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A new analysis of optical excitations in the Hubbard model
Physica B 259—261 (1999) 755—756
A new analysis of optical excitations in the Hubbard model F. Mancini *, N. Perkins, D. Villani Universita% degli Studi di Salerno — Unita% INFM di Salerno, Dipartimento di Scienze Fisiche **E. R. Caianiello++, Via S. Allende, I-84081 Baronissi Salerno, Italy BLTPh, Joint Institute for Nuclear Research, 141980 Dubna, Russia
Abstract A new theoretical analysis of the conductivity tensor is developed by use of the composite operator method for the two-dimensional Hubbard model. With the aid of the Ward—Takahashi identities which phrase the charge conservation law, the calculation of the current—current propagator is reduced hierarchically to the current—charge and charge—charge ones. Besides, the fermionic character of the propagating charges is preserved because of a constraining equation with the content of the Pauli principle which uniquely fixes the single-particle dynamics. 1999 Elsevier Science B.V. All rights reserved. Keywords: Optical conductivity; 2D Hubbard model; Ward—Takahashi identity; Composite operator method
The appearance of optical anomalies in the doped copper oxides, such as a redistribution of the spectral weight upon doping accompanied by an approximately constant integrated conductivity, is one of the most striking features in these materials, and is almost universally observed in the spectra of the systems with strong electron correlations [1,2]. In this paper, we present a new theoretical analysis of the conductivity tensor by use of the composite operator method [3,6,7] for the two-dimensional Hubbard model. By exploiting sum rules after the Ward—Takahashi identities and the Pauli principle, the calculation of the current—current propagator is reduced to the current—charge and charge—charge ones in the framework of a fully self-consistent scheme. In presence of an electromagnetic field, the Hubbard model is described by the Hamiltonian
where we use a standard notation. The hopping matrix elements contain the Peierls factors which guarantee the gauge invariance. In the framework of the linear response theory, the electric conductivity tensor is given by the following expression:
H" [t (A)!kd ]cR(i)c(j)#º n (i)n (i), GH GH j i 6GH7 G
it is immediate to obtain the conservation law ) j(i)# jo(i)/jt"0. The symmetry content of this algebraic equation manifests at the level of observation as relations among matrix elements once a choice of the physical space of states has been made. Indeed, by defining the causal charge and current correlation functions as s (i, j)"1¹[g (i)g (j)]2 where g (i)"(o(i), j (i), j (i)) for ?@ ? @ ? V W a"0, x, y, we can derive a series of Ward—Takahashi
(1)
* Corresponding author. Tel.: #39-89-9653222; fax: #39(0)-89-965275; e-mail: [email protected]. Present address: Serin Physics Laboratory, Rutgers University, Piscataway, NJ 08855-0849, USA.
1 1 1K 2d ! g (k, u), p (k, u)"e ?@ ? ?@ i(u#ig) ?@ i(u#ig)
(2)
where g (k, u) is the retarded current—current correlation ?@ function, with K (i) being the kinetic energy density in the ? a-direction. By defining the charge o(i) and current j(i) densities as o(i)"ecR(i)c(i), j(i)"!iteacR(i)[
!
]c(i),
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 8 6 9 - 2
(3)
756
F. Mancini et al. / Physica B 259—261 (1999) 755—756
identities connecting current—current, charge—current and charge—charge propagators. One of those reads as follows iaus (k, u)"[1!e\ IV?]s (k, u) V #[1!e\ IW?]s (k, u). W In this theoretical framework, we obtain
(4)
aus (k , u) V #e1K (i)2. (5) s (k , u)" V VV V 2[1!cos(k a)] V From the analytical structure of Eq. (5), it is clear that s (k , 0) does not depend on k . It is remarkable that VV V V such a peculiar behavior has been previously found by use of quantum Monte Carlo techniques [4]. For the optical conductivity p (u), we have obtained p (u)"Dp(u)#p (u) auRes0 (k , u) V , D"p lim lim (6) 2[1!cos(k a)] V S IV 1 aus0 (k , u) V p (u)"! Im lim , (7) u 2[1!cos(k a)] V V I D and p (u) being the Drude weight and the incoherent part, respectively, and R indicates the retarded contribution. As emphasized in the beginning, the whole problem of the response to an external electromagnetic field has been reduced to the evaluation of the charge—charge propagator. In the static approximation of the composite operator method the charge—charge correlation function as well as the spin—spin one can be connected to convolutions of single-particle propagators. This occurrence is related to a linearized dynamics together with the choice of occupation-dependent electronic excitations as basic fields [5]. In this way, the electron propagation is described as a repetition of composite excitations which automatically take into account scattering of electrons on spin and charge fluctuations due to strong correlations. Also, the use of a higher order basic field gives the advantage of implementing the Pauli principle and of a clear theoretical understanding of the terms originating from intraband or interband propagation. In fact, it turns out that only intraband excitations contribute to the Drude weight, whereas interband ones build up the incoherent part. It is worth noting that in this context the
charge—charge correlation function satisfies the sum rule
ia(2p)\ dk dus (k, u)"e(n#2d)
(8)
as it should be, being at equal sites 1n(i)2"n#2d, with d the double occupancy. The physical content of Eq. (8) is the one of the Pauli principle because it governs the single-site dynamics of two fermions. In conclusion, we have shown that the operator equation expressing the charge conservation law has as direct corollary the existence of Ward—Takahashi identities, which hierarchically degrade the evaluation of the current—current propagator to the current—charge and charge—charge ones. This approach has been implemented on the static approximation for the two-dimensional Hubbard model in the framework of the composite operator method. Then, a use of general symmetry principles discloses the possibility to realize a fully self-consistent theory for the charge transport in the strongly correlated electronic systems. Further, by recovering such general symmetries the propagation of the charge and their intrinsic dynamics is constrained in a suitable Hilbert space where the charges propagate without degrading in number and avoiding double occupancy configurations unless to form local singlets. Numerical analysis of the derived equations and an extensive comparison with the data for cuprates will be presented elsewhere.
References [1] S. Uchida, Jpn. J. Appl. Phys. 32 (1993) 3784. [2] E. Dagotto, Rev. Mod. Phys. 66 (1994) 763, and references therein. [3] F. Mancini, S. Marra, H. Matsumoto, Physica C 244 (1995) 49. [4] D.J. Scalapino, S.R. White, S.C. Zhang, Phys. Rev. Lett. 68 (1992) 2830. [5] F. Mancini, S. Marra, H. Matsumoto, Physica C 252 (1995) 361. [6] F. Mancini, D. Villani, H. Matsumoto, Phys. Rev. B 57 (1998) 6145. [7] A. Avella, F. Mancini, D. Villani, L. Siurakshina, V.Yu. Yushankhai, Int. J. Mod. Phys. B 12 (1998) 81.
A new analysis of optical excitations in the Hubbard model F. Mancini *, N. Perkins, D. Villani Universita% degli Studi di Salerno — Unita% INFM di Salerno, Dipartimento di Scienze Fisiche **E. R. Caianiello++, Via S. Allende, I-84081 Baronissi Salerno, Italy BLTPh, Joint Institute for Nuclear Research, 141980 Dubna, Russia
Abstract A new theoretical analysis of the conductivity tensor is developed by use of the composite operator method for the two-dimensional Hubbard model. With the aid of the Ward—Takahashi identities which phrase the charge conservation law, the calculation of the current—current propagator is reduced hierarchically to the current—charge and charge—charge ones. Besides, the fermionic character of the propagating charges is preserved because of a constraining equation with the content of the Pauli principle which uniquely fixes the single-particle dynamics. 1999 Elsevier Science B.V. All rights reserved. Keywords: Optical conductivity; 2D Hubbard model; Ward—Takahashi identity; Composite operator method
The appearance of optical anomalies in the doped copper oxides, such as a redistribution of the spectral weight upon doping accompanied by an approximately constant integrated conductivity, is one of the most striking features in these materials, and is almost universally observed in the spectra of the systems with strong electron correlations [1,2]. In this paper, we present a new theoretical analysis of the conductivity tensor by use of the composite operator method [3,6,7] for the two-dimensional Hubbard model. By exploiting sum rules after the Ward—Takahashi identities and the Pauli principle, the calculation of the current—current propagator is reduced to the current—charge and charge—charge ones in the framework of a fully self-consistent scheme. In presence of an electromagnetic field, the Hubbard model is described by the Hamiltonian
where we use a standard notation. The hopping matrix elements contain the Peierls factors which guarantee the gauge invariance. In the framework of the linear response theory, the electric conductivity tensor is given by the following expression:
H" [t (A)!kd ]cR(i)c(j)#º n (i)n (i), GH GH j i 6GH7 G
it is immediate to obtain the conservation law ) j(i)# jo(i)/jt"0. The symmetry content of this algebraic equation manifests at the level of observation as relations among matrix elements once a choice of the physical space of states has been made. Indeed, by defining the causal charge and current correlation functions as s (i, j)"1¹[g (i)g (j)]2 where g (i)"(o(i), j (i), j (i)) for ?@ ? @ ? V W a"0, x, y, we can derive a series of Ward—Takahashi
(1)
* Corresponding author. Tel.: #39-89-9653222; fax: #39(0)-89-965275; e-mail: [email protected]. Present address: Serin Physics Laboratory, Rutgers University, Piscataway, NJ 08855-0849, USA.
1 1 1K 2d ! g (k, u), p (k, u)"e ?@ ? ?@ i(u#ig) ?@ i(u#ig)
(2)
where g (k, u) is the retarded current—current correlation ?@ function, with K (i) being the kinetic energy density in the ? a-direction. By defining the charge o(i) and current j(i) densities as o(i)"ecR(i)c(i), j(i)"!iteacR(i)[
!
]c(i),
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 8 6 9 - 2
(3)
756
F. Mancini et al. / Physica B 259—261 (1999) 755—756
identities connecting current—current, charge—current and charge—charge propagators. One of those reads as follows iaus (k, u)"[1!e\ IV?]s (k, u) V #[1!e\ IW?]s (k, u). W In this theoretical framework, we obtain
(4)
aus (k , u) V #e1K (i)2. (5) s (k , u)" V VV V 2[1!cos(k a)] V From the analytical structure of Eq. (5), it is clear that s (k , 0) does not depend on k . It is remarkable that VV V V such a peculiar behavior has been previously found by use of quantum Monte Carlo techniques [4]. For the optical conductivity p (u), we have obtained p (u)"Dp(u)#p (u) auRes0 (k , u) V , D"p lim lim (6) 2[1!cos(k a)] V S IV 1 aus0 (k , u) V p (u)"! Im lim , (7) u 2[1!cos(k a)] V V I D and p (u) being the Drude weight and the incoherent part, respectively, and R indicates the retarded contribution. As emphasized in the beginning, the whole problem of the response to an external electromagnetic field has been reduced to the evaluation of the charge—charge propagator. In the static approximation of the composite operator method the charge—charge correlation function as well as the spin—spin one can be connected to convolutions of single-particle propagators. This occurrence is related to a linearized dynamics together with the choice of occupation-dependent electronic excitations as basic fields [5]. In this way, the electron propagation is described as a repetition of composite excitations which automatically take into account scattering of electrons on spin and charge fluctuations due to strong correlations. Also, the use of a higher order basic field gives the advantage of implementing the Pauli principle and of a clear theoretical understanding of the terms originating from intraband or interband propagation. In fact, it turns out that only intraband excitations contribute to the Drude weight, whereas interband ones build up the incoherent part. It is worth noting that in this context the
charge—charge correlation function satisfies the sum rule
ia(2p)\ dk dus (k, u)"e(n#2d)
(8)
as it should be, being at equal sites 1n(i)2"n#2d, with d the double occupancy. The physical content of Eq. (8) is the one of the Pauli principle because it governs the single-site dynamics of two fermions. In conclusion, we have shown that the operator equation expressing the charge conservation law has as direct corollary the existence of Ward—Takahashi identities, which hierarchically degrade the evaluation of the current—current propagator to the current—charge and charge—charge ones. This approach has been implemented on the static approximation for the two-dimensional Hubbard model in the framework of the composite operator method. Then, a use of general symmetry principles discloses the possibility to realize a fully self-consistent theory for the charge transport in the strongly correlated electronic systems. Further, by recovering such general symmetries the propagation of the charge and their intrinsic dynamics is constrained in a suitable Hilbert space where the charges propagate without degrading in number and avoiding double occupancy configurations unless to form local singlets. Numerical analysis of the derived equations and an extensive comparison with the data for cuprates will be presented elsewhere.
References [1] S. Uchida, Jpn. J. Appl. Phys. 32 (1993) 3784. [2] E. Dagotto, Rev. Mod. Phys. 66 (1994) 763, and references therein. [3] F. Mancini, S. Marra, H. Matsumoto, Physica C 244 (1995) 49. [4] D.J. Scalapino, S.R. White, S.C. Zhang, Phys. Rev. Lett. 68 (1992) 2830. [5] F. Mancini, S. Marra, H. Matsumoto, Physica C 252 (1995) 361. [6] F. Mancini, D. Villani, H. Matsumoto, Phys. Rev. B 57 (1998) 6145. [7] A. Avella, F. Mancini, D. Villani, L. Siurakshina, V.Yu. Yushankhai, Int. J. Mod. Phys. B 12 (1998) 81.