- Email: [email protected]
Pairing of particles in a one-dimensional Fibonacci lattice within the generalized Hubbard model
ARTICLE IN PRESS
Physica B 359–361 (2005) 762–764 www.elsevier.com/locate/physb
Pairing of particles in a one-dimensional Fibonacci lattice within the generalized Hubbard model J.E. Espinosa, A. Quiroz Fac. de Cs. Fı´sico-Matema´ticas, Posgrado en Optoelectronica B. Univ. Aut. de Puebla, Apdo Post J-17 Puebla, Pue. 72570 Me´xico
Abstract In this work, we analyzed the ground-state symmetry for the cases of two particles in a linear Fibonacci chain, using the generalized Hubbard Hamiltonian and the real-space mapping method. A ground state is obtained when the correlated hopping interactions are included. The two-particle problem is analyzed by looking at the phase diagram for the bound state. r 2005 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.10.Li; 71.27.+a Keywords: Hubbard model; Bound states; Fermiones in reduced dimensions
Since the discovery of the quasicrystalline phase [1] much attention has been devoted to the study of quasiperiodic systems. The Fibonacci lattice, a one-dimensional version of a quasicrystal, has most widely been studied [2]. In this chain the quasiperiodicity could be provided by both the hopping integral or the site energy. In this study, we will introduce it through the hopping integral that will take two values Tð1Þ and Tð2Þ corresponding to a large bond (2) and a short one (1). The Fibonacci sequence is generated by successive application of the substitution rule LðjÞ ¼ Lðj 1ÞLðj 2Þ for jX2; where LðjÞ repreCorresponding author. Fax: +52 222 2295636.
E-mail address: [email protected] (J.E. Espinosa).
sents the generation j: Within the models that attempt to capture the essential physics from correlated electrons systems, the Hubbard model is the simplest approximation used to describe correlations on narrow-band systems [3]. A generalization of this model that includes also the nearest-neighbor interaction-charge interactions is called the Generalized Hubbard Hamiltonian (GHH), which can be written as H¼
X hi;ji;s
ti;j cþ i;s cj;s þ U
X i
ni ; " ni ; # þ
VX ni nj , 2 hi;ji (1)
where hi; ji denotes nearest-neighbor sites, cþ i;s ðci;s Þ is the creation (annihilation) operator with spin
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.219
ARTICLE IN PRESS J.E. Espinosa, A. Quiroz / Physica B 359– 361 (2005) 762–764
s ¼# or " at site i, and ni ¼ ni;" þ ni;# ; where ni;s ¼ cþ i;s cj;s : The parameters U and V are positive because they are direct Coulomb integrals. However, U and V could be negative if attractive indirect interaction through phonons and or other bosonic excitations are included and are stronger than the direct Coulomb repulsion. In Eq. (1), ti;j is the hopping integral. In this paper, we analyzed the bound states of two non-parallel ð"#Þ electrons in a one-dimensional Fibonacci lattice using the GHH. In order to analyze the Hamiltonian (1), we use the mapping method previously reported [4]. In general, this method will map the original many-body problem onto a one-body problem with some ordered site-impurities in an nd-dimensional lattice, n being the number of particles and d the dimensionality of the original system. In Fig. 1 we have a representation of Eq. (1), which can be described by a one-body tight-binding effective Hamiltonian with ordered site and bond-impurities X X H¼ E i bþ T i;j bþ (2) i bi þ i bj , i
i;j
where the operator bþ i creates the many-body states, E i represents the self-energy of two-particle states and T i;j is the hopping amplitude between nearest-neighbor two-particle states. The pairing energy ðDÞ has been calculated from the difference of energies between the lower-most pairing state and the original lower band edge when there is no electron–electron interaction. The electronic pair1 U t
8
V
t
2 V t
t U
t′
t
V
t′
t
t 16
t
0
t
V
t
t 15 t′
4
t′
V
12 t
14
5
0
11 U
ing energy is considered for the case of two electrons with the same and opposite spins in an infinite Fibonacci chain. The numerical diagonalization was carried out in a truncated chain of 630 states, which is the minimum size required so that the physical quantities have no important variation with the matrix size. In Fig. 2, we show the diagram for the pairing energy of two electrons with opposite spins, as function of the one-site ðUÞ attractive interaction of the original Hamiltonian (1). The quasiperiodicity has been generated by choosing the hopping parameters 1.0 and 1.5 for the short and long bonds, respectively, and we also show the results for a periodic chain [5]. In order to observe only the U effect, we let V ¼ 0: In this figure, we can see a hyperbolic growth as a function of U in both systems. We also found that the pairing strength in the quasiperiodicity system is larger than in the periodic one and that for U48 eV; the quasiperiodicity becomes less important and the pairing strength in both systems is very similar. Fig. 3 shows the variation of the pairing energy as a function of the nearestneighbor ðV Þ attractive interaction. We can observe, as well, that the pairing energy of the quasiperiodic lattice is larger than the periodic one and also that as the interaction becomes larger the pairing energy in both structures becomes similar. Finally, Fig. 4 shows the effect of changing the large-bond parameter. As we can see, the pairing
0 t
6
t′
10
9 0
3 0 t
t′
t′
0
7
t′
763
V t
t
13 U
Fig. 1. Geometric representation of the two-particle states for a Fibonacci chain.
Fig. 2. Binding energy ðDÞ as function of U of a quasiperiodic Tð1Þ ¼ 1:0 and Tð2Þ ¼ 1:5 and a periodic Tð1Þ ¼ 1:0 and Tð2Þ ¼ 1:0 linear chains for two antiparallel-spin electrons.
ARTICLE IN PRESS J.E. Espinosa, A. Quiroz / Physica B 359– 361 (2005) 762–764
764
energy in the quasiperiodic lattice increases as the parameter becomes larger, according to the behavior that is seen in Figs. 2 and 3. It is worth mentioning that U; V and T are typical values in eV [3–6]. In summary, we have analyzed the bound states of two electrons in an infinite linear Fibonacci chain. The analysis was done, using the generalized Hubbard Hamiltonian. References
Fig. 3. Binding energy ðDÞ as function of V of a quasiperiodic Tð1Þ ¼ 1:0 and Tð2Þ ¼ 1:5 and a periodic Tð1Þ ¼ 1:0 and Tð2Þ ¼ 1:0 linear chains for two antiparallel-spin electrons.
.
.
.
.
.
. .
.
.
.
.
.
Fig. 4. Variation of the large-bond parameter for a Fibonacci chain.
[1] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [2] T. Fujiwara, M. Kohmoto, T. Tokihiro, Phys. Rev. B 40 (1989) 7413; G.Y. Oh, C.S. Ryu, M.H. Lee, J. Phys.: Condens. Matter 4 (1992) 8187. [3] F. Marsiglio, J.E. Hirsch, Physica C 171 (1990) 554; E.R. Gagliano, A.A. Aligia, Liliana Arranchea, M. Avignon, Phys. Rev. B 51 (1995) 14012; L. Didukh, Hankevych, Phys. Status Solidi B 211 (1999) 703; R. Strack, D. Vollhardt, Phys. Rev. Lett. 70 (1993) 2637. [4] J.E. Espinosa, O. Navarro, M. Avignon, Eur. Phys. J. B 18 (2000) 9. [5] O. Navarro, C. Wang, Solid State Commun. 83 (1992) 473; C. Wang, O. Navarro, R. Oviedo, Mat. Res. Soc. Symp. Proc. 291 (1993) 279. [6] J. Appel, et al., Phys. Rev. B 47 (1993) 2812.
Physica B 359–361 (2005) 762–764 www.elsevier.com/locate/physb
Pairing of particles in a one-dimensional Fibonacci lattice within the generalized Hubbard model J.E. Espinosa, A. Quiroz Fac. de Cs. Fı´sico-Matema´ticas, Posgrado en Optoelectronica B. Univ. Aut. de Puebla, Apdo Post J-17 Puebla, Pue. 72570 Me´xico
Abstract In this work, we analyzed the ground-state symmetry for the cases of two particles in a linear Fibonacci chain, using the generalized Hubbard Hamiltonian and the real-space mapping method. A ground state is obtained when the correlated hopping interactions are included. The two-particle problem is analyzed by looking at the phase diagram for the bound state. r 2005 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.10.Li; 71.27.+a Keywords: Hubbard model; Bound states; Fermiones in reduced dimensions
Since the discovery of the quasicrystalline phase [1] much attention has been devoted to the study of quasiperiodic systems. The Fibonacci lattice, a one-dimensional version of a quasicrystal, has most widely been studied [2]. In this chain the quasiperiodicity could be provided by both the hopping integral or the site energy. In this study, we will introduce it through the hopping integral that will take two values Tð1Þ and Tð2Þ corresponding to a large bond (2) and a short one (1). The Fibonacci sequence is generated by successive application of the substitution rule LðjÞ ¼ Lðj 1ÞLðj 2Þ for jX2; where LðjÞ repreCorresponding author. Fax: +52 222 2295636.
E-mail address: [email protected] (J.E. Espinosa).
sents the generation j: Within the models that attempt to capture the essential physics from correlated electrons systems, the Hubbard model is the simplest approximation used to describe correlations on narrow-band systems [3]. A generalization of this model that includes also the nearest-neighbor interaction-charge interactions is called the Generalized Hubbard Hamiltonian (GHH), which can be written as H¼
X hi;ji;s
ti;j cþ i;s cj;s þ U
X i
ni ; " ni ; # þ
VX ni nj , 2 hi;ji (1)
where hi; ji denotes nearest-neighbor sites, cþ i;s ðci;s Þ is the creation (annihilation) operator with spin
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.219
ARTICLE IN PRESS J.E. Espinosa, A. Quiroz / Physica B 359– 361 (2005) 762–764
s ¼# or " at site i, and ni ¼ ni;" þ ni;# ; where ni;s ¼ cþ i;s cj;s : The parameters U and V are positive because they are direct Coulomb integrals. However, U and V could be negative if attractive indirect interaction through phonons and or other bosonic excitations are included and are stronger than the direct Coulomb repulsion. In Eq. (1), ti;j is the hopping integral. In this paper, we analyzed the bound states of two non-parallel ð"#Þ electrons in a one-dimensional Fibonacci lattice using the GHH. In order to analyze the Hamiltonian (1), we use the mapping method previously reported [4]. In general, this method will map the original many-body problem onto a one-body problem with some ordered site-impurities in an nd-dimensional lattice, n being the number of particles and d the dimensionality of the original system. In Fig. 1 we have a representation of Eq. (1), which can be described by a one-body tight-binding effective Hamiltonian with ordered site and bond-impurities X X H¼ E i bþ T i;j bþ (2) i bi þ i bj , i
i;j
where the operator bþ i creates the many-body states, E i represents the self-energy of two-particle states and T i;j is the hopping amplitude between nearest-neighbor two-particle states. The pairing energy ðDÞ has been calculated from the difference of energies between the lower-most pairing state and the original lower band edge when there is no electron–electron interaction. The electronic pair1 U t
8
V
t
2 V t
t U
t′
t
V
t′
t
t 16
t
0
t
V
t
t 15 t′
4
t′
V
12 t
14
5
0
11 U
ing energy is considered for the case of two electrons with the same and opposite spins in an infinite Fibonacci chain. The numerical diagonalization was carried out in a truncated chain of 630 states, which is the minimum size required so that the physical quantities have no important variation with the matrix size. In Fig. 2, we show the diagram for the pairing energy of two electrons with opposite spins, as function of the one-site ðUÞ attractive interaction of the original Hamiltonian (1). The quasiperiodicity has been generated by choosing the hopping parameters 1.0 and 1.5 for the short and long bonds, respectively, and we also show the results for a periodic chain [5]. In order to observe only the U effect, we let V ¼ 0: In this figure, we can see a hyperbolic growth as a function of U in both systems. We also found that the pairing strength in the quasiperiodicity system is larger than in the periodic one and that for U48 eV; the quasiperiodicity becomes less important and the pairing strength in both systems is very similar. Fig. 3 shows the variation of the pairing energy as a function of the nearestneighbor ðV Þ attractive interaction. We can observe, as well, that the pairing energy of the quasiperiodic lattice is larger than the periodic one and also that as the interaction becomes larger the pairing energy in both structures becomes similar. Finally, Fig. 4 shows the effect of changing the large-bond parameter. As we can see, the pairing
0 t
6
t′
10
9 0
3 0 t
t′
t′
0
7
t′
763
V t
t
13 U
Fig. 1. Geometric representation of the two-particle states for a Fibonacci chain.
Fig. 2. Binding energy ðDÞ as function of U of a quasiperiodic Tð1Þ ¼ 1:0 and Tð2Þ ¼ 1:5 and a periodic Tð1Þ ¼ 1:0 and Tð2Þ ¼ 1:0 linear chains for two antiparallel-spin electrons.
ARTICLE IN PRESS J.E. Espinosa, A. Quiroz / Physica B 359– 361 (2005) 762–764
764
energy in the quasiperiodic lattice increases as the parameter becomes larger, according to the behavior that is seen in Figs. 2 and 3. It is worth mentioning that U; V and T are typical values in eV [3–6]. In summary, we have analyzed the bound states of two electrons in an infinite linear Fibonacci chain. The analysis was done, using the generalized Hubbard Hamiltonian. References
Fig. 3. Binding energy ðDÞ as function of V of a quasiperiodic Tð1Þ ¼ 1:0 and Tð2Þ ¼ 1:5 and a periodic Tð1Þ ¼ 1:0 and Tð2Þ ¼ 1:0 linear chains for two antiparallel-spin electrons.
.
.
.
.
.
. .
.
.
.
.
.
Fig. 4. Variation of the large-bond parameter for a Fibonacci chain.
[1] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [2] T. Fujiwara, M. Kohmoto, T. Tokihiro, Phys. Rev. B 40 (1989) 7413; G.Y. Oh, C.S. Ryu, M.H. Lee, J. Phys.: Condens. Matter 4 (1992) 8187. [3] F. Marsiglio, J.E. Hirsch, Physica C 171 (1990) 554; E.R. Gagliano, A.A. Aligia, Liliana Arranchea, M. Avignon, Phys. Rev. B 51 (1995) 14012; L. Didukh, Hankevych, Phys. Status Solidi B 211 (1999) 703; R. Strack, D. Vollhardt, Phys. Rev. Lett. 70 (1993) 2637. [4] J.E. Espinosa, O. Navarro, M. Avignon, Eur. Phys. J. B 18 (2000) 9. [5] O. Navarro, C. Wang, Solid State Commun. 83 (1992) 473; C. Wang, O. Navarro, R. Oviedo, Mat. Res. Soc. Symp. Proc. 291 (1993) 279. [6] J. Appel, et al., Phys. Rev. B 47 (1993) 2812.