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The magnetization equation for an arbitrary infinite-range spin Hamiltonian
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
Solid State Communications, Vol. 52, No. 7, pp. 689-691, 1984. Printed in Great Britain.
THE MAGNETIZATION EQUATION FOR AN ARBITRARY INFINITE-RANGE SPIN HAMILTONIAN J. Katriei and G.F. Kventsel Department of Chemistry, Technion - Israel Institute of Technology, 32000 Haifa, Israel (Received 27 March 1984 by H. S u h l )
The equation s = --oBo[~olV, HI] V , H / I V s H I is shown to be the magnetization equation for the arbitrary anisotropic infinite-range spin Hamiltonian, H(s). The known equations for the general anisotropic-Heisenberg, and for arbitrary isotropic and axially symmetric spin Hamiltonians are shown to be special cases of this equation. FOR THE INFINITE range Heisenberg Hamiltonian H = (J/2)s 2 the magnetization equation
(1)
s = oBo[[JoJs],
The magnetization equation of an arbitrary axially symmetry infinite-range spin Hamiltonian, which can be written in terms o f s 2 and sz, was recently derived [12]. The total magnetization s and its z component sz are obtained from
and either
a=x,y,z.
#
The special case of an axially symmetric Heisenberg Hamiltonian H = -- [a(s~ + s~) + bs2z] was studied by Vertogen and De Vries [7] and by Lee etal. [8-101, who showed that the only two types of solutions are Sz = 0 o r s x
=sy =0.
or
Sz = s.
N 1 ~. se, i,
ga
ot=x.v,z,
N i=t
which is of the order of magnitude of the elementary spin, satisfies
was treated by Pearce and Thompson [6], who obtained the magnetization equation
(J.s),~ = ~ J~as~;
(~H/~sz)s = 0
For the axially symmetric Heisenberg Hamiltonian these equations reduce to those of [8-10], but in general a solution of an intermediate type, i.e. Sx = sv :# 0 and sz :# 0 is also possible. To treat the general infinite range spin Hamiltonian H(s), we note that
Ot,[J = x , y , z ,
where
(5)
s = aBo[--~o d H / d s l ,
(2)
J's s = oao[~JolJ'sl]'lJ.s I ,
(4)
s = oBol-#odH/ds].
was derived a long time ago. This equation is identical with the mean-field equation obtained for the nearestneighbour Heisenberg Hamiltonian [ 1-3 ]. In recent years, the interest in the study of spin Hamfltonians which are more complex than the Heisenberg Hamiltonian has considerably increased [4, 5]. The equivalence between the mean-field approximation for finite-range spin-Hamiltonians and the exact solutions of the corresponding infinite-range Hamiltonians is assumed to be of general validity. It is therefore desirable to formulate the magnetization equation for infinite-range spin Hamiltonians of as general a form as possible. Let us briefly point out some of the recent developments towards this goal. The anisotropic infinite-range Heisenberg Hamiltonian H = -- ½ ~ saJc, os~j ",O
The generalization of equation (1) to an arbitrary isotropic spin Hamiltonian H = H(s), was shown to be [111
1 Igx, g,} = ~ " igz.
(3)
In the thermodynamic limit the components of the total spin can therefore be considered as commuting variables, which means that sx, gy and gz have a common set of eigenfunctions, with eigenvalues which we denote by s,,, sy, sz. The degeneracy with respect to sz = s~ + s~ + sz2 is given by [111 ~2(N, s) -~ exp [--N(l~s + In C)], where C = shOl/2)/sh [#" (o + ½)],
689
690
AN ARBITRARY INFINITE-RANGE SPIN HAMILTONIAN
and
(14) we shall now show that it entails the various special cases mentioned above. For an isotropic spin ilamiltonian, H(s), VsH = dH/ds • s/s, so that equation (14)becomes s = oBol3oldH/dsll" s/s, from which equation (4) follows. In this case the direction of the magnetization is arbitrary. For an axially symmetric spin Hamiltonian H(s, sz ) we obtain
s = o'Bo[oU]. The canonical partition function is Z = ~ ~(N, s)
exp [-3H(sx, su, sz)].
2~
$
SX,
SZ
Sy,
2
2
2
(Sx+Sy+S
z = S ;~)
To obtain the maximum term we replace H with H* = H - X ( s 2 -
2 -.4
,: -
Z
Z
$
SX,
By,
(O~ s,: OH sy OH s~ + OsH.)
-
Vsn
where ), is a Lagrangian multiplier, write Z =
Vol. 52, No. 7
,
=
- -
•
-
-
,
s ' as
.
.
.
.
s " as
.
,
s
so that
exp [ - 3 H * + I n n ] , Sz
2 2 22 (Sx+~y+Sz-s )
iv ul =
and require Equation (14) can be written in the form
0 as [ - 3 H * + l n ~ 2 ] = 0 I
(6) s,~
=
(OH~as)" (s,~/s)
- o B o ~olV~HI]
•
IVsH I
0 - - [ - 3 H *
=
+ln~]
Osa
0;
a=x,y,z,
(7)
and (s)
From equation (6) we obtain
s = aBo[[3o2Xs],
(9)
- I OH 2), Osa
c~= x , y , z .
(11)
oH~as,, meaning (aH/as,,),, ,~ etc. Substitution of equation (11) in equation (9) and (10) yields
(OH/Os).(sz/s) + (OH/OsA IVsHI '(17)
(a) s,, = sy = O, sz = s:
OH+ OH[ dH(s, Sz=S) 0~ ~s s z = s ds
so that the z equation becomes
s = oBol3oldH/ds[ ].
(18)
( b ) Sx = sy 4: O:
s = -oBo[3olVsHI]" OH/as
(19)
IVsHI (12)
Using equation (19), tile z equation (17), can be written in the form
( 13 )
s~ = sz
and s,~ = - s
•
In this case the x and y equations become
V , H = i OH/Osx + j OH/OSy + k OH/Osz,
OH/Osc,
oBo 13olVsH I1
This set of equations can have solutions of tile following types:
IVsnl =
where 7 , H is the gradient of the spin Hamiltonian with respect to the spin components
s = oBo[3OlV, Ht],
-
(10)
Substitution of equation (10) in equation (8) results in 2), = IVsHI/s.
=
In this case the x and y equations are satisfied trivially and
and from equation (7)
s~ = - - - -
(16)
and .%
=s
a = x,),,
oBo[3olVsHI]
OH
IV, HI
'0s--~ '
IV, HI which is only possible if
Finally, substituting equation (12) in equation (13), we obtain the general magnetization equation s = - o B o [ 3 o l V , H1] • V,H IV,HI"
(14)
As a preliminary illustration of the use of equation
(OH/Osz), = 0.
(20)
Using this result, equation (15) becomes IVsnl = IOH/Osl and, therefore, equation (19) becomes
s = oBo[3010H/Osl].
(21)
Vol. 52, No. 7
AN ARBITRARY INFINITE-RANGE SPIN HAMILTONIAN
Thus, in this case s and Sz are determined by solving the coupled pair of equations (20) and (21 ), which is identical with the appropriate case of equation (5), the other case corresponding to the solution of type a, discussed above. Finally, for the general anisotropic Heisenberg Hamiltonian, equation (2), the magnetization equation, equation (14), reduces to equation (3).
Acknowledgement - Support of this research by the Fund for the Promotion of Research at the Technion and the Technion - V.P.R. Fund is acknowledged. REFERENCES C. Kittel & H. Shore, Phys. Rev. A138, 1165 (1965).
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
691
R.G. Bowers & A. McKerrell, J. Phys. C5, 2392 (1974). P.A.J. Tindemans & H.W. Capel, Physica 72,433 (1974); Physica 75,407 (1974). M.A. Anisimov, E.E. Gorodetskii & V.M. Zaprudskii, Soy. Phys. Usp. 24, 57 (1981). E.L. Nagaev, Soy. Phys. Usp. 25, 31 (1982). P.A. Pearce & C.J. Thompson, Commun. Math. Phys. 41, 191 (1975). G. Vertogen & A.S. De Vries, Physica 59,634 (1972). R. Dekeyser & M.H. Lee, Phys. Rev. BI9,265 (1979). I.M. Kim & M.H. Lee, Phys. Rev. B24, 3961 (1981). M.H. Lee. J. Math. Phys. 23,464(1982). G.F. Kventsel & J. Katriel, J. Appl. Phys. 50, 1820 (1979). J. Katriel & G.F. Kventsel, Phys. Rev. B (in press).
Solid State Communications, Vol. 52, No. 7, pp. 689-691, 1984. Printed in Great Britain.
THE MAGNETIZATION EQUATION FOR AN ARBITRARY INFINITE-RANGE SPIN HAMILTONIAN J. Katriei and G.F. Kventsel Department of Chemistry, Technion - Israel Institute of Technology, 32000 Haifa, Israel (Received 27 March 1984 by H. S u h l )
The equation s = --oBo[~olV, HI] V , H / I V s H I is shown to be the magnetization equation for the arbitrary anisotropic infinite-range spin Hamiltonian, H(s). The known equations for the general anisotropic-Heisenberg, and for arbitrary isotropic and axially symmetric spin Hamiltonians are shown to be special cases of this equation. FOR THE INFINITE range Heisenberg Hamiltonian H = (J/2)s 2 the magnetization equation
(1)
s = oBo[[JoJs],
The magnetization equation of an arbitrary axially symmetry infinite-range spin Hamiltonian, which can be written in terms o f s 2 and sz, was recently derived [12]. The total magnetization s and its z component sz are obtained from
and either
a=x,y,z.
#
The special case of an axially symmetric Heisenberg Hamiltonian H = -- [a(s~ + s~) + bs2z] was studied by Vertogen and De Vries [7] and by Lee etal. [8-101, who showed that the only two types of solutions are Sz = 0 o r s x
=sy =0.
or
Sz = s.
N 1 ~. se, i,
ga
ot=x.v,z,
N i=t
which is of the order of magnitude of the elementary spin, satisfies
was treated by Pearce and Thompson [6], who obtained the magnetization equation
(J.s),~ = ~ J~as~;
(~H/~sz)s = 0
For the axially symmetric Heisenberg Hamiltonian these equations reduce to those of [8-10], but in general a solution of an intermediate type, i.e. Sx = sv :# 0 and sz :# 0 is also possible. To treat the general infinite range spin Hamiltonian H(s), we note that
Ot,[J = x , y , z ,
where
(5)
s = aBo[--~o d H / d s l ,
(2)
J's s = oao[~JolJ'sl]'lJ.s I ,
(4)
s = oBol-#odH/ds].
was derived a long time ago. This equation is identical with the mean-field equation obtained for the nearestneighbour Heisenberg Hamiltonian [ 1-3 ]. In recent years, the interest in the study of spin Hamfltonians which are more complex than the Heisenberg Hamiltonian has considerably increased [4, 5]. The equivalence between the mean-field approximation for finite-range spin-Hamiltonians and the exact solutions of the corresponding infinite-range Hamiltonians is assumed to be of general validity. It is therefore desirable to formulate the magnetization equation for infinite-range spin Hamiltonians of as general a form as possible. Let us briefly point out some of the recent developments towards this goal. The anisotropic infinite-range Heisenberg Hamiltonian H = -- ½ ~ saJc, os~j ",O
The generalization of equation (1) to an arbitrary isotropic spin Hamiltonian H = H(s), was shown to be [111
1 Igx, g,} = ~ " igz.
(3)
In the thermodynamic limit the components of the total spin can therefore be considered as commuting variables, which means that sx, gy and gz have a common set of eigenfunctions, with eigenvalues which we denote by s,,, sy, sz. The degeneracy with respect to sz = s~ + s~ + sz2 is given by [111 ~2(N, s) -~ exp [--N(l~s + In C)], where C = shOl/2)/sh [#" (o + ½)],
689
690
AN ARBITRARY INFINITE-RANGE SPIN HAMILTONIAN
and
(14) we shall now show that it entails the various special cases mentioned above. For an isotropic spin ilamiltonian, H(s), VsH = dH/ds • s/s, so that equation (14)becomes s = oBol3oldH/dsll" s/s, from which equation (4) follows. In this case the direction of the magnetization is arbitrary. For an axially symmetric spin Hamiltonian H(s, sz ) we obtain
s = o'Bo[oU]. The canonical partition function is Z = ~ ~(N, s)
exp [-3H(sx, su, sz)].
2~
$
SX,
SZ
Sy,
2
2
2
(Sx+Sy+S
z = S ;~)
To obtain the maximum term we replace H with H* = H - X ( s 2 -
2 -.4
,: -
Z
Z
$
SX,
By,
(O~ s,: OH sy OH s~ + OsH.)
-
Vsn
where ), is a Lagrangian multiplier, write Z =
Vol. 52, No. 7
,
=
- -
•
-
-
,
s ' as
.
.
.
.
s " as
.
,
s
so that
exp [ - 3 H * + I n n ] , Sz
2 2 22 (Sx+~y+Sz-s )
iv ul =
and require Equation (14) can be written in the form
0 as [ - 3 H * + l n ~ 2 ] = 0 I
(6) s,~
=
(OH~as)" (s,~/s)
- o B o ~olV~HI]
•
IVsH I
0 - - [ - 3 H *
=
+ln~]
Osa
0;
a=x,y,z,
(7)
and (s)
From equation (6) we obtain
s = aBo[[3o2Xs],
(9)
- I OH 2), Osa
c~= x , y , z .
(11)
oH~as,, meaning (aH/as,,),, ,~ etc. Substitution of equation (11) in equation (9) and (10) yields
(OH/Os).(sz/s) + (OH/OsA IVsHI '(17)
(a) s,, = sy = O, sz = s:
OH+ OH[ dH(s, Sz=S) 0~ ~s s z = s ds
so that the z equation becomes
s = oBol3oldH/ds[ ].
(18)
( b ) Sx = sy 4: O:
s = -oBo[3olVsHI]" OH/as
(19)
IVsHI (12)
Using equation (19), tile z equation (17), can be written in the form
( 13 )
s~ = sz
and s,~ = - s
•
In this case the x and y equations become
V , H = i OH/Osx + j OH/OSy + k OH/Osz,
OH/Osc,
oBo 13olVsH I1
This set of equations can have solutions of tile following types:
IVsnl =
where 7 , H is the gradient of the spin Hamiltonian with respect to the spin components
s = oBo[3OlV, Ht],
-
(10)
Substitution of equation (10) in equation (8) results in 2), = IVsHI/s.
=
In this case the x and y equations are satisfied trivially and
and from equation (7)
s~ = - - - -
(16)
and .%
=s
a = x,),,
oBo[3olVsHI]
OH
IV, HI
'0s--~ '
IV, HI which is only possible if
Finally, substituting equation (12) in equation (13), we obtain the general magnetization equation s = - o B o [ 3 o l V , H1] • V,H IV,HI"
(14)
As a preliminary illustration of the use of equation
(OH/Osz), = 0.
(20)
Using this result, equation (15) becomes IVsnl = IOH/Osl and, therefore, equation (19) becomes
s = oBo[3010H/Osl].
(21)
Vol. 52, No. 7
AN ARBITRARY INFINITE-RANGE SPIN HAMILTONIAN
Thus, in this case s and Sz are determined by solving the coupled pair of equations (20) and (21 ), which is identical with the appropriate case of equation (5), the other case corresponding to the solution of type a, discussed above. Finally, for the general anisotropic Heisenberg Hamiltonian, equation (2), the magnetization equation, equation (14), reduces to equation (3).
Acknowledgement - Support of this research by the Fund for the Promotion of Research at the Technion and the Technion - V.P.R. Fund is acknowledged. REFERENCES C. Kittel & H. Shore, Phys. Rev. A138, 1165 (1965).
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
691
R.G. Bowers & A. McKerrell, J. Phys. C5, 2392 (1974). P.A.J. Tindemans & H.W. Capel, Physica 72,433 (1974); Physica 75,407 (1974). M.A. Anisimov, E.E. Gorodetskii & V.M. Zaprudskii, Soy. Phys. Usp. 24, 57 (1981). E.L. Nagaev, Soy. Phys. Usp. 25, 31 (1982). P.A. Pearce & C.J. Thompson, Commun. Math. Phys. 41, 191 (1975). G. Vertogen & A.S. De Vries, Physica 59,634 (1972). R. Dekeyser & M.H. Lee, Phys. Rev. BI9,265 (1979). I.M. Kim & M.H. Lee, Phys. Rev. B24, 3961 (1981). M.H. Lee. J. Math. Phys. 23,464(1982). G.F. Kventsel & J. Katriel, J. Appl. Phys. 50, 1820 (1979). J. Katriel & G.F. Kventsel, Phys. Rev. B (in press).