) and uniform longitudinal field is studied for the first time (H ). Using the effective-field theory (EFT) with correlation in one-site cluster calculation the groundstate phase diagram in the Ω–H plane is determined for the honeycomb (z = 3) and square (z = 4) lattices. It is shown that there is an order–disordered transition line (second-order phase transition) qualitatively in agreement with rigorous results of renormalization group in d = 1. The critical curve in the classical approach is also obtained and the results are compared. 2004 Published by Elsevier B.V. PACS: 64.60.Ak; 64.60.Fr; 68.35.Rh
The quantum antiferromagnetic Ising model (or Ising in transverse field) is among the simplest conceivable classes of quantum models in statistical mechanics to study quantum phase transition [1,2]. Recently, this quantum model with a longitudinal magnetic field was treated numerically [3,4] and the phase diagram at T = 0 (ground state) in the H –Ω plane (H and Ω are the longitudinal and transverse fields, respectively) was obtained in an one-dimensional lattice.
* Corresponding author.
E-mail address: [email protected] (J.R. de Sousa). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2004.07.042
This one-dimensional quantum model does not have any finite temperature phase transition. On the other hand, there is a quantum phase transition at T = 0 where the critical point Ωc (H ) decreases with the longitudinal field increasing and it is null at H = 2J (absence of transverse field). The classical approach (mean field approximation (MFA)) does not give the correct description of the phase diagram: (i) the quantum critical point occurs at Ωc (H = 0) = 2J whereas the accurate value is Ωc (H = 0) = J ; (ii) near the critical longitudinal field Hc = 2J there is a reentrant behavior (two values of critical transverse field) that is qualitatively true.
M.A. Neto, J.R. de Sousa / Physics Letters A 330 (2004) 322–325
The field-induced quantum phase transition has been analyzed intensively in one-dimensional quantum spin models, where a number of different critical aspects in the magnetization process have been observed such as a square-root behavior associated with the excitation gap [5], magnetization plateaux [6] and the cusp singularity [7]. Motived by the absence of study on the transverse Ising antiferromagnetic with a longitudinal magnetic field on higher dimension (d 2), the purpose of this Letter is to treat this quantum model on twodimensional lattices by using the effective-field theory (EFT). The differential operator technique (EFT), proposed by Honmura and Kaneyoshi [8], has been applied in the Ising metamagnet [9,10]. The results obtained by EFT were observed to be in agreement with the Monte Carlo simulation [11]. The EFT is superior to the standard mean-field approximation (MFA) because it correctly accounts for all the single-site kinematic relations between the spin-1/2 operators (i.e., (σiν )2 = 1, with ν = x, y, z). This EFT approach has been applied in many quantum systems, such as the spin-1/2 [12] and spin-1 [13] Heisenberg ferromagnetic (F), mixed spin (S = 1/2, 1) Heisenberg ferromagnetic [14], spin-1/2 Heisenberg antiferromagnetic (AF) [15] and transverse Ising [16] models. The model considered in this Letter is the nearestneighbor Ising antiferromagnet in a mixed transverse and longitudinal magnetic field divided into two equivalent interpenetrating sublattice A and B, that is described by the following Hamiltonian σiz σjz − H σiz − Ω σix , H=J (1) i,j
i
i
where J is the nearest-neighbor exchange coupling, i, j denoted the sum over all pairs of nearestneighbor spins (z) on a d-dimensional lattice (here we treat two-dimensional lattices, z = 3 and 4) and σiν is the ν(= x, z) component of the spin-1/2 Pauli operator at site i. The competition between the antiferromagnetic exchange interaction and the alignment of the local moments with the external field presents interesting properties in the phase diagram. In particular, the model (1) has an antiferromagnetic (ordered) phase (AF) in the presence of a field, with the decreasing staggered magnetization (order parameter) as the field intensity increases, where at T = 0 (ground state) and absence
323
of transverse field (Ω = 0) a second-order transition occurs at critical field Hc = zJ . To treat the model (1) on two-dimensional lattices by the EFT approach, we consider a simple example of one-spin cluster, and the Hamiltonian is given by z z z x H1A = J (2) σ(1+δ)B − H σ1A − Ωσ1A , δ
z
H1B = J
z σ(1+δ)A
z x − H σ1B − Ωσ1B ,
(3)
δ
where A and B denote the sublattice. From the Hamiltonians (2) and (3), we obtain the z , average magnetizations in sublattice A, mA = σ1A z and B, mB = σ1B , using the approximate Callen– Suzuki relation [17], that are given by H − a1A mA = 2 Ω + (H − a1A )2 × tanh β Ω 2 + (H − a1A)2 , (4) H − a1B mB = 2 Ω + (H − a1B )2 2 2 × tanh β Ω + (H − a1B ) , (5) where a1A = J
z
z σ(1+δ)B ,
a1B = J
z
δ
z σ(1+δ)A .
δ
Now, using the identity exp(αDx )F (x) = F (x + α) ∂ (where Dx = ∂x is the differential operator) and the van der Waerden identity for the two-state spin system (i.e., exp(aσiz ) = cosh(a) + σiz sinh(a)) Eqs. (4) and (5) are rewritten as z
z αx + σ(1+δ)B mA = (6) βx F (x) , δ=0 x=0 z
z mB = (7) βx F (x) , αx + σ(1+δ)A δ=0
x=0
with F (x) =
H −x Ω 2 + (H − x)2
tanh β
Ω 2 + (H − x)2 , (8)
where αx = cosh(J Dx ) and βx = sinh(J Dx ).
324
M.A. Neto, J.R. de Sousa / Physics Letters A 330 (2004) 322–325
Eqs. (6) and (7) are analytically unsolvable. The latter has been designed to treat all spin self-correlations exactly while still neglecting all high-order spin correlations on the right-hand sides of Eqs. (6) and (7). Following the same procedure for the thermal average as we did in our previous work [15], we obtain at T = 0 (ground state) mA = mB =
z p=0 z
p
(9)
p
(10)
Ap (H, Ω)mB , Ap (H, Ω)mA ,
p=0
with Ap (H, Ω) =
z! z−p p αx βx Fo (x) , p!(z − p)! x=0
each intensity of the transverse magnetic field Ω. This way, we determine the phase diagram of this quantum model in Ω–H plane for the honeycomb (z = 3) and square (z = 4) lattices, that comprises a fieldinduced antiferromagnetic (AF) phase (ms = 0) at low fields and a paramagnetic (P) phase (ms = 0) at high fields. In the case H = 0 model (1), we have mA = −mB (mo = 0) and the critical behavior reduces to the transverse Ising ferromagnetic analyzed in Ref. [17]. For z = 3 and z = 4, the values of the quantum critical point obtained by EFT are Ωc /J = 1.829 (ΩcMFA /J = 3.00) and Ωc /J = 2.751 (ΩcMFA /J = 4.00), respectively. On the other hand, at the critical point Ω = Ωc
(11)
where the coefficients {Ap (H, Ω)} are obtained by using the relation exp(αDx )Fo (x) = Fo (x + α) and Fo (x) is the function (8) at T = 0. In terms of the uniform m = 12 (mA + mB ) and staggered ms = 12 (mA − mB ) magnetizations, and near the critical point we have ms → 0 and m → mo , the sublattice magnetization mA expanded up to linear order in ms (order parameter) is given by mA X0 (H, Ω, mo ) + X1 (H, Ω, mo )ms ,
(12)
(a)
with X0 (H, Ω, mo ) =
z
p
Ap (H, Ω)mo ,
(13)
p=0
X1 (H, Ω, mo ) = −
z
p−1
pAp (H, Ω)mo
p=0
∂X0 (H, Ω, mo ) ≡− , ∂mo
(14)
using the fact that mA = mo +ms in Eq. (12) we obtain mo = X0 (H, Ω, mo ),
(15)
X1 (H, Ω, mo ) = 1.
(16)
The numerical determination of the phase boundary (second-order phase transition) is obtained solving simultaneously the two expressions above, Eqs. (15) and (16), where we find the values of Hc and mo for
(b) Fig. 1. The ground-state phase diagram of model (1) for a honeycomb (z = 3) (a) and square (z = 4) (b) lattices. The critical line between the AF and P phases obtained by EFT is shown in thin solid line and in the classical approximation (MFA) by dotted line.
M.A. Neto, J.R. de Sousa / Physics Letters A 330 (2004) 322–325
the model becomes gapless and the staggered magnetization vanishes with the classical exponent β = 1/2. In two or more dimensions, this model (1) with H = 0 (transverse Ising model (TIM)) has been studied by EFT [16,17], high-temperature expansions [18], Monte Carlo simulations [19] and renormalization group approach [20]. The quantitative results for the critical field Ωc (and also the critical temperature Tc (Ω = 0)) obtained by EFT are not adequate when compared with rigorous methods (see for example, the series expansion results [18]). With the classical limit (Ω = 0) we obtain the exact value for the critical field Hc = zJ [9,10]. In Fig. 1, the phase diagram in Ω–H plane is shown for the honeycomb (z = 3) and square (z = 4) lattices. We compare our results with those obtained by mean field approximation (MFA). The classical approach (MFA) does not give the correct description of the phase diagram, with the presence of reentrant behavior near H = Hc and the quantum critical point Ωc /zJ = 1.00 is independent of the coordination number (z). This inconsistency in the universality of the critical point and presence of reentrance (true) in the MFA have been analyzed recently by density-matrix renormalization group (DMRG) [4] on an one-dimensional lattice (z = 2), where the critical point Ωc = J is different from the classical value Ωc = 2J . Our results (EFT) for two-dimensional lattice (d = 2) are correct qualitatively in comparison with the phase diagram obtained by DMRG in d = 1. In summary, we present for the first time the critical behavior at T = 0 for the two-dimensional transverse Ising antiferromagnetic in the longitudinal magnetic field. It was shown that the quantum phase transition remains in the TIM in the presence of the uniform longitudinal field. We use the effective-field theory with correlation in an one-site cluster and obtain the state equations. We developed our study for the particular case of honeycomb (z = 3) and square (z = 4) lattices. The qualitative results for the phase diagram in Ω–H plane are in good accordance with those obtained using the DMRG [4] in d = 1.
Acknowledgements J.R.S. would like to thank Klauko P. Mota of the Universidade Federal Fluminense for a critical reading
325
of the manuscript. This work was supported by CNPq, FAPEAM and CAPES (Brazilian agencies).
References [1] B.K. Chakrabarti, A. Dutta, P. Sen, Quantum Ising Models and Phase Transitions in Transverse Ising Systems, Lecture Notes in Physics, vol. M41, Springer-Verlag, Berlin, 1996. [2] S. Sachdev, Quantum Phase Transitions, Cambridge Univ. Press, Cambridge, 1999. [3] P. Sen, Phys. Rev. E 63 (2001) 16112. [4] A.A. Ovchinnikov, D.M. Dmitriev, V.Ya. Krivnov, V.O. Cheranovskii, Phys. Rev. B 68 (2003) 214406. [5] T. Sakai, M. Takahashi, Phys. Rev. B 57 (1998) R8091; K. Okunishi, Y. Hieida, Y. Akutsu, Phys. Rev. B 59 (1999) 6806. [6] M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett. 78 (1997) 1984; K. Totsuka, Phys. Lett. A 228 (1997) 103. [7] K. Okunishi, Y. Hieida, Y. Akutsu, Phys. Rev. B 60 (1999) R6953. [8] R. Honmura, T. Kaneyoshi, J. Phys. C 12 (1979) 3979. [9] M. Zukovic, A. Bobak, J. Magn. Magn. Mater. 170 (1997) 49. [10] M. Zukovic, A. Bobak, T. Idogaki, J. Magn. Magn. Mater. 188 (1998) 52; M. Zukovic, A. Bobak, T. Idogaki, J. Magn. Magn. Mater. 192 (1999) 363. [11] L. Hernández, H.T. Diep, D. Beetrand, Phys. Rev. B 47 (1993) 2602; L. Hernández, H.T. Diep, D. Beetrand, Phys. Rev. B 48 (1993) 15772. [12] T. Idogaki, N. Uryu, Physica A 181 (1992) 173; T. Idogaki, Y. Miyoshi, J.W. Tuchker, J. Magn. Magn. Mater. 154 (1996) 221. [13] T. Idogaki, A. Tanaka, J.W. Tucker, J. Magn. Magn. Mater. 177–181 (1998) 773; Y. Miyoshi, A. Tanaka, J.W. Tucker, T. Idogaki, J. Magn. Magn. Mater. 295 (1999) 110. [14] C. Xu, Q. Jiang, H.X. Cao, Z.Ya. Li, Solid State Commun. 108 (1998) 110. [15] I.G. Araújo, J. Cabral Neto, J.R. de Sousa, Physica A 260 (1998) 150; E.B. Filho, J.R. de Sousa, Phys. Lett. A (2004), in press. [16] N. Benayada, A. Fathi, L. Khaya, J. Magn. Magn. Mater. (2004), in press, and references therein. [17] F.C. Sá Barreto, I.P. Fittipaldi, B. Zeks, Ferroelectrics 39 (1981) 1103. [18] R.J. Elliot, C. Wood, J. Phys. C 4 (1971) 2359. [19] O. Nagai, Y. Yamada, K. Nishino, Y. Miyatake, Phys. Rev. B 35 (1987) 3425. [20] S.M. Oliveira, M.A. Continentino, P.M. Oliveira, Physica A 162 (1990) 458; W.J. Song, C.Z. Yang, Phys. Status Solidi B 186 (1994) 511; W.J. Song, C.Z. Yang, Solid State Commun. 92 (1994) 361; T. Yokota, Phys. Lett. A 171 (1992) 134.
Transverse Ising antiferromagnetic in a longitudinal magnetic field: study of the ground state Minos A. Neto, J. Ricardo de Sousa ∗ Instituto de Ciências Exatas, Departamento de Física, Universidade Federal do Amazonas, 69077-000 Manaus AM, Brazil Received 3 March 2004; received in revised form 14 July 2004; accepted 14 July 2004 Available online 7 August 2004 Communicated by C.R. Doering
Abstract The two-dimensional antiferromagnetic Ising model in a transverse magnetic field (Ω) and uniform longitudinal field is studied for the first time (H ). Using the effective-field theory (EFT) with correlation in one-site cluster calculation the groundstate phase diagram in the Ω–H plane is determined for the honeycomb (z = 3) and square (z = 4) lattices. It is shown that there is an order–disordered transition line (second-order phase transition) qualitatively in agreement with rigorous results of renormalization group in d = 1. The critical curve in the classical approach is also obtained and the results are compared. 2004 Published by Elsevier B.V. PACS: 64.60.Ak; 64.60.Fr; 68.35.Rh
The quantum antiferromagnetic Ising model (or Ising in transverse field) is among the simplest conceivable classes of quantum models in statistical mechanics to study quantum phase transition [1,2]. Recently, this quantum model with a longitudinal magnetic field was treated numerically [3,4] and the phase diagram at T = 0 (ground state) in the H –Ω plane (H and Ω are the longitudinal and transverse fields, respectively) was obtained in an one-dimensional lattice.
* Corresponding author.
E-mail address: [email protected] (J.R. de Sousa). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2004.07.042
This one-dimensional quantum model does not have any finite temperature phase transition. On the other hand, there is a quantum phase transition at T = 0 where the critical point Ωc (H ) decreases with the longitudinal field increasing and it is null at H = 2J (absence of transverse field). The classical approach (mean field approximation (MFA)) does not give the correct description of the phase diagram: (i) the quantum critical point occurs at Ωc (H = 0) = 2J whereas the accurate value is Ωc (H = 0) = J ; (ii) near the critical longitudinal field Hc = 2J there is a reentrant behavior (two values of critical transverse field) that is qualitatively true.
M.A. Neto, J.R. de Sousa / Physics Letters A 330 (2004) 322–325
The field-induced quantum phase transition has been analyzed intensively in one-dimensional quantum spin models, where a number of different critical aspects in the magnetization process have been observed such as a square-root behavior associated with the excitation gap [5], magnetization plateaux [6] and the cusp singularity [7]. Motived by the absence of study on the transverse Ising antiferromagnetic with a longitudinal magnetic field on higher dimension (d 2), the purpose of this Letter is to treat this quantum model on twodimensional lattices by using the effective-field theory (EFT). The differential operator technique (EFT), proposed by Honmura and Kaneyoshi [8], has been applied in the Ising metamagnet [9,10]. The results obtained by EFT were observed to be in agreement with the Monte Carlo simulation [11]. The EFT is superior to the standard mean-field approximation (MFA) because it correctly accounts for all the single-site kinematic relations between the spin-1/2 operators (i.e., (σiν )2 = 1, with ν = x, y, z). This EFT approach has been applied in many quantum systems, such as the spin-1/2 [12] and spin-1 [13] Heisenberg ferromagnetic (F), mixed spin (S = 1/2, 1) Heisenberg ferromagnetic [14], spin-1/2 Heisenberg antiferromagnetic (AF) [15] and transverse Ising [16] models. The model considered in this Letter is the nearestneighbor Ising antiferromagnet in a mixed transverse and longitudinal magnetic field divided into two equivalent interpenetrating sublattice A and B, that is described by the following Hamiltonian σiz σjz − H σiz − Ω σix , H=J (1) i,j
i
i
where J is the nearest-neighbor exchange coupling, i, j denoted the sum over all pairs of nearestneighbor spins (z) on a d-dimensional lattice (here we treat two-dimensional lattices, z = 3 and 4) and σiν is the ν(= x, z) component of the spin-1/2 Pauli operator at site i. The competition between the antiferromagnetic exchange interaction and the alignment of the local moments with the external field presents interesting properties in the phase diagram. In particular, the model (1) has an antiferromagnetic (ordered) phase (AF) in the presence of a field, with the decreasing staggered magnetization (order parameter) as the field intensity increases, where at T = 0 (ground state) and absence
323
of transverse field (Ω = 0) a second-order transition occurs at critical field Hc = zJ . To treat the model (1) on two-dimensional lattices by the EFT approach, we consider a simple example of one-spin cluster, and the Hamiltonian is given by z z z x H1A = J (2) σ(1+δ)B − H σ1A − Ωσ1A , δ
z
H1B = J
z σ(1+δ)A
z x − H σ1B − Ωσ1B ,
(3)
δ
where A and B denote the sublattice. From the Hamiltonians (2) and (3), we obtain the z , average magnetizations in sublattice A, mA = σ1A z and B, mB = σ1B , using the approximate Callen– Suzuki relation [17], that are given by H − a1A mA = 2 Ω + (H − a1A )2 × tanh β Ω 2 + (H − a1A)2 , (4) H − a1B mB = 2 Ω + (H − a1B )2 2 2 × tanh β Ω + (H − a1B ) , (5) where a1A = J
z
z σ(1+δ)B ,
a1B = J
z
δ
z σ(1+δ)A .
δ
Now, using the identity exp(αDx )F (x) = F (x + α) ∂ (where Dx = ∂x is the differential operator) and the van der Waerden identity for the two-state spin system (i.e., exp(aσiz ) = cosh(a) + σiz sinh(a)) Eqs. (4) and (5) are rewritten as z
z αx + σ(1+δ)B mA = (6) βx F (x) , δ=0 x=0 z
z mB = (7) βx F (x) , αx + σ(1+δ)A δ=0
x=0
with F (x) =
H −x Ω 2 + (H − x)2
tanh β
Ω 2 + (H − x)2 , (8)
where αx = cosh(J Dx ) and βx = sinh(J Dx ).
324
M.A. Neto, J.R. de Sousa / Physics Letters A 330 (2004) 322–325
Eqs. (6) and (7) are analytically unsolvable. The latter has been designed to treat all spin self-correlations exactly while still neglecting all high-order spin correlations on the right-hand sides of Eqs. (6) and (7). Following the same procedure for the thermal average as we did in our previous work [15], we obtain at T = 0 (ground state) mA = mB =
z p=0 z
p
(9)
p
(10)
Ap (H, Ω)mB , Ap (H, Ω)mA ,
p=0
with Ap (H, Ω) =
z! z−p p αx βx Fo (x) , p!(z − p)! x=0
each intensity of the transverse magnetic field Ω. This way, we determine the phase diagram of this quantum model in Ω–H plane for the honeycomb (z = 3) and square (z = 4) lattices, that comprises a fieldinduced antiferromagnetic (AF) phase (ms = 0) at low fields and a paramagnetic (P) phase (ms = 0) at high fields. In the case H = 0 model (1), we have mA = −mB (mo = 0) and the critical behavior reduces to the transverse Ising ferromagnetic analyzed in Ref. [17]. For z = 3 and z = 4, the values of the quantum critical point obtained by EFT are Ωc /J = 1.829 (ΩcMFA /J = 3.00) and Ωc /J = 2.751 (ΩcMFA /J = 4.00), respectively. On the other hand, at the critical point Ω = Ωc
(11)
where the coefficients {Ap (H, Ω)} are obtained by using the relation exp(αDx )Fo (x) = Fo (x + α) and Fo (x) is the function (8) at T = 0. In terms of the uniform m = 12 (mA + mB ) and staggered ms = 12 (mA − mB ) magnetizations, and near the critical point we have ms → 0 and m → mo , the sublattice magnetization mA expanded up to linear order in ms (order parameter) is given by mA X0 (H, Ω, mo ) + X1 (H, Ω, mo )ms ,
(12)
(a)
with X0 (H, Ω, mo ) =
z
p
Ap (H, Ω)mo ,
(13)
p=0
X1 (H, Ω, mo ) = −
z
p−1
pAp (H, Ω)mo
p=0
∂X0 (H, Ω, mo ) ≡− , ∂mo
(14)
using the fact that mA = mo +ms in Eq. (12) we obtain mo = X0 (H, Ω, mo ),
(15)
X1 (H, Ω, mo ) = 1.
(16)
The numerical determination of the phase boundary (second-order phase transition) is obtained solving simultaneously the two expressions above, Eqs. (15) and (16), where we find the values of Hc and mo for
(b) Fig. 1. The ground-state phase diagram of model (1) for a honeycomb (z = 3) (a) and square (z = 4) (b) lattices. The critical line between the AF and P phases obtained by EFT is shown in thin solid line and in the classical approximation (MFA) by dotted line.
M.A. Neto, J.R. de Sousa / Physics Letters A 330 (2004) 322–325
the model becomes gapless and the staggered magnetization vanishes with the classical exponent β = 1/2. In two or more dimensions, this model (1) with H = 0 (transverse Ising model (TIM)) has been studied by EFT [16,17], high-temperature expansions [18], Monte Carlo simulations [19] and renormalization group approach [20]. The quantitative results for the critical field Ωc (and also the critical temperature Tc (Ω = 0)) obtained by EFT are not adequate when compared with rigorous methods (see for example, the series expansion results [18]). With the classical limit (Ω = 0) we obtain the exact value for the critical field Hc = zJ [9,10]. In Fig. 1, the phase diagram in Ω–H plane is shown for the honeycomb (z = 3) and square (z = 4) lattices. We compare our results with those obtained by mean field approximation (MFA). The classical approach (MFA) does not give the correct description of the phase diagram, with the presence of reentrant behavior near H = Hc and the quantum critical point Ωc /zJ = 1.00 is independent of the coordination number (z). This inconsistency in the universality of the critical point and presence of reentrance (true) in the MFA have been analyzed recently by density-matrix renormalization group (DMRG) [4] on an one-dimensional lattice (z = 2), where the critical point Ωc = J is different from the classical value Ωc = 2J . Our results (EFT) for two-dimensional lattice (d = 2) are correct qualitatively in comparison with the phase diagram obtained by DMRG in d = 1. In summary, we present for the first time the critical behavior at T = 0 for the two-dimensional transverse Ising antiferromagnetic in the longitudinal magnetic field. It was shown that the quantum phase transition remains in the TIM in the presence of the uniform longitudinal field. We use the effective-field theory with correlation in an one-site cluster and obtain the state equations. We developed our study for the particular case of honeycomb (z = 3) and square (z = 4) lattices. The qualitative results for the phase diagram in Ω–H plane are in good accordance with those obtained using the DMRG [4] in d = 1.
Acknowledgements J.R.S. would like to thank Klauko P. Mota of the Universidade Federal Fluminense for a critical reading
325
of the manuscript. This work was supported by CNPq, FAPEAM and CAPES (Brazilian agencies).
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