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Alloy analogy of the doubly degenerate Hubbard model
Solid State Communications, Vol.21, pp. 837—840, 1977.
Pergamon Press,
Printed in Great Britain
ALLOY ANALOGY OF THE DOUBLY DEGENERATE HUBBARD MODEL C. Lacroix-Lyon-Caen and M. Cyrot Laboratoire de Magnétisme, CNRS, 166X, 38042, Grenoble Cedex, France (Received 18 November 1976 by E.F. Bertaut) The doubly degenerate Hubbard model is studied in the CPA alloy analogy approximation. In the strong coupling limit, a ferromagnetic instability occurs when 1
the ferromagnetism, whereas the alloy analogy describes well the ferromagnetic properties of the degenerate Hubbard model. We consider the following expression for the doubly degenerate Hubbard model (neglecting the spin flip terms): H
=
~
tijCtrnoCima + Ui ~ ~imt~im~
4f,m,o
(1) + U 2
correct results concerning the ferromagnetic instability, in accordance with7 the exact results which we have recently obtained. Brouers and Ducastelle5 already applied the CPA
i,m
~
~imt~irn’~ + (U2 —J) ~
~ii(J~i2(J
i,m * rn
nearest hopping intraatomic where mneighbour is the orbital indiceintegral. (m = 1The or 2); ~ is the Coulomb energies U 1, U2 and J are related by U1 U2 = 2J In the alloy analogy, the motion of the (m, a) elec—
alloy analogy to the degenerate Hubbard model, in the case U1 = U2, U1 and U2 being respectively intraorbital and interorbital coulomb energies, with a nonzero intraatomic exchange energy J; However using their expression of the susceptibility in the paramagnetic state, it can be shown that in the strong coupling limit and when U1 ~ U2, there is no ferromagnetic instability for any7 value electron the exact show of thatthethe groundconcentration; state is ferromagnetic results when the average number of electrons per site, N, is between one and three (excepted N = 2): in the following, we will prove that this result can be obtained when the alloy analogy is done correctly. Mizia8 treated only the intraorbital Coulomb energy terms in the alloy analogy and the interorbital
trons is approximately described by the Hamiltonian: 1a = ~ tuCtmacjma + ~ E~n H 1~0 (2) if
where E~depends on i.the occupation the three otherthe states (m’, a’) of site The different of values of E~and corresponding probabilities are given in Table 1. The same alloy analogy has been done by Brouers and Ducastelle ;5 however in their paper, the probabilities ~x are not correct as can be seen by comparing Table 1 with the table given in their paper: their expressions are correct only if (nji~n~ 20) = (njlQ>(nj2a> etc which is in general not valid. The average Green function is calculated in the 1° Coherent Potential Approximation: Gtm0 = 1 8 N w e~ ~(w) = (3) . . . ,
coulomb energy terms in the Hartree Fock approximation: he found that ferromagnetism occurs for 1111 values ofNifJis sufficiently large. Heiner and Haubenreisser9 treated the sameSchneider, model using the first Hubbard approximation:1 they also obtained ferromagnetism for allN< 1. We will show that in the alloy analogy approximation there is no ferromagnetic instability when N < 1, in the strong coupling limit: both the Hartree—Fock and the first Hubbard approximations increase the effects of correlations and favor
—
—
where G~’0is the partial Green function: Gm° GXm° = 1 +~ E~)Grn~. —
837
(4)
838
ALLOY ANALOGY OF THE HUBBARD MODEL
The average electron numbers <~mo)satisfy the following selfconsistent equations: 8
~rf [_ Im EF
~
Gx~°(w)]dw
(5)
Vol.21, No.8
partially filled. There is no doubly occupied site. With the the density of states (7) the susceptibility can be easily calculated in the paramagnetic state: x = —~— (8) 1—a
where the P~ depend on
7). In the appendix it is shown how expressions for the correlation functions and
{
f + P~ f
zlt~2~ =—p~ EF (—-~-‘mc~t(w)) d~ EF (—
EF $ (—
+ p~t
-~
\
-~-ImG~(w)) dw
)
11m ~
d
f
(—
=
p(E~)
a
=
I ~ _1a~~ sin .
E~.
iT’
+
P1
=
1
—
3N/4.
Thus the susceptibility is finite for all values of N< 1 (in fact this result is independant of the shape of the
and <~jltfli2t> + N—i. However equations (5) and (6) are not sufficient to determine completely the there solution for each value of the the solution: stable state canisbeone found by minimizing the total magnetisation M = (fl124.>. So energy. EF m0(z)dz — (U E= ~ zp 2 —J)[(njltnI2t) + (n114n124.)]
W
IT
EF
+ p~t
X0
density In the of states), case andbands the ground thebandsX=lare state is paramagnetic. pletely filled and1
iT
-
with
5
-KIm GLt) dw
~00
EF
+p~t
f (lImG~)
dw
J_
dw
_U2[(niltnI24.>+]
— U1[(n11tn~j4> + (fli2f~hi24.>I (9) (TheUsing been last taken three the intoexpression terms account must twice (7) be for included inthe the density H~°). since they of states, haveit
—~
.
+ P~— EF ci (— / —Im 1 IT G52 1
ci EF
/
+ p~t
00
dw
IT
)
5 (— -~-i~G~ EF -
can be shown by an expansion of the energy, that the paramagnetic state ((fljrnq> = N/4) corresponds to a maximum of E and the completely ferromagnetic
2
+ P~ J (— — Im G6 -
00
dWJ
(6)
.
Similar expressions can be written for (fljitflj~4.>,
and J -* 00 with fixed value for f/U2 (and U2/U1). Thus the bands11areUsing split an andinitial each elliptic of themdensity contains of exactly states: P~ p states. 2)[W2 — z2] 1/2 for z~< W, it can be 0(z) =easily (2/irWthat the alloy density of states is: shown 2 pm0(z)
=
—~
[P’~°W2 — (z
—
E~)2]~2.
We consider the following two cases: (1) N = 1, (2) 1 2 can be deduced using the electron—hole symmetry. In the case N < 1, only the four bands X = 1 are a (~lma)<
—
(7)
shows the variation of E(M) some values (<1himt) =N/2, <~im4.>= 0)tofora minimum of of E. the Fig. I
electron number. The lowest energy state is always the completely ferromagnetic state. All these results are in good agreement with the 7 the ferromagnetic instability occurs exact results: when 1. We consider first the non degenerate Hubbard model. Using the notations of Hubbard’s paper3 we write the
Vol. 21, No.8
ALLOY ANALOGY OF THE HUBBARD MODEL
839
Table 1. Configurations for the it electrons Configuration of the states
X
Probability Px <(1 —n,14.)(l —nI2t)(l —n~24.))
i.1
2t
2~
1
0
0
0
0
2 3 4 5 6 7 8
0 0 14. 0 14. 14. 1.1.
2t 0 0 2t 2t 0 2t
0 2~ 0 2.1~ 0 21. 21.
U2— J U2 U1
2112 —J U1+U2—J U1 + U2 U1 + 2U2 —J
M/ N 0
0
0.5
Using the CPA expression of 1
1 G°(~)=—~ 1
—.-————---_.
..
Energy Ex
eik~Rj)
N~ Wk~(W)
we have: ~
=
)
Now
2 -0.2
=
equation of motion of the Green function: wG~(w) =
+
~
t~G,~1(w) + UI’~(w) (Al)
k
G°~‘ U
=
~
Fig. 1. Variation of the energy ~=E—(U2 J) (N — 1) as a function of the magnetisation M/N for different values of the electrons number N: Curve 1: N = 1.02; Curve 2: N= 1, 2; Curve 3: N= 1.5; Curve 4: N = 1.9.
—
‘~-~J — =
<~. UT’
JO
<(fl1_0c~~ c70>>.
—
~
Jim [F~(w) + F,~(~)] dw
~:~r1m
+ ntf
l+(~t_ U)G,~’
Im 1 +(~~ — U)Gt1 dwj. (A3)
The first term of(A3)is the number ofup spin electrons on the sites having a down spin electron. Equation (6) is a generalisation of (A3) to the degenerate case. It can easily be shown that equation (6) and the similar equations for the other can be calculated in the same way: it is the number of ~ electrons on the sites having electrons in the states i3 and plus two similar terms. ~‘
REFERENCES I
(A2)
-
HUBBARD J., Proc. R. Soc. A276, 238 (1963).
2.
HUBBARD J., Proc. R. Soc. A277, 237 (1964).
3. 4.
HUBBARD J., Proc. R. Soc. A281,401 (1964). FUKUYAMA H. & EHRENREICH H.,Phys. Rev. B7, 3266 (1973).
840
ALLOY ANALOGY OF THE HUBBARD MODEL
5. 6.
BROUERS F. & DUCASTELLE F., J. Phys. 36, 851(1975). BROUERS F., DUCASTELLE F. & GINER J. (to be published).
7.
LACROIX LYON-CAEN C. & CYROT M. (to be published).
8.
MIZIA J.,Phys. Status Solidi (b) 74,461(1976).
Vol.21, No. 8
9. 10.
SCHNEIDER J., HEINER E. & HAUBENREISSER W., Phys. Status Solidi (b) 53, 553 (1972). VELICKY B., KIRKPATRICK S. & EHRENREICH H., Phys. Rev. 175, 747 (1968).
11.
With the values of Px of reference 5, in the case of a paramagnetic state fl =
Pergamon Press,
Printed in Great Britain
ALLOY ANALOGY OF THE DOUBLY DEGENERATE HUBBARD MODEL C. Lacroix-Lyon-Caen and M. Cyrot Laboratoire de Magnétisme, CNRS, 166X, 38042, Grenoble Cedex, France (Received 18 November 1976 by E.F. Bertaut) The doubly degenerate Hubbard model is studied in the CPA alloy analogy approximation. In the strong coupling limit, a ferromagnetic instability occurs when 1
the ferromagnetism, whereas the alloy analogy describes well the ferromagnetic properties of the degenerate Hubbard model. We consider the following expression for the doubly degenerate Hubbard model (neglecting the spin flip terms): H
=
~
tijCtrnoCima + Ui ~ ~imt~im~
4f,m,o
(1) + U 2
correct results concerning the ferromagnetic instability, in accordance with7 the exact results which we have recently obtained. Brouers and Ducastelle5 already applied the CPA
i,m
~
~imt~irn’~ + (U2 —J) ~
~ii(J~i2(J
i,m * rn
nearest hopping intraatomic where mneighbour is the orbital indiceintegral. (m = 1The or 2); ~ is the Coulomb energies U 1, U2 and J are related by U1 U2 = 2J In the alloy analogy, the motion of the (m, a) elec—
alloy analogy to the degenerate Hubbard model, in the case U1 = U2, U1 and U2 being respectively intraorbital and interorbital coulomb energies, with a nonzero intraatomic exchange energy J; However using their expression of the susceptibility in the paramagnetic state, it can be shown that in the strong coupling limit and when U1 ~ U2, there is no ferromagnetic instability for any7 value electron the exact show of thatthethe groundconcentration; state is ferromagnetic results when the average number of electrons per site, N, is between one and three (excepted N = 2): in the following, we will prove that this result can be obtained when the alloy analogy is done correctly. Mizia8 treated only the intraorbital Coulomb energy terms in the alloy analogy and the interorbital
trons is approximately described by the Hamiltonian: 1a = ~ tuCtmacjma + ~ E~n H 1~0 (2) if
where E~depends on i.the occupation the three otherthe states (m’, a’) of site The different of values of E~and corresponding probabilities are given in Table 1. The same alloy analogy has been done by Brouers and Ducastelle ;5 however in their paper, the probabilities ~x are not correct as can be seen by comparing Table 1 with the table given in their paper: their expressions are correct only if (nji~n~ 20) = (njlQ>(nj2a> etc which is in general not valid. The average Green function is calculated in the 1° Coherent Potential Approximation: Gtm0 = 1 8 N w e~ ~(w) = (3) . . . ,
coulomb energy terms in the Hartree Fock approximation: he found that ferromagnetism occurs for 1111 values ofNifJis sufficiently large. Heiner and Haubenreisser9 treated the sameSchneider, model using the first Hubbard approximation:1 they also obtained ferromagnetism for allN< 1. We will show that in the alloy analogy approximation there is no ferromagnetic instability when N < 1, in the strong coupling limit: both the Hartree—Fock and the first Hubbard approximations increase the effects of correlations and favor
—
—
where G~’0is the partial Green function: Gm° GXm° = 1 +~ E~)Grn~. —
837
(4)
838
ALLOY ANALOGY OF THE HUBBARD MODEL
The average electron numbers <~mo)satisfy the following selfconsistent equations: 8
~rf [_ Im EF
~
Gx~°(w)]dw
(5)
Vol.21, No.8
partially filled. There is no doubly occupied site. With the the density of states (7) the susceptibility can be easily calculated in the paramagnetic state: x = —~— (8) 1—a
where the P~ depend on
7). In the appendix it is shown how expressions for the correlation functions
{
f + P~ f
zlt~2~ =—p~ EF (—-~-‘mc~t(w)) d~ EF (—
EF $ (—
+ p~t
-~
\
-~-ImG~(w)) dw
)
11m ~
d
f
(—
=
p(E~)
a
=
I ~ _1a~~ sin .
E~.
iT’
+
P1
=
1
—
3N/4.
Thus the susceptibility is finite for all values of N< 1 (in fact this result is independant of the shape of the
and <~jltfli2t> +
W
IT
EF
+ p~t
X0
density In the of states), case andbands the ground thebandsX=lare state is paramagnetic. pletely filled and1
iT
-
with
5
-KIm GLt) dw
~00
EF
+p~t
f (lImG~)
dw
J_
dw
_U2[(niltnI24.>+
— U1[(n11tn~j4> + (fli2f~hi24.>I (9) (TheUsing been last taken three the intoexpression terms account must twice (7) be for included inthe the density H~°). since they of states, haveit
—~
.
+ P~— EF ci (— / —Im 1 IT G52 1
ci EF
/
+ p~t
00
dw
IT
)
5 (— -~-i~G~ EF -
can be shown by an expansion of the energy, that the paramagnetic state ((fljrnq> = N/4) corresponds to a maximum of E and the completely ferromagnetic
2
+ P~ J (— — Im G6 -
00
dWJ
(6)
.
Similar expressions can be written for (fljitflj~4.>,
and J -* 00 with fixed value for f/U2 (and U2/U1). Thus the bands11areUsing split an andinitial each elliptic of themdensity contains of exactly states: P~ p states. 2)[W2 — z2] 1/2 for z~< W, it can be 0(z) =easily (2/irWthat the alloy density of states is: shown 2 pm0(z)
=
—~
[P’~°W2 — (z
—
E~)2]~2.
We consider the following two cases: (1) N = 1, (2) 1
—
(7)
shows the variation of E(M) some values (<1himt) =N/2, <~im4.>= 0)tofora minimum of of E. the Fig. I
electron number. The lowest energy state is always the completely ferromagnetic state. All these results are in good agreement with the 7 the ferromagnetic instability occurs exact results: when 1
Vol. 21, No.8
ALLOY ANALOGY OF THE HUBBARD MODEL
839
Table 1. Configurations for the it electrons Configuration of the states
X
Probability Px <(1 —n,14.)(l —nI2t)(l —n~24.))
i.1
2t
2~
1
0
0
0
0
2 3 4 5 6 7 8
0 0 14. 0 14. 14. 1.1.
2t 0 0 2t 2t 0 2t
0 2~ 0 2.1~ 0 21. 21.
U2— J U2 U1
2112 —J U1+U2—J U1 + U2 U1 + 2U2 —J
M/ N 0
0
0.5
Using the CPA expression of 1
1 G°(~)=—~ 1
—.-————---_.
..
Energy Ex
eik~Rj)
N~ Wk~(W)
we have: ~
=
)
Now
2 -0.2
=
equation of motion of the Green function: wG~(w) =
+
~
t~G,~1(w) + UI’~(w) (Al)
k
G°~‘ U
~
Fig. 1. Variation of the energy ~=E—(U2 J) (N — 1) as a function of the magnetisation M/N for different values of the electrons number N: Curve 1: N = 1.02; Curve 2: N= 1, 2; Curve 3: N= 1.5; Curve 4: N = 1.9.
—
‘~-~J — =
<~. UT’
JO
<(fl1_0c~~ c70>>.
—
~
Jim [F~(w) + F,~(~)] dw
~:~r1m
+ ntf
l+(~t_ U)G,~’
Im 1 +(~~ — U)Gt1 dwj. (A3)
The first term of(A3)is the number ofup spin electrons on the sites having a down spin electron. Equation (6) is a generalisation of (A3) to the degenerate case. It can easily be shown that equation (6) and the similar equations for the other
REFERENCES I
(A2)
-
HUBBARD J., Proc. R. Soc. A276, 238 (1963).
2.
HUBBARD J., Proc. R. Soc. A277, 237 (1964).
3. 4.
HUBBARD J., Proc. R. Soc. A281,401 (1964). FUKUYAMA H. & EHRENREICH H.,Phys. Rev. B7, 3266 (1973).
840
ALLOY ANALOGY OF THE HUBBARD MODEL
5. 6.
BROUERS F. & DUCASTELLE F., J. Phys. 36, 851(1975). BROUERS F., DUCASTELLE F. & GINER J. (to be published).
7.
LACROIX LYON-CAEN C. & CYROT M. (to be published).
8.
MIZIA J.,Phys. Status Solidi (b) 74,461(1976).
Vol.21, No. 8
9. 10.
SCHNEIDER J., HEINER E. & HAUBENREISSER W., Phys. Status Solidi (b) 53, 553 (1972). VELICKY B., KIRKPATRICK S. & EHRENREICH H., Phys. Rev. 175, 747 (1968).
11.
With the values of Px of reference 5, in the case of a paramagnetic state fl =