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Variational approach to finite-temperature magnetism
Journal of Magnetism and Magnetic Materials 54-57 (1986) 1231-1232
VARIATIONAL
APPROACH
Yoshiro KAKEHASHI Max
Planck
Institut
1231
TO FINITE-TEMPERATURE
MAGNETISM
and Peter FULDE
ftir Fesrkijrperforschung,
7000 Stuttgart
80. Fed. Rep. Germany
A finite-temperature theory of magnetism is developed which includes local electron correlations and is based on a variational method. It reduces to the Gutzwiller-type ground state at T = 0, and to the static approximation in the high-temperature limit. A large reduction of T, is found in Fe due to correlations.
Recent theories of finite temperature magnetism qualitatively describe itinerant as well as localized features of 3d transition metals [l]. The microscopic theories which determine self-consistently the magnetization, the Curie temperature, and the susceptibility are all based on the static approximation to the functional integral method [2-61. The static approximation is a high-temperature one, and therefore encounters some difficulties at low temperatures. In particular it reduces to the Hartree-Fock one at T = 0. Thus, the magnetic energy is overestimated as it is well known from ground-state investigations. In the actual application of the static approximation this theoretical problem is avoided by introducing phenomenologically a reduced Coulomb interaction i&r. In this paper we present a theory which improves the shortcomings of the static approximation mentioned above, and discuss the correlation effects at finite temperatures by choosing Fe as an example. The correlated motion of electrons certainly exists even above T, because the related energy gain is much larger than T,. Therefore one can adiabatically take account of the correlation energy which is missing in the static approximation. This is made possible by the variational principle for the free energy. In the following we adopt the single-band Hubbard model for brevity. According to the functional integral method the free energy F can be expressed as follows by an effective energy functional E( 5, T) with the static field variables .$=(t,. t2,...,tN) where N is the number of sites,
For any trial free-energy functional write down Feynman’s inequality FrF,+(E(S.
7--&(5,
E,(t,
T)),,
T) one can
(2)
where ( - jt E /rn,d5,I(-)exp(-pE,(5, T))/ /[II,d<,]exp( -P&(5, T)). 6 is the free energy for the trial energy functional E,([, T). We choose the energy functional E, so that the ground state energy agrees with that of a Gutzwiller-type of wave function f&(5, n(t)>
T)=
0304-8853/86/$03.50
E,,(U)
+
(Q(v)fiQb~))o~.
0 Elsevier
Science
(3) Publishers
Q(v) =
-~AW4)2)oE”2~(l -vr(t)O,)> I
or=h - (n,doE)(nlL- (flzJo&.
(4 (5)
Here E,,(tT) is the energy functional in the static approximation and fi = H - (H),[. The projection operator Q(q) of the Gutzwiller type describes the correlated electron motion [7,8]. The average ( - )ot is taken with respect to the one-electron state with given random exchange fields 5. n,, is the electron number operator with spin u on site i. The correlation parameters TJ(5) = ( TJ,, nZ,. , qN) are determined by variation from eq. (2).
The present free energy reduces to the Gutzwiller-type correlated ground state at T = 0 and agrees with that of the static approximation in the high-temperature limit. Therefore it will certainly improve the static approximation. Other thermodynamical quantities are derived from F,. We show in fig. 1 as a numerical example the results for Fe. The single-site approximation has been made in this calculation. The electron correlations reduce the Curie temperature by a factor of three. This is due to the reduction of the energy difference between the ferromagnetic ground state and the paramagnetic state above T,. The susceptibility follows a Curie-Weiss law. The Curie constant is approximately the same as in the static approximation. The effects of correlations on the magnetization and susceptibility can be simulated by the static approximation with an effective Coulomb interaction U,tr( = 0.7U) (see fig. 1). However, the degree of localization due to electron correlations is underestimated in this case. Thus the reduced magnetization vs. temperature curve considerably deviates from a Brillouin curve. The amplitude of the local moment is underestimated, and the charge fluctuation is overestimated as shown in fig. 1. This shortcoming becomes severe when one is trying to explain the cohesive properties at finite temperatures [9]. In summary we have developed a finite temperature B.V.
theory which is consistent with the Gutzwiller-type of many-body theory at T= 0. and also with the static approxlmntion at high temperatures. Such a theory ih needed in order to explain consistently various thermodynamic quantities.
(l”B)
06
04
004
[II t!lectron Correlation 02
0
0
0010
Fig. 1. Magnetization tude
of
local
(m),
moment
0020
inverse susceptibility m.
and
charge
’ 9 I/(( 6n )-) for Fe as a function of temperature.
002
PI VI
0
[41
2T/W
x- ‘. amplifluctuation
The d electron number II = 7.2/5, the d band width W = 0.45 Ry. the Coulomb the different quanintegral U = 0.52 Ry are assumed. -: tities within the variational approach. : (m} in the xtatic : the different quantities in the static approximation. approximation but with c/,r, = 0.694U. The scale on the r.h.s. refers to x ‘. The model density of states is shown in the inset together with the definition of d-band width W.
[51 [61 171 [X] [9]
and Magnetism in Narrow Band SW terns. ed. T. Moriya (Springer-Verlag. Berlin. 1981). 1. Hubhard. Phys. Rev. B 19 (1979) 2626. B 20 (1979) 4.5X4. H. Hasegawa. J. Phys. Sot. Japan 46 (1979) 1504. 49 ( 19X0) 17X. K. l!\ami and 7. MorIya. J. Magn. Magn. Mat. 20 (19X0) 171. T’. Moriya and H. Hasegawa, J. Phyh. Sot. Japan 48 (19X0) 1490. Y. Kakehashs. J. Phys. Sot. Japan 50 (19X1) 1.505, 3620. M.C. Gutzwiller. Phys. Rev. 134 (1964) A293. 137 (1965) A 1726. G. Stollhoff and P. Fulde. Z. Phys. B 29 (197X) 231: J. C‘hem. Phys. 73 (1980) 454X. Y. Kakehashi and J.H. Samson. in preparation.
VARIATIONAL
APPROACH
Yoshiro KAKEHASHI Max
Planck
Institut
1231
TO FINITE-TEMPERATURE
MAGNETISM
and Peter FULDE
ftir Fesrkijrperforschung,
7000 Stuttgart
80. Fed. Rep. Germany
A finite-temperature theory of magnetism is developed which includes local electron correlations and is based on a variational method. It reduces to the Gutzwiller-type ground state at T = 0, and to the static approximation in the high-temperature limit. A large reduction of T, is found in Fe due to correlations.
Recent theories of finite temperature magnetism qualitatively describe itinerant as well as localized features of 3d transition metals [l]. The microscopic theories which determine self-consistently the magnetization, the Curie temperature, and the susceptibility are all based on the static approximation to the functional integral method [2-61. The static approximation is a high-temperature one, and therefore encounters some difficulties at low temperatures. In particular it reduces to the Hartree-Fock one at T = 0. Thus, the magnetic energy is overestimated as it is well known from ground-state investigations. In the actual application of the static approximation this theoretical problem is avoided by introducing phenomenologically a reduced Coulomb interaction i&r. In this paper we present a theory which improves the shortcomings of the static approximation mentioned above, and discuss the correlation effects at finite temperatures by choosing Fe as an example. The correlated motion of electrons certainly exists even above T, because the related energy gain is much larger than T,. Therefore one can adiabatically take account of the correlation energy which is missing in the static approximation. This is made possible by the variational principle for the free energy. In the following we adopt the single-band Hubbard model for brevity. According to the functional integral method the free energy F can be expressed as follows by an effective energy functional E( 5, T) with the static field variables .$=(t,. t2,...,tN) where N is the number of sites,
For any trial free-energy functional write down Feynman’s inequality FrF,+(E(S.
7--&(5,
E,(t,
T)),,
T) one can
(2)
where ( - jt E /rn,d5,I(-)exp(-pE,(5, T))/ /[II,d<,]exp( -P&(5, T)). 6 is the free energy for the trial energy functional E,([, T). We choose the energy functional E, so that the ground state energy agrees with that of a Gutzwiller-type of wave function f&(5, n(t)>
T)=
0304-8853/86/$03.50
E,,(U)
+
(Q(v)fiQb~))o~.
0 Elsevier
Science
(3) Publishers
Q(v) =
-~AW4)2)oE”2~(l -vr(t)O,)> I
or=h - (n,doE)(nlL- (flzJo&.
(4 (5)
Here E,,(tT) is the energy functional in the static approximation and fi = H - (H),[. The projection operator Q(q) of the Gutzwiller type describes the correlated electron motion [7,8]. The average ( - )ot is taken with respect to the one-electron state with given random exchange fields 5. n,, is the electron number operator with spin u on site i. The correlation parameters TJ(5) = ( TJ,, nZ,. , qN) are determined by variation from eq. (2).
The present free energy reduces to the Gutzwiller-type correlated ground state at T = 0 and agrees with that of the static approximation in the high-temperature limit. Therefore it will certainly improve the static approximation. Other thermodynamical quantities are derived from F,. We show in fig. 1 as a numerical example the results for Fe. The single-site approximation has been made in this calculation. The electron correlations reduce the Curie temperature by a factor of three. This is due to the reduction of the energy difference between the ferromagnetic ground state and the paramagnetic state above T,. The susceptibility follows a Curie-Weiss law. The Curie constant is approximately the same as in the static approximation. The effects of correlations on the magnetization and susceptibility can be simulated by the static approximation with an effective Coulomb interaction U,tr( = 0.7U) (see fig. 1). However, the degree of localization due to electron correlations is underestimated in this case. Thus the reduced magnetization vs. temperature curve considerably deviates from a Brillouin curve. The amplitude of the local moment is underestimated, and the charge fluctuation is overestimated as shown in fig. 1. This shortcoming becomes severe when one is trying to explain the cohesive properties at finite temperatures [9]. In summary we have developed a finite temperature B.V.
theory which is consistent with the Gutzwiller-type of many-body theory at T= 0. and also with the static approxlmntion at high temperatures. Such a theory ih needed in order to explain consistently various thermodynamic quantities.
(l”B)
06
04
004
[II t!lectron Correlation 02
0
0
0010
Fig. 1. Magnetization tude
of
local
(m),
moment
0020
inverse susceptibility m.
and
charge
’ 9 I/(( 6n )-) for Fe as a function of temperature.
002
PI VI
0
[41
2T/W
x- ‘. amplifluctuation
The d electron number II = 7.2/5, the d band width W = 0.45 Ry. the Coulomb the different quanintegral U = 0.52 Ry are assumed. -: tities within the variational approach. : (m} in the xtatic : the different quantities in the static approximation. approximation but with c/,r, = 0.694U. The scale on the r.h.s. refers to x ‘. The model density of states is shown in the inset together with the definition of d-band width W.
[51 [61 171 [X] [9]
and Magnetism in Narrow Band SW terns. ed. T. Moriya (Springer-Verlag. Berlin. 1981). 1. Hubhard. Phys. Rev. B 19 (1979) 2626. B 20 (1979) 4.5X4. H. Hasegawa. J. Phys. Sot. Japan 46 (1979) 1504. 49 ( 19X0) 17X. K. l!\ami and 7. MorIya. J. Magn. Magn. Mat. 20 (19X0) 171. T’. Moriya and H. Hasegawa, J. Phyh. Sot. Japan 48 (19X0) 1490. Y. Kakehashs. J. Phys. Sot. Japan 50 (19X1) 1.505, 3620. M.C. Gutzwiller. Phys. Rev. 134 (1964) A293. 137 (1965) A 1726. G. Stollhoff and P. Fulde. Z. Phys. B 29 (197X) 231: J. C‘hem. Phys. 73 (1980) 454X. Y. Kakehashi and J.H. Samson. in preparation.