- Email: [email protected]
The ferromagnetic planar model in the presence of uniaxial single-site anisotropy
3 February 1997 PHYSICS LETTERS A
Physics Letters A 225 ( 1997) 3 1l-3 14
ELSEVIER
The ferromagnetic planar model in the presence of uniaxial single-site anisotropy Y.Q.Ma CCAST (World Laboratory). PO. Box 8730, Beijing 100080, China Department of Physics, Nanjing University, Nanjing 210093, China ’
Received 22 October 1996; revised manuscript received 20 November 1996; accepted for publication 21 November 1996 Communicated by V.M. Agranovich
Abstract Considering spin operators in terms of boson operators, we employ the coherent state path integral approach to study the Heisenberg planar XY model in the presence of uniaxial anisotropy of the form -D( S/)’ with the x direction parallel to the easy axis. The critical temperatures TCbelow which a ferromagnetic phase appears, are obtained as a function of the anisotropy D, and the effects of the anisotropy on the phase transition of the model are examined. We find that in the limits D = 0 and 00, our results give finite values of the critical temperatures 6, contrary to those obtained in the molecular field approximation. We show the consistency between our results and those obtained by Lovesey and Balcar. PACS: 64.60.Cn; 75.10.-b; 75.30.G~ Keywords: Phase transition; Heisenberg planar model; Anisotropy
For many years much theoretical and experimental work has been performed to study the anisotropic Heisenberg model [ l-51. Especially, extensive studies have been made of the Heisenberg or Ising model in the presence of uniaxial single-site anisotropy. This model was introduced to describe a number of magnetic materials, e.g. the rutile antiferromagnets MnF2 and FeF2 [6,7]. Most studies of this model have focused on analyzing the effects of an additional anisotropic field on the phase transition and the low-temperature behavior of the pure Heisenberg or Ising systems considered. While research on this anisotropic model has established the important role of anisotropy regarding their magnetic ordering properties, there is still a considerable controversy ’ Mailing address.
on many basic issues. Devlin [ 81, using the Green function formalism, showed that there is a finite critical temperature as the anisotropy becomes infinite, contrary to previous predictions in the context of the decoupling Green function schemes. On the other hand, Lovesey and Balcar [9] recently investigated the anisotropy effects on the isotropic Heisenberg and uniaxial anisotropic Ising models by use of the coupled mode theory. Unlike the molecular field approximation result in which planar anisotropy reduces the critical temperature to below the value for the isotropic Heisenberg model, they proved that for a finite value of the anisotropy parameter, the critical temperature of the model with a planar anisotropy is intermediate between the values for the isotropic and uniaxial anisotropy models. For extremely large values of anisotropy the critical temperatures remain
0375~9601/97/$17.00 Copyright 0 1997 Published by Elsevier Science B.V. All rights resewed. PII SO375-96Ol(96)00888-2
312
Y.Q. Ma/Physics
Letters A 225 (1997) 311-314
finite, whereas in the molecular field approximation this limit has no real physical meaning because in this approximation the critical temperatures for the planar and uniaxial anisotropy cases are unbounded [ 10,111. It is therefore of great interest to examine these claims with other methods. In the present paper, we use the coherent state path integral method to investigate the Heisenberg planar XY model with arbitrary spin S and the inclusion of a planar uniaxial anisotropy term -D( Sf )* with the x direction parallel to the easy axis. We directly consider the planar anisotropy effects on the Heisenberg planar XY system, and obtain a theoretical construct which demonstrates the interplay between the isotropic XY model and the uniaxial anisotropic Ising model. As D ranges from 0 to cc, we can go from the isotropic XY to the Ising model with the single-ion uniaxial anisotropy in which all spins are confined to the x direction. The XY model is equivalent to a hard-core boson model, and thinking of the spin problem in terms of bosons and vice versa has proven to be a fruitful way to understand the physics of these problems [ 121. We will start by introducing boson operators to map the spin system into a hard-core boson system plus a boson constraint, and then use the coherent state path integral to solve this equivalent boson problem exactly. We calculate the transition temperature Tc as function of the anisotropy parameter D. Our results agree with those obtained by Lovesey and Balcar [ 93. The Heisenberg planar XY model with uniaxial anisotropy D can be described by the following Hamiltonian [ 131,
In the boson language, the spin operators in each lattice site are replaced by the boson creation and annihilation operators at and a, and Eq. ( 1) with (2) can be rewritten as [ 141 ‘If = -2SC
Jijataj
- &SD C(LZ.~CZ!+ aiai)
U -
i
i\/zsC
Hi(a/
+ ai)
i
+2S(~~-fD)Catn~-ps*N_fsDN, i
(3)
where N is the total number of sites, and a Lagrange multiplier p enforces the mean constraint on the amplitude of the spins N-’ CE, [ (Sf )* + (l$‘)*] = S( S+ I), which now becomes a boson constraint. Actually, the Hamiltonian (3) is an extended hard-core boson system with boson hopping Jij and chemical potential ,u. In the absence of the anisotropy D = 0, the model with S = i reduces to a hard-core Bose system which has been extensively studied in the context of a superfluid-Mott insulator transition [ 151. The Fourier transformation of Eq. (3) is given by - ;D - J(k)]aiq
1-I = 2Sc[p k -
$0
x(U:a:,
+ aka_k)
k -
;V%x(Hkai
+ H_kak)
- ,!.d*N
-
;SDN,
k
(4) where Uk, Hk are the Fourier transforms of the operators and the external field, respectively. The interactions J(k) are given by
ij
-DE(Sf)*-cH,Sf, i
(1)
i
J(k)
and the constraint Si’Si=S(S+
1) 9
(2)
where Si = (Sf, Sy) is the XY-spin operator at site i. The exchange Jij is the interaction between sites i and j, and Hi is an external magnetic field. The cases D = 0 and cc correspond to the isotropic ferromagnetic XY model and the ferromagnetic Ising model in which the anisotropy suppresses fluctuations in the y components of the spins, respectively.
=25&cosk,. LGl
(5)
where J is the strength of the nearest-neighbor interaction, and d is the spatial dimension of the lattice. The Hamiltonian (4) is quadratic in the boson creation and annihilation operators CZ~and ak, but contains terms involving the combination aka_k and its Hermitian adjoint, in addition to the diagonal terms ‘&k. In order to obtain the partition function Z = Trexp( -PH), we use the coherent state functional
Y.Q. Ma /Physics
Letters A 225 (1997) 31 l-314
integral representation [ 161 in the Matsubara “imaginary time” formulation. The advantage is that in the coherent state path-integral, the boson operator will become a c-number, and the trace operation in the partition function can be performed explicitly. After the Hamiltonian ‘H is diagonalized by a Bogoliubov transformation, the resulting free energy per site, is obtained after integration over f=-(l/@V)lnZ, the bosonic variables f = &
xln2sinhPSak
H2 4(/_~ - 2Jd - D)
-
k
-pS(S+
(6)
1) -&SD,
313
ordered ferromagnetic one (M # 0) when the energy gap A vanishes. The condition A = 0 yields ,u=2dJfD
(D>O),
= 2dJ
(D < 0).
(11)
We will consider the case D 3 0 without loss of generality. The paramagnetic-ferromagnetic phase boundary is determined by solving Eq. (9) for the mean constraint in the limit ,u = 2Jd + D. Note that the critical region is dominated by the long wavelength limit Ikl + 0 [ 161. Expanding the coth term in Eq. (9), we obtain the critical temperature T, below which the ferromagnetic phase appears,
where T,=S(S+l) @k = {[,u - ;D - J(k)12
- iD2}“2
(7)
is the dispersion relation. p is the inverse of the temperature T. The Lagrange multiplier ,u is determined by minimizing the free energy with respect to it,
af /+ = 0,
s x
c
k {[p
p-
_ &D _ J(k)]2 H2
+ 4(,u - 2dJ - D)2
_ $D2}‘/2
-S(S+
X
c
,u - ;D - J(k) _ ;D _ J(k)]2 _ .!D2}‘/2
cothPSwk
’
In the absence of anisotropy, Eq. (9) shows a zerotemperature magnetization M(T = 0) = S, which is the expected result. The excitation energy spectrum ok = { [ ,u - i D - ;D2}‘i2
exhibits an energy gap
A=[(,u-;D-~~J)~-~D~]“~.
+ 1)
= I(0)
+ I(D/2dJ)
J(k)12-
$D2 >
(12)
’ extended
Watson
inte-
l
x+1-y(k)’
(13)
(8)
(9)
J(k)12
4dJS(S
Z(x)?$k
1) =o.
1) -;
k {[p
[;D+2dJ-
cothPSwk
We introduce the magnetization M defined by M = -af/G’H. The equation of state can then be written as M2=S(Sf
ck
where I(x) is the standard gral [ 171 defined by
+D - J(k)
-1
;D + 2dJ - J(k)
(10)
The system indicates a second-order phase transition from a disordered paramagnetic phase (M = 0) to an
with y(k) = d-’ Cz=, cos k,. The integral I (0) converges for the dimension d > 2 which is therefore the lower critical dimension, in accord with the MerminWagner theorem [ 181. In the absence of anisotropy, Eq. (12) gives the critical temperature T, for the isotropic XY model, T,(D=O)
=2dJS(S+l)1(0)-‘.
(14)
This result is in agreement with that of the coupled mode theory [ 91 except for an overall factor of 5. This numerical discrepancy is attributed by the fact that in Lovesey and Balcar’s model all three spin degrees of freedom fully participate in its static and dynamic properties, and they demand (Sr Sy ) = $S( S + 1) , a = x, y, or z, whereas in the present case the only contribution to the spin constraint comes from two spin degrees of freedom in the X-Y plane, i.e., (( Sf)2 + ( Sy)2) = S( Sf 1) [ 131. It is obvious that fluctuations of the out of plane spin z-component have the effect of lowering the critical temperature T, below which the
314
Y.Q. Ma/Physics
in-plane ferromagnetic hand, for large values Ising system with the along the x direction, Tc becomes 7. _
4dJS(SS
’ - I(0)
1)
+ 4dJ/D
’
Letters A 225 (1997) 31 I-314
ordering appears. On the other of D the model approaches an single-ion uniaxial anisotropy and the transition temperature
(15)
When D > 1 the transition temperature T, approaches 4dJS( S + 1)/Z(O), which is the same as that of the easy-z uniaxial anisotropic Ising limit [ 9 1. The fact that the critical temperatures T, (D = 0) and T, (D -t co) remain finite, is in agreement with the conclusion of Lovesey and Balcar [ 91. Since Z(x) is a monotonically decreasing function of its argument, for general values of D, from Eq. (12) we find that the critical temperature T, smoothly increases with increasing the strength of anisotropy D. The physical reason is that an uniaxial single-site anisotropy with the x direction parallel to the easy axis breaks the rotational O(2) symmetry of the isotropic XY ferromagnet in the XY spin plane, and has the tendency to transform the isotropic XY system into an Ising system in which all spins are confined to the x direction; a finite singlesite anisotropy effectively enhances the exchange interaction in the x direction, which leads to an effective in-plane exchange anisotropy. In the D -+ 0~)limit all spins are parallel or antiparallel to the x direction and the model becomes an Ising one. We conclude that the critical temperature of the model with plane anisotropy is intermediate between the values for the isotropic XY and uniaxial anisotropic Ising models, similar to the result obtained by Lovesey and Balcar [93. The Heisenberg planar XY model including the effects of anisotropy has been presented in the boson space and solved exactly by using the coherent state path integral method. We have discussed aspects of the phase transition of the model with arbitrary spin S and uniaxial anisotropy. It is shown that the critical temperature Tc increases with an increase of the anisotropy parameter D, and remains finite even for the anisotropy parameter D -+ 00. The result is
consistent with that of Lovesey and Balcar within the coupled mode theory. Our paper provides a simple method, allowing a straightforward extension to more complex spin systems and related boson ones. This work was supported by the Doctoral Foundation of Chinese Education Commission, the Natural Nature Science Foundation of China, and the Fifth Huo Ying-dong Teacher’s Foundation.
References 111 R.M. White, Quantum theory of magnetism (Springer, Berlin, 1983).
121 K.H. Fischer and J.A. Hertz, Spin glasses (Cambridge Univ. Press, Cambridge, 1991) [31 D.H. Ryan, ed., Recent progress in random magnets (World Scientific, Singapore, 1992). [41 G.E. Granroth et al., Phys. Rev. Lett. 77 (1996) 1616. [51 D.R. Taylor and W.J.L. Buyers, Phys. Rev. B 54 (1996) R3734. [61 R.A. Cowley, Methods of experimental physics, Vol. 23, Part C (Academic Press, Orlando, FL, 1987). I71 M.F. Collins, Magnetic critical scattering (Oxford Univ. Press, New York, 1989). [81 J.F. Devlin, Phys. Rev. B 4 (1971) 136. 191 S.W. Lovesey and E. Balcar, J. Phys. Condens. Matter 7 (1995) 2147. [lOI SW. Lovesey, Condensed matter physics: dynamic correlations, 2nd Ed. (Benjamin, Menlo Park, CA, 1986). IllI S.W. Lovesey, Theory of neutron scattering by condensed matter, 3rd Ed. (Oxford Univ. Press, Oxford, 1987). [I21 MIA. Fisher, PB. Weichman, G. Grinstein and D.S.Fisher, Phys. Rev. B 40 ( 1989) 546. (13 See, for example, K. Binder and D.W. Heermann, Monte Carlo simulation in statistical physics (Springer, Berlin. 1988). 114,I D.D. Bet&, in: Phase transitions and critical phenomena, Vol. 3, eds. C. Domb and MS. Green (Academic Press, New York, 1974). I15 E Pazmandi. G. Zimanyi and R. Scalettar, Cond-Mat. preprint no. 9602158, and references therein. [IhI J.W. Negele and H. Orland, Quantum many-particle systems ( Addision-Wesley, New York, 1988), [I71 G.N. Watson, Quart. J. Math (Oxford) 10 ( 1939) 266. IIS1 N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133.
Physics Letters A 225 ( 1997) 3 1l-3 14
ELSEVIER
The ferromagnetic planar model in the presence of uniaxial single-site anisotropy Y.Q.Ma CCAST (World Laboratory). PO. Box 8730, Beijing 100080, China Department of Physics, Nanjing University, Nanjing 210093, China ’
Received 22 October 1996; revised manuscript received 20 November 1996; accepted for publication 21 November 1996 Communicated by V.M. Agranovich
Abstract Considering spin operators in terms of boson operators, we employ the coherent state path integral approach to study the Heisenberg planar XY model in the presence of uniaxial anisotropy of the form -D( S/)’ with the x direction parallel to the easy axis. The critical temperatures TCbelow which a ferromagnetic phase appears, are obtained as a function of the anisotropy D, and the effects of the anisotropy on the phase transition of the model are examined. We find that in the limits D = 0 and 00, our results give finite values of the critical temperatures 6, contrary to those obtained in the molecular field approximation. We show the consistency between our results and those obtained by Lovesey and Balcar. PACS: 64.60.Cn; 75.10.-b; 75.30.G~ Keywords: Phase transition; Heisenberg planar model; Anisotropy
For many years much theoretical and experimental work has been performed to study the anisotropic Heisenberg model [ l-51. Especially, extensive studies have been made of the Heisenberg or Ising model in the presence of uniaxial single-site anisotropy. This model was introduced to describe a number of magnetic materials, e.g. the rutile antiferromagnets MnF2 and FeF2 [6,7]. Most studies of this model have focused on analyzing the effects of an additional anisotropic field on the phase transition and the low-temperature behavior of the pure Heisenberg or Ising systems considered. While research on this anisotropic model has established the important role of anisotropy regarding their magnetic ordering properties, there is still a considerable controversy ’ Mailing address.
on many basic issues. Devlin [ 81, using the Green function formalism, showed that there is a finite critical temperature as the anisotropy becomes infinite, contrary to previous predictions in the context of the decoupling Green function schemes. On the other hand, Lovesey and Balcar [9] recently investigated the anisotropy effects on the isotropic Heisenberg and uniaxial anisotropic Ising models by use of the coupled mode theory. Unlike the molecular field approximation result in which planar anisotropy reduces the critical temperature to below the value for the isotropic Heisenberg model, they proved that for a finite value of the anisotropy parameter, the critical temperature of the model with a planar anisotropy is intermediate between the values for the isotropic and uniaxial anisotropy models. For extremely large values of anisotropy the critical temperatures remain
0375~9601/97/$17.00 Copyright 0 1997 Published by Elsevier Science B.V. All rights resewed. PII SO375-96Ol(96)00888-2
312
Y.Q. Ma/Physics
Letters A 225 (1997) 311-314
finite, whereas in the molecular field approximation this limit has no real physical meaning because in this approximation the critical temperatures for the planar and uniaxial anisotropy cases are unbounded [ 10,111. It is therefore of great interest to examine these claims with other methods. In the present paper, we use the coherent state path integral method to investigate the Heisenberg planar XY model with arbitrary spin S and the inclusion of a planar uniaxial anisotropy term -D( Sf )* with the x direction parallel to the easy axis. We directly consider the planar anisotropy effects on the Heisenberg planar XY system, and obtain a theoretical construct which demonstrates the interplay between the isotropic XY model and the uniaxial anisotropic Ising model. As D ranges from 0 to cc, we can go from the isotropic XY to the Ising model with the single-ion uniaxial anisotropy in which all spins are confined to the x direction. The XY model is equivalent to a hard-core boson model, and thinking of the spin problem in terms of bosons and vice versa has proven to be a fruitful way to understand the physics of these problems [ 121. We will start by introducing boson operators to map the spin system into a hard-core boson system plus a boson constraint, and then use the coherent state path integral to solve this equivalent boson problem exactly. We calculate the transition temperature Tc as function of the anisotropy parameter D. Our results agree with those obtained by Lovesey and Balcar [ 93. The Heisenberg planar XY model with uniaxial anisotropy D can be described by the following Hamiltonian [ 131,
In the boson language, the spin operators in each lattice site are replaced by the boson creation and annihilation operators at and a, and Eq. ( 1) with (2) can be rewritten as [ 141 ‘If = -2SC
Jijataj
- &SD C(LZ.~CZ!+ aiai)
U -
i
i\/zsC
Hi(a/
+ ai)
i
+2S(~~-fD)Catn~-ps*N_fsDN, i
(3)
where N is the total number of sites, and a Lagrange multiplier p enforces the mean constraint on the amplitude of the spins N-’ CE, [ (Sf )* + (l$‘)*] = S( S+ I), which now becomes a boson constraint. Actually, the Hamiltonian (3) is an extended hard-core boson system with boson hopping Jij and chemical potential ,u. In the absence of the anisotropy D = 0, the model with S = i reduces to a hard-core Bose system which has been extensively studied in the context of a superfluid-Mott insulator transition [ 151. The Fourier transformation of Eq. (3) is given by - ;D - J(k)]aiq
1-I = 2Sc[p k -
$0
x(U:a:,
+ aka_k)
k -
;V%x(Hkai
+ H_kak)
- ,!.d*N
-
;SDN,
k
(4) where Uk, Hk are the Fourier transforms of the operators and the external field, respectively. The interactions J(k) are given by
ij
-DE(Sf)*-cH,Sf, i
(1)
i
J(k)
and the constraint Si’Si=S(S+
1) 9
(2)
where Si = (Sf, Sy) is the XY-spin operator at site i. The exchange Jij is the interaction between sites i and j, and Hi is an external magnetic field. The cases D = 0 and cc correspond to the isotropic ferromagnetic XY model and the ferromagnetic Ising model in which the anisotropy suppresses fluctuations in the y components of the spins, respectively.
=25&cosk,. LGl
(5)
where J is the strength of the nearest-neighbor interaction, and d is the spatial dimension of the lattice. The Hamiltonian (4) is quadratic in the boson creation and annihilation operators CZ~and ak, but contains terms involving the combination aka_k and its Hermitian adjoint, in addition to the diagonal terms ‘&k. In order to obtain the partition function Z = Trexp( -PH), we use the coherent state functional
Y.Q. Ma /Physics
Letters A 225 (1997) 31 l-314
integral representation [ 161 in the Matsubara “imaginary time” formulation. The advantage is that in the coherent state path-integral, the boson operator will become a c-number, and the trace operation in the partition function can be performed explicitly. After the Hamiltonian ‘H is diagonalized by a Bogoliubov transformation, the resulting free energy per site, is obtained after integration over f=-(l/@V)lnZ, the bosonic variables f = &
xln2sinhPSak
H2 4(/_~ - 2Jd - D)
-
k
-pS(S+
(6)
1) -&SD,
313
ordered ferromagnetic one (M # 0) when the energy gap A vanishes. The condition A = 0 yields ,u=2dJfD
(D>O),
= 2dJ
(D < 0).
(11)
We will consider the case D 3 0 without loss of generality. The paramagnetic-ferromagnetic phase boundary is determined by solving Eq. (9) for the mean constraint in the limit ,u = 2Jd + D. Note that the critical region is dominated by the long wavelength limit Ikl + 0 [ 161. Expanding the coth term in Eq. (9), we obtain the critical temperature T, below which the ferromagnetic phase appears,
where T,=S(S+l) @k = {[,u - ;D - J(k)12
- iD2}“2
(7)
is the dispersion relation. p is the inverse of the temperature T. The Lagrange multiplier ,u is determined by minimizing the free energy with respect to it,
af /+ = 0,
s x
c
k {[p
p-
_ &D _ J(k)]2 H2
+ 4(,u - 2dJ - D)2
_ $D2}‘/2
-S(S+
X
c
,u - ;D - J(k) _ ;D _ J(k)]2 _ .!D2}‘/2
cothPSwk
’
In the absence of anisotropy, Eq. (9) shows a zerotemperature magnetization M(T = 0) = S, which is the expected result. The excitation energy spectrum ok = { [ ,u - i D - ;D2}‘i2
exhibits an energy gap
A=[(,u-;D-~~J)~-~D~]“~.
+ 1)
= I(0)
+ I(D/2dJ)
J(k)12-
$D2 >
(12)
’ extended
Watson
inte-
l
x+1-y(k)’
(13)
(8)
(9)
J(k)12
4dJS(S
Z(x)?$k
1) =o.
1) -;
k {[p
[;D+2dJ-
cothPSwk
We introduce the magnetization M defined by M = -af/G’H. The equation of state can then be written as M2=S(Sf
ck
where I(x) is the standard gral [ 171 defined by
+D - J(k)
-1
;D + 2dJ - J(k)
(10)
The system indicates a second-order phase transition from a disordered paramagnetic phase (M = 0) to an
with y(k) = d-’ Cz=, cos k,. The integral I (0) converges for the dimension d > 2 which is therefore the lower critical dimension, in accord with the MerminWagner theorem [ 181. In the absence of anisotropy, Eq. (12) gives the critical temperature T, for the isotropic XY model, T,(D=O)
=2dJS(S+l)1(0)-‘.
(14)
This result is in agreement with that of the coupled mode theory [ 91 except for an overall factor of 5. This numerical discrepancy is attributed by the fact that in Lovesey and Balcar’s model all three spin degrees of freedom fully participate in its static and dynamic properties, and they demand (Sr Sy ) = $S( S + 1) , a = x, y, or z, whereas in the present case the only contribution to the spin constraint comes from two spin degrees of freedom in the X-Y plane, i.e., (( Sf)2 + ( Sy)2) = S( Sf 1) [ 131. It is obvious that fluctuations of the out of plane spin z-component have the effect of lowering the critical temperature T, below which the
314
Y.Q. Ma/Physics
in-plane ferromagnetic hand, for large values Ising system with the along the x direction, Tc becomes 7. _
4dJS(SS
’ - I(0)
1)
+ 4dJ/D
’
Letters A 225 (1997) 31 I-314
ordering appears. On the other of D the model approaches an single-ion uniaxial anisotropy and the transition temperature
(15)
When D > 1 the transition temperature T, approaches 4dJS( S + 1)/Z(O), which is the same as that of the easy-z uniaxial anisotropic Ising limit [ 9 1. The fact that the critical temperatures T, (D = 0) and T, (D -t co) remain finite, is in agreement with the conclusion of Lovesey and Balcar [ 91. Since Z(x) is a monotonically decreasing function of its argument, for general values of D, from Eq. (12) we find that the critical temperature T, smoothly increases with increasing the strength of anisotropy D. The physical reason is that an uniaxial single-site anisotropy with the x direction parallel to the easy axis breaks the rotational O(2) symmetry of the isotropic XY ferromagnet in the XY spin plane, and has the tendency to transform the isotropic XY system into an Ising system in which all spins are confined to the x direction; a finite singlesite anisotropy effectively enhances the exchange interaction in the x direction, which leads to an effective in-plane exchange anisotropy. In the D -+ 0~)limit all spins are parallel or antiparallel to the x direction and the model becomes an Ising one. We conclude that the critical temperature of the model with plane anisotropy is intermediate between the values for the isotropic XY and uniaxial anisotropic Ising models, similar to the result obtained by Lovesey and Balcar [93. The Heisenberg planar XY model including the effects of anisotropy has been presented in the boson space and solved exactly by using the coherent state path integral method. We have discussed aspects of the phase transition of the model with arbitrary spin S and uniaxial anisotropy. It is shown that the critical temperature Tc increases with an increase of the anisotropy parameter D, and remains finite even for the anisotropy parameter D -+ 00. The result is
consistent with that of Lovesey and Balcar within the coupled mode theory. Our paper provides a simple method, allowing a straightforward extension to more complex spin systems and related boson ones. This work was supported by the Doctoral Foundation of Chinese Education Commission, the Natural Nature Science Foundation of China, and the Fifth Huo Ying-dong Teacher’s Foundation.
References 111 R.M. White, Quantum theory of magnetism (Springer, Berlin, 1983).
121 K.H. Fischer and J.A. Hertz, Spin glasses (Cambridge Univ. Press, Cambridge, 1991) [31 D.H. Ryan, ed., Recent progress in random magnets (World Scientific, Singapore, 1992). [41 G.E. Granroth et al., Phys. Rev. Lett. 77 (1996) 1616. [51 D.R. Taylor and W.J.L. Buyers, Phys. Rev. B 54 (1996) R3734. [61 R.A. Cowley, Methods of experimental physics, Vol. 23, Part C (Academic Press, Orlando, FL, 1987). I71 M.F. Collins, Magnetic critical scattering (Oxford Univ. Press, New York, 1989). [81 J.F. Devlin, Phys. Rev. B 4 (1971) 136. 191 S.W. Lovesey and E. Balcar, J. Phys. Condens. Matter 7 (1995) 2147. [lOI SW. Lovesey, Condensed matter physics: dynamic correlations, 2nd Ed. (Benjamin, Menlo Park, CA, 1986). IllI S.W. Lovesey, Theory of neutron scattering by condensed matter, 3rd Ed. (Oxford Univ. Press, Oxford, 1987). [I21 MIA. Fisher, PB. Weichman, G. Grinstein and D.S.Fisher, Phys. Rev. B 40 ( 1989) 546. (13 See, for example, K. Binder and D.W. Heermann, Monte Carlo simulation in statistical physics (Springer, Berlin. 1988). 114,I D.D. Bet&, in: Phase transitions and critical phenomena, Vol. 3, eds. C. Domb and MS. Green (Academic Press, New York, 1974). I15 E Pazmandi. G. Zimanyi and R. Scalettar, Cond-Mat. preprint no. 9602158, and references therein. [IhI J.W. Negele and H. Orland, Quantum many-particle systems ( Addision-Wesley, New York, 1988), [I71 G.N. Watson, Quart. J. Math (Oxford) 10 ( 1939) 266. IIS1 N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133.