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Dyson equation for Heisenberg ferromagnet
PHYSICS LETTERS
Volume 43A, number 6
23 April 1973
DYSON EQUATION FOR HEISENBERG FERROMAGNET N.M. PLAKLDA Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Moscow, USSR
Received 7 March 1973 An exact representation for the self-energy operator of the two-time Green function for the transverse spin components for the Heisenberg ferromagnet is obtained.
There is a large number of approximate calculations for the Heisenberg model of a ferromagnet, based on the two-time Green functions [l] in the literature (see e.g. [2] ), the validity of which is usyally difficult to examine. At the same time a diagram method for the spin operators [3] permits, in principle, to evaluate various contributions to the Green functions. In the present paper an exact Dyson equation for the two-time Green function will be obtained and the correspondence between some approximate decoupling for the self-energy part and the calculations in the diagram method [3] will be established. The method of the irreducible Green functions, proposed for the phonon Green functions in our recent papers [4] which is close to the methods in [5,6], will be used. Consider a ferromagnet which can be described by the Heisenberg Hamiltonian H = - 3 ~ij Jij Sisi, Jij = J(rj - rj) and write down the equation of motion for the two-time commutator type Green function Gik(t-f’) = ((Sf(t);Si(t’)) in the form:
where the usual notation identity: (B($)l
is used [ 11. Now we define the irreducible
bS,>) =((lS;Sf-LS)lSf)
part (ir) of the Green function
in (1) by the
-(AijSf-AjiS))ISk))
(2)
where the coefficients A ij are defined by the condition: ( [B(i),
Sk] > = 0.
(3)
Performing the commutation &jkweget(i#j): Aij =Aji =(2(SfSJ)
in (3) with the help of (2) and equating the terms at the equal 6-functions
+(SiSf>)\2(SZ).
8,
and
(4)
The irreducible Green function (B?)(t); Sk (t’)) in (2) is obtained by differentiating it with respect to the second time C’ [4,5] and by introdzcing the irreducible part for the right-hand side operators at time f ’ by analogy with (2,3). Then defining the “zero order” Green function in the Hartree-Fock approximation by the equation [7] wG~O~(O) = 2(Sz)6ik
+ CJijAij i
{Go
- Go”,},
(5)
(1) can be written in the matrix form as G = Go t G”PGo, P = Pik (a). From the Dyson equation in the form G=G”+GoHGwegettheequationP=H+TIGoPfr om which it follows that the self-energy operator Hik (w) is defined by the proper part of the operator Pik (a):
481
Volume
43A, number
6
PHYSICS
LETTERS
23 April
1973
(6) where the proper (p) part of the irreducible Green function has no parts connected by one GO-line. The solution of the Dyson equation by the Fourier transformation can be written in the form Gik(o)
=N-’
C 4
exp [iq(ri - rk)] 2(Sz) [W - Eq - 2(Sz) Ill]-’
The energy of spin excitations
=N-’ CJijAij ij
Eq
in the Hartree-Fock
(1 -exp[iq(ri
where Jq , K 4+, KY
-ri)])
approximation
=(S’)(J,
- Jq) +N-’
(7) (5) is given by C
(Jql - Jq_4r) (Ki?
t Z?K~)/~(S’),
4’
are the Fourier components
of the exchange energy .Q and the correlation
(8)
functions
K I+ = (SLY+), K42 = ((ST - CS”)) (Sz - CS”))) respectively. Eq. (8) was obtained in the first order theory by decotpling [k.:. 7, 2rj and bry diagram method [3], where the first term defines the energy of spin waves in RPA (Tyablikov decoupling [l] ), the second and the third ones decribe an elastic scattering by spin waves (- Ki+) and by the fluctuations of the z-spin components (-Kg”‘). The second and higher order carrections are described by the frequency dependent part of the self-energy in (7) which can be written in the form: 2(SZ) II,(w)
=J _m
dw’ {exp (w’/O)w--WI
x N-1l$ expb&i
- rk)l Aj&
- dt 1) J 2;;exp(-iw’t) _m
ItB+w(t) [k
B(!f))(P)/2(SZ)}. '1
(9)
It gives rise to the damping of the spin waves which can be evaluated by the decoupling of the irreducible two-time correlation function in (9). E.g., an inelastic scattering of the spin waves by the fluctuations of the z-spin components is given from the diagram consideration by the two-time decoupling ( {S:(t) Sk (t)]“” 2(sz)
II
=N-’
{Sfsp(“)) c
w KY:(t) KiJT(t), (Jq, - J,_,fT
which gives the result of [3] :
(w - E~‘_~)-~ K”4;:
4’ if we use approximations: KY(t) z Ky(t=O) and - Im [o - Eq - 2LSz) II4 (wtie)] -’ = rrS (w - e4) for the calculation of K.-+ (t) by the Green function (7). Higher order corrections can be obtained by introducing some representations Qor Sf operators in (9) which will be discussed elsewhere. The author thanks Drs. Yu.G. Rudoj and Yu.A. Tserkovnikov
for useful discussions.
[l] S.V. Tyablikov, Methods in the quantum theory of magnetism (Plenum Press, New York, 1967). [2] Yu.G. Rudoj and Yu.A. Tserkovnikov, Teor. i Mat. Fiz. 14 (1973) 102. [3] V.G. Vaks, AI. Larkin and S.A. Pkin, Zh. Eksp. i Teor. Fiz. 53 (1967) 281, 1089; Yu.A. Izyumov and F.A. Kassan-Ogly, Fiz. Met. i Metalloved. 26 (1968) 385,30 (1970) 225. [4] N.M. Plakida, Teor. i Mat. Fiz. 5 (1970) 147; 12 (1972) 135; H. Konwent and N.M. Plakida, Phys. Lett. 37A (1971) 173. [5] Yu.A. Tserkovnikov, Teor. i Mat. Fiz. 7 (1971) 250; M.W.C. Dharmawardana and C. Mavroyannis, Phys. Rev. Bl (1970) 1166. [6] M. Ichiyanagi, J. Phys. Sot. Jap. 32 (1972) 604. [7] V. Mubayi and R.V. Lange, Phys. Rev. 178 (1969) R.P. Kenan, Phys. Rev. Bl (1970) 3205.
482
882;
Volume 43A, number 6
23 April 1973
DYSON EQUATION FOR HEISENBERG FERROMAGNET N.M. PLAKLDA Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Moscow, USSR
Received 7 March 1973 An exact representation for the self-energy operator of the two-time Green function for the transverse spin components for the Heisenberg ferromagnet is obtained.
There is a large number of approximate calculations for the Heisenberg model of a ferromagnet, based on the two-time Green functions [l] in the literature (see e.g. [2] ), the validity of which is usyally difficult to examine. At the same time a diagram method for the spin operators [3] permits, in principle, to evaluate various contributions to the Green functions. In the present paper an exact Dyson equation for the two-time Green function will be obtained and the correspondence between some approximate decoupling for the self-energy part and the calculations in the diagram method [3] will be established. The method of the irreducible Green functions, proposed for the phonon Green functions in our recent papers [4] which is close to the methods in [5,6], will be used. Consider a ferromagnet which can be described by the Heisenberg Hamiltonian H = - 3 ~ij Jij Sisi, Jij = J(rj - rj) and write down the equation of motion for the two-time commutator type Green function Gik(t-f’) = ((Sf(t);Si(t’)) in the form:
where the usual notation identity: (B($)l
is used [ 11. Now we define the irreducible
bS,>) =((lS;Sf-LS)lSf)
part (ir) of the Green function
in (1) by the
-(AijSf-AjiS))ISk))
(2)
where the coefficients A ij are defined by the condition: ( [B(i),
Sk] > = 0.
(3)
Performing the commutation &jkweget(i#j): Aij =Aji =(2(SfSJ)
in (3) with the help of (2) and equating the terms at the equal 6-functions
+(SiSf>)\2(SZ).
8,
and
(4)
The irreducible Green function (B?)(t); Sk (t’)) in (2) is obtained by differentiating it with respect to the second time C’ [4,5] and by introdzcing the irreducible part for the right-hand side operators at time f ’ by analogy with (2,3). Then defining the “zero order” Green function in the Hartree-Fock approximation by the equation [7] wG~O~(O) = 2(Sz)6ik
+ CJijAij i
{Go
- Go”,},
(5)
(1) can be written in the matrix form as G = Go t G”PGo, P = Pik (a). From the Dyson equation in the form G=G”+GoHGwegettheequationP=H+TIGoPfr om which it follows that the self-energy operator Hik (w) is defined by the proper part of the operator Pik (a):
481
Volume
43A, number
6
PHYSICS
LETTERS
23 April
1973
(6) where the proper (p) part of the irreducible Green function has no parts connected by one GO-line. The solution of the Dyson equation by the Fourier transformation can be written in the form Gik(o)
=N-’
C 4
exp [iq(ri - rk)] 2(Sz) [W - Eq - 2(Sz) Ill]-’
The energy of spin excitations
=N-’ CJijAij ij
Eq
in the Hartree-Fock
(1 -exp[iq(ri
where Jq , K 4+, KY
-ri)])
approximation
=(S’)(J,
- Jq) +N-’
(7) (5) is given by C
(Jql - Jq_4r) (Ki?
t Z?K~)/~(S’),
4’
are the Fourier components
of the exchange energy .Q and the correlation
(8)
functions
K I+ = (SLY+), K42 = ((ST - CS”)) (Sz - CS”))) respectively. Eq. (8) was obtained in the first order theory by decotpling [k.:. 7, 2rj and bry diagram method [3], where the first term defines the energy of spin waves in RPA (Tyablikov decoupling [l] ), the second and the third ones decribe an elastic scattering by spin waves (- Ki+) and by the fluctuations of the z-spin components (-Kg”‘). The second and higher order carrections are described by the frequency dependent part of the self-energy in (7) which can be written in the form: 2(SZ) II,(w)
=J _m
dw’ {exp (w’/O)w--WI
x N-1l$ expb&i
- rk)l Aj&
- dt 1) J 2;;exp(-iw’t) _m
ItB+w(t) [k
B(!f))(P)/2(SZ)}. '1
(9)
It gives rise to the damping of the spin waves which can be evaluated by the decoupling of the irreducible two-time correlation function in (9). E.g., an inelastic scattering of the spin waves by the fluctuations of the z-spin components is given from the diagram consideration by the two-time decoupling ( {S:(t) Sk (t)]“” 2(sz)
II
=N-’
{Sfsp(“)) c
w KY:(t) KiJT(t), (Jq, - J,_,fT
which gives the result of [3] :
(w - E~‘_~)-~ K”4;:
4’ if we use approximations: KY(t) z Ky(t=O) and - Im [o - Eq - 2LSz) II4 (wtie)] -’ = rrS (w - e4) for the calculation of K.-+ (t) by the Green function (7). Higher order corrections can be obtained by introducing some representations Qor Sf operators in (9) which will be discussed elsewhere. The author thanks Drs. Yu.G. Rudoj and Yu.A. Tserkovnikov
for useful discussions.
[l] S.V. Tyablikov, Methods in the quantum theory of magnetism (Plenum Press, New York, 1967). [2] Yu.G. Rudoj and Yu.A. Tserkovnikov, Teor. i Mat. Fiz. 14 (1973) 102. [3] V.G. Vaks, AI. Larkin and S.A. Pkin, Zh. Eksp. i Teor. Fiz. 53 (1967) 281, 1089; Yu.A. Izyumov and F.A. Kassan-Ogly, Fiz. Met. i Metalloved. 26 (1968) 385,30 (1970) 225. [4] N.M. Plakida, Teor. i Mat. Fiz. 5 (1970) 147; 12 (1972) 135; H. Konwent and N.M. Plakida, Phys. Lett. 37A (1971) 173. [5] Yu.A. Tserkovnikov, Teor. i Mat. Fiz. 7 (1971) 250; M.W.C. Dharmawardana and C. Mavroyannis, Phys. Rev. Bl (1970) 1166. [6] M. Ichiyanagi, J. Phys. Sot. Jap. 32 (1972) 604. [7] V. Mubayi and R.V. Lange, Phys. Rev. 178 (1969) R.P. Kenan, Phys. Rev. Bl (1970) 3205.
482
882;