- Email: [email protected]
The Néel order for a frustrated antiferromagnetic Heisenberg model: beyond linear spin-wave theory
23 October 2000
Physics Letters A 275 Ž2000. 481–485 www.elsevier.nlrlocaterpla
The Neel ´ order for a frustrated antiferromagnetic Heisenberg model: beyond linear spin-wave theory Qingshan Yuan a,b,1 a
Max-Planck-Institut f ur Str. 38, 01187 Dresden, Germany ¨ Physik komplexer Systeme, Nothnitzer ¨ b Pohl Institute of Solid State Physics, Tongji UniÕersity, Shanghai 200092, PR China Received 28 August 2000; accepted 13 September 2000 Communicated by J. Flouquet
Abstract Within Dyson–Maleev ŽDM. transformation and self-consistent mean-field treatment, the Neel ´ orderrdisorder transition is studied for an antiferromagnetic Heisenberg model which is defined on a square lattice with a nearest neighbour exchange J1 and a next-nearest neighbour exchange J2 along only one of the diagonals. It is found that the Neel ´ order may exist up to J2rJ1 s 0.572, beyond its classically stable regime. This result qualitatively improves that from linear spin-wave theory based on Holstein–Primakoff transformation. q 2000 Elsevier Science B.V. All rights reserved. PACS: 75.10.-b; 75.10.Jm
The two-dimensional Ž2D. antiferromagnetic Heisenberg models have attracted great interest in recent years, partly because of the fact that the parent compounds of the high temperature superconducting materials are excellent realizations of quasi-2D quantum antiferromagnets w1x. While the unfrustrated Heisenberg model has been well understood, much attention has been paid to the frustrated models such as square-lattice nearest and next-nearest neighbour interaction Žso called J1 –J2 . model, triangular lattice model and Kagome´ lattice model etc. w2–7x. In these frustrated Heisenberg models the property of the ground state, whether magnetically ordered or disordered, is a subject of considerable interest. For exE-mail address: [email protected] ŽQ. Yuan.. 1 Fax: q49-821-598 3652.
ample, the J1 –J2 model takes on Neel ´ order at small J2rJ1 and collinear order at large J2rJ1 , which are seperated by a region of disordered state w2x. Very recently a S s 1r2 Heisenberg model, which defined on a square lattice with a nearest neighbour antiferromagnetic exchange J1 and a next-nearest neighbour exchange J2 along only one of the diagonals of the lattice as shown in Fig. 1, has been proposed w8,9x. Its Hamiltonian is written as H s J1 Ý S i P S j q J 2 ² ij :
Ý
Sl P Sm ,
Ž 1.
² lm :
where the notation ² ij : denote nearest neighbour bonds and ² lm: denote next-nearest neighbour bonds along only one diagonal. Topologically this model is equivalent to the Heisenberg antiferromagnet on an anisotropic triangular lattice w10,11x. In special cases
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 6 2 5 - 3
482
Q. Yuan r Physics Letters A 275 (2000) 481–485
by Merino et al. w9x to discuss the possible ordered and disordered states. It is helpful to repeat some details here for later discussion. First, for convenience the spin at each site is rotated along its reference direction characterized by the angle u i . Such a rotation may be accomplished by the following transformation for spin operators Fig. 1. The 2D square lattice with nearest neighbour interaction J1 and next-nearest neighbour inteaction J2 along only one of the diagonals.
Six s sin u i Sˆiz q cos u i Sˆix Siy s Sˆiy
J2 s 0, J1 s J2 and J1 s 0, it will recover to unfrustrated square lattice model, isotropic triangular lattice model and decoupled spin chains, respectively. Therefore this model provides a way of interpolating between several well-known one and two-dimensional models. One can study the role of frustration in going from one to two dimensions. On the other hand, this model is of direct relevance to the magnetic phases of some quasi-2D organic superconductors. It has been argued that this model may describe the spin degree of freedom of the insulating phase of the layered molecular crystals k-ŽBEDT–TTF. 2 X w12x. The parameter J2rJ1 for these materials is suggested to be ; 0.3–1 and the magnetic frustration will play an important role. Classically, the ground state of the model can be derived straightforwardly as a function of the ratio J2rJ1 if we assume that the spins lie in the xz plane and are described by a spiral form S i s SŽsin u i ,0,cos u i . as shown in Fig. 1. Here the angle u i s q P r i and the wavevector q s Ž q,q . defines a relative orientation of the spins. Minimization of the classical energy with respect to q gives the result that the ground state take on Neel ´ order Ži.e., q s p . for J2rJ1 F 1r2, and spiral order with q s arccosŽyJ1r2 J2 . for J2rJ1 ) 1r2. The quantum model has also been studied numerically and analytically. The series expansions were adopted by Zheng et al. w8x for numerical calculation. It was found that the Neel ´ order persists up to J2rJ1 s 0.7. In the region 0.7 F J2rJ1 F 0.9 there is no magnetic order and for larger values of J2rJ1 there is incommensurate or spiral order. It is interesting to note that the Neel ´ order exists beyond its classical result Ž J2rJ1 s 0.5.. Analytically the standard and simple linear spin-wave theory ŽLSW. based on Holstein–Primakoff ŽHP. transformation was used
Siz s cos u i Sˆiz y sin u i Sˆix For new spins Sˆ i the reference ground state becomes ferromagnetic. The rotated Hamiltonian becomes H s J1 Ý cos u i j Sˆix Sˆjx q Sˆiz Sˆjz
ž
² ij :
/
qsin u i j Sˆiz Sˆjx y Sˆix Sˆjz q Sˆiy Sˆjy
ž
q J2
/
cos u l m Sˆlx Sˆmx q Sˆlz Sˆmz
Ý ² lm :
ž
/
qsin u l m Sˆlz Sˆmx y Sˆlx Sˆmz q Sˆly Sˆmy .
ž
/
Ž 2.
with u i j s u i y u j s q, u l m s u l y um s 2 q. Then with HP transformation the above Hamiltonian with only quadratic terms kept may be diagonalized in the momentum space. The final dispersion relation for the spin excitation is
'
v k s w J Ž kqq. q J Ž kyq. x r2y J Ž q . 4 w J Ž k . y J Ž q. x
with J Žk. s J1Žcos k x q cos k y . q J2 cosŽ k x q k y .. By calculation of the magnetization ² Sˆiz : Žsee also the dashed line in Fig. 2 later. it was suggested by the authors in Ref. w9x that a possible Neel ´ orderrdisorder transition happens at the point J2rJ1 , 0.5. However, we want to point out the inherent limitation for LSW theory here. From the spectrum v k , it is easy to find that, in order to ensure the argument in the square root always positive in the whole Brillouin zone, the parameter q has to be set as p Žcharacterizing Neel ´ order. in the region J2rJ1 - 0.5, but must not be set as p in the region J2rJ1 ) 0.5. This means that within LSW theory the Neel ´ ordered state can never appear beyond its classically stable region, which is in contrast with the numerical result
Q. Yuan r Physics Letters A 275 (2000) 481–485
483
p , u l m s 2p and then use the DM transformation for A and B two sublattices in the following form: † † ˆy † ˆz Sˆq i s Ž 1 y a i a i . a i , S i s a i , S i s 1r2 y a i a i † † † ˆq ˆz Sˆy j s bj Ž 1 y bj bj . , S j s bj , S j s 1r2 y bj bj ,
where aŽ a† ., bŽ b† . are bosonic operators for sublattices A and B, respectively. Then the Hamiltonian Ž2. is transformed into Fig. 2. The magnetization m calculated from our treatment Žsolid line. and that from LSW theory Ždashed line. as a function of J2 r J1. It goes to zero at about J2 r J1 s 0.499 within LSW theory, but up to J2 r J1 s 0.572 within our treatment.
H s yJ1 Ý Ž 1r2 y a†i a i . Ž 1r2 y b†j bj . ² ij :
q Ž 1 y a†i a i . a i bj q a†i b†j Ž 1 y b†j bj . r2 w8x. As also pointed out by the authors in Ref. w9x themselves, the interaction between spin waves becomes very large at the transition point and it may lead to a completely different picture for the states. So it is very necessary to go beyond the LSW theory to see how the above result will be modified, which is the purpose of this Letter. Instead of HP transformation, the Dyson–Maleev ŽDM. transformation will be adopted to avoid 1rS expansion. It has been also recognized that the DM transformation must be prefered if one needs really to go beyond LSW theory within a perturbation scheme w13,14x. In the treatment of the so called J1 –J2 model, it has been shown that the spin-wave theory based on DM transformation gives perfectly consistent result as that from numerical calculation; much better than that from LSW theory w3x. In this work we will not discuss the possible spiral state for large J2rJ1 , but focus on the Neel ´ orderrdisorder transition in the intermediate J2rJ1 region, especially on the problem whether the Neel ´ order may appear beyond its classical region of stability which is one of the most interesting topics for this model. Technically, the Hamiltonian under DM transformation have terms as high as sixth order when spiral state is considered Ži.e., for general q ., which are relatively complicated to treat. When only Neel ´ state is considered the transformed Hamiltonian has no term higher than fourth order, which can be easily treated by mean-field ŽMF. theory or perturbation theory. Explicitly, one may apply the DM transformation onto the original Hamiltonian Ž1.; or begin with the rotated Hamiltonian Ž2. by setting u i j s
q J2
½
Ý Ž 1r2 y a†l a l .Ž 1r2 y a†m a m . ² lm :gA
q Ž 1 y a†l a l . a l a†m q Ž 1 y a†m a m . a m a†l r2 q
Ý ² lm :gB
a
™b5 .
Ž 3.
To diagonalize the above Hamiltonian, we treat the quartic terms with a self-consistent MF theory. For those terms which could be decoupled in two ways, we will combine the two kinds of decoupling form together through a weight factor l as introduced by Chu and Shen w15x. For example, the term like a†i a i b†j bj will be decoupled in the way a†i a i b†j bj , l ² a†i a i : b†j bj q a†i a i ² b†j bj : y ² a†i a i : ² b†j bj : q Ž 1 y l . ² a†i b†j : a i bj q a†i b†j ² a i bj : y² a†i b†j : ² a i bj : . The parameter 0 F l F 1 reflects the competition between two decoupling ways. Its value may be decided by minimization of the energy, which was found to be 1r2 in Ref. w15x for unfrustrated lattice; or it is required to be equal to 1r2 in order to keep the symmetry of the Hamiltonian before and after decoupling w16,17x. We will take the value l s 1r2 throughout our calculations. With definition of several parameters: u s ² a†i a i : s ² b†i bi :, Õ s ² a†i b†j : s
Q. Yuan r Physics Letters A 275 (2000) 481–485
484
² a i bj :, w s ² a†l a m : s ² b†l bm :, we may obtain a quadratic Hamiltonian in the momentum space: † H s Ý C k Ž a†k a k q b†k b k . q E k Ž a k byk q a†k byk .
k
q const ,
Ž 4.
C k s 2 J1 Ž 1 y u q Õ . y J 2 Ž 1 y u q w . = 1 y cos Ž k x q k y . , Ek s yJ1 Ž 1 y u q Õ . Ž cos k x q cos k y . , N
const s
J1 Ž y1 q 2 u 2 q 2 Õ 2 y 4 uÕ .
2
qJ2 Ž 1 y 2 u 2 y 2 w 2 q 4 uw . r2 , where the summation is over half of the original Brillouin zone and N is total number of lattice sites. Under a Bogoliubov transformation † a k s cosh lk a k q sinh lk byk † † byk s sinh l k a k q cosh l k byk
with tanh2 l k s yC krEk , the Hamiltonian Ž4. may be diagonalized into H s Ý v k a†k a k q b†k b k q 1 y C k q const Ž 5 .
ž
k
/
(
with the excitation spectrum v k s C k2 y Ek2 . Correspondingly the self-consistent equations for u, Õ, w are expressed as us
1
Ý N k
Ck
(
Ý N k
ws
1
Ý N k
1 y 2
C k2 y E k2
1 Õsy
Fig. 3. The self-consistent results for the parameters Õ and w.
Ek cos k x
(C y E 2 k
2 k
,
,
C k cos Ž k x q k y .
(C y E 2 k
2 k
.
The magnetization is simply given by m s 1r2 y u and the ground state energy is E0 s Ýk Ž v k y C k . q const. We show the numerical results for the above self-consistent equations in Figs. 2 and 3. The magnetization m Žderived from the parameter u. as a function of J2rJ1 is given by the solid line in Fig. 2,
which is the main result in this Letter. It is found that the magnetization does not vanish until J2rJ1 , 0.572, which gives the Neel ´ orderrdisorder transition point. As comparison, the result from LSW theory is plotted by the dashed line; the magnetization goes to zero immediately before the classical value J2rJ1 s 0.5, see also Ref. w11x. As expected, the current result is closer to the numerical one from series expansions and most importantly, it qualitatively improves the result from LSW theory. As we discussed before, the LSW theory is impossible to deduce a Neel ´ ordered state beyond J2rJ1 s 0.5. In our treatment the interaction between spin waves is actually partly considered, then the fact that Neel ´ order may exist beyond its classically stable region is shown. It is quite possible that the transition point will shift to larger value if the residual interaction between spin waves is included. This may be also hinted from Fig. 3, where the parameter Õ, which represents the antiferromagnetic correlation between the two original nearest-neighbour spins, is large near the transition point. For completeness, the ground state energy in the region 0 - J2rJ1 - 0.57 is also shown in Fig. 4 by the solid line, which is close to the result from LSW theory Žthe dashed line.. In the small J2rJ1 region the energy calculated here is a little lower than that of LSW theory, and becomes a little higher with increase of J2rJ1. For J2rJ1 ) 0.5 the energy within LSW theory is derived from spiral state; there is a cusp at point J2rJ1 s 0.5 w9x. In the current case the state is still Neel ´ ordered and the energy changes smoothly. In summary, within Dyson–Maleev ŽDM. transformation and self-consistent mean-field treatment,
Q. Yuan r Physics Letters A 275 (2000) 481–485
485
Acknowledgements The author would like to thank J. Merino for helpful discussion. This work was supported in part by the Chinese NSF.
References Fig. 4. The ground state energy calculated from our treatment Žsolid line. and that from LSW theory Ždashed line. as a function of J2 r J1 Žsee text..
we have studied an antiferromagnetic Heisenberg model on a square lattice which includes a nearest neighbour exchange J1 and a next-nearest neighbour exchange J2 along only one of the diagonals. This model should be of direct relevance to some layered organic superconductors. In this work we focus on the discussion of Neel ´ orderrdisorder transition for not large J2rJ1. It is found that the Neel ´ order may exist up to J2rJ1 s 0.572, beyond its classically stable regime. This property is consistent with numerical finding from series expansions, which is one of the most interesting features for this model. Especially, because the interaction between spin waves is partly considered in our treatment, the result derived here qualitatively improves that from LSW theory based on Holstein–Primakoff transformation. It is certainly necessary to continue this work to study the large J2rJ1 region where a spiral order will appear, so that a whole phase diagram may be constructed.
w1x S. Chakravarty, B.I. Halperin, D.R. Nelson, Phys. Rev. B 39 Ž1989. 2344. w2x J. Oitmma, Z. Weihong, Phys. Rev. B 54 Ž1996. 3022, and references therein. w3x J.H. Hu, C.S. Ting, Phys. Rev. B 42 Ž1990. 6861. w4x N.B. Ivanov, J. Richter, J. Phys.: Cond. Matter 6 Ž1994. 3785. w5x F. Mila, D. Poilblanc, C. Bruder, Phys. Rev. B 43 Ž1991. 7891. w6x R.R.P. Singh, D.A. Huse, Phys. Rev. Lett. 68 Ž1992. 1766. w7x P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, P. Sindzingre, Phys. Rev. B 56 Ž1997. 2521. w8x Z. Weihong, R.H. McKenzie, R.R.P. Singh, Phys. Rev. B 59 Ž1999. 14367. w9x J. Merino, R.H. McKenzie, J.B. Marston, C.H. Chung, J. Phys.: Cond. Matter 11 Ž1999. 2965. w10x U. Bhaumik, I. Bose, Phys. Rev. B 58 Ž1998. 73. w11x A.E. Trumper, Phys. Rev. B 60 Ž1999. 2987. w12x R.H. McKenzie, Comments Cond. Matter Phys. 18 Ž1998. 309. w13x C.M. Canali, S.M. Girvin, Phys. Rev. B 45 Ž1992. 7127. w14x C.M. Canali, S.M. Girvin, M. Wallin, Phys. Rev. B 45 Ž1992. 10131. w15x D.P. Chu, J.L. Shen, Phys. Rev. B 44 Ž1991. 4689. w16x M.R. Li, Y.J. Wang, C.D. Gong, Z. Phys. B 102 Ž1997. 129. w17x J. Yang, J.L. Shen, H.Q. Lin, J. Phys. Soc. Jpn. 68 Ž1999. 2384.
Physics Letters A 275 Ž2000. 481–485 www.elsevier.nlrlocaterpla
The Neel ´ order for a frustrated antiferromagnetic Heisenberg model: beyond linear spin-wave theory Qingshan Yuan a,b,1 a
Max-Planck-Institut f ur Str. 38, 01187 Dresden, Germany ¨ Physik komplexer Systeme, Nothnitzer ¨ b Pohl Institute of Solid State Physics, Tongji UniÕersity, Shanghai 200092, PR China Received 28 August 2000; accepted 13 September 2000 Communicated by J. Flouquet
Abstract Within Dyson–Maleev ŽDM. transformation and self-consistent mean-field treatment, the Neel ´ orderrdisorder transition is studied for an antiferromagnetic Heisenberg model which is defined on a square lattice with a nearest neighbour exchange J1 and a next-nearest neighbour exchange J2 along only one of the diagonals. It is found that the Neel ´ order may exist up to J2rJ1 s 0.572, beyond its classically stable regime. This result qualitatively improves that from linear spin-wave theory based on Holstein–Primakoff transformation. q 2000 Elsevier Science B.V. All rights reserved. PACS: 75.10.-b; 75.10.Jm
The two-dimensional Ž2D. antiferromagnetic Heisenberg models have attracted great interest in recent years, partly because of the fact that the parent compounds of the high temperature superconducting materials are excellent realizations of quasi-2D quantum antiferromagnets w1x. While the unfrustrated Heisenberg model has been well understood, much attention has been paid to the frustrated models such as square-lattice nearest and next-nearest neighbour interaction Žso called J1 –J2 . model, triangular lattice model and Kagome´ lattice model etc. w2–7x. In these frustrated Heisenberg models the property of the ground state, whether magnetically ordered or disordered, is a subject of considerable interest. For exE-mail address: [email protected] ŽQ. Yuan.. 1 Fax: q49-821-598 3652.
ample, the J1 –J2 model takes on Neel ´ order at small J2rJ1 and collinear order at large J2rJ1 , which are seperated by a region of disordered state w2x. Very recently a S s 1r2 Heisenberg model, which defined on a square lattice with a nearest neighbour antiferromagnetic exchange J1 and a next-nearest neighbour exchange J2 along only one of the diagonals of the lattice as shown in Fig. 1, has been proposed w8,9x. Its Hamiltonian is written as H s J1 Ý S i P S j q J 2 ² ij :
Ý
Sl P Sm ,
Ž 1.
² lm :
where the notation ² ij : denote nearest neighbour bonds and ² lm: denote next-nearest neighbour bonds along only one diagonal. Topologically this model is equivalent to the Heisenberg antiferromagnet on an anisotropic triangular lattice w10,11x. In special cases
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 6 2 5 - 3
482
Q. Yuan r Physics Letters A 275 (2000) 481–485
by Merino et al. w9x to discuss the possible ordered and disordered states. It is helpful to repeat some details here for later discussion. First, for convenience the spin at each site is rotated along its reference direction characterized by the angle u i . Such a rotation may be accomplished by the following transformation for spin operators Fig. 1. The 2D square lattice with nearest neighbour interaction J1 and next-nearest neighbour inteaction J2 along only one of the diagonals.
Six s sin u i Sˆiz q cos u i Sˆix Siy s Sˆiy
J2 s 0, J1 s J2 and J1 s 0, it will recover to unfrustrated square lattice model, isotropic triangular lattice model and decoupled spin chains, respectively. Therefore this model provides a way of interpolating between several well-known one and two-dimensional models. One can study the role of frustration in going from one to two dimensions. On the other hand, this model is of direct relevance to the magnetic phases of some quasi-2D organic superconductors. It has been argued that this model may describe the spin degree of freedom of the insulating phase of the layered molecular crystals k-ŽBEDT–TTF. 2 X w12x. The parameter J2rJ1 for these materials is suggested to be ; 0.3–1 and the magnetic frustration will play an important role. Classically, the ground state of the model can be derived straightforwardly as a function of the ratio J2rJ1 if we assume that the spins lie in the xz plane and are described by a spiral form S i s SŽsin u i ,0,cos u i . as shown in Fig. 1. Here the angle u i s q P r i and the wavevector q s Ž q,q . defines a relative orientation of the spins. Minimization of the classical energy with respect to q gives the result that the ground state take on Neel ´ order Ži.e., q s p . for J2rJ1 F 1r2, and spiral order with q s arccosŽyJ1r2 J2 . for J2rJ1 ) 1r2. The quantum model has also been studied numerically and analytically. The series expansions were adopted by Zheng et al. w8x for numerical calculation. It was found that the Neel ´ order persists up to J2rJ1 s 0.7. In the region 0.7 F J2rJ1 F 0.9 there is no magnetic order and for larger values of J2rJ1 there is incommensurate or spiral order. It is interesting to note that the Neel ´ order exists beyond its classical result Ž J2rJ1 s 0.5.. Analytically the standard and simple linear spin-wave theory ŽLSW. based on Holstein–Primakoff ŽHP. transformation was used
Siz s cos u i Sˆiz y sin u i Sˆix For new spins Sˆ i the reference ground state becomes ferromagnetic. The rotated Hamiltonian becomes H s J1 Ý cos u i j Sˆix Sˆjx q Sˆiz Sˆjz
ž
² ij :
/
qsin u i j Sˆiz Sˆjx y Sˆix Sˆjz q Sˆiy Sˆjy
ž
q J2
/
cos u l m Sˆlx Sˆmx q Sˆlz Sˆmz
Ý ² lm :
ž
/
qsin u l m Sˆlz Sˆmx y Sˆlx Sˆmz q Sˆly Sˆmy .
ž
/
Ž 2.
with u i j s u i y u j s q, u l m s u l y um s 2 q. Then with HP transformation the above Hamiltonian with only quadratic terms kept may be diagonalized in the momentum space. The final dispersion relation for the spin excitation is
'
v k s w J Ž kqq. q J Ž kyq. x r2y J Ž q . 4 w J Ž k . y J Ž q. x
with J Žk. s J1Žcos k x q cos k y . q J2 cosŽ k x q k y .. By calculation of the magnetization ² Sˆiz : Žsee also the dashed line in Fig. 2 later. it was suggested by the authors in Ref. w9x that a possible Neel ´ orderrdisorder transition happens at the point J2rJ1 , 0.5. However, we want to point out the inherent limitation for LSW theory here. From the spectrum v k , it is easy to find that, in order to ensure the argument in the square root always positive in the whole Brillouin zone, the parameter q has to be set as p Žcharacterizing Neel ´ order. in the region J2rJ1 - 0.5, but must not be set as p in the region J2rJ1 ) 0.5. This means that within LSW theory the Neel ´ ordered state can never appear beyond its classically stable region, which is in contrast with the numerical result
Q. Yuan r Physics Letters A 275 (2000) 481–485
483
p , u l m s 2p and then use the DM transformation for A and B two sublattices in the following form: † † ˆy † ˆz Sˆq i s Ž 1 y a i a i . a i , S i s a i , S i s 1r2 y a i a i † † † ˆq ˆz Sˆy j s bj Ž 1 y bj bj . , S j s bj , S j s 1r2 y bj bj ,
where aŽ a† ., bŽ b† . are bosonic operators for sublattices A and B, respectively. Then the Hamiltonian Ž2. is transformed into Fig. 2. The magnetization m calculated from our treatment Žsolid line. and that from LSW theory Ždashed line. as a function of J2 r J1. It goes to zero at about J2 r J1 s 0.499 within LSW theory, but up to J2 r J1 s 0.572 within our treatment.
H s yJ1 Ý Ž 1r2 y a†i a i . Ž 1r2 y b†j bj . ² ij :
q Ž 1 y a†i a i . a i bj q a†i b†j Ž 1 y b†j bj . r2 w8x. As also pointed out by the authors in Ref. w9x themselves, the interaction between spin waves becomes very large at the transition point and it may lead to a completely different picture for the states. So it is very necessary to go beyond the LSW theory to see how the above result will be modified, which is the purpose of this Letter. Instead of HP transformation, the Dyson–Maleev ŽDM. transformation will be adopted to avoid 1rS expansion. It has been also recognized that the DM transformation must be prefered if one needs really to go beyond LSW theory within a perturbation scheme w13,14x. In the treatment of the so called J1 –J2 model, it has been shown that the spin-wave theory based on DM transformation gives perfectly consistent result as that from numerical calculation; much better than that from LSW theory w3x. In this work we will not discuss the possible spiral state for large J2rJ1 , but focus on the Neel ´ orderrdisorder transition in the intermediate J2rJ1 region, especially on the problem whether the Neel ´ order may appear beyond its classical region of stability which is one of the most interesting topics for this model. Technically, the Hamiltonian under DM transformation have terms as high as sixth order when spiral state is considered Ži.e., for general q ., which are relatively complicated to treat. When only Neel ´ state is considered the transformed Hamiltonian has no term higher than fourth order, which can be easily treated by mean-field ŽMF. theory or perturbation theory. Explicitly, one may apply the DM transformation onto the original Hamiltonian Ž1.; or begin with the rotated Hamiltonian Ž2. by setting u i j s
q J2
½
Ý Ž 1r2 y a†l a l .Ž 1r2 y a†m a m . ² lm :gA
q Ž 1 y a†l a l . a l a†m q Ž 1 y a†m a m . a m a†l r2 q
Ý ² lm :gB
a
™b5 .
Ž 3.
To diagonalize the above Hamiltonian, we treat the quartic terms with a self-consistent MF theory. For those terms which could be decoupled in two ways, we will combine the two kinds of decoupling form together through a weight factor l as introduced by Chu and Shen w15x. For example, the term like a†i a i b†j bj will be decoupled in the way a†i a i b†j bj , l ² a†i a i : b†j bj q a†i a i ² b†j bj : y ² a†i a i : ² b†j bj : q Ž 1 y l . ² a†i b†j : a i bj q a†i b†j ² a i bj : y² a†i b†j : ² a i bj : . The parameter 0 F l F 1 reflects the competition between two decoupling ways. Its value may be decided by minimization of the energy, which was found to be 1r2 in Ref. w15x for unfrustrated lattice; or it is required to be equal to 1r2 in order to keep the symmetry of the Hamiltonian before and after decoupling w16,17x. We will take the value l s 1r2 throughout our calculations. With definition of several parameters: u s ² a†i a i : s ² b†i bi :, Õ s ² a†i b†j : s
Q. Yuan r Physics Letters A 275 (2000) 481–485
484
² a i bj :, w s ² a†l a m : s ² b†l bm :, we may obtain a quadratic Hamiltonian in the momentum space: † H s Ý C k Ž a†k a k q b†k b k . q E k Ž a k byk q a†k byk .
k
q const ,
Ž 4.
C k s 2 J1 Ž 1 y u q Õ . y J 2 Ž 1 y u q w . = 1 y cos Ž k x q k y . , Ek s yJ1 Ž 1 y u q Õ . Ž cos k x q cos k y . , N
const s
J1 Ž y1 q 2 u 2 q 2 Õ 2 y 4 uÕ .
2
qJ2 Ž 1 y 2 u 2 y 2 w 2 q 4 uw . r2 , where the summation is over half of the original Brillouin zone and N is total number of lattice sites. Under a Bogoliubov transformation † a k s cosh lk a k q sinh lk byk † † byk s sinh l k a k q cosh l k byk
with tanh2 l k s yC krEk , the Hamiltonian Ž4. may be diagonalized into H s Ý v k a†k a k q b†k b k q 1 y C k q const Ž 5 .
ž
k
/
(
with the excitation spectrum v k s C k2 y Ek2 . Correspondingly the self-consistent equations for u, Õ, w are expressed as us
1
Ý N k
Ck
(
Ý N k
ws
1
Ý N k
1 y 2
C k2 y E k2
1 Õsy
Fig. 3. The self-consistent results for the parameters Õ and w.
Ek cos k x
(C y E 2 k
2 k
,
,
C k cos Ž k x q k y .
(C y E 2 k
2 k
.
The magnetization is simply given by m s 1r2 y u and the ground state energy is E0 s Ýk Ž v k y C k . q const. We show the numerical results for the above self-consistent equations in Figs. 2 and 3. The magnetization m Žderived from the parameter u. as a function of J2rJ1 is given by the solid line in Fig. 2,
which is the main result in this Letter. It is found that the magnetization does not vanish until J2rJ1 , 0.572, which gives the Neel ´ orderrdisorder transition point. As comparison, the result from LSW theory is plotted by the dashed line; the magnetization goes to zero immediately before the classical value J2rJ1 s 0.5, see also Ref. w11x. As expected, the current result is closer to the numerical one from series expansions and most importantly, it qualitatively improves the result from LSW theory. As we discussed before, the LSW theory is impossible to deduce a Neel ´ ordered state beyond J2rJ1 s 0.5. In our treatment the interaction between spin waves is actually partly considered, then the fact that Neel ´ order may exist beyond its classically stable region is shown. It is quite possible that the transition point will shift to larger value if the residual interaction between spin waves is included. This may be also hinted from Fig. 3, where the parameter Õ, which represents the antiferromagnetic correlation between the two original nearest-neighbour spins, is large near the transition point. For completeness, the ground state energy in the region 0 - J2rJ1 - 0.57 is also shown in Fig. 4 by the solid line, which is close to the result from LSW theory Žthe dashed line.. In the small J2rJ1 region the energy calculated here is a little lower than that of LSW theory, and becomes a little higher with increase of J2rJ1. For J2rJ1 ) 0.5 the energy within LSW theory is derived from spiral state; there is a cusp at point J2rJ1 s 0.5 w9x. In the current case the state is still Neel ´ ordered and the energy changes smoothly. In summary, within Dyson–Maleev ŽDM. transformation and self-consistent mean-field treatment,
Q. Yuan r Physics Letters A 275 (2000) 481–485
485
Acknowledgements The author would like to thank J. Merino for helpful discussion. This work was supported in part by the Chinese NSF.
References Fig. 4. The ground state energy calculated from our treatment Žsolid line. and that from LSW theory Ždashed line. as a function of J2 r J1 Žsee text..
we have studied an antiferromagnetic Heisenberg model on a square lattice which includes a nearest neighbour exchange J1 and a next-nearest neighbour exchange J2 along only one of the diagonals. This model should be of direct relevance to some layered organic superconductors. In this work we focus on the discussion of Neel ´ orderrdisorder transition for not large J2rJ1. It is found that the Neel ´ order may exist up to J2rJ1 s 0.572, beyond its classically stable regime. This property is consistent with numerical finding from series expansions, which is one of the most interesting features for this model. Especially, because the interaction between spin waves is partly considered in our treatment, the result derived here qualitatively improves that from LSW theory based on Holstein–Primakoff transformation. It is certainly necessary to continue this work to study the large J2rJ1 region where a spiral order will appear, so that a whole phase diagram may be constructed.
w1x S. Chakravarty, B.I. Halperin, D.R. Nelson, Phys. Rev. B 39 Ž1989. 2344. w2x J. Oitmma, Z. Weihong, Phys. Rev. B 54 Ž1996. 3022, and references therein. w3x J.H. Hu, C.S. Ting, Phys. Rev. B 42 Ž1990. 6861. w4x N.B. Ivanov, J. Richter, J. Phys.: Cond. Matter 6 Ž1994. 3785. w5x F. Mila, D. Poilblanc, C. Bruder, Phys. Rev. B 43 Ž1991. 7891. w6x R.R.P. Singh, D.A. Huse, Phys. Rev. Lett. 68 Ž1992. 1766. w7x P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, P. Sindzingre, Phys. Rev. B 56 Ž1997. 2521. w8x Z. Weihong, R.H. McKenzie, R.R.P. Singh, Phys. Rev. B 59 Ž1999. 14367. w9x J. Merino, R.H. McKenzie, J.B. Marston, C.H. Chung, J. Phys.: Cond. Matter 11 Ž1999. 2965. w10x U. Bhaumik, I. Bose, Phys. Rev. B 58 Ž1998. 73. w11x A.E. Trumper, Phys. Rev. B 60 Ž1999. 2987. w12x R.H. McKenzie, Comments Cond. Matter Phys. 18 Ž1998. 309. w13x C.M. Canali, S.M. Girvin, Phys. Rev. B 45 Ž1992. 7127. w14x C.M. Canali, S.M. Girvin, M. Wallin, Phys. Rev. B 45 Ž1992. 10131. w15x D.P. Chu, J.L. Shen, Phys. Rev. B 44 Ž1991. 4689. w16x M.R. Li, Y.J. Wang, C.D. Gong, Z. Phys. B 102 Ž1997. 129. w17x J. Yang, J.L. Shen, H.Q. Lin, J. Phys. Soc. Jpn. 68 Ž1999. 2384.