- Email: [email protected]
Study of the spin-32 Hubbard–Kondo lattice model by means of the Composite Operator Method
ARTICLE IN PRESS
Physica B 378–380 (2006) 700–701 www.elsevier.com/locate/physb
Study of the spin-32 Hubbard–Kondo lattice model by means of the Composite Operator Method A. Avella, F. Mancini Dipartimento di Fisica ‘‘E.R. Caianiello’’ - Unita` Cnism di Salerno, INFM Universita` degli Studi di Salerno, I-84081 Baronissi (SA), Italy
Abstract We study the spin-32 Hubbard–Kondo lattice model by means of the Composite Operator Method, after applying a Holstein–Primakov transformation. The spin and particle dynamics in the ferromagnetic state are calculated by taking into account strong on-site correlations between electrons and antiferromagnetic exchange among 32 spins, together with usual Hund coupling between electrons and spins. r 2006 Elsevier B.V. All rights reserved. PACS: 75.25.þz; 75.30.Et; 75.30.Vn; 71.45.Lr Keywords: Spin-32 Hubbard–Kondo lattice model; Composite Operator Method; Holstein–Primakov transformation
The revival in the study of manganites has led to experimental re-examination of their different properties. One of the puzzling features is the strong deviation of the spin-wave dispersion from the typical Heisenberg behavior. In particular, an unexpected softening at the zone boundary has been observed [1]. These observations are very important as they indicate that some aspects of spin dynamics in manganites have not been satisfactorily understood yet [2,3]. According to this, we have decided to investigate the spin dynamics of the ferromagnetic state of the spin-32 Hubbard–Kondo lattice model by means of the Composite Operator Method (COM) [4]. The Hamiltonian under analysis reads as X X ð2dtaij mdij Þcy ðiÞcðjÞ þ U n" ðiÞn# ðiÞ H¼ ij
JH
X i
sðiÞ SðiÞ þ dJ AF
X
i
SðiÞ Sa ðiÞ,
ð1Þ
i
where i is a vector of the d-dimensional lattice and i ¼ ði; tÞ, m is the chemical potential, cy ðiÞ ¼ ðcy" ðiÞ; cy# ðiÞÞ is the
electronic creation operator in spinorial notation, t is the hopping amplitude, aij is the nearest-neighbor projector, U is the on-site Coulomb interaction, ns ðiÞ ¼ cys ðiÞcs ðiÞ, J H is the Hund coupling, sðiÞ ¼ 12 cy ðiÞrcðiÞ, r is the Pauli matrices, SðiÞ is the core 32-spin, and J AF is the antiferromagnetic coupling. We have used the notation fa ðiÞ ¼ P j aij fði; tÞ. In order to avoid the difficulties related to the high value of the core spin, we p have used the transfor-ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Holstein–Primakoff pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mation: Sþ ðiÞ ¼ 2S na ðiÞaðiÞ, S ðiÞ ¼ ay ðiÞ 2S na ðiÞ, Sz ðiÞ ¼ S na ðiÞ, where S ¼ 32, aðiÞ is a spinless bosonic destruction operator and na ðiÞ ¼ ay ðiÞaðiÞ. Then, we have ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi decided to approximate the non-linear term 2S na ðiÞ to the first order in dna ðiÞ=ð2S na Þ where dna ðiÞ ¼ na ðiÞ hna ðiÞi and na ¼ hna ðiÞi. It is worth noticing that this approximation preserves all properties related to the angular momentum algebra of the core spin. The transformed Hamiltonian reads as X X ð2dtaij mdij Þcy ðiÞcðjÞ þ U n" ðiÞn# ðiÞ H¼ ij
JH
X i
Corresponding author. Tel.: +39 089 965418; fax: +39 089 965275.
E-mail address: [email protected] (A. Avella). URL: http://www.sa.infn.it/Homepage.asp?avella. 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.227
i
½S na dna ðiÞsz ðiÞ
X 1 dna ðiÞ J HA sþ ðiÞay ðiÞ 1 þ h:c: 2 2A2 i
ARTICLE IN PRESS A. Avella, F. Mancini / Physica B 378–380 (2006) 700–701
2dJ AF ðS na Þ þ dJ AF A2
X
X
dna ðiÞ
i
ay ðiÞaa ðiÞ,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where A ¼ 2S na . Within the framework of the COM, we have chosen two operatorial bases to study the spin and particle dynamics: 1 0 x" ðiÞ ! B Z ðiÞ C aðiÞ B " C C (3) BðiÞ ¼ ; cðiÞ ¼ B B x# ðiÞ C, sþ ðiÞ A @ Z# ðiÞ where xðiÞ ¼ ½1 nðiÞcðiÞ and ZðiÞ ¼ nðiÞcðiÞ. Then, we have linearized the equations of motion by projecting the currents onto the basis: X q i Bði; tÞ ffi eB ði; jÞBð j; tÞ, (4) qt j
1 a ðC þ C aF 22 þ C aF 33 þ C aF 44 Þ, m F 11
(15)
p~ 2 ¼
1 haðiÞs ðiÞi 1 C B12 ¼ , A m A m
(16)
p~ 3 ¼
1 hdna ðiÞsz ðiÞi , m A2
(17)
p~ 4 ¼
1 hdna ðiÞaðiÞs ðiÞi , m A3
(18)
p~ 5 ¼
1 hdna ðiÞnðiÞi , m A2
(19)
D0 ¼ 12 ðC aF 11 C aF 22 þ C aF 33 C aF 44 Þ,
(20)
Dz ¼ 12 ðC aF 11 C aF 22 C aF 33 þ C aF 44 Þ,
(21)
p ¼ hð14 na ðiÞnðiÞ þ sa ðiÞ sðiÞÞi hðc" ðiÞc# ðiÞÞa cy# ðiÞcy" ðiÞi,
X q i cði; tÞ ffi eF ði; jÞcð j; tÞ, qt j
(5)
eB;F ði; jÞ ¼
X
mB;F ði; lÞI 1 B;F ðl; jÞ,
(6)
l
q y mB ði; jÞ ¼ i Bði; tÞB ðj; tÞ , qt q mF ði; jÞ ¼ i cði; tÞcy ð j; tÞ , qt
I B ði; jÞ ¼ Bði; tÞBy ðj; tÞ ,
cði; tÞcy ðj; tÞ
(7)
(8) (9)
.
(10)
This procedure assures that the neglected component of the current is orthogonal to the chosen basis. According to this, we have obtained the corresponding retarded Green’s functions in the pole approximation G B;F ðo; kÞ ¼
X
sðiÞ B;F ðkÞ
i
o E ðiÞ B;F ðkÞ þ id
,
(11)
where the energies E ðiÞ B;F ðkÞ are the eigenvalues of the energy matrices eB;F ðkÞ ¼ F½eB;F ði; jÞ and the spectral densities sðiÞ B;F ðkÞ can be computed in terms of the normalization matrices I B;F and of the eigenvectors of the energy matrices [4]. F is the Fourier transform. The parameters appearing in the expressions of mB;F and I B;F have the following definitions: n ¼ hnðiÞi ¼ 2 ðC F 11 þ C F 22 þ C F 33 þ C F 44 Þ,
(12)
m ¼ hsz ðiÞi ¼ 12 ðC F 44 C F 22 Þ,
(13)
na ¼ C B11 1,
(14)
ð22Þ
waz ¼ hna ðiÞsz ðiÞi, hcb ðiÞcyg ðiÞi,
where
I F ði; jÞ ¼
p~ 1 ¼ ð2Þ
i
701
(23) C aF bg
hcab ðiÞcyg ðiÞi,
¼ and C Bbg ¼ where C F bg ¼ hBb ðiÞByg ðiÞi. According to the prescriptions of the COM [4], the parameters that cannot be computed by their definitions (p~ 3 , p~ 4 , p~ 5 , p and waz ) would be fixed by the following relations C 11 ¼ C 33 ; C 34 ¼ 0;
C 12 ¼ 0,
(24)
C 44 ¼ C B22 ,
(25)
coming from the algebra and by one more relation coming from the request that the hydrodynamic limit should be satisfied (i.e., the existence of a sound mode). All these equations (definitions and constraints) form a coupled system that should be computed self-consistently. The results of these calculations will be presented elsewhere. In conclusion, we have reported the solution for the Hubbard–Kondo model in the presence of antiferromagnetic coupling between the core spin and the framework of the composite operator method in the pole approximation. The model has been first mapped through the Holstein– Primakov transformation that has been then approximated to the first order in the number fluctuation operator. The spin dynamics has been fully determined and will be analyzed numerically. The authors acknowledge the support by Cost-Action P16. References [1] [2] [3] [4]
H.Y. Hwang, et al., Phys. Rev. Lett. 80 (1998) 1316. G. Khaliullin, P. Kilian, Phys. Rev. B 61 (2000) 3494. F. Mancini, N.B. Perkins, N.M. Plakida, Phys. Lett. A 284 (2001) 286. F. Mancini, A. Avella, Adv. Phys. 53 (2004) 537.
Physica B 378–380 (2006) 700–701 www.elsevier.com/locate/physb
Study of the spin-32 Hubbard–Kondo lattice model by means of the Composite Operator Method A. Avella, F. Mancini Dipartimento di Fisica ‘‘E.R. Caianiello’’ - Unita` Cnism di Salerno, INFM Universita` degli Studi di Salerno, I-84081 Baronissi (SA), Italy
Abstract We study the spin-32 Hubbard–Kondo lattice model by means of the Composite Operator Method, after applying a Holstein–Primakov transformation. The spin and particle dynamics in the ferromagnetic state are calculated by taking into account strong on-site correlations between electrons and antiferromagnetic exchange among 32 spins, together with usual Hund coupling between electrons and spins. r 2006 Elsevier B.V. All rights reserved. PACS: 75.25.þz; 75.30.Et; 75.30.Vn; 71.45.Lr Keywords: Spin-32 Hubbard–Kondo lattice model; Composite Operator Method; Holstein–Primakov transformation
The revival in the study of manganites has led to experimental re-examination of their different properties. One of the puzzling features is the strong deviation of the spin-wave dispersion from the typical Heisenberg behavior. In particular, an unexpected softening at the zone boundary has been observed [1]. These observations are very important as they indicate that some aspects of spin dynamics in manganites have not been satisfactorily understood yet [2,3]. According to this, we have decided to investigate the spin dynamics of the ferromagnetic state of the spin-32 Hubbard–Kondo lattice model by means of the Composite Operator Method (COM) [4]. The Hamiltonian under analysis reads as X X ð2dtaij mdij Þcy ðiÞcðjÞ þ U n" ðiÞn# ðiÞ H¼ ij
JH
X i
sðiÞ SðiÞ þ dJ AF
X
i
SðiÞ Sa ðiÞ,
ð1Þ
i
where i is a vector of the d-dimensional lattice and i ¼ ði; tÞ, m is the chemical potential, cy ðiÞ ¼ ðcy" ðiÞ; cy# ðiÞÞ is the
electronic creation operator in spinorial notation, t is the hopping amplitude, aij is the nearest-neighbor projector, U is the on-site Coulomb interaction, ns ðiÞ ¼ cys ðiÞcs ðiÞ, J H is the Hund coupling, sðiÞ ¼ 12 cy ðiÞrcðiÞ, r is the Pauli matrices, SðiÞ is the core 32-spin, and J AF is the antiferromagnetic coupling. We have used the notation fa ðiÞ ¼ P j aij fði; tÞ. In order to avoid the difficulties related to the high value of the core spin, we p have used the transfor-ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Holstein–Primakoff pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mation: Sþ ðiÞ ¼ 2S na ðiÞaðiÞ, S ðiÞ ¼ ay ðiÞ 2S na ðiÞ, Sz ðiÞ ¼ S na ðiÞ, where S ¼ 32, aðiÞ is a spinless bosonic destruction operator and na ðiÞ ¼ ay ðiÞaðiÞ. Then, we have ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi decided to approximate the non-linear term 2S na ðiÞ to the first order in dna ðiÞ=ð2S na Þ where dna ðiÞ ¼ na ðiÞ hna ðiÞi and na ¼ hna ðiÞi. It is worth noticing that this approximation preserves all properties related to the angular momentum algebra of the core spin. The transformed Hamiltonian reads as X X ð2dtaij mdij Þcy ðiÞcðjÞ þ U n" ðiÞn# ðiÞ H¼ ij
JH
X i
Corresponding author. Tel.: +39 089 965418; fax: +39 089 965275.
E-mail address: [email protected] (A. Avella). URL: http://www.sa.infn.it/Homepage.asp?avella. 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.227
i
½S na dna ðiÞsz ðiÞ
X 1 dna ðiÞ J HA sþ ðiÞay ðiÞ 1 þ h:c: 2 2A2 i
ARTICLE IN PRESS A. Avella, F. Mancini / Physica B 378–380 (2006) 700–701
2dJ AF ðS na Þ þ dJ AF A2
X
X
dna ðiÞ
i
ay ðiÞaa ðiÞ,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where A ¼ 2S na . Within the framework of the COM, we have chosen two operatorial bases to study the spin and particle dynamics: 1 0 x" ðiÞ ! B Z ðiÞ C aðiÞ B " C C (3) BðiÞ ¼ ; cðiÞ ¼ B B x# ðiÞ C, sþ ðiÞ A @ Z# ðiÞ where xðiÞ ¼ ½1 nðiÞcðiÞ and ZðiÞ ¼ nðiÞcðiÞ. Then, we have linearized the equations of motion by projecting the currents onto the basis: X q i Bði; tÞ ffi eB ði; jÞBð j; tÞ, (4) qt j
1 a ðC þ C aF 22 þ C aF 33 þ C aF 44 Þ, m F 11
(15)
p~ 2 ¼
1 haðiÞs ðiÞi 1 C B12 ¼ , A m A m
(16)
p~ 3 ¼
1 hdna ðiÞsz ðiÞi , m A2
(17)
p~ 4 ¼
1 hdna ðiÞaðiÞs ðiÞi , m A3
(18)
p~ 5 ¼
1 hdna ðiÞnðiÞi , m A2
(19)
D0 ¼ 12 ðC aF 11 C aF 22 þ C aF 33 C aF 44 Þ,
(20)
Dz ¼ 12 ðC aF 11 C aF 22 C aF 33 þ C aF 44 Þ,
(21)
p ¼ hð14 na ðiÞnðiÞ þ sa ðiÞ sðiÞÞi hðc" ðiÞc# ðiÞÞa cy# ðiÞcy" ðiÞi,
X q i cði; tÞ ffi eF ði; jÞcð j; tÞ, qt j
(5)
eB;F ði; jÞ ¼
X
mB;F ði; lÞI 1 B;F ðl; jÞ,
(6)
l
q y mB ði; jÞ ¼ i Bði; tÞB ðj; tÞ , qt q mF ði; jÞ ¼ i cði; tÞcy ð j; tÞ , qt
I B ði; jÞ ¼ Bði; tÞBy ðj; tÞ ,
cði; tÞcy ðj; tÞ
(7)
(8) (9)
.
(10)
This procedure assures that the neglected component of the current is orthogonal to the chosen basis. According to this, we have obtained the corresponding retarded Green’s functions in the pole approximation G B;F ðo; kÞ ¼
X
sðiÞ B;F ðkÞ
i
o E ðiÞ B;F ðkÞ þ id
,
(11)
where the energies E ðiÞ B;F ðkÞ are the eigenvalues of the energy matrices eB;F ðkÞ ¼ F½eB;F ði; jÞ and the spectral densities sðiÞ B;F ðkÞ can be computed in terms of the normalization matrices I B;F and of the eigenvectors of the energy matrices [4]. F is the Fourier transform. The parameters appearing in the expressions of mB;F and I B;F have the following definitions: n ¼ hnðiÞi ¼ 2 ðC F 11 þ C F 22 þ C F 33 þ C F 44 Þ,
(12)
m ¼ hsz ðiÞi ¼ 12 ðC F 44 C F 22 Þ,
(13)
na ¼ C B11 1,
(14)
ð22Þ
waz ¼ hna ðiÞsz ðiÞi, hcb ðiÞcyg ðiÞi,
where
I F ði; jÞ ¼
p~ 1 ¼ ð2Þ
i
701
(23) C aF bg
hcab ðiÞcyg ðiÞi,
¼ and C Bbg ¼ where C F bg ¼ hBb ðiÞByg ðiÞi. According to the prescriptions of the COM [4], the parameters that cannot be computed by their definitions (p~ 3 , p~ 4 , p~ 5 , p and waz ) would be fixed by the following relations C 11 ¼ C 33 ; C 34 ¼ 0;
C 12 ¼ 0,
(24)
C 44 ¼ C B22 ,
(25)
coming from the algebra and by one more relation coming from the request that the hydrodynamic limit should be satisfied (i.e., the existence of a sound mode). All these equations (definitions and constraints) form a coupled system that should be computed self-consistently. The results of these calculations will be presented elsewhere. In conclusion, we have reported the solution for the Hubbard–Kondo model in the presence of antiferromagnetic coupling between the core spin and the framework of the composite operator method in the pole approximation. The model has been first mapped through the Holstein– Primakov transformation that has been then approximated to the first order in the number fluctuation operator. The spin dynamics has been fully determined and will be analyzed numerically. The authors acknowledge the support by Cost-Action P16. References [1] [2] [3] [4]
H.Y. Hwang, et al., Phys. Rev. Lett. 80 (1998) 1316. G. Khaliullin, P. Kilian, Phys. Rev. B 61 (2000) 3494. F. Mancini, N.B. Perkins, N.M. Plakida, Phys. Lett. A 284 (2001) 286. F. Mancini, A. Avella, Adv. Phys. 53 (2004) 537.