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On instabilities of electron systems in the mean field approximation
Solid State Communications, Vol. 25, pp. 101—103, 1978.
Pergamon Press.
Printed in Great Britain
ON INSTABIUTIES OF ELECTRON SYSTEMS IN THE MEAN FIELD APPROXIMATION* B. Bulka Ferromagnetics Laboratory, Institute of Molecular Physics of the Polish Academy of Sciences, 60—179 Poznarl, Smoluchowskiego 17/19, Poland (Received 27 June 1977 by P. G. de-Gennes) Various instabilities of quasi-one-dimensional electron systems are examined in the mean field approximation. In addition to the charge density wave and the superconducting states, the antiferromagnetic state, described by the different orbitals for different spins and the spin density methods, is also considered. A phase diagram of these states is constructed.
wave vector transfers (~ Q and 0), respectively. g4, represents a small wave vector transfer scattering of electrons from the same side of the FS, while g38 is due to Umldapp processes. s is equal 0 for a = a’ or 1 for a = a’, thus we distinguish interactions of electrons with the same and the different spins. Ifg,,3 = g~1,then we have the case considered by Horovitz [2]. We have investigated stability conditions for the charge density wave (CDW), the antiferromagnetism
RECENTLY extensive theoretical investigations of various instabilities of electrons in quasi-one-dimensional conductors have been performed [1, 2]. Horovitz [2], using the extended Nambu representation and the Hartree—Fock approximation, gave conditions of stability of various phases of an electron system. We have considered a system of electrons described by the Hamiltonian, which is generalization of the one discussed by Horovitz [2] H
=
—
ek÷Q,2(ck+Q,2Ock+Q,20—ck_Q,2Uck_Q,20)
~
+~
~
+!
(E~2scZ
+
—
—
+~
+ Q/2uCk4_
+ Q/2
aCk
+
Q/2aCk2_ Q,2U’ck3+ Q/2a’
Q/2qCk2_ Q/2U’Ck3_ Q12u’
g38c~ + Q/2aC~4— Q/2aCk2+ Q,20’Ck3_ Q/2a’
+ h.c.) + h.c.
}
+ h.c.)
(Q ~—Q)~
(~~4sck÷Q/2ack+Q/2ack+Q/2a~ck+Q/2a~ +
where the prime above the sum sign restricts wave vector summation to Ik,I
(I)
described by the different orbitals for different spins (DODS) and the spin density wave (SDW) methods (cf [4]), the singlet (S) and triplet (T) superconducting states. We introduce a vector field +
+
=
+
(Ck + Q/2+, C_k_ Q/2—’ Ck — Q/2+’ C_k + Q/2)’
(3)
Ek÷Q/2
with Q = (ir/a, ir/a, 2kF) (a is a lattice constant). This is the condition for nesting of the Fermi surface (FS) [3]. The couplings g~,g 28 describe the scattering of electrons from the opposite sides of the FS with large and small
analogically to the Nambu formalism. In this representation we may find the stability equation of the CDW,
*
I 1
.
.
This work was supported by the Pohsh Academy of Sciences on project MR.I.9. 101
the DODS, the singlet and triplet superconducting states = =
(—grn —g11 +g20 —g31)I(A~Dw), (—g10 +g11 +g~~
(4) (5)
102 I
INSTABILITIES OF ELECTRON SYSTEMS
=
(—gii
=
(gii —g~)I(~),
where I(L~.)
=JI
Vol. 25, No. 2
u
(6)
—g21)I(A~),
/
(7)
//
AF
2)~2/kBT] [(~2 + deN(e) tanh 8(c2 + L1 ~2)i/2
/
///
7”
(8)
7’
/
/
/
If we replace the Pauh matrices r 1 r2 which are used in the Nambu formalism (see [2]) we get a change m the phase factor of the order parameters and a change of a sign in front ofg31 in the equations (4) and (5) Let us consider the part of the Hamiltoman due to the Umklapp processes of electrons with different spins
T
—~
V31
g3~ ~Ck~
=
Q/2+Ck4_Q/2_Ck2+ 012_Ck3_
0/2- +
(9)
V31
//
V /
//
/ /
,~‘
~ \s
\
v”~
~ \/\~
h.c.
and, for example, the stability of the CDW state. The linearisation procedure gives the Hamiltonian
//
C~D~/ / /
‘~
/
\~~/\ \
// /
/
//
\
~
/ /
//
//
/
//////
Fig. 1. The stability phase diagram for the extended Hubbard model for one electron per site and J = 0 (the stability regions of the SDW and the DODS states are the same and are denoted by AF).
=
1k!
+
+
x {1~kCk+Q/2GCk_Q,2G + L~kCkQ/2oCk+Q/2G}
~
(10)
U
~
\
~
\
DOD~
~<
~SDW
but k Q/2 = k + Q/2 implies that the parameter in the second term is equal to that one in the first term thus the order parameter can be taken as purely real We can not describe the SDW state in the represen tation (3) However using the following representation —
(c~÷~
=
C~ Q~24’ C~ 0/2-
Ck_ 0/2-)
i
C DW
(11)
we get the stability for both the SDW state 1
+g3j)I(~s~~)
(g21
=
“
(12)
and the CDW and the DODS states, the equations (4) and (5). In order to illustrate the above procedure let us consider the one-dimensional extended Hubbard Hamiltonian +
H
=
~
+
tCu,C,+i~+
U~
i,a
/ // / Fig. 2. The stability phase diagram for the extended Hubbard model for one electron per site and J = 1/4 K. 1 +~ ~
+
1
1
-~--
2
+
—f ~
(K
~ S,
c~c~ct+ lG~~ ii,’
~
—
2
/1,
~
i
~
~‘
+
1
1
~
q
—J) cos qa
C~+q 0C~0C~_q,aCk’,u
~(U + K cos
cos qa
~
C~’~ q, aCk,aCZ
+ ~,
-
- ~,
(13)
k, k’, q
where U is the intraatomic Coulomb interaction, K and J are the interatomic Coulomb and exchange interactions. For the case of a nearly half-filled electronic band we can determine the coupling constants.
L~~Ck~C~~ +
12(K q
c~+c~+c~_c~_
/
+
s
qa)
X Ck+q,aCk,aC~_ q,_aCk,_u
J—K
g10
=
g20
=
N K—J
,
g1i g21
U+J—K =
N U+K+J
=
N
‘
g31
U+J—K =
N
(14)
Vol. 25, No.2
INSTABILITIES OF ELECTRON SYSTEMS
thus the equations (4)—(7) and (12), for the Hamiltonian (13), give the stability conditions of these phases. For one electron per site and J = 0 we have the stability criterion of the CDW or of the DODS states identical to those given by Cabib and Callen [6]. We would like to emphasise the difference of the stability conditions of the antiferromagnetic phase described by the DODS and by the SDW [equations (5) and (12)].
1.
Figures 1 and 2 are examples of the stability phase diagrams for one electron per site (n = I) for J = 0 and J = 1/4 K respectively. In Figure 2 the stability regions of the DODS and the SDW state are shown, they are separated from each other by the ordinate axis. Acknowledgement The author would like to express his sincere thanks to Prof. J. Morkowski for helpful discussions and advice throughout the work. —
REFERENCES For example BYCHKOV Yu.A., GORKOV L.P. & DZYALOSHINSKY I.E., Zh. Eksp. Teor. Fiz. 50, 738 (1966); GUTFREUND H. & KELMM R.A., Phys. Rev. BI4, 1073 (1973).
2.
HOROVITZ B., Solid State Commun. 18, 445 (1976).
3.
HOROVITZ B., GUTFREUND H. & WEGER M., Phys. Rev. B12, 3174 (1975).
4.
BERGGREN K.F. & JOHANSSON B.,Int. J. Quantum Chem. 2,483 (1968). CABIB D. & CALLEN E.,Phys. Rev. B12, 5249 (1975).
5.
103
Pergamon Press.
Printed in Great Britain
ON INSTABIUTIES OF ELECTRON SYSTEMS IN THE MEAN FIELD APPROXIMATION* B. Bulka Ferromagnetics Laboratory, Institute of Molecular Physics of the Polish Academy of Sciences, 60—179 Poznarl, Smoluchowskiego 17/19, Poland (Received 27 June 1977 by P. G. de-Gennes) Various instabilities of quasi-one-dimensional electron systems are examined in the mean field approximation. In addition to the charge density wave and the superconducting states, the antiferromagnetic state, described by the different orbitals for different spins and the spin density methods, is also considered. A phase diagram of these states is constructed.
wave vector transfers (~ Q and 0), respectively. g4, represents a small wave vector transfer scattering of electrons from the same side of the FS, while g38 is due to Umldapp processes. s is equal 0 for a = a’ or 1 for a = a’, thus we distinguish interactions of electrons with the same and the different spins. Ifg,,3 = g~1,then we have the case considered by Horovitz [2]. We have investigated stability conditions for the charge density wave (CDW), the antiferromagnetism
RECENTLY extensive theoretical investigations of various instabilities of electrons in quasi-one-dimensional conductors have been performed [1, 2]. Horovitz [2], using the extended Nambu representation and the Hartree—Fock approximation, gave conditions of stability of various phases of an electron system. We have considered a system of electrons described by the Hamiltonian, which is generalization of the one discussed by Horovitz [2] H
=
—
ek÷Q,2(ck+Q,2Ock+Q,20—ck_Q,2Uck_Q,20)
~
+~
~
+!
(E~2scZ
+
—
—
+~
+ Q/2uCk4_
+ Q/2
aCk
+
Q/2aCk2_ Q,2U’ck3+ Q/2a’
Q/2qCk2_ Q/2U’Ck3_ Q12u’
g38c~ + Q/2aC~4— Q/2aCk2+ Q,20’Ck3_ Q/2a’
+ h.c.) + h.c.
}
+ h.c.)
(Q ~—Q)~
(~~4sck÷Q/2ack+Q/2ack+Q/2a~ck+Q/2a~ +
where the prime above the sum sign restricts wave vector summation to Ik,I
(I)
described by the different orbitals for different spins (DODS) and the spin density wave (SDW) methods (cf [4]), the singlet (S) and triplet (T) superconducting states. We introduce a vector field +
+
=
+
(Ck + Q/2+, C_k_ Q/2—’ Ck — Q/2+’ C_k + Q/2)’
(3)
Ek÷Q/2
with Q = (ir/a, ir/a, 2kF) (a is a lattice constant). This is the condition for nesting of the Fermi surface (FS) [3]. The couplings g~,g 28 describe the scattering of electrons from the opposite sides of the FS with large and small
analogically to the Nambu formalism. In this representation we may find the stability equation of the CDW,
*
I 1
.
.
This work was supported by the Pohsh Academy of Sciences on project MR.I.9. 101
the DODS, the singlet and triplet superconducting states = =
(—grn —g11 +g20 —g31)I(A~Dw), (—g10 +g11 +g~~
(4) (5)
102 I
INSTABILITIES OF ELECTRON SYSTEMS
=
(—gii
=
(gii —g~)I(~),
where I(L~.)
=JI
Vol. 25, No. 2
u
(6)
—g21)I(A~),
/
(7)
//
AF
2)~2/kBT] [(~2 + deN(e) tanh 8(c2 + L1 ~2)i/2
/
///
7”
(8)
7’
/
/
/
If we replace the Pauh matrices r 1 r2 which are used in the Nambu formalism (see [2]) we get a change m the phase factor of the order parameters and a change of a sign in front ofg31 in the equations (4) and (5) Let us consider the part of the Hamiltoman due to the Umklapp processes of electrons with different spins
T
—~
V31
g3~ ~Ck~
=
Q/2+Ck4_Q/2_Ck2+ 012_Ck3_
0/2- +
(9)
V31
//
V /
//
/ /
,~‘
~ \s
\
v”~
~ \/\~
h.c.
and, for example, the stability of the CDW state. The linearisation procedure gives the Hamiltonian
//
C~D~/ / /
‘~
/
\~~/\ \
// /
/
//
\
~
/ /
//
//
/
//////
Fig. 1. The stability phase diagram for the extended Hubbard model for one electron per site and J = 0 (the stability regions of the SDW and the DODS states are the same and are denoted by AF).
=
1k!
+
+
x {1~kCk+Q/2GCk_Q,2G + L~kCkQ/2oCk+Q/2G}
~
(10)
U
~
\
~
\
DOD~
~<
~SDW
but k Q/2 = k + Q/2 implies that the parameter in the second term is equal to that one in the first term thus the order parameter can be taken as purely real We can not describe the SDW state in the represen tation (3) However using the following representation —
(c~÷~
=
C~ Q~24’ C~ 0/2-
Ck_ 0/2-)
i
C DW
(11)
we get the stability for both the SDW state 1
+g3j)I(~s~~)
(g21
=
“
(12)
and the CDW and the DODS states, the equations (4) and (5). In order to illustrate the above procedure let us consider the one-dimensional extended Hubbard Hamiltonian +
H
=
~
+
tCu,C,+i~+
U~
i,a
/ // / Fig. 2. The stability phase diagram for the extended Hubbard model for one electron per site and J = 1/4 K. 1 +~ ~
+
1
1
-~--
2
+
—f ~
(K
~ S,
c~c~ct+ lG~~ ii,’
~
—
2
/1,
~
i
~
~‘
+
1
1
~
q
—J) cos qa
C~+q 0C~0C~_q,aCk’,u
~(U + K cos
cos qa
~
C~’~ q, aCk,aCZ
+ ~,
-
- ~,
(13)
k, k’, q
where U is the intraatomic Coulomb interaction, K and J are the interatomic Coulomb and exchange interactions. For the case of a nearly half-filled electronic band we can determine the coupling constants.
L~~Ck~C~~ +
12(K q
c~+c~+c~_c~_
/
+
s
qa)
X Ck+q,aCk,aC~_ q,_aCk,_u
J—K
g10
=
g20
=
N K—J
,
g1i g21
U+J—K =
N U+K+J
=
N
‘
g31
U+J—K =
N
(14)
Vol. 25, No.2
INSTABILITIES OF ELECTRON SYSTEMS
thus the equations (4)—(7) and (12), for the Hamiltonian (13), give the stability conditions of these phases. For one electron per site and J = 0 we have the stability criterion of the CDW or of the DODS states identical to those given by Cabib and Callen [6]. We would like to emphasise the difference of the stability conditions of the antiferromagnetic phase described by the DODS and by the SDW [equations (5) and (12)].
1.
Figures 1 and 2 are examples of the stability phase diagrams for one electron per site (n = I) for J = 0 and J = 1/4 K respectively. In Figure 2 the stability regions of the DODS and the SDW state are shown, they are separated from each other by the ordinate axis. Acknowledgement The author would like to express his sincere thanks to Prof. J. Morkowski for helpful discussions and advice throughout the work. —
REFERENCES For example BYCHKOV Yu.A., GORKOV L.P. & DZYALOSHINSKY I.E., Zh. Eksp. Teor. Fiz. 50, 738 (1966); GUTFREUND H. & KELMM R.A., Phys. Rev. BI4, 1073 (1973).
2.
HOROVITZ B., Solid State Commun. 18, 445 (1976).
3.
HOROVITZ B., GUTFREUND H. & WEGER M., Phys. Rev. B12, 3174 (1975).
4.
BERGGREN K.F. & JOHANSSON B.,Int. J. Quantum Chem. 2,483 (1968). CABIB D. & CALLEN E.,Phys. Rev. B12, 5249 (1975).
5.
103