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Long-range orders in the doubly degenerate Hubbard model
Solid State Communications. Vol. 33, pp. 437-439. Pergamon Press Ltd. 1980. Printed in Great Britain. LONG-RANGE ORDERS IN THE DOUBLY DEGENERATE HUBBARD MODEL J. Mizia Gasthaf Mair, A-4880 St. Georgen im Attergau, Austria (Received 20 September 1979 by E. F. Bertaut)
The doubly degenerate Hubbard model is studied in the CPA alloy analogy approximation corrected by Lacroix-Lyon-Caen and Cyrot. The static susceptibility method is used. In the strong scattering limit, a ferromagnetic instability for 1
LACROIX-LYON-CAEN AND CYROT [l] have recently proposed a new method for the calculation of the magnetic instabilities in the framework of the alloy method. Their essential idea lies in the fact that in the expressions for probabilities one has: (nmonm~o~)f (n,,>Ot,~O~), etc., where m and nmo are the orbital We consider the approximate two-band Hamiltonian index and the particle number operator respectively. [l] (m = 1 or 2) and adopt the same notation as [I]. They have shown by the total energy calculation that The intraatomic Coulomb energies are treated in the the ferromagnetic instability occurs for 1 = (n2+u) = 3 n, for ferromagnetism, the energy vs magnetization curve becomes flat. So the problem reduces to the looking for of the denominator zero points. The general expression for x can be written (nl *J = (n2d = n, on a-sites for antiferromagas (nl*,> = h2*J = nr on&sites I netismt 2&P(Q) (1) ’ = 1 -K(N)’ (ni*o) = (nz*to) = n+ on o-sites for charge-orderedwhere K(N) is given by the derivative with respect to (nl,,) = (n25u) = n_ on&sites 1 state. magnetization M
-jpyw2
N+
(5) Im [Go(z) -Gsu(z)]
Now, following the classical CPA method of Brouers and Ducastelle [3] and assuming for simplicity (I, = U2 and J + 0 one gets with the help of equations (5) the following results for the probabilities of the subbands X=landX=2:
dz
Im G”(z) dz 437
438
LONG-RANGE
ORDERS IN THE DOUBLY DEGENERATE
HUBBARD MODEL
Vol. 33, No. 4
06-
04
I
02-
0.2
0
kYkk%Gk
0
Fig. 1. The results for quantities calculated in the classical alloy analogy approximation: (a) static susceptibility denominator for ferromagnetism; (b) denominator for the (o, /3) long-range order; (c) molecular field constant value required for producing (a, /3) order in units of the half-bandwidth. Subbands are filing up at N = 1.272 and 2. = (1 --&)(I
-n+)(l
P2’ = n*(l -n#
-n_)
+ 2n,(l
-n_)(l
for ferro- and antiferro P:
-n+) magnetism,
= (1 - n*)3 for charge-ordered
Pz’ = 3n*(l
-n*>*
1
state. (6)
With these probabilities one gets N = 1.272 and 2 for the insulating points of the subbands X = 1 and X = 2, respectively. For (cr, 0) long-range orders the density of states can be calculated analytically by the method published in [4]. The main idea is that using the Brouers [S ] transformation diagonalizing the two-sublattice Hamiltonian, in the strong scattering limit for ZZand for the semi-elliptic initial state density one obtains a final analytical formula for the densities, which in W = 1 units are G, = 2[z_ - (ZZ -fkPy*], G, = 2[z+ - (z: -P;/_&)““],
(7)
where z* = z-E:
04
06
08
10
I 12
!O
N N
P:
02
= z-Eh+Kn,
Fig. 2. The same quantities as in Fig. 1 calculated in the improved alloy analogy approximation. The subbands are filling up at N = 1 and 2. Above Ei = Eh - Kn, are the energies of f a electrons in any “Weiss field” proportional to mag netization. This field is a characteristic result of the Hartree-Fock approximation, but here it is artificially added to the Hamiltonian in order to investigate how much it is necessary to get alignment. In Fig. 1 the susceptibility denominators vs N for ferromagnetism and (a, /3) alignments calculated on the basis of the above formula are drawn. As follows from this figure the classical CPA does not lead to any order and in the case of (cr, /3) alignment not even to any enhancement to the susceptibility. This fact is well known for ferromagnetism [3,6, 71. For antiferromagnetism it was also established for arbitrary N in [4,8] and additionally proved by Ducastelle [9] for the halffilled band. It seems that in CPA method the situation in the half-filled subbands is more favourable for antiferromagnetism. The curve c in Fig. 1 shows the magnitude of molecular field constant K required to produce (a, 0) order. At the points N = 0.9 16, 1.657 and symmetrically 2.343 and 3.084 constant K decreases to zero which means that in this case the energy difference between paramagnetic and (a, 0) orders toward to zero for order parameter M + 0. Let us consider now the improved [l] formalism. For N < 1 and 1
forN<
and PP = 1 -n, (8)
- 2n_, + (n, _,n,_,)
1
Vol. 33, No. 4
LONG-RANGE
= Pi”
(n,_,n,_,)
7
ORDERS IN THE DOUBLY DEGENERATE
(- i\ Im G;” dz
(9) Integration
in (9) is performed
over the X = 2 sub-
band. Taking equations (9) and (5) (where n, G n,,) and substituting them for the equations (4) or (7) (8) one gets with the help of (2) and (3) the results for the denominators of the magnetic susceptibility shown in Fig. 2. For example, in the case of ferromagnetism and 1 < N < 2 after differentiating Pf from equation (9) with respect to M, one arrives at the following denominator DF = l-4~m~~_?p”(Pp,~)de-4~m~~ + co
--f
EF X
!,P”(&‘, 4 de
HUBBARD MODEL
439
cerning the existence of ferromagnetism for I
ifKrOforh= andA=
I
The points where (cr, 0) order is easily reached by the molecular field are once more localized at the half-filled subbands (now atN = 0.8, 1.33,2.66,3.2).
-I=_
X
I
Acknowledgements - I would like to express my great thanks to Dr Asta Richter from Dresden for preliminary discussions suggesting this problem and also to Prof. M. Cyrot from Grenoble for explaining many details of his corrected CPA method.
1
p-“(P;u,e’)de’
-4$ye&
-c;
x [ PW, e)de
-e;
REFERENCES
47 = 1 - 5 arcsin &
I
1. -4
lim -!MdoaM
2.
-R EF
P-“(p~u,e)de--4
X
a
lim -M-0 aM
EF
,o”(P,“, e) de = 0, (10) -6; where Pf is the paramagnetic limit of Pp. As is seen in Fig. 2 the results presented confirm the Lacroix-Lyon-Caen and Cyrot statement conX
J^
3. 4. 5. 6. 7. 8. 9.
C. Lacroix-Lyon-Caen & M. Cyrot, Solid State Commun 21,837 (1977). A.E.K. Dowson, Solid State Commun. 27,933 (1978). F. Brouers & F. Ducastelle, J. Phys. 36,85 1 (1975). J. Mizia, Phys. Status Solidi (b) 84,449 (1977). F. Brouers, Phys Status Solidi (b) 76,145 (1976). H. Fukuyama & H. Ehrenreich, Phys. Rev. B7, 3266 (1973). J. Mizia. Phvs. Status Solidi lbl74.461 (1976). F. Broukrs,F. Ducastelle & J. &er, J. Phys. $7, 1427 (1976). F. Ducastelle, Phys, Lett. 64A, 229 (1977).
The doubly degenerate Hubbard model is studied in the CPA alloy analogy approximation corrected by Lacroix-Lyon-Caen and Cyrot. The static susceptibility method is used. In the strong scattering limit, a ferromagnetic instability for 1
LACROIX-LYON-CAEN AND CYROT [l] have recently proposed a new method for the calculation of the magnetic instabilities in the framework of the alloy method. Their essential idea lies in the fact that in the expressions for probabilities one has: (nmonm~o~)f (n,,>Ot,~O~), etc., where m and nmo are the orbital We consider the approximate two-band Hamiltonian index and the particle number operator respectively. [l] (m = 1 or 2) and adopt the same notation as [I]. They have shown by the total energy calculation that The intraatomic Coulomb energies are treated in the the ferromagnetic instability occurs for 1
-jpyw2
N+
(5) Im [Go(z) -Gsu(z)]
Now, following the classical CPA method of Brouers and Ducastelle [3] and assuming for simplicity (I, = U2 and J + 0 one gets with the help of equations (5) the following results for the probabilities of the subbands X=landX=2:
dz
Im G”(z) dz 437
438
LONG-RANGE
ORDERS IN THE DOUBLY DEGENERATE
HUBBARD MODEL
Vol. 33, No. 4
06-
04
I
02-
0.2
0
kYkk%Gk
0
Fig. 1. The results for quantities calculated in the classical alloy analogy approximation: (a) static susceptibility denominator for ferromagnetism; (b) denominator for the (o, /3) long-range order; (c) molecular field constant value required for producing (a, /3) order in units of the half-bandwidth. Subbands are filing up at N = 1.272 and 2. = (1 --&)(I
-n+)(l
P2’ = n*(l -n#
-n_)
+ 2n,(l
-n_)(l
for ferro- and antiferro P:
-n+) magnetism,
= (1 - n*)3 for charge-ordered
Pz’ = 3n*(l
-n*>*
1
state. (6)
With these probabilities one gets N = 1.272 and 2 for the insulating points of the subbands X = 1 and X = 2, respectively. For (cr, 0) long-range orders the density of states can be calculated analytically by the method published in [4]. The main idea is that using the Brouers [S ] transformation diagonalizing the two-sublattice Hamiltonian, in the strong scattering limit for ZZand for the semi-elliptic initial state density one obtains a final analytical formula for the densities, which in W = 1 units are G, = 2[z_ - (ZZ -fkPy*], G, = 2[z+ - (z: -P;/_&)““],
(7)
where z* = z-E:
04
06
08
10
I 12
!O
N N
P:
02
= z-Eh+Kn,
Fig. 2. The same quantities as in Fig. 1 calculated in the improved alloy analogy approximation. The subbands are filling up at N = 1 and 2. Above Ei = Eh - Kn, are the energies of f a electrons in any “Weiss field” proportional to mag netization. This field is a characteristic result of the Hartree-Fock approximation, but here it is artificially added to the Hamiltonian in order to investigate how much it is necessary to get alignment. In Fig. 1 the susceptibility denominators vs N for ferromagnetism and (a, /3) alignments calculated on the basis of the above formula are drawn. As follows from this figure the classical CPA does not lead to any order and in the case of (cr, /3) alignment not even to any enhancement to the susceptibility. This fact is well known for ferromagnetism [3,6, 71. For antiferromagnetism it was also established for arbitrary N in [4,8] and additionally proved by Ducastelle [9] for the halffilled band. It seems that in CPA method the situation in the half-filled subbands is more favourable for antiferromagnetism. The curve c in Fig. 1 shows the magnitude of molecular field constant K required to produce (a, 0) order. At the points N = 0.9 16, 1.657 and symmetrically 2.343 and 3.084 constant K decreases to zero which means that in this case the energy difference between paramagnetic and (a, 0) orders toward to zero for order parameter M + 0. Let us consider now the improved [l] formalism. For N < 1 and 1
forN<
and PP = 1 -n, (8)
- 2n_, + (n, _,n,_,)
1
Vol. 33, No. 4
LONG-RANGE
= Pi”
(n,_,n,_,)
7
ORDERS IN THE DOUBLY DEGENERATE
(- i\ Im G;” dz
(9) Integration
in (9) is performed
over the X = 2 sub-
band. Taking equations (9) and (5) (where n, G n,,) and substituting them for the equations (4) or (7) (8) one gets with the help of (2) and (3) the results for the denominators of the magnetic susceptibility shown in Fig. 2. For example, in the case of ferromagnetism and 1 < N < 2 after differentiating Pf from equation (9) with respect to M, one arrives at the following denominator DF = l-4~m~~_?p”(Pp,~)de-4~m~~ + co
--f
EF X
!,P”(&‘, 4 de
HUBBARD MODEL
439
cerning the existence of ferromagnetism for I
ifKrOforh= andA=
I
The points where (cr, 0) order is easily reached by the molecular field are once more localized at the half-filled subbands (now atN = 0.8, 1.33,2.66,3.2).
-I=_
X
I
Acknowledgements - I would like to express my great thanks to Dr Asta Richter from Dresden for preliminary discussions suggesting this problem and also to Prof. M. Cyrot from Grenoble for explaining many details of his corrected CPA method.
1
p-“(P;u,e’)de’
-4$ye&
-c;
x [ PW, e)de
-e;
REFERENCES
47 = 1 - 5 arcsin &
I
1. -4
lim -!MdoaM
2.
-R EF
P-“(p~u,e)de--4
X
a
lim -M-0 aM
EF
,o”(P,“, e) de = 0, (10) -6; where Pf is the paramagnetic limit of Pp. As is seen in Fig. 2 the results presented confirm the Lacroix-Lyon-Caen and Cyrot statement conX
J^
3. 4. 5. 6. 7. 8. 9.
C. Lacroix-Lyon-Caen & M. Cyrot, Solid State Commun 21,837 (1977). A.E.K. Dowson, Solid State Commun. 27,933 (1978). F. Brouers & F. Ducastelle, J. Phys. 36,85 1 (1975). J. Mizia, Phys. Status Solidi (b) 84,449 (1977). F. Brouers, Phys Status Solidi (b) 76,145 (1976). H. Fukuyama & H. Ehrenreich, Phys. Rev. B7, 3266 (1973). J. Mizia. Phvs. Status Solidi lbl74.461 (1976). F. Broukrs,F. Ducastelle & J. &er, J. Phys. $7, 1427 (1976). F. Ducastelle, Phys, Lett. 64A, 229 (1977).